Handbook of Materials Modeling

HANDBOOK OF MATERIALS MODELING HANDBOOK OF MATERIALS MODELING Part A. Methods Editor Sidney Yip, Massachusetts Instit...

Author: Sidney Yip

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HANDBOOK OF MATERIALS MODELING

HANDBOOK OF MATERIALS MODELING Part A. Methods Editor Sidney Yip, Massachusetts Institute of Technology

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN-10 1-4020-3287-0 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-10 1-4020-3286-2 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3287-5 (HB) Springer Dordrecht, Berlin, Heidelberg, New York ISBN-13 978-1-4020-3286-8 (e-book) Springer Dordrecht, Berlin, Heidelberg, New York

Published by Springer, P.O. Box 17, 3300 AA Dordrecht, The Netherlands.

Printed on acid-free paper

All Rights Reserved

© 2005 Springer No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed in The Netherlands

CONTENTS PART A – METHODS Preface

xii

List of Subject Editors

ix

List of Contributors

xi

Detailed Table of Contents

xxix

Introduction

1

Chapter 1.

Electronic Scale

7

Chapter 2.

Atomistic Scale

449

Chapter 3.

Mesoscale/Continuum Methods

1069

Chapter 4.

Mathematical Methods

1215

PART B – MODELS Preface

xii

List of Subject Editors

ix

List of Contributors

xi

Detailed Table of Contents

xxix

Chapter 5.

Rate Processes

1565

Chapter 6.

Crystal Defects

1849

Chapter 7.

Microstructure

2081

Chapter 8.

Fluids

2409

Chapter 9.

Polymers and Soft Matter

2553

Plenary Perspectives

2657

Index of Contributors

2943

Index of Keywords

2947 v

PREFACE This Handbook contains a set of articles introducing the modeling and simulation of materials from the standpoint of basic methods and studies. The intent is to provide a compendium that is foundational to an emerging field of computational research, a new discipline that may now be called Computational Materials. This area has become sufficiently diverse that any attempt to cover all the pertinent topics would be futile. Even with a limited scope, the present undertaking has required the dedicated efforts of 13 Subject Editors to set the scope of nine chapters, solicit authors, and collect the manuscripts. The contributors were asked to target students and non-specialists as the primary audience, to provide an accessible entry into the field, and to offer references for further reading. With no precedents to follow, the editors and authors were only guided by a common goal – to produce a volume that would set a standard toward defining the broad community and stimulating its growth. The idea of a reference work on materials modeling surfaced in conversations with Peter Binfield, then the Reference Works Editor at Kluwer Academic Publishers, in the spring of 1999. The rationale at the time already seemed quite clear – the field of computational materials research was taking off, powerful computer capabilities were becoming increasingly available, and many sectors of the scientific community were getting involved in the enterprise. It was felt that a volume that could articulate the broad foundations of computational materials and connect with the established fields of computational physics and computational chemistry through common fundamental scientific challenges would be timely. After five years, none of the conditions have changed; the need remains for a defining reference volume, interest in materials modeling and simulation is further intensifying, the community continues to grow. In this work materials modeling is treated in 9 chapters, loosely grouped into two parts. Part A, emphasizing foundations and methodology, consists of three chapters describing theory and simulation at the electronic, atomistic, and mesoscale levels, and a chapter on analysis-based methods. Part B is more concerned with models and basic applications. There are five chapters describing basic problems in materials modeling and simulation, rate-dependent phenomena, crystal defects, microstructure, fluids, polymers and soft matter. In vii

viii

Preface

addition this part contains a collection of commentaries on a range of issues in materials modeling, written in a free-style format by experienced individuals with definite views that could enlighten the future members of the community. See the opening Introduction for further comments on modeling and simulation and an overview of the Handbook contents. Any organizational undertaking of this magnitude cans only be a collective effort. Yet the fate of this volume would not be so certain without the critical contributions from a few individuals. My gratitude goes to Liesbeth Mol, Peter Binfield’s successor at Springer Science + Business Media, for continued faith and support, Ju Li and Xiaofeng Qian for managing the websites and manuscript files, and Tim Kaxrias for stepping in at a critical stage of the project. To all the authors who found time in your hectic schedules to write the contributions, I am deeply appreciative and trust you are not disappointed. To the Subject Editors I say the Handbook is a reality only because of your perseverance and sacrifices. It has been my good fortune to have colleagues who were generous with advice and assistance. I hope this work motivates them even more to continue sharing their knowledge and insights in the work ahead. Sidney Yip Department of Nuclear Science and Engineering, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

LIST OF SUBJECT EDITORS Martin Bazant, Massachusetts Institute of Technology (Chapter 4) Bruce Boghosian, Tufts University (Chapter 8) Richard Catlow, Royal Institution, UK (Chapter 6) Long-Qing Chen, Pennsylvania State University (Chapter 7) William Curtin, Brown University (Chapter 1, Chapter 2, Chapter 4) Tomas Diaz de la Rubia, Lawrence Livermore National Laboratory (Chapter 6) Nicolas Hadjiconstantinou, Massachusetts Institute of Technology (Chapter 8) Mark F. Horstemeyer, Mississippi State University (Chapter 3) Efthimios Kaxiras, Harvard University (Chapter 1, Chapter 2) L. Mahadevan, Harvard University (Chapter 9) Dimitrios Maroudas, University of Massachusetts (Chapter 4) Nicola Marzari, Massachusetts Institute of Technology (Chapter 1) Horia Metiu, University of California Santa Barbara (Chapter 5) Gregory C. Rutledge, Massachusetts Institute of Technology (Chapter 9) David J. Srolovitz, Princeton University (Chapter 7) Bernhardt L. Trout, Massachusetts Institute of Technology (Chapter 1) Dieter Wolf, Argonne National Laboratory (Chapter 6) Sidney Yip, Massachusetts Institute of Technology (Chapter 1, Chapter 2, Chapter 6, Plenary Perspectives)

ix

LIST OF CONTRIBUTORS Farid F. Abraham IBM Almaden Research Center, San Jose, California [email protected] P20

Robert Averback Accelerator Laboratory, P.O. Box 43 (Pietari Kalmin k. 2), 00014, University of Helsinki, Finland; Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Illinois, USA [email protected] 6.2

Francis J. Alexander Los Alamos National Laboratory, Los Alamos, NM, USA [email protected] 8.7

D.J. Bammann Sandia National Laboratories, Livermore, CA, USA [email protected] 3.2

N.R. Aluru Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 8.3

K. Barmak Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA [email protected] 7.19

Filippo de Angelis Istituto CNR di Scienze e Tecnologie Molecolari ISTM, Dipartimento di Chimica, Universitá di Perugia, Via Elce di Sotto $, I-06123, Perugia, Italy [email protected] 1.4

Stefano Baroni DEMOCRITOS-INFM, SISSA-ISAS, Trieste, Italy [email protected] 1.10

Emilio Artacho University of Cambridge, Cambridge, UK [email protected] 1.5

Rodney J. Bartlett Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA [email protected] 1.3

Mark Asta Northwestern University, Evanston, IL, USA [email protected] 1.16

Corbett Battaile Sandia National Laboratories, Albuquerque, NM, USA [email protected] 7.17

xi

xii Martin Z. Bazant Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.1, 4.10 Noam Bernstein Naval Research Laboratory, Washington, DC, USA [email protected] 2.24 Kurt Binder Institut fuer Physik, Johannes Gutenberg Universitaet Mainz, Staudinger Weg 7, 55099 Mainz, Germany [email protected] P19 Peter E. Blöhl Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany [email protected] 1.6 Bruce M. Boghosian Department of Mathematics, Tufts University, Bromfield-Pearson Hall, Medford, MA 02155, USA [email protected] 8.1 Jean Pierre Boon Center for Nonlinear Phenomena and Complex Systems, Université Libre de Bruxelles, 1050-Bruxelles, Belgium [email protected] P21

List of contributors Russel Caflisch University of California at Los Angeles, Los Angeles, CA, USA [email protected] 7.15 Wei Cai Department of Mechanical Engineering, Stanford University, Stanford, CA 94305-4040, USA [email protected] 2.21 Roberto Car Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, NJ, USA [email protected] 1.4 Paolo Carloni International School for Advanced Studies (SISSA/ISAS) and INFM Democritos Center, Trieste, Italy [email protected] 1.13 Emily A. Carter Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] 1.8

Iain D. Boyd University of Michigan, Ann Arbor, MI, USA [email protected] P22

C.R.A. Catlow Davy Faraday Laboratory, The Royal Institution, 21 Albemarle Street, London W1S 4BS, UK; Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK [email protected] 2.7, 6.1

Vasily V. Bulatov Lawrence Livermore National Laboratory, University of California, Livermore, CA 94550, USA [email protected] P7

Gerbrand Ceder Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 1.17, 1.18

List of contributors

xiii

Alan V. Chadwick Functional Materials Group, School of Physical Sciences, University of Kent, Canterbury, Kent CT2 7NR, UK [email protected] 6.5

Marvin L. Cohen University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USA [email protected] 1.2

Hue Sun Chan University of Toronto, Toronto, Ont., Canada [email protected] 5.16

John Corish Department of Chemistry, Trinity College, University of Dublin, Dublin 2, Ireland [email protected] 6.4

James R. Chelikowsky University of Minnesota, Minneapolis, MN, USA [email protected] 1.7 Long-Qing Chen Department of Materials Science and Engineering, Penn State University, University Park, PA 16802, USA [email protected] 7.1 I-Wei Chen Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104-6282, USA [email protected] P27 Sow-Hsin Chen Department of Nuclear Engineering, MIT, Cambridge, MA 02139, USA [email protected] P28 Christophe Chipot Equipe de dynamique des assemblages membranaires, Unité mixte de recherche CNRS/UHP 7565, Institut nancéen de chimie moléculaire, Université Henri Poincaré, BP 239, 54506 Vanduvre-lès-Nancy cedex, France 2.26 Giovanni Ciccotti INFM and Dipartimento di Fisica, Università “La Sapienza,” Piazzale Aldo Moro, 2, 00185 Roma, Italy [email protected] 2.17, 5.4

Peter V. Coveney Centre for Computational Science, Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK [email protected] 8.5 Jean-Paul Crocombette CEA Saclay, DEN-SRMP, 91191 Gif/Yvette cedex, France [email protected] 2.28 Darren Crowdy Department of Mathematics, Imperial College, London, UK [email protected] 4.10 Gábor Csányi Cavendish Laboratory, University of Cambridge, UK [email protected] P16 Nguyen Ngoc Cuong Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.15 Christoph Dellago Institute of Experimental Physics, University of Vienna, Vienna, Austria [email protected] 5.3

xiv J.D. Doll Department of Chemistry, Brown University, Providence, RI, USA Jimmie [email protected] 5.2 Patrick S. Doyle Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 9.7

List of contributors Diana Farkas Department of Materials Science and Engineering, Virginia Tech, Blacksburg, VA 24061, USA [email protected] 2.23 Clemens J. Först Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany [email protected] 1.6

Weinan E Department of Mathematics, Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544-1000, USA [email protected] 4.13

Glenn H. Fredrickson Department of Chemical Engineering & Materials, The University of California at Santa, Barbara Santa Barbara, CA, USA [email protected] 9.9

Jens Eggers School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK [email protected] 4.9

Daan Frenkel FOM Institute for Atomic and Molecular Physics, Amsterdam, The Netherlands [email protected] 2.14

Pep Espanol ˜ Dept. Física Fundamental, Universidad Nacional de Educación a Distancia, Aptdo. 60141, E-28080 Madrid, Spain [email protected] 8.6 J.W. Evans Ames Laboratory - USDOE, and Department of Mathematics, Iowa State University, Ames, Iowa, 50011, USA [email protected] 5.12 Denis J. Evans Research School of Chemistry, Australian National University, Canberra, ACT, Australia [email protected] P17 Michael L. Falk University of Michigan, Ann Arbor, MI, USA [email protected] 4.3

Julian D. Gale Nanochemistry Research Institute, Department of Applied Chemistry, Curtin University of Technology, Perth, 6845, Western Australia [email protected] 1.5, 2.3 Giulia Galli Lawrence Livermore National Laboratory, CA, USA [email protected] P8 Venkat Ganesan Department of Chemical Engineering, The University of Texas at Austin, Austin, TX, USA [email protected] 9.9 Alberto García Universidad del País Vasco, Bilbao, Spain [email protected] 1.5

List of contributors C. William Gear Princeton University, Princeton, NJ, USA [email protected] 4.11 Timothy C. Germann Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 2.11 Eitan Geva Department of Chemistry, University of Michigan, Ann Arbor, Michigan 48109-1055, USA [email protected] 5.9 Nasr M. Ghoniem Department of Mechanical and Aerospace Engineering, University of California, Los Angeles, CA 90095-1597, USA [email protected] 7.11, P11, P30 Paolo Giannozzi Scuola Normale Superiore and National Simulation Center, INFM-DEMOCRITOS, Pisa, Italy [email protected] 1.4, 1.10 E. Van der Giessen University of Groningen, Groningen, The Netherlands [email protected] 3.4 Daniel T. Gillespie Dan T Gillespie Consulting, 30504 Cordoba Place, Castaic, CA 91384, USA [email protected] 5.11 George Gilmer Lawrence Livermore National Laboratory, P.O. box 808, Livermore, CA 94550, USA [email protected] 2.10

xv William A. Goddard III Materials and Process Simulation Center, California Institute of Technology, Pasadena, CA 91125, USA [email protected] P9 Axel Groß Physik-Department T30, TU München, 85747 Garching, Germany [email protected] 5.10 Peter Gumbsch Institut für Zuverlässigkeit von Bauteilen und Systemen izbs, Universität Karlsruhe (TH), Kaiserstr. 12, 76131Karlsruhe, Germany and Fraunhofer Institut für Werkstoffmechanik IWM, Wöhlerstr. 11, D-79194 Freiburg, Germany [email protected] P10 Fran¸cois Gygi Lawrence Livermore National Laboratory, CA, USA P8 Nicolas G. Hadjiconstantinou Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA [email protected] 8.1, 8.8 J.P. Hirth Ohio State and Washington State Universities, 114 E. Ramsey Canyon Rd., Hereford, AZ 85615, USA [email protected] P31 K.M. Ho Ames Laboratory-U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA 1.15

xvi


Wesley P. Hoffman Air Force Research Laboratory, Edwards, CA, USA [email protected] P37

C.S. Jayanthi Department of Physics, University of Louisville, Louisville, KY 40292 [email protected] P39

Wm.G. Hoover Department of Applied Science, University of California at Davis/Livermore and Lawrence Livermore National Laboratory, Livermore, California, 94551-7808 [email protected] P34

Raymond Jeanloz University of California, Berkeley, CA, USA [email protected] P25

M.F. Horstemeyer Mississippi State University, Mississippi State, MS, USA [email protected] 3.1, 3.5 Thomas Y. Hou California Institute of Technology, Pasadena, CA, USA [email protected] 4.14 Hanchen Huang Department of Mechanical, Aerospace and Nuclear Engineering, Rensselaer Polytechnic Institute, 110 8th Street, Troy, NY 12180-3590, USA [email protected] 2.30 Gerhard Hummer National Institutes of Health, Bethesda, MD, USA [email protected] 4.11 M. Saiful Islam Chemistry Division, SBMS, University of Surrey, Guildford GU2 7XH, UK [email protected] 6.6 Seogjoo Jang Chemistry Department, Brookhaven National Laboratory, Upton, New York 11973-5000, USA [email protected] 5.9

Pablo Jensen Laboratoire de Physique de la Matiére Condensée et des Nanostructures, CNRS and Université Claude Bernard Lyon-1, 69622 Villeurbanne Cédex, France [email protected] 5.13 Yongmei M. Jin Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA [email protected] 7.12 Xiaozhong Jin Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 8.3 J.D. Joannopoulos Francis Wright Davis Professor of Physics, Department of Physics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] P4 Javier Junquera Rutgers University, New Jersey, USA [email protected] 1.5 João F. Justo Escola Politécnica, Universidade de São Paulo, São Paulo, Brazil [email protected] 2.4

List of contributors Hideo Kaburaki Japan Atomic Energy Research Institute, Tokai, Ibaraki, Japan [email protected] 2.18 Rajiv K. Kalia Collaboratory for Advanced Computing and Simulations, Department of Physics & Astronomy, University of Southern California, 3651 Watt Way, VHE 608, Los Angeles, CA 90089-0242, USA [email protected] 2.25 Raymond Kapral Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ont. M5S 3H6, Canada [email protected] 2.17, 5.4 Alain Karma Northeastern University, Boston, MA, USA [email protected] 7.2 Johannes Kästner Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany [email protected] 1.6 Markos A. Katsoulakis Department of Mathematics and Statistics, University of Massachusetts - Amherst, Amherst, MA 01002, USA [email protected] 4.12 Efthimios Kaxiras Department of Nuclear Science and Engineering and Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 2.1, 8.4

xvii Ronald J. Kerans Air Force Research Laboratory, Materials and Manufacturing Directorate, Wright-Patterson Air Force Base, Ohio, USA [email protected] P38 Ioannis G. Kevrekidis Princeton University, Princeton, NJ, USA [email protected] 4.11 Armen G. Khachaturyan Department of Ceramic and Materials Engineering, Rutgers University, 607 Taylor Road, Piscataway, NJ 08854, USA [email protected] 7.12 T.A. Khraishi University of New Mexico, Albuquerque, NM, USA [email protected] 3.3 Seong Gyoon Kim Kunsan National University, Kunsan 573-701, Korea [email protected] 7.3 Won Tae Kim Chongju University, Chongju 360-764, Korea [email protected] 7.3 Michael L. Klein Center for Molecular Modeling, Chemistry Department, University of Pennsylvania, 231 South 34th Street, Philadelphia, PA 19104-6323, USA [email protected] 2.26 Walter Kob Laboratoire des Verres, Université Montpellier 2, 34095 Montpellier, France [email protected] P24

xviii David A. Kofke University at Buffalo, The State University of New York, Buffalo, New York, USA [email protected] 2.14 Maurice de Koning University of São Paulo, São Paulo, Brazil [email protected] 2.15 Anatoli Korkin Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA 1.3 Kurt Kremer MPI for Polymer Research, D-55021 Mainz, Germany [email protected] P5

List of contributors C. Leahy Department of Physics, University of Louisville, Louisville, KY 40292, USA P39 R. LeSar Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 7.14 Ju Li Department of Materials Science and Engineering, Ohio State University, Columbus, OH, USA [email protected] 2.8, 2.19, 2.31 Xiantao Li Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] 4.13

Carl E. Krill III Materials Division, University of Ulm, Albert-Einstein-Allee 47, D-89081 Ulm, Germany [email protected] 7.6

Gang Li Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 8.3

Ladislas P. Kubin LEM, CNRS-ONERA, 29 Av. de la Division Leclerc, BP 72, 92322 Chatillon Cedex, France [email protected] P33

Vincent L. Lignères Department of Chemistry, Princeton University, Princeton, NJ 08544, USA 1.8

D.P. Landau Center for Simulational Physics, The University of Georgia, Athens, GA 30602, USA [email protected] P2 James S. Langer Department of Physics, University of California, Santa Barbara, CA 93106-9530, USA [email protected] 4.3, P14

Turab Lookman Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA [email protected] 7.5 Steven G. Louie Department of Physics, University of California at Berkeley and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA [email protected] 1.11


xix

John Lowengrub University of California, Irvine, California, USA [email protected] 7.8

Richard M. Martin University of Illinois at Urbana, Urbana, IL, USA [email protected] 1.5

Gang Lu Division of Engineering and Applied Science, Harvard University, Cambridge, Massachusetts, USA [email protected] 2.20

Georges Martin ´ Commissariat a` l’Energie Atomique, Cab. H.C., 33 rue de la Fédération, 75752 Paris Cedex 15, France [email protected] 7.9

Alexander D. MacKerell, Jr. Department of Pharmaceutical Sciences, School of Pharmacy, University of Maryland, 20 Penn Street, Baltimore, MD, 21201, USA [email protected] 2.5

Nicola Marzari Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 1.1, 1.4

Alessandra Magistrato International School for Advanced Studies (SISSA/ISAS) and INFM Democritos Center, Trieste, Italy 1.13

Wayne L. Mattice Department of Polymer Science, The University of Akron, Akron, OH 44325-3909 [email protected] 9.3

L. Mahadevan Division of Engineering and Applied Sciences, Department of Organismic and Evolutionary Biology, Department of Systems Biology, Harvard University Cambridge, MA 02138, USA [email protected] Dionisios Margetis Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 4.8

V.G. Mavrantzas Department of Chemical Engineering, University of Patras, Patras, GR 26500, Greece [email protected] 9.4 D.L. McDowell Georgia Institute of Technology, Atlanta, GA, USA [email protected] 3.6, 3.9

E.B. Marin Sandia National Laboratories, Livermore, CA, USA [email protected] 3.5

Michael J. Mehl Center for Computational Materials Science, Naval Research Laboratory, Washington, DC, USA [email protected] 1.14

Dimitrios Maroudas University of Massachusetts, Amherst, MA, USA [email protected] 4.1

Horia Metiu University of California, Santa Barbara, CA, USA [email protected] 5.1

xx R.E. Miller Carleton University, Ottawa, ON, Canada [email protected] 2.13 Frederick Milstein Mechanical Engineering and Materials Depts., University of California, Santa Barbara, CA, USA [email protected] 4.2 Y. Mishin George Mason University, Fairfax, VA, USA [email protected] 2.2 Francesco Montalenti INFM, L-NESS, and Dipartimento di Scienza dei Materiali, Università degli Studi di Milano-Bicocca, Via Cozzi 53, I-20125 Milan, Italy [email protected] 2.11 Dane Morgan Massachusetts Institute of Technology, Cambridge MA, USA [email protected] 1.18 John A. Moriarty Lawrence Livermore National Laboratory, University of California, Livermore, CA 94551-0808 [email protected] P13 J.W. Morris, Jr. Department of Materials Science and Engineering, University of California, Berkeley, CA, USA [email protected] P18 Raymond D. Mountain Physical and Chemical Properties Division, Chemical Science and Technology Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899-8380, USA [email protected] P23

List of contributors Marcus Muller ¨ Department of Physics, University of Wisconsin, Madison, WI 53706-1390, USA [email protected] 9.5 Aiichiro Nakano Collaboratory for Advanced Computing and Simulations, Department of Computer Science, University of Southern California, 3651 Watt Way, VHE 608, Los Angeles, CA 90089-0242, USA [email protected] 2.25 A. Needleman Brown University, Providence, RI, USA [email protected] 3.4 Abraham Nitzan Tel Aviv University, Tel Aviv, 69978, Israel [email protected] 5.7 Kai Nordlund Accelerator Laboratory, P.O. Box 43 (Pietari Kalmin k. 2), 00014, University of Helsinki, Finland; Department of Materials Science and Engineering, University of Illinois at Urbana-Champaign, Illinois, USA 6.2 G. Robert Odette Department of Mechanical Engineering and Department of Materials, University of California, Santa Barbara, CA, USA [email protected] 2.29 Shigenobu Ogata Osaka University, Osaka, Japan [email protected] 1.20

List of contributors Gregory B. Olson Department of Materials Science and Engineering, Northwestern University, Evanston, IL, USA [email protected] P3 Pablo Ordejón Instituto de Materiales, CSIC, Barcelona, Spain [email protected] 1.5 Tadeusz Pakula Max Planck Institute for Polymer Research, Mainz, Germany and Department of Molecular Physics, Technical University, Lodz, Poland [email protected] P35 Vijay Pande Department of Chemistry and of Structural Biology, Stanford University, Stanford, CA 94305-5080, USA [email protected] 5.17 I.R. Pankratov Russian Research Centre, “Kurchatov Institute”, Moscow 123182, Russia [email protected] 7.10 D.A. Papaconstantopoulos Center for Computational Materials Science, Naval Research Laboratory, Washington, DC, USA [email protected] 1.14 J.E. Pask Lawrence Livermore National Laboratory, Livermore, CA, USA [email protected] 1.19 Anthony T. Patera Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.15

xxi Mike Payne Cavendish Laboratory, University of Cambridge, UK [email protected] P16 Leonid Pechenik University of California, Santa Barbara, CA, USA [email protected] 4.3 Joaquim Peiró Department of Aeronautics, Imperial College, London, UK [email protected] 8.2 Simon R. Phillpot Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611, USA [email protected] 2.6, 6.11 G.P. Potirniche Mississippi State University, Mississippi State, MS, USA [email protected] 3.5 Thomas R. Powers Division of Engineering, Brown University, Providence, RI, USA thomas [email protected] 9.8 Dierk Raabe Max-Planck-Institut für Eisenforschung, Max-Planck-Str. 1, D-40237 Düsseldorf, Germany [email protected] 7.7, P6 Ravi Radhakrishnan Department of Bioengineering, University of Pennsylvania, Philadelphia, PA, USA [email protected] 5.5

xxii Christian Ratsch University of California at Los Angeles, Los Angeles, CA, USA [email protected] 7.15 John R. Ray 1190 Old Seneca Road, Central, SC 29630, USA [email protected] 2.16 William P. Reinhardt University of Washington Seattle, Washington, USA [email protected] 2.15 Karsten Reuter Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany [email protected] 1.9 J.M. Rickman Department of Materials Science and Engineering, Lehigh University, Bethlehem, PA 18015, USA [email protected] 7.14, 7.19

List of contributors Tomonori Sakai Centre for Computational Science, Queen Mary, University of London, Mile End Road, London E1 4NS, UK 8.5 Deniel Sánchez-Portal Donostia International Physics Center, Donostia, Spain [email protected] 1.5 Joachim Sauer Institut für Chemie, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099 Berlin, Germany 1.12 Avadh Saxena Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA [email protected] 7.5 Matthias Scheffler Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany [email protected] 1.9

Angel Rubio Departamento Física de Materiales and Unidad de Física de Materiales Centro Mixto CSIC-UPV, Universidad del País Vasco and Donosita Internacional Physics Center (DIPC), Spain [email protected] 1.11

Klaus Schulten Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 5.15

Robert E. Rudd Lawrence Livermore National Laboratory, University of California, L-045 Livermore, CA 94551, USA [email protected] 2.12

Steven D. Schwartz Departments of Biophysics and Biochemistry, Albert Einstein College of Medicine, New York, USA [email protected] 5.8

Gregory C. Rutledge Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA [email protected] 9.1

Robin L.B. Selinger Physics Department, Catholic University, Washington, DC 20064, USA [email protected] 2.23

List of contributors Marcelo Sepliarsky Instituto de Física Rosario, Facultad de Ciencias Exactas, Ingenieria y Agrimensura, Universidad Nacional de Rosario, 27 de Febreo 210 Bis, (2000) Rosario, Argentina [email protected] 2.6 Alessandro Sergi Chemical Physics Theory Group, Department of Chemistry, University of Toronto, Toronto, Ont. M5S 3H6, Canada [email protected] 2.17, 5.4 J.A. Sethian Department of Mathematics, University of California, Berkeley, CA, USA [email protected] 4.6 Michael J. Shelley Courant Institute of Mathematical Sciences, New York University, New York, NY, USA [email protected] 4.7 C. Shen The Ohio State University, Columbus, Ohio, USA [email protected] 7.4 Spencer Sherwin Department of Aeronautics, Imperial College, London, UK [email protected] 8.2 Marek Sierka Institut für Physikalische Chemie, Lehrstuhl für Theoretische Chemie, Universität Karlsruhe, Kaiserstraße 12, D-76128 Karlsruhe, Germany [email protected] 1.12 Asimina Sierou University of Cambridge, Cambridge, UK [email protected] 9.6

xxiii Grant D. Smith Department of Materials Science and Engineering, Department of Chemical Engineering, University of Utah, Salt Lake City, Utah, USA [email protected] 9.2 Frédéric Soisson CEA Saclay, DMN-SRMP, 91191 Gif-sur-Yuette, France [email protected] 7.9 José M. Soler Universidad Autónoma de Madrid, Madrid, Spain [email protected] 1.5 Didier Sornette Institute of Geophysics and Planetary Physics and Department of Earth and Space Science, University of California, Los Angeles, California, USA and CNRS and Université des Sciences, Nice, France [email protected] 4.4 David J. Srolovitz Princeton Materials Institute and Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA [email protected] 7.1, 7.13 Marcelo G. Stachiotti Instituto de Física Rosario, Facultad de Ciencias Exactas, Ingenieria y Agrimensura, Universidad Nacional de Rosario, 27 de Febreo 210 Bis, (2000) Rosario, Argentina [email protected] 2.6 Catherine Stampfl Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany; School of Physics, The University of Sydney, Sydney 2006, Australia [email protected] 1.9

xxiv


H. Eugene Stanley Center for Polymer Studies and Department of Physics Boston, University, Boston, MA 02215, USA [email protected] P36

Meijie Tang Lawrence Livermore National Laboratory, P.O. Box 808, Livermore, CA 94550 [email protected] 2.22

P.A. Sterne Lawrence Livermore National Laboratory, Livermore, CA, USA [email protected] 1.19

Mounir Tarek Equipe de dynamique des assemblages membranaires, Unité mixte de recherche CNRS/UHP 7565, Institut nancéien de chimie moléculaire, Université Henri Poincaré, BP 239, 54506 Vanduvre-lès-Nancy cedex, France 2.26

Howard A. Stone Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA 01238, USA [email protected] 4.8 Marshall Stoneham Centre for Materials Research, and London Centre for Nanotechnology, Department of Physics and Astronomy, University College London, Gower Street, London WC1E 6BT, UK [email protected] P12 Sauro Succi Istituto Applicazioni Calcolo, National Research Council, viale del Policlinico, 137, 00161, Rome, Italy [email protected] 8.4 E.B. Tadmor Technion-Israel Institute of Technology, Haifa, Israel [email protected] 2.13 Emad Tajkhorshid Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 5.15

DeCarlos E. Taylor Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA [email protected] 1.3 Doros N. Theodorou School of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechniou Street, Zografou Campus, 157 80 Athens, Greece [email protected] P15 Carl V. Thompson Department of Materials Science and Engineering, M.I.T., Cambridge, MA 02139, USA [email protected] P26 Anna-Karin Tornberg Courant Institute of Mathematical Sciences, New York University, New York, NY, USA [email protected] 4.7 S. Torquato Department of Chemistry, PRISM, and Program in Applied & Computational Mathematics, Princeton University, Princeton, NJ 08544, USA [email protected] 4.5, 7.18

List of contributors Bernhardt L. Trout Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 5.5 Mark E. Tuckerman Department of Chemistry, Courant Institute of Mathematical Science, New York University, New York, NY 10003, USA [email protected] 2.9 Blas P. Uberuaga Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 2.11, 5.6 Patrick T. Underhill Department of Chemical Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA 9.7 V.G. Vaks Russian Research Centre, “Kurchatov Institute”, Moscow 123182, Russia [email protected] 7.10 Priya Vashishta Collaboratory for Advanced Computing and Simulations, Department of Chemical Engineering and Materials Science, University of Southern California, 3651 Watt Way, VHE 608, Los Angeles, CA 90089-0242, USA [email protected] 2.25 A. Van der Ven Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA 1.17 Karen Veroy Massachusetts Institute of Technology, Cambridge, MA, USA [email protected] 4.15

xxv Alessandro De Vita King’s College London, UK, Center for Nanostructured, Materials (CENMAT) and DEMOCRITOS National Simulation Center, Trieste, Italy alessandro.de [email protected] P16 V. Vitek Department of Materials Science and Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA [email protected] P32 Dionisios G. Vlachos Department of Chemical Engineering, Center for Catalytic Science and Technology, University of Delaware, Newark, DE 19716, USA [email protected] 4.12 Arthur F. Voter Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] 2.11, 5.6 Gregory A. Voth Department of Chemistry and Henry Eyring Center for Theoretical Chemistry, University of Utah, Salt Lake City, Utah 84112-0850, USA [email protected] 5.9 G.Z. Voyiadjis Louisiana State University, Baton Rouge, LA, USA [email protected] 3.8 Dimitri D. Vvedensky Imperial College, London, United Kingdom [email protected] 7.16 Göran Wahnström Chalmers University of Technology and Göteborg University Materials and Surface Theory, SE-412 96 Göteborg, Sweden [email protected] 5.14

xxvi


Duane C. Wallace Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA [email protected] P1

Brian D. Wirth Department of Nuclear Engineering, University of California, Barkeley, CA, USA [email protected] 2.29

Axel van de Walle Northwestern University, Evanston, IL, USA [email protected] 1.16

Dieter Wolf Materials Science Division, Argonne National Laboratory, Argonne, IL 60439, USA [email protected] 6.7, 6.9, 6.10, 6.11, 6.12, 6.13

Chris G. Van de Walle Materials Department, University of California, Santa Barbara, California, USA [email protected] 6.3

C.Z. Wang Ames Laboratory-U.S. DOE and Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA [email protected] 1.15

Y. Wang The Ohio State University, Columbus, Ohio, USA [email protected] 7.4

Yu U. Wang Department of Materials Science and Engineering, Virginia Tech., Blacksburg, VA 24061, USA [email protected] 7.12

Hettithanthrige S. Wijesinghe Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA [email protected] 8.8

Chung H. Woo The Hong Kong Polytechnic University, Hong Kong SAR, China [email protected] 2.27 Christopher Woodward Northwestern University, Evanston, Illinois, USA [email protected] P29 S.Y. Wu Department of Physics, University of Louisville, Louisville, KY 40292, USA [email protected] P39 Yang Xiang Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong [email protected] 7.13 Sidney Yip Department of Physics, Harvard University, Cambridge, MA 02138, USA [email protected] 2.1, 2.10, 6.7, 6.8, 6.11 M. Yu Department of Physics, University of Louisville, Louisville, KY 40292, USA P39

List of contributors H.M. Zbib Washington State University, Pullman, WA, USA [email protected] 3.3 Fangqiang Zhu Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA [email protected] 5.15

xxvii M. Zikry North Carolina State University, Raleigh, NC, USA [email protected] 3.7

DETAILED TABLE OF CONTENTS PART A – METHODS Chapter 1. Electronic Scale 1.1

Understand, Predict, and Design Nicola Marzari 1.2 Concepts for Modeling Electrons in Solids: A Perspective Marvin L. Cohen 1.3 Achieving Predictive Simulations with Quantum Mechanical Forces Via the Transfer Hamiltonian: Problems and Prospects Rodney J. Bartlett, DeCarlos E. Taylor, and Anatoli Korkin 1.4 First-Principles Molecular Dynamics Roberto Car, Filippo de Angelis, Paolo Giannozzi, and Nicola Marzari 1.5 Electronic Structure Calculations with Localized Orbitals: The Siesta Method Emilio Artacho, Julian D. Gale, Alberto García, Javier Junquera, Richard M. Martin, Pablo Ordejón, Deniel Sánchez-Portal, and José M. Soler 1.6 Electronic Structure Methods: Augmented Waves, Pseudopotentials and the Projector Augmented Wave Method Peter E. Blöchl, Johannes Kästner, and Clemens J. Först 1.7 Electronic Scale James R. Chelikowsky 1.8 An Introduction to Orbital-Free Density Functional Theory Vincent L. Lignères and Emily A. Carter 1.9 Ab Initio Atomistic Thermodynamics and Statistical Mechanics of Surface Properties and Functions Karsten Reuter, Catherine Stampfl, and Matthias Scheffler 1.10 Density-Functional Perturbation Theory Paolo Giannozzi and Stefano Baroni

xxix

9 13

27

59

77

93 121 137

149 195

xxx

Detailed table of contents

1.11 Quasiparticle and Optical Properties of Solids and Nanostructures: The GW-BSE Approach Steven G. Louie and Angel Rubio 1.12 Hybrid Quantum Mechanics/Molecular Mechanics Methods and their Application Marek Sierka and Joachim Sauer 1.13 Ab Initio Molecular Dynamics Simulations of Biologically Relevant Systems Alessandra Magistrato and Paolo Carloni 1.14 Tight-Binding Total Energy Methods for Magnetic Materials and Multi-Element Systems Michael J. Mehl and D.A. Papaconstantopoulos 1.15 Environment-Dependent Tight-Binding Potential Models C.Z. Wang and K.M. Ho 1.16 First-Principles Modeling of Phase Equilibria Axel van de Walle and Mark Asta 1.17 Diffusion and Configurational Disorder in Multicomponent Solids A. Van der Ven and G. Ceder 1.18 Data Mining in Materials Development Dane Morgan and Gerbrand Ceder 1.19 Finite Elements in Ab Initio Electronic-Structure Calculations J.E. Pask and P.A. Sterne 1.20 Ab Initio Study of Mechanical Deformation Shigenobu Ogata

215

241

259

275 307 349

367 395 423 439

Chapter 2. Atomistic Scale 2.1 2.2 2.3 2.4 2.5 2.6

2.7 2.8

Introduction: Atomistic Nature of Materials Efthimios Kaxiras and Sidney Yip Interatomic Potentials for Metals Y. Mishin Interatomic Potential Models for Ionic Materials Julian D. Gale Modeling Covalent Bond with Interatomic Potentials João F. Justo Interatomic Potentials: Molecules Alexander D. MacKerell, Jr. Interatomic Potentials: Ferroelectrics Marcelo Sepliarsky, Marcelo G. Stachiotti, and Simon R. Phillpot Energy Minimization Techniques in Materials Modeling C.R.A. Catlow Basic Molecular Dynamics Ju Li

451 459 479 499 509

527 547 565

Detailed table of contents 2.9 2.10 2.11

2.12

2.13 2.14 2.15 2.16

2.17 2.18

2.19 2.20

2.21 2.22

2.23 2.24 2.25

2.26

Generating Equilibrium Ensembles Via Molecular Dynamics Mark E. Tuckerman Basic Monte Carlo Models: Equilibrium and Kinetics George Gilmer and Sidney Yip Accelerated Molecular Dynamics Methods Blas P. Uberuaga, Francesco Montalenti, Timothy C. Germann, and Arthur F. Voter Concurrent Multiscale Simulation at Finite Temperature: Coarse-Grained Molecular Dynamics Robert E. Rudd The Theory and Implementation of the Quasicontinuum Method E.B. Tadmor and R.E. Miller Perspective: Free Energies and Phase Equilibria David A. Kofke and Daan Frenkel Free-Energy Calculation Using Nonequilibrium Simulations Maurice de Koning and William P. Reinhardt Ensembles and Computer Simulation Calculation of Response Functions John R. Ray Non-Equilibrium Molecular Dynamics Giovanni Ciccotti, Raymond Kapral, and Alessandro Sergi Thermal Transport Process by the Molecular Dynamics Method Hideo Kaburaki Atomistic Calculation of Mechanical Behavior Ju Li The Peierls–Nabarro Model of Dislocations: A Venerable Theory and its Current Development Gang Lu Modeling Dislocations Using a Periodic Cell Wei Cai A Lattice Based Screw-Edge Dislocation Dynamics Simulation of Body Center Cubic Single Crystals Meijie Tang Atomistics of Fracture Diana Farkas and Robin L.B. Selinger Atomistic Simulations of Fracture in Semiconductors Noam Bernstein Multimillion Atom Molecular-Dynamics Simulations of Nanostructured Materials and Processes on Parallel Computers Priya Vashishta, Rajiv K. Kalia, and Aiichiro Nakano Modeling Lipid Membranes Christophe Chipot, Michael L. Klein, and Mounir Tarek

xxxi

589 613

629

649 663 683 707

729 745

763 773

793 813

827 839 855

875 929

xxxii


2.27 Modeling Irradiation Damage Accumulation in Crystals Chung H. Woo 2.28 Cascade Modeling Jean-Paul Crocombette 2.29 Radiation Effects in Fission and Fusion Reactors G. Robert Odette and Brian D. Wirth 2.30 Texture Evolution During Thin Film Deposition Hanchen Huang 2.31 Atomistic Visualization Ju Li

959 987 999 1039 1051

Chapter 3. Mesoscale/Continuum Methods 3.1 3.2

3.3 3.4 3.5 3.6 3.7 3.8 3.9

Mesoscale/Macroscale Computational Methods M.F. Horstemeyer Perspective on Continuum Modeling of Mesoscale/Macroscale Phenomena D.J. Bammann Dislocation Dynamics H.M. Zbib and T.A. Khraishi Discrete Dislocation Plasticity E. Van der Giessen and A. Needleman Crystal Plasticity M.F. Horstemeyer, G.P. Potirniche, and E.B. Marin Internal State Variable Theory D.L. McDowell Ductile Fracture M. Zikry Continuum Damage Mechanics G.Z. Voyiadjis Microstructure-Sensitive Computational Fatigue Analysis D.L. McDowell

1071

1077 1097 1115 1133 1151 1171 1183 1193

Chapter 4. Mathematical Methods 4.1 4.2

4.3

4.4

Overview of Chapter 4: Mathematical Methods Martin Z. Bazant and Dimitrios Maroudas Elastic Stability Criteria and Structural Bifurcations in Crystals Under Load Frederick Milstein Toward a Shear-Transformation-Zone Theory of Amorphous Plasticity Michael L. Falk, James S. Langer, and Leonid Pechenik Statistical Physics of Rupture in Heterogeneous Media Didier Sornette

1217

1223

1281 1313

Detailed table of contents 4.5 4.6

4.7 4.8 4.9 4.10 4.11 4.12

4.13 4.14

4.15

Theory of Random Heterogeneous Materials S. Torquato Modern Interface Methods for Semiconductor Process Simulation J.A. Sethian Computing Microstructural Dynamics for Complex Fluids Michael J. Shelley and Anna-Karin Tornberg Continuum Descriptions of Crystal Surface Evolution Howard A. Stone and Dionisios Margetis Breakup and Coalescence of Free Surface Flows Jens Eggers Conformal Mapping Methods for Interfacial Dynamics Martin Z. Bazant and Darren Crowdy Equation-Free Modeling for Complex Systems Ioannis G. Kevrekidis, C. William Gear, and Gerhard Hummer Mathematical Strategies for the Coarse-Graining of Microscopic Models Markos A. Katsoulakis and Dionisios G. Vlachos Multiscale Modeling of Crystalline Solids Weinan E and Xiantao Li Multiscale Computation of Fluid Flow in Heterogeneous Media Thomas Y. Hou Certified Real-Time Solution of Parametrized Partial Differential Equations Nguyen Ngoc Cuong, Karen Veroy, and Anthony T. Patera

xxxiii

1333

1359 1371 1389 1403 1417 1453

1477 1491

1507

1529

PART B – MODELS Chapter 5. Rate Processes 5.1 5.2 5.3 5.4 5.5

5.6

Introduction: Rate Processes Horia Metiu A Modern Perspective on Transition State Theory J.D. Doll Transition Path Sampling Christoph Dellago Simulating Reactions that Occur Once in a Blue Moon Giovanni Ciccotti, Raymond Kapral, and Alessandro Sergi Order Parameter Approach to Understanding and Quantifying the Physico-Chemical Behavior of Complex Systems Ravi Radhakrishnan and Bernhardt L. Trout Determining Reaction Mechanisms Blas P. Uberuaga and Arthur F. Voter

1567 1573 1585 1597

1613 1627

xxxiv 5.7 5.8

5.9 5.10

5.11 5.12

5.13 5.14 5.15

5.16 5.17

Detailed table of contents Stochastic Theory of Rate Processes Abraham Nitzan Approximate Quantum Mechanical Methods for Rate Computation in Complex Systems Steven D. Schwartz Quantum Rate Theory: A Path Integral Centroid Perspective Eitan Geva, Seogjoo Jang, and Gregory A. Voth Quantum Theory of Reactive Scattering and Adsorption at Surfaces Axel Groß Stochastic Chemical Kinetics Daniel T. Gillespie Kinetic Monte Carlo Simulation of Non-Equilibrium Lattice-Gas Models: Basic and Refined Algorithms Applied to Surface Adsorption Processes J.W. Evans Simple Models for Nanocrystal Growth Pablo Jensen Diffusion in Solids Göran Wahnström Kinetic Theory and Simulation of Single-Channel Water Transport Emad Tajkhorshid, Fangqiang Zhu, and Klaus Schulten Simplified Models of Protein Folding Hue Sun Chan Protein Folding: Detailed Models Vijay Pande

1635

1673 1691

1713 1735

1753 1769 1787

1797 1823 1837

Chapter 6. Crystal Defects 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Point Defects C.R.A. Catlow Point Defects in Metals Kai Nordlund and Robert Averback Defects and Impurities in Semiconductors Chris G. Van de Walle Point Defects in Simple Ionic Solids John Corish Fast Ion Conductors Alan V. Chadwick Defects and Ion Migration in Complex Oxides M. Saiful Islam Introduction: Modeling Crystal Interfaces Sidney Yip and Dieter Wolf

1851 1855 1877 1889 1901 1915 1925

Detailed table of contents 6.8 6.9 6.10

6.11 6.12 6.13

Atomistic Methods for Structure–Property Correlations Sidney Yip Structure and Energy of Grain Boundaries Dieter Wolf High-Temperature Structure and Properties of Grain Boundaries Dieter Wolf Crystal Disordering in Melting and Amorphization Sidney Yip, Simon R. Phillpot, and Dieter Wolf Elastic Behavior of Interfaces Dieter Wolf Grain Boundaries in Nanocrystalline Materials Dieter Wolf

xxxv

1931 1953

1985 2009 2025 2055

Chapter 7. Microstructure 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8

7.9

7.10 7.11 7.12 7.13

Introduction: Microstructure David J. Srolovitz and Long-Qing Chen Phase-Field Modeling Alain Karma Phase-Field Modeling of Solidification Seong Gyoon Kim and Won Tae Kim Coherent Precipitation – Phase Field Method C. Shen and Y. Wang Ferroic Domain Structures using Ginzburg–Landau Methods Avadh Saxena and Turab Lookman Phase-Field Modeling of Grain Growth Carl E. Krill III Recrystallization Simulation by Use of Cellular Automata Dierk Raabe Modeling Coarsening Dynamics using Interface Tracking Methods John Lowengrub Kinetic Monte Carlo Method to Model Diffusion Controlled Phase Transformations in the Solid State Georges Martin and Frédéric Soisson Diffusional Transformations: Microscopic Kinetic Approach I.R. Pankratov and V.G. Vaks Modeling the Dynamics of Dislocation Ensembles Nasr M. Ghoniem Dislocation Dynamics – Phase Field Yu U. Wang, Yongmei M. Jin, and Armen G. Khachaturyan Level Set Dislocation Dynamics Method Yang Xiang and David J. Srolovitz

2083 2087 2105 2117 2143 2157 2173

2205

2223 2249 2269 2287 2307

xxxvi


7.14 Coarse-Graining Methodologies for Dislocation Energetics and Dynamics J.M. Rickman and R. LeSar 7.15 Level Set Methods for Simulation of Thin Film Growth Russel Caflisch and Christian Ratsch 7.16 Stochastic Equations for Thin Film Morphology Dimitri D. Vvedensky 7.17 Monte Carlo Methods for Simulating Thin Film Deposition Corbett Battaile 7.18 Microstructure Optimization S. Torquato 7.19 Microstructural Characterization Associated with Solid–Solid Transformations J.M. Rickman and K. Barmak

2325 2337 2351 2363 2379

2397

Chapter 8. Fluids 8.1 8.2

8.3

8.4 8.5

8.6 8.7

8.8

Mesoscale Models of Fluid Dynamics Bruce M. Boghosian and Nicolas G. Hadjiconstantinou Finite Difference, Finite Element and Finite Volume Methods for Partial Differential Equations Joaquim Peiró and Spencer Sherwin Meshless Methods for Numerical Solution of Partial Differential Equations Gang Li, Xiaozhong Jin, and N.R. Aluru Lattice Boltzmann Methods for Multiscale Fluid Problems Sauro Succi, Weinan E, and Efthimios Kaxiras Discrete Simulation Automata: Mesoscopic Fluid Models Endowed with Thermal Fluctuations Tomonori Sakai and Peter V. Coveney Dissipative Particle Dynamics Pep Español The Direct Simulation Monte Carlo Method: Going Beyond Continuum Hydrodynamics Francis J. Alexander Hybrid Atomistic–Continuum Formulations for Multiscale Hydrodynamics Hettithanthrige S. Wijesinghe and Nicolas G. Hadjiconstantinou

2411

2415

2447 2475

2487 2503

2513

2523

Chapter 9. Polymers and Soft Matter 9.1 9.2

Polymers and Soft Matter L. Mahadevan and Gregory C. Rutledge Atomistic Potentials for Polymers and Organic Materials Grant D. Smith

2555 2561

Detailed table of contents 9.3 9.4 9.5 9.6 9.7

9.8 9.9

Rotational Isomeric State Methods Wayne L. Mattice Monte Carlo Simulation of Chain Molecules V.G. Mavrantzas The Bond Fluctuation Model and Other Lattice Models Marcus Müller Stokesian Dynamics Simulations for Particle Laden Flows Asimina Sierou Brownian Dynamics Simulations of Polymers and Soft Matter Patrick S. Doyle and Patrick T. Underhill Mechanics of Lipid Bilayer Membranes Thomas R. Powers Field-Theoretic Simulations Venkat Ganesan and Glenn H. Fredrickson

xxxvii

2575 2583 2599 2607

2619 2631 2645

Plenary Perspectives P1 P2 P3 P4 P5 P6

P7

Progress in Unifying Condensed Matter Theory Duane C. Wallace The Future of Simulations in Materials Science D.P. Landau Materials by Design Gregory B. Olson Modeling at the Speed of Light J.D. Joannopoulos Modeling Soft Matter Kurt Kremer Drowning in Data – A Viewpoint on Strategies for Doing Science with Simulations Dierk Raabe Dangers of “Common Knowledge” in Materials Simulations Vasily V. Bulatov

Quantum Simulations as a Tool for Predictive Nanoscience Giulia Galli and François Gygi P9 A Perspective of Materials Modeling William A. Goddard III P10 An Application Oriented View on Materials Modeling Peter Gumbsch P11 The Role of Theory and Modeling in the Development of Materials for Fusion Energy Nasr M. Ghoniem

2659 2663 2667 2671 2675

2687

2695

P8

2701 2707 2713

2719

xxxviii


P12 Where are the Gaps? Marshall Stoneham P13 Bridging the Gap between Quantum Mechanics and Large-Scale Atomistic Simulation John A. Moriarty P14 Bridging the Gap between Atomistics and Structural Engineering J.S. Langer P15 Multiscale Modeling of Polymers Doros N. Theodorou P16 Hybrid Atomistic Modelling of Materials Processes Mike Payne, Gábor Csányi, and Alessandro De Vita P17 The Fluctuation Theorem and its Implications for Materials Processing and Modeling Denis J. Evans P18 The Limits of Strength J.W. Morris, Jr. P19 Simulations of Interfaces between Coexisting Phases: What Do They Tell us? Kurt Binder P20 How Fast Can Cracks Move? Farid F. Abraham P21 Lattice Gas Automaton Methods Jean Pierre Boon P22 Multi-Scale Modeling of Hypersonic Gas Flow Iain D. Boyd P23 Commentary on Liquid Simulations and Industrial Applications Raymond D. Mountain P24 Computer Simulations of Supercooled Liquids and Glasses Walter Kob P25 Interplay between Materials Theory and High-Pressure Experiments Raymond Jeanloz P26 Perspectives on Experiments, Modeling and Simulations of Grain Growth Carl V. Thompson P27 Atomistic Simulation of Ferroelectric Domain Walls I-Wei Chen

2731

2737

2749 2757 2763

2773 2777

2787 2793

2805 2811

2819 2823

2829

2837 2843


xxxix

P28 Measurements of Interfacial Curvatures and Characterization of Bicontinuous Morphologies Sow-Hsin Chen

2849

P29 Plasticity at the Atomic Scale: Parametric, Atomistic, and Electronic Structure Methods Christopher Woodward P30 A Perspective on Dislocation Dynamics Nasr M. Ghoniem P31 Dislocation-Pressure Interactions J.P. Hirth P32 Dislocation Cores and Unconventional Properties of Plastic Behavior V. Vitek P33 3-D Mesoscale Plasticity and its Connections to Other Scales Ladislas P. Kubin P34 Simulating Fluid and Solid Particles and Continua with SPH and SPAM Wm.G. Hoover P35 Modeling of Complex Polymers and Processes Tadeusz Pakula P36 Liquid and Glassy Water: Two Materials of Interdisciplinary Interest H. Eugene Stanley P37 Material Science of Carbon Wesley P. Hoffman P38 Concurrent Lifetime-Design of Emerging High Temperature Materials and Components Ronald J. Kerans P39 Towards a Coherent Treatment of the Self-Consistency and the Environment-Dependency in a Semi-Empirical Hamiltonian for Materials Simulation S.Y. Wu, C.S. Jayanthi, C. Leahy, and M. Yu

2865 2871 2879

2883 2897

2903 2907

2917 2923

2929

2935

INTRODUCTION Sidney Yip Department of Nuclear Science and Engineering, Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 (USA)

The way a scientist looks at the materials world is changing dramatically. Advances in the synthesis of nanostructures and in high-resolution microscopy are allowing us to create and probe assemblies of atoms and molecules at a level that was unimagined only a short time ago – the prospect of manipulating materials for device applications, one atom at a time, is no longer a fantasy. Being able to see and touch the materials up close means that we are more interested than ever in understanding their properties and behavior at the atomic level. Another factor which contributes to the present state of affairs is the advent of large-scale computation, once a rare and highly sophisticated resource accessible only to a few privileged scientists. In the past few years materials modeling, in the broad sense of theory and simulation in integration with experiments, has emerged as a field of research with unique capabilities, most notably the ability to analyze and predict a very wide range of physical structures and phenomena. Some would now say the modeling approach is becoming an equal partner to theory and experiment, the traditional methods of scientific inquiry. There are certain problems in the fundamental description of matter, previously regarded as intractable, now are amenable to simulation and analysis. The ab initio calculation of solid-state properties using electronic-structure methods and the direct estimation of free energies based on statistical mechanical formulations are just two examples where predictions are being made without input from experiments. Because materials modeling draws from all the disciplines in science and engineering, it greatly benefits from cross fertilization within a multidisciplinary community. There is recognition that Computational Materials is just as much a field as Computational Physics or Chemistry; it offers a robust framework for focused scientific studies and exchanges, from the introduction of new university curricula to the formation of centers for collaborative research among academia, corporate and government laboratories. A basic appeal to all members of the growing community 1 S. Yip (ed.), Handbook of Materials Modeling, 1–5. c 2005 Springer. Printed in the Netherlands.

2

S. Yip

is the challenge and opportunity of solving problems that are fundamental in nature and yet have great technological impact, problems spanning the disciplines of physics, chemistry, engineering and biology. Multiscale modeling has come to symbolize the emerging field of computational materials research. The idea is to link simulation models and techniques across the micro-to-macro length and time scales, with the goal of analyzing and eventually controlling the outcome of critical materials processes. Invariably these are highly nonlinear, inhomogeneous, or non-equilibrium phenomena in nature. In this paradigm, electronic structure would be treated by quantum mechanical calculations, atomistic processes by molecular dynamics or Monte Carlo simulations, mesoscale microstructure evolution by methods such as finite-element, dislocation dynamics, or kinetic Monte Carlo, and continuum behavior by field equations central to continuum elasticity and computational fluid dynamics. The vision of multiscale modeling is that by combining these different methods, one can deal with complex problems in a much more comprehensive manner than when the methods are used individually [1]. “Modeling is the physicalization of a concept, simulation is its computational realization.”

This is an oversimplified statement. On the other hand, it is a way to articulate the intellectual character of the present volume. This Handbook is certainly about modeling and simulation. Many would agree that conceptually the process of modeling ought to be distinguished from the act of simulation. Yet there seems to be no consensus on how the two terms should be used to show that each plays an essential role in computational research. Here we suggest a brief all-purpose definition (admittedly lacking specificity). By concept we have in mind an idea, an idealization, or a picture of a system (a scenario of a process) which has the connotation of functionality. For an example consider the subway map of Boston. Although it gives no information about the city streets, its purpose is to display the connectivity of the stations – few would dispute that for the given purpose it is a superb physical construct enabling any person to navigate from point A to point B [2]. So it is with our twopart definition; it is first a thoughtfully simplified representation of an object to be studied, a phenomenon, or a process (modeling), then it is the means with which to investigate the model (simulation). Notice also that when used together modeling and simulation implies an element of coordination between what is to be studied and how the study is to be conducted.

Length/Time Scales in Materials Modeling Many physical phenomena have significant manifestations on more than one level of length or time scale. For example, wave propagation and

Introduction

3

attenuation in a fluid can be described at the continuum level using the equations of fluid dynamics, while the determination of shear viscosity and thermal conductivity is best treated at the level of molecular dynamics. While each level has its own set of relevant phenomena, an even more powerful description would result if the microscopic treatment of transport could be integrated into the calculation of macroscopic flows. Generally speaking, one can identify four distinct length (and corresponding time) scales where materials phenomena are typically studied. As illustrated in Fig. 1, the four regions may be referred to as electronic structure, atomistic, microstructure, and continuum. Imagine a piece of material, say a crystalline solid. The smallest length scale of interest is about a few angstroms (10−8 cm). On this scale one deals directly with the electrons in the system which are governed by the Schrödinger equation of quantum mechanics. The techniques that have been developed for solving this equation are extremely computationally intensive, as a result they can be applied only to small simulation systems, at present no more than about 300 atoms. On the other hand, these calculations are theoretically the most rigorous; they are particularly valuable for developing and validating more approximate but computationally more efficient descriptions. The scale at the next level, spanning from tens to about a thousand angstroms, is called atomistic. Here discrete particle simulation techniques, molecular dynamics (MD) and Monte Carlo (MC), are well developed,

Figure 1. Length scales in materials modeling showing that many applications in our physical world take place on the micron scale and higher, while our basic understanding and predictive ability lie at the microscopic levels.

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requiring the specification of an empirical classical interatomic potential function with parameters fitted to experimental data and electronic-structure calculations. The most important feature of atomistic simulation is that one can now study a system of large number of atoms, at present as many as 109 . On the other hand, because the electrons are ignored atomistic simulations are not as reliable as ab initio calculations. Above the atomistic level the relevant length scale is a micron (104 angstroms). Whether this level should be called microscale or mesoscale is a matter for which convention has not been clearly established. The simulation technique commonly in use is finite-element calculations (FEM). Because many useful properties of materials are governed by the microstructure in the system, this is perhaps the most critical level for materials design. However, the information required to carry out such calculations, for example, the stiffness matrix, or any material-specific physical parameters, has to be provided from either experiment or calculations at the atomistic or ab initio level. To a large extend, the same can be said for the continuum-level methods, such as computational fluid dynamics (CFD) and continuum elasticity (CE). The parameters needed to perform these calculations have to be supplied externally. There are definite benefits when simulation techniques at different scales can be linked. Continuum or finite-element methods are often most practical for design calculations. They require parameters or properties which cannot be generated within the methods themselves. Also they cannot provide the atomic-level insights needed for design. For these reasons continuum and finite element calculations should be coupled to atomistic and ab initio methods. It is only when methods at different scales are effectively integrated that one can expect materials modeling to give fundamental insight as well as reliable predictions across the scales. The efficient bridging of the scales in Fig. 1 is a significant challenge in the further development of multiscale modeling. The classification of materials modeling and simulation in terms of length and time scales is but one way of approaching the subject. The point of Fig. 1 is to emphasize the theoretical and computational methods that have been developed to describe the properties and behavior of physical systems, but it does not address other equally important issues, those of applications. One might imagine discussing materials modeling through a matrix of methods and applications which could be useful for displaying their connection and particular suitability. This would be quite difficult to carry out at present because there are not enough clear-cut case studies in the literature to make the construction of such a matrix meaningful. From the standpoint of knowing what methods are best suited for certain problems, materials modeling is a field still in its infancy.

Introduction

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An Overview of the Handbook The Handbook is laid out in 9 chapters, dealing with modeling and simulation methods (Part A) and models for specific areas of studies (Part B). In Part A the first three chapters describe modeling concepts and simulation techniques at the electronic (Chapter 1), atomistic (Chapter 2), and mesoscale (Chapter 3) levels, in the spirit of Fig. 1. In contrast Chapter 4 describes a variety of methods based on mathematical analysis. The chapters in Part B focus on systems in which basic studies have been carried out. Chapter 5 treats rate processes where time-scale problems are just as important and challenging as length-scale problems. The next four chapters cover a range of physical structures, crystal defects (Chapter 6) and microstructure (Chapter 7) in solids, various models and methods for fluid simulation (Chapter 8), and models of polymer and soft matter (Chapter 9). In each chapter there are other significant topics which have not been included; for these we recommend the readers consult the references given in each article. Each chapter begins with an introduction which serves to connect the individual articles in the chapter with the broad themes that are relevant to our growing community. While no single chapter attempts to be inclusive in treating the many important aspects of materials modeling, even with restrictions to fundamental methods and models, hopefully, the entire Handbook is a first step in that direction. The Handbook also has a special section which we call Plenary Perspectives. This is a collection of commentaries by recognized authorities in the materials modeling or related fields. Each author was invited to write briefly on a topic that would give the readers, especially the students, insight on different issues in materials modeling. Together with the 9 chapters these perspectives are meant to inform the future workers coming into this exciting field.

References [1] S. Yip, “Synergistic science,” Nature Mater., 3, 1–3, 2003. [2] M. Ashby, “Modelling of materials problems,” J. Comput.-Aided Mater. Des., 3, 95–99, 1996.

1.1 UNDERSTAND, PREDICT, AND DESIGN Nicola Marzari Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

Electronic-structure approaches are changing dramatically the way much theoretical and computational research is done. This success derives from the ability to characterize from first-principles many material properties with an accuracy that complements or even augments experimental observations. This accuracy can extend beyond the properties for which a real-life experiment is either feasible or just cost-effective, and it is based on our ability to compute and understand the quantum-mechanical behavior of interacting electrons and nuclei. Density-functional theory, for which the Nobel prize in chemistry was awarded in 1998, has been instrumental to this success, together with the availability of computers that are now routinely able to deal with the complexity of realistic problems. The extent of such revolution should not be underestimated, notwithstanding the many algorithmic and theoretical bottlenecks that await resolution, and the existence of hard problems rarely amenable to direct simulations. Since ab-initio methods combine fundamental predictive power with atomic resolution, they provide a quantitatively-accurate first step in the study and characterization of new materials, and the ability to describe with unprecedented control molecular architectures exactly at those scales (hundreds to thousands of atoms) where some of the most promising and undiscovered properties are to be engineered. In the current effort to control and design the properties of novel molecules, materials, and devices, firstprinciples approaches constitute thus a unique and very powerful instrument. Complementary strategies emerge: • Insight: First-principles simulations provide a unique connection between microscopic and macroscopic properties. When partnered with experimental tools – from spectroscopies to microscopies – they can deliver unique insight and understanding on the detailed arrangements of atoms 9 S. Yip (ed.), Handbook of Materials Modeling, 9–11. c 2005 Springer. Printed in the Netherlands.

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and molecules, and on their relation to the observed phenomena. Gedanken computational experiments can be used to prove or probe cause-effect relationships in ways that are different, and novel, compared with our established approaches. • Control: Microscopic simulations provide an unprecedented degree of control on the systems studied. While macroscopic behavior often emerges from complexity – thus explaining all the ongoing efforts in overcoming the time- and length-scale limitations – fundamental understanding needs to be built from the bottom-up, under the carefully controlled condition of a computational experiment. Simulations can offer early and accurate insights on complex materials that are challenging to control or characterize. • Design: Quantitatively accurate predictions of materials’ properties provide us with an unprecedented freedom, a “magic wand” that can be used with ingenuity to try and engineer novel material properties. Intuitions can often be rapidly validated, shifting and focusing appropriately the synthetic challenge to the later stages, once a promising class of materials has been identified. • Optimization: Finally, the systematic exploration of material properties inside or across different classes of materials can highlight the potential for absolute or differential improvements. Stochastic techniques such as data mining and optimization then identify the most promising candidates, narrowing down the field of structures to be targeted in real-life testing. While the extent and scope of this emerging discipline are nothing short of revolutionary, researchers in the field face key challenges that are worth remembering: achieving thermodynamical accuracy, bridging length-scales, and overcoming time-scales limitations. It is unlikely that an overarching solution to these problems will appear, and much of the art of modeling goes into solving these challenges for the problem at hand. It is nevertheless important to remark the role of correlations: whenever the typical correlation lengths become smaller then the size of the simulation box (e.g., for a liquid studied in periodic-boundary conditions), the system studied becomes virtually infinite, and the finite-size bias irrelevant. The articles presented in this volume offer a glimpse on the panorama of electronic-structure modeling; in such distinguished company, it would be inappropriate for me to condense such diverse and exciting contributions into a few sentences. I will leave the science to the authors, and conclude with a few statements on future developments. The continuous improvement in the price vs. performance ratio for commodity CPUs is now widely apparent. Whereas computational resources seem never enough, and the desire of a longer and bigger simulation is always looming, we are now in the position where even a single desktop is sufficient to

Understand, predict, and design

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sustain research of world-class quality (of course, human resources are even more precious, and human ingenuity can be sometimes light-heartedly traded for sheer computational power). This availability of computer power is now combined with the availability of state-of-the-art computer packages – some of them freely distributed and developed under a shared-community, public-license model akin to that, e.g., of Linux. The net result has been that “computational laboratories” around the world have been increasing in capability with a speed comparable to Moore’s law, their hardware and software infrastructures replicated almost at the flick of a switch. Some conclusions can be attempted: • The geographic distribution of researchers in this field might change significantly. World-class science can now be done inexpensively and extensively, and knowhow and human resources become almost exclusively the most precious commodities. • Publicly available electronic structure packages take the role of internationally shared infrastructures; in perfect analogy with the way brick-andmortar facilities (such as synchrotrons) serve many groups in different countries. It could even be argued that investment in “computational infrastructures” (electronic-structure packages) can have comparable benefits, and a remarkable cost structure. • While these technologies become faster, more robust, and prettier, they also become more and more complex, often requiring years of training to be mastered – content and expertise could also be developed and freely shared following similar public-license models. The last point brings us back to one of the greatest challenges, and one for which we hope this Handbook will bring a positive contribution: how to avoid trading contents for form, critical thinking for indiscriminate simulations. In T.S. Eliot’s words: “The last temptation is the greatest treason: To do the right deed for the wrong reason.”

1.2 CONCEPTS FOR MODELING ELECTRONS IN SOLIDS: A PERSPECTIVE Marvin L. Cohen University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USA

1.

The Electron’s Central Role

It’s clear that an understanding of the behavior of electrons in solids is essential for explaining and predicting solid state properties. Electrons provide the glue holding solids together, and hence they are central in determining structural, mechanical and vibrational properties. Under the influence of electromagnetic fields, electrical current transport involves electron transport for most solids. Optical properties for many ranges of frequency are dominated by electronic transitions. Understanding superconductivity, magnetism, dielectric properties, ferroelectricity, and most properties of solids requires a detailed knowledge of “electronic structure” which is the term associated with the study of electronic energy levels, but more broadly a general label for the subfield of condensed matter physics which is focused on the properties of electrons in solids. In the end, modeling, simulating, calculating, and computing refer to producing equations, numbers or pictures which describe, explain, and predict properties. So this general area has always had a mixed set of goals. Theoretical researchers vary in their emphasis on these goals. For example, some theorists are focused on explaining phenomena with the simplest possible models containing the fundamental physics. A good example is the Bardeen–Cooper– Schrieffer (BCS) [1] theory of superconductivity which is one of the great achievements of 20th century physics. This theory brought new concepts, but the modeling of the electrons forming Cooper pairs considered electrons in free electron states because calculating normal-state properties for particular solids was not very far along in 1957. As a result, computing transition

13 S. Yip (ed.), Handbook of Materials Modeling, 13–26. c 2005 Springer. Printed in the Netherlands.

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temperatures for specific solids using BCS theory was, and still is, difficult; and, for some researchers, this was viewed at the time as a defect in the fundamental theory, which it was not. There are theorists interested in numerical precision. They continually push at the forefront of computer science and applied mathematics to develop consistent approaches that can deal with properties of clusters, molecules, and complex solids with many atoms in a unit cell. Sometimes these researchers have strong overlap with computer scientists and engineers and even get involved in hardware development. Perhaps the largest and most dominant group of researchers in modeling solids at this time are theorists motivated by particular experimental properties or phenomena. Unlike the researchers interested only in phenomena, they are trying to calculate these properties for “real materials.” For these theorists, it is essential that interactions among electrons and ionic cores not be replaced by a constant (as in the BCS model), and electrons are not viewed as completely free or as atomic states. They want the appropriate description of the electronic states for the material at hand and a computational approach to calculate measured properties. Successful comparisons with experiments is the goal, and it is the degree of accuracy in these comparisons which measures the worth of the calculation rather than numerical precision. In the papers presented in this volume, the reader will find authors with research goals having varying degrees of “accuracy for explaining and predicting properties” versus “calculational precision” as a primary goal. Irrespective of motivation, an essential component for modeling is the conceptual base. In other words, the way we picture solids on a microscopic or nanoscopic level.

2.

Conceptual Base

Under pressure, gases made of atoms can condense to become liquids with molecular units of clusters or atoms, and then, with more pressure, they generally transform into solids. So most models of solids involve a picture of atoms interacting to form a periodic array of ions with electrons in various geometric configurations. Modern electron charge density plots [2] have influenced our mental images of covalent, ionic and metallic bonding using contour maps and pictures of dense dots to represent electrons confined in bonds appropriate for covalent or ionic semiconductors or spread out charge maps to represent electrons in metals. As an example, Fig. 1 shows the electronic charge density in the (110) plane for carbon and silicon both in the diamond structure. The bond lengths are 1.54 Å and 2.35 Å, respectively. It has been said that carbon is the basis of biology while silicon is the basis of geology, and it is the nature of the covalent bonds in these two systems which determines these properties. As

Concepts for modeling electrons in solids: a perspective

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Valence charge density (110 plan)

Figure 1. Contour maps of the valence electron charge density of C and Si in the diamond structure to illustrate a visual perception of covalent bonding.

shown in the figure, the carbon bond has two maxima while there is essentially one for silicon. The electrons in carbon can form sp2 hybrids for three-fold coordination and multiple bonds while elemental silicon at ambient pressures and temperatures forms sp3 bonds and is tetrahedrally coordinated. If solids are made of atoms, then it is the job of those modeling electronic behavior to illustrate this evolution of electrons from being localized around ions to the formation of covalent and metallic bonds. For this purpose, the old atomic models of Thomson and Newton work well pictorially. Thomson’s plum pudding model resembled our modern picture of jellium with a positive smeared out background representing the ions and then electrons existing in this background. Unlike jellium where the electrons are smeared out, Thompson’s electrons were plums. Hence, the essential difference is that the electrons in the jellium model are treated quantum mechanically and despite the fact that they can be excited out of the metal and look like Thomson’s plums, inside the metal they are itinerant. The resulting jellium model works for many properties of metals. In contrast to Thomson’s atomic model, Newton’s atoms had hooks, and it takes little imagination to see how these atoms with interlocking hooks can be used to form the basis of covalent and ionic crystals. However, again we need to show how the electrons can become hooks and form covalent or ionic bonds, and this requires quantum mechanics.

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Our modern quantum atom description is based on wavefunctions which yield probabilities for electron density. So, we can determine “exactly where an electron probably is.” This brings up the challenge of Dirac [3] posed after the development of quantum theory: “The underlying physical laws necessary for a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.” It is probably safe to say that to some extent we have answered Dirac’s challenge and we can now model electrons in some solids. Modern computing machines and new algorithms for solving complex equations have been an important ingredient, but just as important and probably more so is the conceptual base or modern “picture” of a solid that is inherently quantum mechanical.

3.

Standard Model

Since solids are made of atoms, why not start with atomic wavefunctions and perturb them. This works; it is the tight binding model which has had great success especially for systems where electrons are not “too itinerant.” Methods like this represent a natural path for quantum chemists who start from atoms and study molecules. This is also a logical path for doing computations of finite small systems like clusters or nanostructures. Another approach is to think of the free electron metal where each atom contributes its electrons to the soup of electrons in a solid. Perturbations on this model, such as the nearly free electron model, represent a very successful approach. Both of these very different paths will be represented in this volume and both are useful. The latter approach is conceptually the more difficult because in some sense it starts from a gas of electrons instead of electrons bound to atoms, but it has had widespread use and leads to very useful methods. One generally restricts the basis set to plane waves which are appropriate for free electrons, but there are other approaches. So in this model, sometimes referred to as the “Standard Model,” one can visualize an array of positive cores in a background sea of valence electrons coming from the atoms. In the plane wave pseudopotential version of this model, there are two types of particles: valence electrons and positive cores. For a study of a particular solid, one arranges the cores in a periodic array and uses a plane wave basis set for the quantum mechanical calculations. The particles interact in the following way. Core–core interactions which can be viewed as point-like Coulombic objects which can be represented by Madelung type sums to give accurate descriptions of these interactions. The electron–core


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interaction is modeled using pseudopotentials [4, 5] and the electron–electron interactions are dealt with using density functional theory [6]. It is amazing how robust this model is when one considers the fact that for over 50 years beginning with approaches like the OPW [7] and APW [8] methods, researchers struggled with the band structure dilemma of how to describe electrons which are atomic-like near the cores and free electron-like between the cores. The conceptual breakthrough was the pseudopotential which accounted for the Pauli forces near the cores and led to weak effective potentials. Early versions were empirical [9] and fit to optical data, but eventually it became possible to construct pseudopotentials from first principles. Further discussion of pseudopotentials will be given in this volume. A convenient approach using the standard model is to calculate total energies [10] for model solids where the atoms are arranged in different configurations and only atomic information such as atomic numbers and masses are used as input. Hence different candidate crystal structures can be compared at varying volumes or pressures to explain the stability of observed structures or predict new ones. Here we find a major application of this method since in addition to structural stability, properties such as lattice constants, elastic constants, bulk moduli, vibrational spectra, and even electron–phonon and anharmonic properties of solids can be evaluated. The techniques connected to this method have evolved and they too will be discussed in this volume. Using plane waves or other basis sets and even tight binding schemes, there appears to be consensus in this area. Particularly dramatic early successes were the successful predictions of new high pressure crystalline phases of Si and Ge, and the successful prediction of superconductivity in high pressure phases of Si [11]. A more recent success is a detailed explanation of the unusual superconducting properties of MgB2 [12].

4.

Now and Later

So what are the modern challenges? If in fact we have to some extent answered Dirac’s challenge of 75 years ago, what’s next? A few obvious areas at this point for future exploration and development are: studies of electron behavior and transport in confined or small systems; development of better order N methods for calculating electronic properties so that more complex systems can be addressed; further development of theories designed to study excited states for optical and related properties; and the evaluation of the effects of strong electron correlation. In addition, more semi-empirical models should be developed since they were important in the past, and there is reason to believe these will contribute to future development.

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Confinement

It is clear that confinement sets the energy scale whether we are considering protons in nuclei, electrons in atoms or clusters, and to some extent, electrons in nano and macro materials. In the latter case, there are confinement scales set by the overall object size and by the components such as atoms or unit cells. One gets a good sense of how this works when considering shell models for nuclei or for alkali metal clusters [13, 14]. The so-called magic numbers emerge for the number of atoms in a cluster and stability of energy shells. The energy shell structure can influence overall structure and properties. For macrosystems, it is the atoms, their spacings, and the unit cell which set the energy scales. For confinement in macrosystems, their large sizes lead to such small energy splittings that the available energy states appear continuous even at the lowest attainable temperatures. However, size effects for small systems and surfaces can bring in a new scale and methods such as the supercell method [15] can be used to address situations like this where translational symmetry is lost. Clusters are good examples of systems where confinement effects can be dominant. Here, supercell techniques can be used, but real space methods, such as those described in this volume, can cover a wide range of situations where size matters. Nanotubes, peapods, atomic chains, quantum dots, large molecules, network systems, polymers, fullerenes, etc. are all examples of systems where electron confinement can lead to significant alterations in wavefunctions and hence properties. Transport is a particularly interesting field of study on the nanoscale. There are a number of research groups focused on the formulation of a transport theory for electron conduction through molecules and nanosystems. Here the vexing problem of contacts must be dealt with, and, for chains of atoms, questions related to even and odd numbers of atoms are relevant. Because the nanoscale is of interest to physicists, chemists, biologists, engineers, materials scientists, and computer scientists, there has been a great deal of synergy between these disciplines and surprising demonstrations of the commonality of the problems facing researchers in these fields. One example is molecular motors. The problem of understanding friction in molecular motors with nanotube bearings is not very different from similar questions posed by biologists studying friction in biomotors. Another example is the application of nanostructures for devices. Figure 2 shows the merging of an (8,0) semiconducting carbon nanotube with a (7,1) metallic carbon nanotube. This is achieved by inserting a defect between them with adjacent five-member and seven-member rings of carbon atoms. The result is a Schottky barrier whose properties are determined just by the action of a handful of atoms at the interface.


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Figure 2. A schematic drawing of Schottky barrier composed of semiconducting (8,0) and metallic (7,1) carbon nanotubes.

6.

Methods

Many researchers are exploring so-called “order N ” methods for attacking large or complex systems. As mentioned before, real space methods also appear promising. Researchers have developed new schemes for attempting to do inversions of matrices employing methods that resemble a “divide and conquer” approach. Schematically, a large matrix can be cut down through different point sampling into smaller units. The developments in this area are encouraging, and the collaborations between mathematicians doing numerical analysis and theoretical physicists and chemists appear to be productive. Another approach is to acknowledge that most problems on solids are multi-scale problems. A multi-scale approach can be most simply illustrated

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by an example where one calculates microscopic parameters and uses them along with semi-empirical models at a larger scale. Many sophisticated versions of this approach have been developed in recent years. Some of this very interesting research is described in detail in this volume.

7.

Excited States

Generally the problem which arises when excited states of solids are considered is that many of the standard methods used to compute the effects of electron–electron interactions use the local density approximation (LDA) which is not directly applicable for calculating excited state properties. For example, in the total energy LDA approach [10], ground state properties such as lattice constants and mechanical properties are determined quite accurately. However, in an optical process, photons create electron–hole pairs in the solid which influence the excited state properties of the many electron system. When band gaps of semiconductors are evaluated from energy bands obtained using the LDA methods, there is an underestimate of the band gap typically by a factor of about two. In some cases metallic behavior is predicted for systems known to be semiconductors. The so called “band gap” problem was of central concern when applications of the “standard model,” which were so successful for ground state properties, became clearly unusable for computing band gaps. The overall topology of the energy bands was approximately right and in agreement with empirical models and experimental data where checks were possible, but the details were wrong. Early suggestions such as the “scissors model” where levels were artificially shifted by adding a constant energy to the calculated bandgap were considered to be “band aids” and not cures. Although this is still an active area of research, there are methods for evaluating quasiparticle energies. One of the most successful is the GW method [16] which works for a broad class of solids. Two major ingredients in this approach are the inclusion of electron self-energy effects and the modulation of the charge density in the crystal. This latter feature allows for the effects on exchange and correlation energies arising from the concentrations of electrons into bonds as an example. Another feature of the properties of the excited state which must be addressed is the role of electron–hole interactions. Two of the most dramatic effects are the formation of excitons and the alteration of oscillator strengths arising from electron–hole interactions. Again, this is an active area of research, but a workable theory is available [17] where the Bethe–Salpeter approach for two particle scatterings is adopted and applied along with the GW machinery. Forces in the excited state and other special features arising


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from considering these interactions can be calculated. Comparisons between this method and others, such as time dependent density functional approaches [18], quantum Monte Carlo methods and more quantum chemistry oriented approaches are yielding new insights into this area. It appears that research in this field will remain active for some time as there are many possible applications.

8.

Strongly Correlated Electrons

At this time, it is commonly believed that a forefront field of condensed matter theory is the study of strongly correlated electrons. However, as in the case of defining biophysics, the image of what is meant by this field of study varies with individuals. As was described at the beginning of this article, there are theorists attempting to use simplified models to get the essence of the physics associated with problems related to strongly correlated electron systems. A prime example is the large amount of research devoted to the study of superconductivity in copper oxide systems. Here it is clear why theorists are motivated. Electron correlation effects are important, there is no consensus yet on the underlying electron pairing mechanism, and the normal state and superconducting properties are very interesting. So the application of models such as the Hubbard Model has attracted a large number of theoretical researchers. Many interesting proposals for explaining the electronic properties of the oxides using Hubbard-like models have been advanced. At present, this is an active field, but as mentioned before, there is still no general agreement on “the” appropriate description of these systems, and in general, there is a lack of definitive proof of good theoretical–experimental agreement. The more ab initio approaches designed for specific materials are beginning to make some impact on this area. Despite the known shortcomings of applying band structure calculations based on a density functional approach to materials of this kind, these were among the most useful calculations for interpreting experiments like photoelectron spectroscopy aimed at determining electronic structure. The Fermi surface topology and other electronic characteristics were explored with considerable success through experimental– theoretical comparisons along with reasonable empirical adjustments to the electronic structure calculations. Currently, efforts are underway for a more frontal assault on this problem. By combining local spin density calculations together with Hubbard-like terms to account for electron–electron repulsion, more realistic electronic structure calculations are being done. Variations and improvements on these “LSDA + U” approaches [19] including the use of pseudopotentials appear to be promising. And it is possible that the more first

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principles, materials-motivated approach may make important contributions to the conceptual development of this field.

9.

Empirical Models

Just as the atomic models of Thompson and Newton described earlier help to form a basis for the conceptual picture of electronic behavior, other empirical and semi-empirical models had a considerable effect on the the development of this field of study. The Thomas–Fermi model which allowed calculation of electron screening effects, Slater’s and Wigner’s formulas for evaluating the effects of exchange and correlation gave important insight into the role of these many body effects. Free electron and nearly free electron models were extremely important as were empirical tight binding models for estimating band structure effects. An example which illustrates the transition from an empirical model designed to explain experimental data into a first-principles approach is the Empirical Pseudopotential Method (EPM). In this approach [9], a few form factors (usually three per atom) of the potential in the unit cell are fit to yield band structures consistent with experimental measurements. For example, three band gaps in the optical spectrum of Si or Ge can be used to fix the potential for these atoms, and then the electronic band structure and other properties can be computed with a high degree of accuracy. When applying the EPM, the pseudopotential is taken to be the total potential a valence electron experiences; it combines the electron–ion and electron– electron interactions. In the course of fitting these potentials, the problem of how the optical properties of semiconductors were related to interband transitions was solved in the 1960s and 1970s. In addition, a great deal was learned about the pseudopotential. It was found that pseudopotentials were “transferable.” Pseudopotentials constructed for InAs, InSb and GaAs could be used to extract As, In, Sb and Ga pseudopotentials. In fact, the extracted In, Ga As, and Sb pseudopotentials were transferable between compounds and even worked well to give the electronic structure of these metals and semi-metals. So it became clear that each atom had its own transferable potential, and at least to a first approximation, these could be extracted from experiment and applied widely. In addition to learning about the transferability of the pseudopotentials, their general form and properties gave a great deal of information which was used when first-principles potentials were developed. So this empirical approach which is still used not only provided an accessible and flexible calculational tool, it also provided ideas and facts for use in developing the fundamental theory. The resulting band structures were also accurate. Figure 3 shows a comparison between the predicted EPM band structures of GaAs


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Figure 3. A comparison of the predicted pseudopotential band structure for occupied energy bands in GaAs together with the experimental bands determined by Angular Resolved Photoemission Spectroscopy.

and the subsequent experimentally determined data using Angular Resolved Photoemission Spectroscopy. Another example involved bulk moduli of semiconductors and insulators. The first principles approach using total energy calculations as a function of volume E(V ) allows the determination of elastic constants and, in particular, the bulk modulus B. These calculations are fairly extensive and hence costly. Another approach based on concepts introduced by Phillips [20] yields a connection between spectral properties of semiconductors and insulators and their structural or bonding properties. By exploiting [21] these concepts, a simple formula can be derived for B which requires only the bond length d, and the integers I = 0, 1, 2 to indicate a group IV, III–V, or II–VI compounds. The resulting formula B = (1972 − 220 I) d−3.5 gives calculated values for B to within a few percent of the experimental values. Again, not only is this semi-empirical approach valuable because the calculation can be done on a hand calculator in a few seconds, it also give

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insight into the nature of compressibilities. For example, one can make estimates and explore limits of B for aids in predicting the existence of superhard solids [22].

10.

Future

As Yogi Berra stated, “Predictions are hard to make, especially about the future.” However, it is clear that this area of physics will expand. Multi-scale methods [23] to study materials assembled from fundamental building blocks that are understood at the micro or nano level will continue to be an active field with interest coming from materials science, chemistry, and physics. Problems like understanding the nature of growth, diffusion, amorphous materials, and even non-equilibrium processes can be addressed. Molecular dynamics [24] can also be used to attack problems of this kind [25]. Real space methods [26, 27] will also continue to impact this area of research. The general interest in clusters and how they develop properties associated with bulk properties and the study of the evolution of material properties as size changes will demand new methods and concepts. As mentioned in the section on excited states, there has been considerable progress in determining optical properties from first-principles theory for solids. There has also been progress on the calculation of optical properties for clusters and nanocrystals. These approaches [18] are sometimes labeled as time dependent LDA or TDLDA. Growth in this area is also expected. A frontier has always been the study of increasingly more complex solids. Many materials can be described in terms of unit cells with a finite number of atoms. Computational problems arise as the number of atoms increases. Here hardware development helps, and it is impressive how much progress continues to be made in extending the complexity of systems that can be studied. However, the appetite for considering more complex systems is large particularly at the border where this field of science merges with biophysics. Complex molecules and systems like DNA are coming into the range of study where researchers expect precision on the level of what has been achieved for crystals. Clearly this is an area of important research with a bright future as is nanoscience and quantum computation where we may possibly learn new things about quantum mechanics. As mentioned earlier, the frontier of correlated electrons remains, and many feel that present theory is up to the challenge. If success is achieved in this area and our ability to treat more complex systems is enhanced, it may be possible to predict new states of matter. I would expect that this phase of discovery, if it is in the cards for theorists, will be preceded by the development of semiempirical theories like the EPM. With good models and general knowledge of effects such as polarizability [28] one may be able to predict phenomena


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on the level of magnetism, superconductivity, and the quantum Hall effects. However, this may be a long way off, so we still need experimentalists.

Acknowledgments This work was supported by National Science Foundation Grant No. DMR00-87088 and by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering, US Department of Energy under contract No. DE-AC03-76SF00098.

References [1] J. Bardeen, L.N. Cooper, and JR., Schrieffer, “Theory of superconductivity,” Phys. Rev., 108, 1175–1204, 1957. [2] J.P. Walter and M.L. Cohen, “Electronic charge densities in semiconductors,” Phys. Rev. Lett., 26, 17–19, 1971. [3] P.A.M. Dirac, “Quantum mechanics of many-electron systems,” Proc. R. Soc. (London), A123, 714–733, 1929. [4] E. Fermi, “On the pressure shift of the higher levels of a spectral line series,” Nuovo Cimente, 11, 157, 1934. [5] J.C. Phillips and L. Kleinman, “New method for calculating wave functions in crystals and molecules,” Phys. Rev., 116, 287–294, 1959. [6] W. Kohn and L.J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev., 140, A1133–A1138, 1965. [7] C. Herring, “A new method for calculating wave functions in crystals,” Phys. Rev., 57, 1169–1177, 1940. [8] J.C. Slater, “Wave functions in a periodic potential,” Phys. Rev., 51, 846–851, 1937. [9] M.L. Cohen and T.K. Bergstresser, “Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zincblende structures,” Phys. Rev., 141, 789–796, 1966. [10] M.L. Cohen, “Pseudopotentials and total energy calculations,” Phys. Scripta, T1, 5–10, 1982. [11] K.J. Chang, M.L. Cohen, J.M. Mignot, G. Chouteau, and G. Martinez, “Superconductivity in high-pressure metallic phases of Si,” Phys. Rev. Lett., 54, 2375–2378, 1985. [12] H.J. Choi, D. Roundy, H. Sun, M.L. Cohen, and S.G. Louie, “The origin of the anomalous superconducting properties of MgB2 ,” Nature, 418, 758, 2002. [13] W.D. Knight, K. Clemenger, W.A. de Heer, W.A. Saunders, M.Y. Chou, and M.L. Cohen, “Electronic shell structure and abundances of sodium clusters,” Phys. Rev. Lett., 52, 2141–2143, 1984. [14] W.A. de Heer, W.D. Knight, M.Y. Chou, and M.L. Cohen, “Electronic shell structure and metal clusters,” In: H. Ehrenreich and D. Turnbull, (eds.), Solid State Physics, vol. 40, Academic Press, New York, p. 93, 1987. [15] M.L. Cohen, M. Schlüter, J.R. Chelikowsky, and S.G. Louie, “Self-consistent pseudopotential method for localized configurations: molecules,” Phys. Rev. B, 12, 5575–5579, 1975.

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M.L. Cohen [16] M.S. Hybertsen and S.G. Louie, “First-principles theory of quasiparticles: calculation of band gaps in semiconductors and insulators,” Phys. Rev. Lett., 55, 1418–1421, Phys. Rev. B, 34, 5390–5413, 1986. [17] M. Rohlfing and S.G. Louie, “Electron–hole exitations in semiconductors and insulators,” Phys. Rev. Lett., 81, 2312–2315, 1998, Phys. Rev. B, 62, 4927–4944, 2000. [18] I. Vasiliev, S. Ogut, and J.R. Chelikowsky, “First-principles density-functional calculations for optical spectra of clusters and nanocrystals,” Phys. Rev. B, 65, 115416, 2002. [19] V.I. Anisimov, J. Zaanen, and O.K. Andersen, “Band theory and Mott insulators: Hubbard U instead of Stoner I,” Phys. Rev. B, 44, 943–954, 1991. [20] J.C. Phillips, Bonds and Bands in Semiconductors, Academic Press, New York, 1973. [21] M.L. Cohen, “Calculation of bulk moduli of diamond and zinc-blende solids,” Phys. Rev. B, 32, 7988–7991, 1985. [22] A.Y. Liu and M.L. Cohen, “Prediction of new low compressibility solids,” Science, 245, 841, 1989. [23] N. Choly and E. Kaxiras, “Fast method for force computations in electronic structure calculations,” Phys. Rev. B, 67, 155101, 2003. [24] R. Carr and M. Parrinello, “Variational quantum Monte Carlo nonlocal pseudopotential approach to solids: cohesive and structural properties of diamond,” Phys. Rev. Lett., 61, 1631–1634, 1988. [25] S. Yip, “Nanocrystaline metals – Mapping plasticity,” Nature Mater., 3, 11, 2004. [26] J.R. Chelikowsky, N. Troullier, and Y. Saad, “The finite-difference-pseudopotential method: electronic structure calculations without a basis,” Phys. Rev. Lett., 72, 1240–1243, 1994. [27] M.M.G. Alemany, M. Jain, J.R. Chelikowsky, and L. Kronik, “A real space pseudopotential method for computing the electronic properties of periodic systems,” Phys. Rev. B, 69, 075101, 2004. [28] I. Souza, J. Iniguez, D. Vanderbilt, “Dynamics of berry-phase polarization in timedependent electric fields,” Phys. Rev. B, 69, 085106, 2004. [29] M.L. Cohen, and J.R. Chelikowsky, Electronic Structure and Optical Properties of Semiconductors, Springer-Verlag, Berlin, 1988. [30] C. Kittel, Introduction to Solid State Physics, seventh edition, Wiley, New York, 1996. [31] J.C. Phillips, Bonds and Bands in Semiconductors, Acadamic Press, New York, 1973. [32] P.Y. Yu and M. Cardona, Fundamentals of Semiconductors, Springer, Berlin, 1996.

1.3 ACHIEVING PREDICTIVE SIMULATIONS WITH QUANTUM MECHANICAL FORCES VIA THE TRANSFER HAMILTONIAN: PROBLEMS AND PROSPECTS Rodney J. Bartlett, DeCarlos E. Taylor, and Anatoli Korkin Quantum Theory Project, Departments of Chemistry and Physics, University of Florida, Gainesville, FL 32611, USA

1.

Prologue

According to the Westmoreland report [1], “in the next ten years, molecularly based modeling will profoundly affect how new chemistry, biology, and materials physics are understood, communicated, and transformed to technology, both intellectually and in commercial applications. It creates new ways of thinking – and of achieving.” Computer modeling of materials can potentially have an enormous impact in designing or identifying new materials, how they fracture or decompose, what their optical properties are, and how these and other properties can be modified. However, materials’ simulations can be no better than the forces provided by the potentials of interaction among the atoms involved in the material. Today, these are almost invariably classical, analytical, two- or threebody potentials, because only such potentials permit the very rapid generation of forces required by large-scale molecular dynamics. Furthermore, while such potentials have been laboriously developed over many years, adding new species frequently demands another long-term effort to generate potentials for the new interactions. Most simulations also depend upon idealized crystalline (periodic) symmetry, making it more difficult to describe the often more technologically important amorphous materials. If we also want to observe bond breaking and formation, optical properties, and chemical reactions, we must have a quantum mechanical basis for our simulations. This requires a multi-scale philosophy, where a quantum mechanical core is tied to a classical 27 S. Yip (ed.), Handbook of Materials Modeling, 27–57. c 2005 Springer. Printed in the Netherlands.

28

R.J. Bartlett et al.

atomistic region, which in turn is embedded in a continuum of some sort, like a reaction field or a finite-element region. It is now well-known that ab initio quantum chemistry has achieved the quality of being “predictive” to within established small error bars for most properties of isolated, relatively small molecules, making it far easier to obtain requisite information about molecules from applications of theory, than to attempt complicated and expensive experimental observation. In fact, applied quantum chemistry as implemented in many widely used computer programs, ACES II [2], GAUSSIAN, MOLPRO, MOLCAS, QCHEM, etc, has now attained the status of a tool that is complimentary to those of X-ray structure determination and NMR and IR spectra in the routine determination of the structure and spectra of molecules. However, there is an even greater need for the computer simulations of complex materials to be equally predictive. Unlike molecules, which can usually be characterized in detail by spectral and other means, materials are far more complex and cannot usually be investigated experimentally under similarly controlled conditions. They have to be studied at elevated temperatures and under non-equilibrium conditions. Frequently, the application of the material might be meant for extreme situations that might not even be accessible in a laboratory. Hence, if we use more economical computer models to learn how to suitably modify a material to achieve an objective, our materials simulations must be “predictive,” to trust both the qualitative and quantitative consequences of the simulations. Besides the predictive aspect, another theme that permeates our work with materials is “chemistry.” By chemistry we mean that unlike the idealized systems that have been the focus of most of the simulation work in materials science, we want to consider the essential interactions among many different molecular species; and, in particular, under stress. As an example, a long unsolved problem in materials is why water will cause forms of silica to weaken by several orders of magnitude compared to their dry forms [3–5] while ammonia with silica shows a different behavior. A proper, quantum mechanically based simulation should reflect these differences, qualitatively and quantitatively. The third theme of our work is that by virtue of using a quantum mechanical (QM) core in multi-scale simulations, unlike all the simulations based upon classical potentials, we have quantum state specificity. In a problem like etching silica with CF4 , which generates the ething agent, CF3 , a classical potential − · cannot distinguish between CF+ 3 , CF3 , and CF3 , yet obviously the chemistry will be very different. Furthermore, we also have need for the capability to use excited electronic states in our simulations, to include species like CF∗3 , e.g., or to distinguish between different modes of fractures of the silica target, such as radical dissociation as opposed to ionic dissociation. Conventionally, the only quantum mechanically based multi-scale dynamics simulations that would permit as many as 500–1000 atoms in the QM region were based upon the tight-binding (TB) method, density functional theory

Achieving predictive simulations with quantum mechanical forces

29

(DFT) being used only for smaller QM regions. TB is a pervasive term that covers everything from crude, non-self-consistent descriptions like extended Hückel theory [6], to quasi-self-consistent schemes based upon Mulliken or other point charges [7], to a long history of solid state efforts [8, 9], to TB with three-body terms [10]. The poorest of these do not introduce overlap, selfconsistency, nor explicit consideration of the nuclear–nuclear repulsion terms that would be essential in any ab initio approach; so in general such methods cannot correctly describe bond breaking, where charge transfer is absolutely essential. However, there have been significant improvements on several fronts in the recent TB literature [11, 12] which are helping to rectify these failings. The alternative approach to TB is that based upon the semi-empirical quantum chemistry tradition starting with Pariser and Parr [13, 14], Dewar et al. [15, 16] and Pople et al. [17, 18], and being extended on several fronts by Stewart [19–21], Thiel [22], Merz [23], Repasky et al. [24], and TubertBrohman et al. [25]. These “neglect of differential overlap methods,” of which the most flexible is the NDDO method, meaning “neglect of diatomic differential overlap” will be our initial focus. Like TB methods, the Hamiltonian is greatly simplified but not necessarily by limiting all interactions to nearest neighbors, but instead to operationally limiting interactions to mostly diatomic units in molecules. We will address some of the details later, but for most of our purposes, the particular form for the “transfer Hamiltonian” will be at our disposal and suitable forms with rigorous justification are a prime objective of our research. It might be asked why a “Hamiltonian” instead of a potential energy surface? Fitting the latter especially while including the plethora of bond-breaking regions, is virtually impossible for even simple molecules. Highly parameterized molecular mechanics (MM) methods [26] can do a good job of generating a potential energy surface near equilibrium for well-defined and unmodified molecular units; but bond breaking and formation is outside the scope of MM. So our objective, instead of the PES (potential energy surface), is to create a “transfer Hamiltonian” that permit the very rapid determination of, in principle, all the properties of a molecule; and especially the forces on a PES for steps of the MD. The transfer Hamiltonian gives us a way to subsum most of the complications of a PES in a very convenient package that will yield the energy and first and second derivatives upon command. This has been done to some degree in rate constant applications for several atom molecules where the complication is the need for multi-dimensional PES information [27–29]. Here, we conceive of the transfer Hamiltonian as a way to get all the relevant properties of a molecule including its electronic density, and related properties like dipole moments, and its photoelectron, electronic, and vibrational spectra. Except for the latter, these are purely “electronic” properties, which depend solely on the electronic Schrödinger equation. These should be distinguished from forces and the PES itself, which are properties of the total energy.

30


The distinction between the two has been at the heart of the principal dilemma in simplified or semi-emprirical theory, where a set of parameters that give the total energy are not able to describe electronic properties equally well. It is also critical that the Hamiltonian be computed very rapidly to accomodate MD applications, and a form for it needs to be determined such that we retain the accuracy of the forces and other properties that would come from ab initio correlated theory. This is more an objective than a fait-accompli, but we will discuss how to try to accomplish this in this contribution. Our approach is to appeal to the highest level of ab initio quantum chemistry, namely coupled-cluster (CC) theory, to use as a basis for a “transfer Hamiltonian” that embed the accurate, predictive quality CC forces taken from suitable clusters into it, but in an operator that is of very low rank, making it possible to do fully self-consistent calculations on ∼500–1000 atoms undergoing MD. Hence, as long as a phenomena is accessible to MD, and if the transfer Hamiltonian forces retain the accuracy of CC theory, we should be able to retain the predictive quality of the CC method in materials simulations; and if we can also describe the electronic properties accurately, we have everything that the Schrödinger equation could tell us about our system. In addition, we have no problem with changing atoms or adding new molecules to our simulations, as our transfer Hamiltonian is applicable to any system once trained to ensure its proper description. We will also develop the transfer Hamiltonian approach from DFT considerations in the following to show the essential consistency between the wavefunction and density functional methods. Our emphasis on predictability, chemistry, and state specificity, offers a novel perspective in the field; and the tools we are developing, all tied together with highly flexible software, sets the stage for the kinds of simulations that will lead to reliable materials design. As the Westmoreland report further states, ‘The top needs required by industry are methods that are “bigger, better, faster;” (with) more extensive validation, and multiscale techniques.’

2.

Introduction

Our objective is predictive simulations of materials. The critical element in any such simulation are the forces that drive the molecular dynamics. For a reliable description of bond breaking, as in fracture or chemical reaction, or to distinguish between a free radical and a cation or anion, to be electronic state specific; or to account for optical spectra; the forces must be obtained from a quantum mechanical method. Today’s entirely first-principles, quantum chemical methods are “predictive” for small molecules in the sense that with a suitable level of electron correlation, notably with coupled-cluster (CC) theory [30], and large enough basis sets [30, 31]; or to a lesser extent, density functional theory (DFT) [32–34] the results for molecular structure, spectra,


31

energetics and the associated atomic forces required for these quantities and for reaction paths are competitive with experiment. In particular, these highly correlated methods offer accurate results for transient molecules and other experimentally inaccessible species, and particularly reaction paths that can seldom be known from solely experimental considerations. In terms of ab initio theory, the established paradigm of results from converging, correlated methods is MP2b.c

The T1 generates all single excitations, i.e., T1 |0= a,i tia ai from the vacuum, usually HF (but could equally well be the Kohn–Sham determinant), meaning excitation of an electron from an occupied orbital to an unoccupied one. We use the convention that i, j, k, l represent orbitals occupied in the Fermi vacuum, while a, b, c, d are unoccupied, and p, q, r, s are unspecified. T2 does the same for the double excitations, and T3 the triple excitations. Continuation through Tn for n electrons will give the full CI solution. Multiplying the Schrödinger equations from the left by exp(−T ), the critical quantity in CC theory is the similarity transformed Hamiltonian, exp(−T )H exp(T ) = H

(7)

where the Schrodinger equation becomes, H |0 = E|0

(8)

|0 is the Fermi vacuum, or an independent particle wavefunction, but E(R) = 0|H |0 is the exact energy at a given geometry, and the exact forces subject to atomic displacement are ∇ E(R) = F(R)

(9)

The effects of electron correlation are contained in the cluster amplitudes, whose equations at a given R are Q n H |0 = 0 ab abc abc where Q1 = |ai ai |, Q 2 = |ab i j i j |, Q 3 = |i j k i j k |+ · · · . Q1 projections give the equations for {tai }, and similarly for the other amplitudes. Limiting ourselves to single and double excitations, we have CCSD which is a highly correlated, accurate wavefunction. Consideration of triples provides, CCSDT, the state-of-the-art; while for practical application, its non-iterative forms CCSD[T] and its improved modification, CCSD[T]; is currently considered the “gold standard” for most molecular studies [36, 43].

36


Regardless of choice of excitation, H may be written in secondquantization as 1 pq † † p H = h q p† q + grs p q s r + III + IV + · · · 2

(10)

where summation of repeated indices is assumed and III and IV indicate threeand four-body operators. The indices can indicate either atomic or molecular pq = pq|rs = ( pr|qs) = d1 d2φ ∗p (1)φr (1)g12 φq∗ orbitals. More explicitly, grs (2)φs (2) where the latter two-electron integral indicates the interaction between the electron distributions associated with electrons 1 and 2, respectively. We use g12 instead of r−1 12 because in the generalized form for H there may be additional operators of two-electron type besides just the familiar integrals. Such one- and two-electron quantitites further separated into one, two, and more atomic centers, are the quantitites that will have to be computed or in the case of simplified theories, approximated, to provide the results we require. At this point, we have an explicitly correlated, many-particle theory. It is important to distinguish this from an effective one-particle theory as in DFT or Hartree–Fock, which are much easier to apply to complicated systems. To make this connection, we choose to reformulate the many-particle theory into an effective one-particle form. This is accomplished by insisting that the energy variation δ E = 0, which means the derivative of E with respect to the orbitals that will compose the single determinant, |, vanish. As our expressions for tab.. i j.. , the CC equations, will depend upon the integrals over these orbitals, and consequently H ; this procedure is iterative. As any such variation of a determinant can be written in the form | = exp(T1 )|0, the single excitation projection of H has to vanish, ai |H |0 = 0 = a|hT |i

(11) (12)

where we introduce the “transfer Hamiltonian” operator, hT . Since this matrix element vanishes between the occupied orbital, i, and the unoccupied orbital, a, we can use the resolution of the identity 1= j |j j | + b |bb| to rewrite this equation in the familiar form, hT |i =

λ j i | j = i |i

(13)

j

where the first form retains the off-diagonal Lagrangian multipliers, while the second is canonical. The above can equally well be done for HF-SCF theory, except hT = f= t + v + J− K =h + J− K , wherewe have the kinetic-energy operator, the electron–nuclear attraction term − Z A /|r − R A |, combined together into the one-particle element of Eq. (13); the Coulomb repulsion and


37

the non-local exchange operator, repectively. The Hartree–Fock effective one particle operator, J − K = j d2φ ∗j (2)(1 − P12 )φ j (2), and there would be no correlation in the Fock operator. In that case, i provides the negative of the Koopmans’ estimate of ionization potentials, and a the Koopmans’ approximation to the electron affinities. For the correlated hT , which is the one-particle theory originally due to Brueckner [45, 46], all single excitations vanish from the exact wavefunction, and as a consequence, we have maximum overlap of the Brueckner determinant with the exact wavefunction, | B ||. In general, Brueckner theory is not Hermitian, but in any order of perturbation theory we can insist upon its hermiticity, i.e., i|hT |a = 0, and that will be sufficient for our purposes. The specific form for the transfer Hamiltonian matrix element is a|hT |i = a| f|i +

1 a j ||cbticbj − k j ||ib tkjab 2

(14)

where summation over repeated indices is implied. Keeping the form of the hT operator in the a|hT |i matrix element the same, when a is replaced by an occupied orbital, m, we have m|hT |i = m| f|i +

1 m j ||cbticbj − k j ||ibtkjmb 2

(15)

Then, we have the Hartree–Fock-like equations but now for the correlated one-particle operator, hT , represented in the basis set, |χ, where S = χ|χ is the overlap matrix, hT C = SC

(16)

and the (molecular) orbitals are |φ = |χC. The Brueckner determinant, B , is composed of the lowest n occupied MOs, |φ0 = |χC0 In particular, the matrix elements for the transfer Hamiltonian in terms of the atomic orbital basis set are

µ

µα µ|h T |ν = h ν + Pαβ (g µα νβ − g βν )

Pµν = cµi ciν

(17) (18)

(summation of repeated indices is assumed),where Pνµ is the density matrix for µ the Brueckner determinant. Hence, subject to modified definitions for h ν and µα g νβ , which we will assume are renormalized to include the critical parts of the three- and higher-electron effects, we have the matrix which contains the exact ionization potentials for the system.

38

R.J. Bartlett et al. The total energy, E = B |H | B =

i|h|i +

1 1 i j ||i j + i j ||abtiab j 2 i, j 4 i, j,a,b

(19)

i|h|i +

1 i j |g 12 |i j 2 i, j

(20)

i

=

i

1 T rP(h+hT ) 2 1 −1 = r12 + T2 ||abab|| 2 a,b

(21)

= g 12

(22)

†

is also written in terms of the reference density matrix P = C0 C0 , evaluated from the occupied orbital coefficients, C0 . The quantityt hT = differs from the form in Eq. (15), because of the absence of the third term on the RHS. This term is an orbital relaxation term that only pertains to the ionization potentials, as there we would need to allow the system to relax after the ionization. Hence, this cannot contribute to the ground state energy, and its manifestation of that is that the total energy cannot be written in terms of the exact ionization potentials in Eq. (13), but can be written in terms of an approxi mation introduced by hT . The analytical forces for MD can be written eas T includes all electron correlation. Once h µ and g µα ily, as well. Notice h νβ ν are specified, which need to be viewed as quantities to be determined to reproduce the reference results from ab initio correlated calculations, we obtain self-consistent solutions for the correlated, effective, one-particle Hamiltonian. The self-consistency is essential in accounting for bond-breaking and associated charge rearrangement. The overlap matrix is included for generality, but as is often done in NDDO type theories, enforcing the ZDO approximation removes it. Another way to view this is to assume the parameters are based upon using the orthonormal expansion basis, |χ = |χS−1/2 which gives hT = S−1/2 hT S−1/2 . Developing this expression to include low-order in some S terms permits us to still retain the simpler and computationally faster orthogonal form of the eigenvalue equation, yet introduce what is sometimes called “Pauli repulsion” in the semi-empricial community [22]. A self-consistent solution provides the coefficients, C and the reference orbital energies, ,which as we discussed, are not the exact Ip’s that would come from including the contributions of the tmb j k amplitudes, which contain three-hole line and one-particle line. Such terms arise in the generalized EOM or Fock space CC theory for ionized, electron attached, and excited states. In lowest order, tmb j k =mb|| j k/( j +k −b −m ).


4.

39

Transfer Hamiltonian: Density Functional Viewpoint

The DFT approach to the hT starts from a different premise that is actually simpler, since DFT is already exact in an independent particle form, unlike the usual many-particle theory above. As is well known, we have the Sham oneparticle Hamiltonian [32] whose first n eigenvectors give the exact density, h S = t + v + J + Vx + V h S |i = i |i. h S C = SC ρ(1) = φi (1)φi∗ (1) = χµ (1)Pµν χ∗ν (1) i

(23) (24) (25) (26)

µ,ν

†

and like the above, the density matrix is P = C0 C0 . The highest-occupied MO, n, has the property that n = −Ip(n). However, solving these equations does not provide an energy until we know the functional E xc [ρ], from which we know that δ E xc [ρ]/δρ(1) = Vxc (1), to close the cycle. The objective of DFT is to get the density, ρ, first; and then all other ground state properties follow; in particular, the energy and forces we need for MD. The transfer Hamiltonian in this case will be defined by the condition that ρCCSD = ρKS . Satisfying this condition means that we could obtain a Vxc from this density by using the ZMP method [47], but our approach is simply to parameterize the elements in h S = hT in analogy with that in semi-empirical quantum chemistry or TB such that the density condition is satisfied. This should specify Vxc , and indeed, the other terms in hT , which is then sufficient to obtain the forces, {∂ E(R)/∂X A }. Note this bypasses the need to use an explicit Exc [ρ],but, of course, that would always be an option. We can also bypass any explicit treatment of the kinetic energy operator by virtue of parametrization of h = t + v as in the semi-empirical approach discussed below. Besides the density condition, we also have the option to use the force condition in the sense that the forces can be obtained from CC theory, and then their values directly used to obtain the parameterized version of h S = hT . Ideally, the parameters will be able to describe both the densities and the forces, although this raises the issue of the long-term inability of semi-empirical methods to describe structures and spectra with the same parameters, discussed further in the last section. As our objective is to be able to define a hT that will satisfy many of the essential elements of ab initio theory, some of interest besides the forces are the density, and the ionization potential and electron affinity. The latter define the Mulliken electronegativity, E N = (I − A)/2, which should help to ensure that our calculations correctly describe the charge distribution in a system and the density. We also know the correct long-range behavior of the √ density is determined by the homo ionization potential, ρ (r) ∝ exp(−2 2I )r, which is a property of exact DFT. If the density is right, then we also know

40


that we will get the correct dipole moments for the molecules involved, and this is likely to be critical if we hope to correctly describe polar systems like water, along with their hydrogen bonding.

5.

What About Semi-Empirical Methods?

Before embarking upon a particular form for the transfer Hamiltonian that must inevitably be semi-empiricial or TB type, we can ask what kind of accuracy is possible with such methods. In an recent paper on PM5, a parameterized NDDO Hamiltonian, [20, 21] Stewart reports that the PM5 heats of formation for over ∼1000 molecules composed of H, C, N, O, F, S, Cl, Br, and I have a mean absolute deviation (MAD) of 4.6 kcal/mol, nearly the same as DFT using BLYP or BPW91. The errors of PM3 are slightly larger (5.2) and AM1 (7.2). The largest errors are 27.2, (PM5), 35.1, (PM3), 54.8, (AM1) and 55.7 for BLYP and 34.5 for BPW91. Using a TZ instead of a DZ basis for the latter gives some improvement in the worst cases. For Jorgensen’s reparameterized PM3 and MNDO methods, referred to as PDDG [22, 25], the MAD heats of formation for 662 molecules limited to H, C, N, and O are reduced from 8.4 to 5.2, and with some extra PDDG additions, from 4.4 to 3.2 kcal/mol. For geometries, PDDG gets bond lengths to a MAD of 0.016 Å, 2.3◦ bond angle, and 29.0◦ dihedral angle. The principal Ip is typically within ∼0.5 eV – though it can be off by several – which is some 3% more accurate than PM3 and 12% less accurate than PM5. For dipole moments, the MAD is 0.24 Debye. There is less information about transition states and activation barriers, but these methods have seen extensive use for such problems in chemistry. Recent TB work termed SCC-DFTB for self-consistent charge density functional TB [11] is based upon DFT rather than HF and is less empirical, but still simplified using similar approximations for two-center interactions as in NDDO, discussed below. It is developed for solids as well as molecules. For the latter, in 63 organic examples the MAD deviations in bond lengths are 0.012 Å, and angles, 1.80◦ . For heats of reaction, in 36 example molecules composed of H, C, N, O the MAD is 12.5 kcal/mol compared to 11.1 for DFT-LSD. On the other hand, we can have dramatic failures. None of these new semi-empirical methods yet even treat Si, much less heavier elements of the sort that are important in many materials applications. To quote just one example, in comparisons of nine Zn complexes with B3LYP and CCSD(T), “MNDO/d failed the case study” and the errors compared to ab initio or DFT were dramatic.” The authors [48] say “No one semiempiricial model is applicable for the calculations of the whole variety of structures found in Zn chemistry.”


6.

41

Forms for Tranfer Hamiltonian

Our objective is to model hT for the particular phenomena of interest and for chosen representative systems (i.e. unlike normal semi-empirical theory we do not expect the parameters to describe many elements at once) in a way that permits the routine, self-consistent treatment of a very large number of the same kinds of atoms. We also recognize that the traditional approaches are built upon approximating the HF-SCF one-particle Hamiltonian, f, not the more exact DFT or Brueckner approach discussed above. Also, traditionally, only a minimum basis set of an s orbital on H, and one s and a set of p orbtials are used on the other atoms, until d orbtials are occupied. Thinking more like ab initio theory, we do not presuppose such restrictions, but will use polarization functions and potentially double zeta sets of s and p orbitals on all atoms. We recognize the attraction of a transfer Hamiltonian that (1) consists solely of atomic parameters; and (2), is essentially two-atom in form, as all threeand four-center contributions are excluded. This is the fundamental premise of all neglect of differential overlap approximations [15, 17, 19]. Hence, as a first realization, guided by many years of semi-empirical quantum chemistry, we choose the “neglect of diatomic differential overlap” (NDDO) Hamiltonian, µ|hT |ν =

αµν δuv +

µ∈ A

+

µ=α,ν=β µ,ν∈ A

µ∈ A,ν∈B

−

µ= /β∈ A, ν= /α∈B

Pαβ (µα|νβ) −

µβ=να,µ= /β µ,β∈A

Pαβ (µβ|να)

1 (βu + βv )Sνµ + Pαβ (µα|νβ) 2 µ=α∈ A,v,β∈B,

Pαβ (µβ|να)

ν=β∈B,µ,α∈ A

(27)

consisting of atomic and diatomic units. αµµ is a purely atomic quantity that represents the one-particle part of the energy of an electron in its atomic orbital. We would have different values for s, p, d, . . . orbitals, collectively indicated as αA . The one-center, two-electron terms for atom A are separated into coulomb and exchange terms and weighted by the density matrix. No explicit correlation operator as in DFT is yet considered. Instead modifications (parameterizations) of the coulomb and exchange terms are viewed as potentially accomplishing the same objective. βu is an atomic parameter indicative of each orbital type (s,p,d) on atom A and Sµν is the overlap integral between, formally, two atomic orbitals on atoms A and B. A Slater type orbital on atom A is χA =rAn−1 exp (−ζA )Yl,m (ϑA, ϕA ), and the overlap integral, Sµν (ζA, ζB ) depends upon ζA and ζB , so it is entirely determined by what the atoms are. So it, too, consists of atomic parameters.

42


The terms which include density matrix elements account for the twoelectron repulsion terms which depend upon the purely one-center two-electron µν integral type, (µA νA |µA νA ) = γAA . A typical choice for the two-center, twoelectron term then becomes [49, 50]

2 (µA νA |µB νB ) ∝ rAB + (cAuv + cBuv )2

−1/2

(28)

where rAB = RAB + qi and the additive terms cuv are numerically determined such that the two-center repulsion integral goes to the proper one-center limiting value. RAB is the distance bewteen atoms A and B, but differs from rAB due to the multipole method used to compute the two-electron integral. For (sA sA | pB pB ), a monopole and quadrupole are used for the p electron distribution while a monopole is used for the s distribution. The radial extent of the multipoles is given by qi = q p B, and is a function of the atomic orbital exponent ζB on atom B. This form for the two-electron integrals assumes the correct long-range (1/R) behavior. More general forms for the two-center, two-electron integrals combine such contributions together from several multipoles to distingush (ss|ss) from (ss|dd), etc. [19, 51]. This set of approximations defines the NDDO form of the matrix elements of hT between two atomic orbitals. Now we have to consider the nuclear repulsion contribution to the ene rgy, A,B ZA Z B /RAB . Importantly, and unlike in ab initio theory, the effective atomic number, ZA ,which is chosen initially to be equal to the number of valence electrons being contributed by atom A, is also made a function of all RAB in the system. This introduces several new parameters into the calculation, justified roughly by some ideas of electron screening. The AM1 choice [16] for the latter reflects screening of the effective nuclear charge with the parameterized form

E CR = Z A Z B (sA sA |s B sB ) 1 + e(−dA RAB ) + e(−dB RAB ) Z Z + A B RAB

k

aAk e

−bA (RAB −CkA )2

+

aBk e

−bB (RAB −CkB )2

(29)

k

These core repulsion (CR) parameters, d, b, a and C account for the nuclear repulsion, which means they contribute to total energies and forces, but not to purely electronic results. The latter depend upon the electronic parameters βA, γAA , αA , . . . . In our work, both sets are specified via a genetic algorithm to ensure that correlated CCSD results are obtained for representative systems, tailored to the phenomena of interest. Looking at the above approximations, we see that we retain only one and two-center two-electron integrals. In principle, we can have a three-center one-electron integral from µA |Z C /|r − RC νB , but in NDDO, such terms are excluded as well. Any approximation of hT that is to be tied to ab initio


43

results, has to have the property of “saturation.” To achieve this, we insist that our form for hT be fundamentally short range. We see from the above, that our hT depends on two-center interactions, but unlike TB, not just those for the nearest neighbor atoms but for all the two-body interactions in the system. This short-range character helps to saturate the atomic parameters for comparatively small example systems that are amendable to ab initio correlated methods. Then once the atomic parameters are obtained, and found to be unchanged to within a suitable tolerance when redetermined for larger clusters, they define a saturated, self-consistent, correlated, effective one-particle Hamiltonian that can be readily solved for quite large systems to rapidly determine the forces required for MD. We also have easy access to the secondderivatives (Hessians) for definitive saddle point determination, vibrational frequencies, and interpolation between calculations at different points for MD. Using H2 O as an example for saturation, we can obtain the cartesian force matrix for the monomer by insisting that our simplified Hamiltonian provide the same force curves as a function of intra-atomic separation for breaking the O–H bond with the other degrees of freedom being optimum (i.e. a distinguished reaction path). Call this matrix FA. From FA we use a GA to obtain the Hamiltonian parameters that, in turn, determine h and g elements that make our transfer Hamiltonian reproduce these values. The more meaningful gradient norm |F | is used in practice rather than the individual cartesian elements. Now consider two water molecules interacting. The principal new element is the dihedral angle that orients one monomer relative to the other, but the H-bonding and dipole–dipole interaction will cause some small change when we break an O–H bond in the dimer. Our first approximation to FAB =FA +FB + VAB . Then by changing our parameters to accomodate the dimer bond breaking, we get slightly modified h and g elements in the transfer hamiltonian. VAC, VBC This makes FAB = FA + FB . Going to a third unit, we would add VABC, perturbations and repeat the process to define FABC = FA + FB + FC . Since these atomic based interactions will rapidly fall off with distance, we expect that relatively quickly we would have a saturated set of parameters for the bond breaking in water with a relatively small number of clusters. We can obviously look at other properties, too, such as dipole moments, cluster structures, etc., to assess their degree of saturation with our hT parameters. If we fail to achieve a satisfactory saturation, then we have to pursue more flexible, or more accurate forms of transfer Hamiltonians. It is essential to identify the terms that matter, and the DFT form provides complimentary input to the wavefunction approach in this regard. Also, unlike most semi-empirical methods we do not limit ourselves to a minimum basis set. The general level we would anticipate is CCSD with a double-zeta + polarization basis, while dropping the core electrons. This is viewed as the quality of ab initio result that we would pursue for complicated molecules.

44


In addition, following the equation-of-motion (EOM) CC approach [52], we insist that H Rk |0 = ωk Rk |0

(30)

where Rk exp (T )|0 = k and ωk is the excitation energy for any ionized, Ik, electron-attached, Ak, or excited state. In other words, this provides Ips and Eas that tie to the Mulliken electronegativity, to help to ensure that our transfer Hamiltonian represents the correct charge distribution and density size. Furthermore, whereas forces and geometries are highly sensitive to the corerepulsion parameters, properties like I and A are sensitive to the electronic parameters in the transfer Hamiltonian. The transfer Hamiltonian procedure is far more general than the particular choice of Hamiltonian chosen here, since we can choose any expansion of H or hT that is formally correct and include elements to be computed or parameters to be determined, to define a transfer Hamiltonian. Furthermore, we can insist that it satisfy suitable exact and consistency conditions such as having the correct asymptotic or scaling behavior. Other desirable conditions might include the satisfaction of the virial and Hellman–Feynman theorems. We can also choose to do many of the terms like the one-center ones, ab initio, and keep those values fixed subsequently. Then, our simplified forms 12 (βu +βv )Sνµ and that of Eq. (29), are the only ones where there is an electronic dependence upon geometry. Adding this dependence to that from the core–core repulsions, has to provide the forces that drive the MD. We can explore many other practical approximations such as supressing self-consistency by setting P = 1, and impose the restriction that only nearest neighbor two-atom interactions be retained, to extract a non-self-consistent TB Hamiltonian that should be very fast in application. We can obviously make many other choices and create, perhaps, a series of improving approximations to the ab initio results that parallel their computational demands.

7.

Numerical Illustrations

As an illustration of the procedure, consider the prototype system for an Si–O–Si bond as in silica, pyrosilicic acid (Fig. 6). This molecule has been frequently used as a simple model for silica. We are interested in the Si-O bond rupture. Hence, we perform a series of CCSD calculations as a function of the Si–O distance all the way to the separated radical units, ·Si(OH)3 and ·O–Si(OH)3 , relaxing all other degrees of freedom at each point (while avoiding any hydrogen bonding which would be artificial for silica) using now wellknown CC analytical gradient techniques [36]. For each point we compute the


45

O

O H

H O Si

Si O

O O

H

H

O H

H

Figure 6. Structure of pyrosilicic acid.

Figure 7. Comparison of forces from standard semi-empirical theory (AMI) and the transfer Hamiltonian (TH-CCSD) with coupled-cluster (CCSD) results for dissociation of pyrosilicic acid into neutral fragments.

gradient norm of the forces for the 3N cartesian coordinates, q I , (3 per atom 2 1/2 and use the genetic algorithm PIKAIA [53] A), |F| = 3N I [(∂ E/∂q I ) ] to minimize the difference between |F(CCSD)-F(hT )| for the transfer Hamiltonian and the CCSD solution. This is shown in Fig. 7. Since forces drive the MD, their determination is more relevant for the problem than the potential energy curves, themselves. For this case, we find that fixing the parameters in our transfer Hamiltonian that are associated with the core-repulsion

46


function is sufficient, leaving the electronic parameters at the standard values for the AM1 method. As seen in Fig. 7, these new parameters are responsible for removing AM1s too large repulsion at short Si–O distances and erroneous behavior shortly beyond the equilibrium point. Hence, to a small tolerance, the transfer Hamiltonian provides the same forces as that in the highly sophisticated ab initio CCSD method. In a second study, QM forces permit the description of different electronic states. As an example, for this system we can also separate pyrosilicic acid into charged fragments, Si(OH)3+ and O–Si(OH)3− , and in a material undergoing bond-breaking, we would expect to take multiple paths such as this. A classical potential has no such capability. Figure 8 shows the curve and once again we obtain a highly accurate representation from the transfer Hamiltonian, with the same parameters obtained for the radical dissociation. Hence, our transfer Hamiltonian has the capability of describing the effects of these different electronic states in simulations, which besides enabling reliable descriptions of bond-breaking, should have an essential role if a materials’ optical properties are of interest. Figure 9 shows the integrated force curves to illustrate that even though the parameters were determined from the forces, the associated potential energy surfaces are also accurate compared to the reference CCSD results, and more accurate than the conventional AM1 results. The latter has an error of ∼0.4 eV between the neutral and charged paths compared to the CCSD results. We have also investigated the parameter saturation. Moving to trisilicic acid we obtain the reference results wihout any further change in our parameters.

Figure 8. Comparison of forces from standard semi-empirical theory (AM1) and the Transfer Hamiltonian (TH-CCSD) with coupled-cluster (CCSD) results for dissociation of pyrosilicic acid into charged fragments.


47

Figure 9. Comparison of PES for dissociation of pyrosilicic acid. Each curve is labeled by the Hamiltonian used and the dissociation path followed.

The correct description of complicated phenomena in materials requires that the approach be able to describe, accurately, a wealth of different valence states and coordination states of the relevant atoms involved. For example, the surface structure of silica is known to show three, four, and five coordinate Si atoms. Hence, a critical test of the ability of the hT is how well its form can account for the observed structure of such species with the same parameters already determined for bond breaking. In Figs. 10 and 11, we show comparisons of the hT results for some Six O y molecules with DFT (B3LYP), various two-body classical potentials [54, 55], and a three-body potential [56] frequently used in simulations, and molecular mechanics [26]. The reference values are from CCSD(T), which are virtually the same as the experimental values when available. The hT results are competitive with DFT and superior to all classical forms, including even MM with standard parameterization. The latter is usually quite accurate for molecular structures at equilibrium geometries, but not necessarily for SiO2 . MM methods do not attempt to describe bond breaking. The comparative timings using the various methods are shown in Table 2 for two different sized systems, pyrosilicic acid and a 108-atom SiO2 nanorod [57]. The 216-atom version is shown in Fig. 12. The hT procedure is about 3.5 orders of magnitude faster than the gaussian basis B3LYP DFT results, which is another ∼3.5 orders of magnitude faster than CCSD[ACESII]. The 108 atom nanorod is clearly well beyond the capacity of CCSD ab initio calculations, but even the DFT result (in this case with a plane wave basis using the BO-LSD-MD (GGA) program, is excessive, while the hT is again three to four orders of magnitude faster. With streamlining of programs, we expect that this can still be significantly improved.

48


Figure 10. Error in computed Six O y equilibrium bond lengths relative to CCSD(T) using various potentials.

Figure 11. Error in computed Six O y equilibrium bond angles relative to CCSD(T) using various potentials.


49

Table 2. Comparative timings for electronic structure calculations (IBM RS/6000) Pyrosilicic acid Method CCSD DFT T h BKS

CPU time (s) 8656 375 0.17 0.001

108-atom nanorod Method CCSD DFT T h BKS

CPU time (s) N/A 85,019 43 0.02

Finally, to illustrate the results of a simulation we consider the 216-atom SiO2 system of Fig. 12, subject to a uniaxial stress, using various classical potentials and that for our QM transfer Hamiltonian. The equilibrated nanorod was subjected to uniaxial tension by assigning a fixed velocity (25 m/s) in the loading direction to the 15 atoms in the caps at each end of the rod. The stress was computed by summing the forces in the end caps and dividing by the projected cross sectional area at each time step. The simulations evolved for (approximately) 10 ps where the system temperature was maintained at 1 K by velocity rescaling. Figure 13 shows the computed stress–strain curves. The main differences between the classical potentials and their QM potentials seems to be the differnce at the maximum and the long tail indicating surface reconstruction. The QM potential shows the expected brittle fracture, perhaps a little more than the classical potentials. The transfer Hamiltonian, retains self-consistency, state specificity, and permits readily adding other molecules to simulations after ensuring that they, too, reflect the reference ab initio values for their various interactions. Hence, the transfer Hamiltonian built upon NDDO or more general forms, would seem to offer a practical approach to moving toward the objective of predictive simulations. In Fig. 14 we show the same kind of information about bond-breaking in water, showing the substantial superiority of the hT results compared to standard AM1. A well-known failing of semi-empirical methods is their inability to correctly describe H-bonding. In Fig. 15 we compare the equilibrium structure of the water dimer obtained from the hT , ab initio MBPT(2), and standard semi-empirical theory. It provides the quite hard to describe water dimer in excellent agreement with the first-principles calculations, contrary to AM1 which leads to errors in the donor–acceptor O–H bond of 0.15 Å. In this example, we have to change the electronic parameters along with the corecore repulsion. We would expect this to be the case for most applications. In the future, we hope we can develop the hT to the point that we will have an accurate, QM, description of water and its interactions with other species.

50


Figure 12. Silica nanorod containing 216 atoms.


Figure 13. potentials.

51

Stress–strain curve for 216-atom silica nanorod with classical and quantum

Figure 14. Comparison of forces for O–H bond breaking in water monomer.

52


Figure 15. Structure of water dimer using transfer Hamiltonian, MBPT(2), and standard AM1 Hamiltonian. Bond lengths in angstroms and angles in degrees.

8.

Future

This article calls for some expectations for the future. We have little doubt that the future will demand QM potentials and forces in simulations. It seems to be the single most critical, unsolved, requirement if we aspire toward “predictive” quality. If we could use high-level CC forces in simulations for realistic systems, we would be as confident of our results – as long as the phenomena of interest is amendable to classical MD – as we would be for the determination of molecular properties at that level of theory and basis. Of course, in many cases we cannot run MD for long enough time periods to allow some phenomena to manifest themselves, perhaps forcing more of a kinetic Monte Carlo time extension at that point. We clearly also need much accelerated MD methods regardless of the choice of forces. Like the above NDDO and TB methods, DFT as used in practice, is also a “semi-empirical” theory, as methods like B3LYP now use many parameters to define their functionals and potentials. Even the bastion of state-of-the-art ab initio correlated methods – coupled-cluster theory – is not exact because it depends upon a basis set, as shown in the examples in the introduction. Since even DFT cannot generally be used in MD simulations involving more than ∼300 atoms, to make progress in this field demands that we have “simplified” methods that we can argue retain ab initio or DFT accuracy but now for


53

>1000 atoms, and that can be readily tied to simulations. In this article, we have suggested a procedure for doing so. We showed that the many-electron CC theory could be reformulated into a single determinant form, but at the cost λδη of having a procedure to reliably introduce the quantites we called gνµ ,gλδ µν , gµν , etc. These are complicated quantities that in an ab initio calculation would depend upon one- and two-electron integrals over the basis functions and the cluster amplitudes in T . We could directly compute these elements from ab initio CC methods, to assess their more detailed importance and behavior, and expect to do so. But we prefer, initially, to obtain most of these elements from consideration of a smaller set of quantities and parameters like those in NDDO, or perhaps in TB; and investigatewhether those limited numbers of parameters will be capable of fixing hT = µ,ν |µµ|hT |νν| to the required accuracy. We believe in ensuring that hT has the correct long- and short-range behavior, including the united atom and the separated atom limits. We also want to make sure that the proper balance between the core–core repulsions and the electronic energy is maintained. In our opinion, this is the origin of the age-old problem in semi-empirical theory, that there needs to be different parameters for the total energy, forces, transition states, and those for purely electronic parameters like the electronic density, or photo-electron, or electronic spectra. The same features are observed in solid state applications where the accuracy of cohesive energies and lattice parameters does not transfer to the band structure. Such electronic properties do not depend upon the core– core repulsion at all, yet for many of the total energy properties, as we saw for SiO2 , only the core repulsion parameters need to be changed to get agreement with CCSD. This is not surprising. For total energies and forces, we are fitting the difference between two large numbers, which is much easier to fit than the much larger electronic energy, itself. It would be nice to develop a method that fully accounts for whatever the appropriate cancellation of the core–core effects with the electronic effects from the beginning. Only an ability to describe both reliably will pay the dividends of a truly predictive theory. DFT, MP2, and even higher level methods will continue to progress using local criteria [41], linear scaling, various density fitting tricks [58] and a wealth of other schemes; but regardless, if we can make a transfer Hamiltonian that is already ∼4–5 orders of magnitude faster than DFT, retain and transfer the predictive quality of ab initio or DFT results for clusters to very large molecules, there will always be a need to describe much larger systems accurately and smaller systems faster. In fact, it might be argued, that if such a procedure can be created that will be able to correctly reproduce high-level ab initio results for representative clusters – and fulfill the saturation property we emphasized – the final results might well exceed those from a purely ab initio or DFT method for ∼1000 atoms. The compromises made to make such large molecule applications possible, even at one geometry, forces

54


restricting the basis sets, or number of grid points, or other assorted elements to acommodate the size of system. In principle, the transfer Hamiltonian would not be similarly compromised. Its compromises lie elsewhere.

Acknowledgments This work was support by the National Science Foundation under grant numbers DMR-9980015 and DMR-0325553.

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[54] S. Tsuneyuki, H. Aoki, M. Tsukada, and Y. Matsui, “First-principle interatomic potential of silica applied to molecular dynamics,” Phys. Rev. Lett., 61, 869, 1988. [55] B.W.H van Beest, G.J. Kramer, and R.A. van Santen, “Force fields for silicas and aluminophosphates based on ab initio calculations,” Phys. Rev. Lett., 64, 1955, 1990. [56] P. Vashishta, R.K. Kalia, J.P. Rino, and I. Ebbsjo, “Interaction potential for SiO2 – a molecular dynamics study of structural correlations,” Phys. Rev. B, 41, 12197, 1990. [57] T. Zhu, J. Li, S. Yip, R.J. Bartlett, S.B. Trickey and N.H. de Leeuw, “Deformation and fracture of a SiO2 nanorod,” Mol. Simul., 29, 671, 2003. [58] M. Schutz and M.R. Manby, “Linear scaling local coupled cluster theory with density fitting. Part I: 4-external integrals,” Phys. Chem. – Chem. Phys., 5, 3349, 2003.

1.4 FIRST-PRINCIPLES MOLECULAR DYNAMICS Roberto Car1 , Filippo de Angelis2 , Paolo Giannozzi3, and Nicola Marzari4 1 Department of Chemistry and Princeton Materials Institute, Princeton University, Princeton, NJ, USA 2 Istituto CNR di Scienze e Tecnologie Molecolari ISTM, Dipartimento di Chimica, Universitá di Perugia, Via Elce di Sotto 8, I-06123, Perugia, Italy 3 Scuola Normale Superiore and National Simulation Center, INFM-DEMOCRITOS, Pisa, Italy 4 Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

Ab initio or first-principles methods have emerged in the last two decades as a powerful tool to probe the properties of matter at the microscopic scale. These approaches are used to derive macroscopic observables under the controlled condition of a “computational experiment,” and with a predictive power rooted in the quantum-mechanical description of interacting atoms and electrons. Density-functional theory (DFT) has become de facto the method of choice for most applications, due to its combination of reasonable scaling with system size and good accuracy in reproducing most ground state properties. Such an electronic-structure approach can then be combined with classical molecular dynamics to provide an accurate description of thermodynamic properties and phase stability, atomic dynamics, and chemical reactions, or as a tool to sample the features of a potential energy surface. In a molecular-dynamics (MD) simulation the microscopic trajectory of each individual atom in the system is determined by integration of Newton’s equations of motion. In classical MD, the system is considered composed of massive, point-like nuclei, with forces acting between them derived from empirical effective potentials. Ab initio MD maintains the same assumption of treating atomic nuclei as classical particles; however, the forces acting on them are considered quantum mechanical in nature, and are derived from an electronic-structure calculation. The approximation of treating quantummechanically only the electronic subsystem is usually perfectly appropriate, due to the large difference in mass between electrons and nuclei. Nevertheless, nuclear quantum effects can be sometimes relevant, especially for light 59 S. Yip (ed.), Handbook of Materials Modeling, 59–76. c 2005 Springer. Printed in the Netherlands.

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elements such as hydrogen; classical or ab initio path integral approaches can then be applied, albeit at a higher computational cost. The use of Newton’s equations of motion for the nuclear evolution implies that vibrational degrees of freedom are not quantized, and will follow a Boltzmann statistics. This approximation becomes fully justified only for temperatures comparable with the highest vibrational level in the system considered. In the following, we will describe the combined approach of Car and Parrinello to determine the simultaneous “on-the-fly” evolution of the (Newtonian) nuclear degrees of freedom and of the electronic wavefunctions, as implemented in a modern density-functional code [1] based on plane-waves basis sets, and with the electron–ion interactions described by ultrasoft pseudopotentials [2].

1.

Total Energies and the Ultrasoft Pseudopotential Method

Within DFT, the ground-state energy of a system of Nv electrons, whose one-electron Kohn–Sham (KS) orbitals are φi , is given by E tot [{φi }, {R I }] =

i

+

h2 ¯ 2 φi − ∇ + VNL φi + E H [n] + E xc [n] 2m ion dr Vloc (r)n(r) + U ({R I }),

(1)

where the i index runs over occupied KS orbitals (Nv /2 for closed-shell systems) and n(r) is the electron density. E H [n] is the Hartree energy defined as: E H [n] =

e2 2

dr dr

n(r)n(r ) , |r − r |

(2)

E xc [n] is the exchange and correlation energy, R I are the coordinates of the I th nucleus, {R I } is the set of all nuclear coordinates, and U ({R I }) is the nuclear Coulomb interaction energy. In typical first-principles MD implementations, pseudopotentials (PPs) are used to describe the interaction between the valence electrons and the ionic core, which includes the nucleus and the core electrons. The use of PPs allows to simplify the many-body electronic problem by avoiding an explicit description of the core electrons, which in turn results in a greatly reduced number of orbitals and allows the use of plane waves as a basis set. In the following, we will consider the general case of ultrasoft PPs [2], which includes as a special case norm-conserving PPs [3] in separable form. The PP is composed of ion , given by a sum of atom-centred radial potentials: a local part Vloc ion Vloc (r) =

I

I Vloc ( |r − R I | )

(3)

First-principles molecular dynamics

61

and a nonlocal part VNL :

VNL =

(0) I Dnm |βn βmI |,

(4)

nm,I (0) characterize the PP and are where the functions βnI and the coefficients Dnm specific for each atomic species. For simplicity, we will consider only a single atomic species in the following. The βnI functions, centred at site R I , depend on the nuclear positions via

βnI (r) = βn (r − R I ).

(5)

βn here is a combination of an angular momentum eigenfunction in the angular variables times a radial function which vanishes outside the core region; the indices n and m in Eq. (4) run over the total number Nβ of these functions. The electron density entering Eq. (1) is given by n(r) =



|φi (r)|2 +

i

 I Q nm (r)φi |βnI βmI |φi ,

(6)

nm,I

where the sum runs over occupied KS orbitals. The augmentation functions I (r) = Q nm (r − R I ) are localized in the core. The ultrasoft PP is fully Q nm I (0) (r), Dnm , Q nm (r), and βn (r). The functions determined by the quantities Vloc Q nm (r) are related to atomic orbitals via Q nm (r) = ψnae∗ (r)ψmae (r) − ψnps∗ (r) ψmps (r), where ψ ae are the all-electron atomic orbitals (not necessarily bound), and ψ ps are the corresponding pseudo-orbitals. The Q nm (r) themselves can be smoothed for computational convenience, by taking a truncated multipole expansion [4]. For the case of norm-conserving PPs the Q nm (r) are identically zero. The KS orbitals obey generalized orthonormality conditions φi | S({R I }) |φ j = δi j ,

(7)

where S is a Hermitian overlap operator given by S=1+

qnm |βnI βmI |,

(8)

nm,I

and

qnm =

dr Q nm (r).

(9)

The orthonormality condition (7) is consistent with the conservation of the charge dr n(r) = Nv . Note that the overlap operator S depends on nuclear positions through the |βnI .

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The ground-state orbitals φi that minimize the total energy (1) subject to the constraints (7) are given by δ E tot = i Sφi (r), δφi∗ (r)

(10)

where the i are Lagrange multipliers. Equation (10) yields the KS equations H |φi = i S|φi ,

(11)

where H , the KS Hamiltonian, is defined as H =−

h¯ 2 2 I Dnm |βnI βmI |. ∇ + Veff + 2m nm,I

(12)

Here, Veff is a screened effective local potential ion (r) + VH (r) + µxc (r), Veff (r) = Vloc

(13)

µxc (r) is the exchange-correlation potential µxc (r) =

δ E xc [n] , δn(r)

(14)

and VH (r) is the Hartree potential VH (r) = e

2

dr

n(r ) . |r − r |

(15)

I appearing in Eq. (12) are defined as The “screened” coefficients Dnm I Dnm

=

(0) Dnm

+

I dr Veff (r)Q nm (r).

(16)

I The Dnm depend on the KS orbitals through Veff (Eq. (13)) and the charge density n(r) (Eq. (6)). Since the KS Hamiltonian in Eq. (11) depends on the KS orbitals φi via the charge density, the solution of Eq. (11) is achieved by an iterative self-consistent field procedure.

2.

First-Principles Molecular Dynamics: Born–Oppenheimer and Car–Parrinello

We will assume here that all nuclei (together with their core electrons) can be treated as classical particles; furthermore, we consider only systems for which a separation between the classical motion of the atoms and the quantum motion of the electrons can be achieved, i.e., systems satisfying the


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Born–Oppenheimer adiabatic approximation. For any given ionic configurations, it is possible to calculate the self-consistent electronic ground state, and consequently the forces acting on the ions by virtue of the Hellmann– Feynman theorem. The knowledge of the ionic forces allows then to evolve the nuclear trajectories in time, using any of the algorithms developed in classical mechanics for finite-differences solution of Newton’s equations of motion (two of the most popular choices are Verlet algorithms and Gear predictor– corrector approaches). Born–Oppenheimer MD strives for an accurate evolution of the ions by alternatively converging the electronic wavefunctions to full selfconsistency, for a given set of nuclear coordinates, and then evolving by a finite time step the ions according to the quantum mechanical forces acting on them. A practical algorithms could be summarized as such: • self-consistent solution of the KS equations for a given ionic configuration {R I }; • calculation of the forces acting on the nuclei via the Hellmann–Feynman theorem; • integration of the Newton’s equations of motion for the nuclei; • update of the ionic configuration. This way, the nuclei move on the Born–Oppenheimer surface, i.e., with the electrons in their ground state for any instantaneous configuration of the {R I }. An efficient implementation of this class of algorithms relies on efficient selfconsistent minimization schemes for the electronic wavefunctions, and on accurate extrapolations of the electronic ground-state from one step to the other. The time step itself will only be limited by the need to integrate accurately the highest ionic frequencies. In addition, due to the impossibility of reaching perfect electronic selfconsistency, a drift of the constant of motion is unavoidable, and long simulations require the use of a thermostat to compensate. On the other hand, the Car–Parrinello approach [5] combines “on-thefly” the simultaneous classical MD evolution of the atomic nuclei with the determination of the ground-state wavefunction for the electrons. A (fictitious) dynamics for the electronic degrees of freedom is introduced, defining a classical Lagrangian for the combined electronic and ionic degrees of freedom L=µ

i

dr |φ˙i (r)|2 +

1 ˙ 2 − E tot ({φi }, {R I }); MI R I 2 I

(17)

the wavefunctions above are subject to the set of orthonormality constraints Ni j ({φi }, {R I }) = φi |S({R I })|φ j − δi j = 0.

(18)

Here, µ is a mass parameter coupled to the electronic degrees of freedom, M I are the masses of the atoms, and E tot and S were given in Eqs. (1) and (8),

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respectively. The first term in Eq. (17) plays the role of a kinetic energy associated to the electronic degrees of freedom. The orthonormality constraints (18) are holonomic and do not lead to energy dissipation in a MD run. The Euler equations of motion generated by the Lagrangian of Eq. (17) under the constraints (18) are: µφ¨ i = −

δ E tot + i j Sφ j , δφi∗ j

¨ I = − ∂ E tot + FI = MI R ∂R I

ij

(19)

∂S i j φi ∂R

I

φj .

(20)

where i j are Lagrange multipliers enforcing orthogonality. If the system is in the electronic ground state corresponding to the nuclear configuration at that time step, the forces acting on the electronic degrees of freedom µφï =0 vanish and Eq. (19) reduces to the KS equations (10) or (11). A unitary rotation brings the matrix into diagonal form: i j = i δi j . Similarly, the equilibrium nuclear configuration is achieved when the atomic forces F I in Eq. (20) vanish. In deriving explicit expressions for the forces, Eq. (20), one should keep in mind that the electron density also I depends on R I through Q nm and βnI . Introducing the quantities I = ρnm

φi |βnI βmI |φi ,

(21)

i

and I = ωnm

i j φ j |βnI βmI |φi ,

(22)

ij

we arrive at the expression FI = − −

∂U − ∂R I nm

dr

ion ∂ Vloc n(r) − ∂R I

dr Veff (r)

I ∂ω I I ∂ρnm Dnm + qnm nm , ∂R I ∂R I nm

I ∂ Q nm (r) nm

∂R I

I ρnm

(23)

I and Veff have been defined in Eqs. (16) and (13), respectively. The where Dnm last term of Eq. (23) gives the constraint contribution to the forces. We underline that the dynamical evolution for the electronic degrees of freedom should not be construed as representing the true electron dynamics; rather it represent a dynamical system of fictitious degree of freedom adiabatically decoupled from the moving ions, but driven to follow closely the ionic dynamics, with small and oscillatory departures from what would be the exact Born–Oppenheimer ground-state energy. As a consequence, even


65

the Car–Parrinello dynamics for the nuclei becomes in principle inequivalent to the Born–Oppenheimer dynamics. However, suitable choices for the computational parameters used in the simulation exist, and are such that the two dynamics give the same macroscopic observables. The full self-consistency cycle of the Born–Oppenheimer dynamics can be dispensed for, at a great computational advantage only marginally offset by the need to use shorter timesteps to integrate the fast electronic degrees of freedom. The adiabatic separation can be understood on the basis of the following argument [6, 7]. The fictitious electronic dynamics, once close to the ground state, can be described as a superposition of harmonic oscillators whose frequencies are given by:

2( j − i ) ωi j = µ

1/2

,

(24)

where i is the KS eigenvalue of the ith occupied orbital and j is the KS eigenvalue of the j th unoccupied orbital. For a system with an energy gap E g , the lowest frequency can be estimated to be ωmin = (2E g /µ)1/2. If ωmin is much larger than the highest frequency appearing in the nuclear motion, there is a large separation between electronic and nuclear frequencies. Under such conditions, the electronic motion is adiabatically decoupled from the nuclear motion and there is negligible energy transfer from nuclear to electronic degrees of freedom. This is a nonobvious result, since both dynamics are classical and subject to the equipartion of energy, and it is the key to understand when and why the Car–Parrinello dynamics works. For typical E g values, in the order of a few electronvolts, the electronic mass parameter µ can be chosen relatively large, in the order of 300–500 amu or even more, without any loss of adiabaticity. The time step of the simulation can be chosen as the largest compatible with the resulting electronic dynamics. Larger values of µ allow the use of larger time steps, but the requirement of adiabaticity sets an upper limit to µ. Time steps of a fraction of a femtosecond are typically accessible. The electronic dynamics is faster than the nuclear dynamics and averages out the error on forces that is present because the system is never at the instantaneous electronic ground state, but only close to it (the system has to be brought close to the electronic ground state at the beginning of the dynamics). In such conditions, the resulting nuclear dynamics is very close to the true Born–Oppenheimer dynamics, and the electronic dynamics is stable (with negligible energy transfer from the nuclei) even for long simulation times. Moreover, the Car–Parrinello dynamics is computationally more convenient than the Born–Oppenheimer dynamics, because the latter requires a high accuracy in self-consistency in order to provide the needed accuracy on the forces. The Car–Parrinello dynamics does not provide accurate instantaneous forces, but it provides accurate average nuclear trajectories.

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2.1.

Equations of Motion and Orthonormality Constraints

In Car–Parrinello implementations equations of motion (19) and (20) are discretized using the standard-Verlet or the velocity-Verlet algorithm. The following discussion, including the treatment of the R I -dependence of the orthonormality constraints, applies to the standard Verlet algorithm, and using the Fourier acceleration scheme of Tassone et al. [8]. (In this approach the fictitious electronic mass is generally represented by an operator , chosen in such a way to reduce the highest electronic frequencies.∗ ) From the knowledge of the electronic orbitals at time t and t − t, the orbitals at t + t are given, in the standard Verlet, by φi (t + t) = 2φi (t) − φi (t − t) 



δ E tot i j (t + t) S(t)φ j (t); −(t)2 −1  ∗ − δφi j

(25)

where t is the time step, and S(t) indicates the operator S evaluated for nuclear positions R I (t). Similarly the nuclear coordinates at time t + t are given by: R I (t + t) = 2R I (t) − R I (t − t) − 

(t)2 MI



∂ S(t) ∂ E tot φ j (t) . × − i j (t + t) φi (t) ∂R I ∂R I ij

(26)

The orthonormality conditions must be imposed at each time-step: φi (t + t)|S(t + t)|φ j (t + t) = δi j ,

(27)

leading to the following matrix equation: A + λB + B † λ† + λCλ† = 1

(28)

where the unknown matrix λ is related to the matrix of Lagrange multipliers at time t + t via λ = (t)2 ∗ (t + t). In Eq. (28), the dagger indicates ∗ When using plane waves, a convenient choice for the matrix elements of such operator is

G,G = max(µ, µ((h¯ 2 G 2 )/(2m E c )))δG,G , where G, G are the wave vector of PWs, E c is a cutoff (typically

a few Ry) which defines the threshold for Fourier acceleration. The fictitious electron mass depends on G as the kinetic energy for large G, it is constant for small G. This scheme allows us to use larger steps with negligible computational overhead.


67

Hermitian conjugate (λ = λ† ). The matrices A, B, and C are given by: Ai j = φ¯i |S(t + t)|φ¯ j , Bi j = −1 S(t)φi (t)|S(t + t)|φ¯ j , Ci j = −1 S(t)φi (t)|S(t + t)| −1 S(t)φ j (t),

(29)

with φ¯ i = 2φi (t) − φi (t − t) − (t)2 −1

δ E tot(t) . δφi∗

(30)

The solution of Eq. (28) in the ultrasoft PP case is not obvious, because Eq. (26) is not a closed expression for R I (t + t). The problem is that (t + t) appearing in Eq. (26) depends implicitly on R I (t + t) through S(t + t). Consequently, it is in principle necessary to solve iteratively for R I (t + t) in Eq. (26). A simple solution to this problem was provided in Laasonen et al. [4]. (t + t) is extrapolated using two previous values: i(0) j (t + t) = 2i j (t) − i j (t − t).

(31)

4 Equation (26) is used to find R(0) I (t +t), which is correct to O(t ). From (0) (1) R I (t +t) we can obtain a new set i j (t +t) and repeat the procedure until convergence is achieved. It turns out that in most practical applications the procedure converges at the very first iteration. Thus, the operations described above are generally executed only once per time step. The solution of Eq. (28) is found using a modified version [4, 9] of the iterative procedure of Car and Parrinello [10]. The matrix B is decomposed into hermitian (Bh ) and antihermitian (Ba ) parts,

B = Bh + Ba ,

(32)

and the solution is obtained by iteration: λ(n+1) Bh + Bhλ(n+1) = 1 − A − λ(n) Ba − Ba† λ(n) − λ(n) Cλ(n) .

(33)

The initial guess λ(0) can be obtained from λ(0) Bh + Bh λ(0) = 1 − A.

(34)

Here, the Ba - and C-dependent terms are neglected because they are of higher order in t (Ba vanishes for vanishing t). Equations (34) and (33) have the same structure: λBh + Bhλ = X

(35)

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where X a Hermitian matrix. Equation (35) can be solved exactly by finding the unitary matrix U that diagonalizes Bh : U † BhU = D, where Di j = di δi j . The solution is obtained from (U † λU )i j = (U † XU )i j /(di + d j ).

(36)

When X = 1 − A, Eq. (36) yields the starting λ(0), while λ(n+1) is obtained from λ(n) by solving Eq. (36) with X given by Eq. (33). This iterative procedure usually converges in very few steps (ten or less).

3.

Plane-Wave Implementation

In most standard implementations, first-principles MD schemes employ a plane-wave (PW) basis set. An advantage of PWs is that they do not depend on atomic positions and are free of basis-set superposition errors. Total energies and forces on the atoms can be calculated using computationally efficient Fast Fourier transform (FFT) techniques and Pulay forces [11] vanish because PWs do not depend on atomic positions. Finally, the convergence of a calculation can be controlled in a simple way, since it depends only upon the number of PWs included in the expansion of the electron density. The dimension of a PW basis set is controlled by a cutoff in the kinetic energy of the PWs. A disadvantage of PWs is their extremely slow convergence in describing core states, which can however be circumvented by the use of PPs. Ultrasoft PPs allow to efficiently deal with this difficulty also in systems containing transition metals or first-row elements O, N, F whose 3d and 2p orbitals, respectively, are very contracted. The use of a PW basis set implies that periodic boundary conditions are imposed. Systems not having translational symmetry in one or more directions, have to be placed into a suitable periodically repeated box (a “supercell”). Let {R} be the translation vectors of the periodically repeated supercell. The corresponding reciprocal lattice vectors {G} obey the conditions Ri · G j = 2π n, with n an integer number. The KS orbitals can be expanded in a plane-wave basis up to a kinetic energy cutoff E cwf : 1 φ j,k (r) = √ φ j,k (G)e−i(k+G)·r , G∈{G wf }

(37)

c

where is the volume of the cell, {Gcwf} is the set of G vectors satisfying the condition h¯ 2 |k + G|2 < E cwf , 2m

(38)

and k is the Bloch vector of the electronic states. In crystals, one must use a grid of k-points dense enough to sample the Brillouin zone (the unit cell of the


69

reciprocal lattice). In molecules, liquids and in general if the simulation cell is large enough, the Brillouin zone can be sampled using only the k = 0 () point. An advantage of this choice is that the orbitals can be taken to be real in r-space. In the following we will drop the k vector index. Functions in real space and their Fourier transforms will be denoted by the symbols, if this does not originate ambiguity. The φ j (G)s are the actual electronic variables in the fictitious dynamics. The calculation of H φ j and of the forces acting on the ions are the basic ingredients of the computation. Scalar products φ j |βnI and their spatial derivatives are typically evaluated in G-space. An important advantage of I are easily working in G-space is that atom-centred functions like βnI and Q nm evaluated at any atomic position: βnI (G) = βn (G)e−iG·R I .

(39)

Thus,

φ j |βnI =

φ ∗j (G)βn (G)e−iG·R I

(40)

G∈{Gcwf }

and

∂β I n φj = −i ∂R I

Gφ ∗j (G)βn (G)e−iG·R I .

(41)

G∈{Gcwf }

The kinetic energy term is diagonal in G-space and is easily calculated:

− ∇ 2 φ j (G) = G 2 φ j (G).

(42)

In summary, the kinetic and nonlocal PP terms in H φ j are calculated in G-space, while the local potential term Veff φ j , that could be calculated in G-space, is more convenient determined using a ‘dual space’ technique, switching from G- to r-space with FFTs, and performing the calculation in the space where it is least expensive. In practice, the KS orbitals are first Fourier-transformed to r-space; then, (Veff φ j )(r) = Veff (r)φ j (r) is calculated in r-space, where Veff is diagonal; finally (Veff φ j )(r) is Fourier-transformed back to (Veff φ j )(G). In order to use FFT, the r-space is discretized by a uniform grid spanning the unit cell: f (m 1 , m 2 , m 3 ) ≡ f (rm 1 ,m 2 ,m 3 ),

rm 1 ,m 2 ,m 3 = m 1

a1 a2 a3 + m2 + m3 , N1 N2 N3 (43)

where a1 , a2 , a3 are lattice basis vectors, the integer index m 1 runs from 0 to N1 − 1, and similarly for m 2 and m 3 . In the following we will assume

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for simplicity that N1 , N2 , N3 are even numbers. The FFT maps a discrete periodic function in real space f (m 1 , m 2 , m 3 ) into a discrete periodic function in reciprocal space f˜(n 1 , n 2 , n 3 ) (where n 1 runs from 0 to N1 − 1, and similarly for n 2 and n 3 ), and vice versa. The link between G-space components and FFT indices is: f˜(n 1 , n 2 , n 3 ) ≡ f (Gn1 ,n2 ,n3 ), n 1

n 1

n 1

Gn1 ,n2 ,n3 = n 1 b1 + n 2 b2 + n 3 b3

(44)

n 1

≥ 0, n 1 = + N1 if < 0, and similarly for n 2 and n 3 . where n 1 = if The FFT dimensions N1 , N2 , N3 must be big enough to include all non negligible Fourier components of the function to be transformed: ideally the Fourier component corresponding to n 1 = N1 /2, and similar for n 2 and n 3 , should vanish. In the following, we will refer to the set of indices n 1 , n 2 , n 3 and to the corresponding Fourier components as the “FFT grid”. The soft part of the charge density n soft(r) = j |φ j (r)|2 contains Fourier components up to a kinetic energy cutoff E csoft = 4E cwf . This is evident from the formula: n soft(G) =

G ∈{Gcwf }

j

φ ∗j (G − G )φ j (G ).

(45)

In the case of norm-conserving PPs, the entire charge density is given by n soft(r). Veff should be expanded up to the same E csoft cutoff since all the Fourier components of Veff φ j up to E cwf are required. Let us call {Gcsoft} the set of G-vectors such that h¯ 2 G < E csoft . (46) 2m The soft part of the charge density is calculated in r-space, by Fouriertransforming φ j (G) into φ j (r) and summing over the occupied states. The exchange-correlation potential µxc (r), Eq. (14), is a function of the local charge density and – for gradient-corrected functionals – of its gradient at point r: µxc (r) = Vxc (n(r), |∇n(r)|).

(47)

The gradient ∇n(r) is conveniently calculated from the charge density in G-space, using (∇n)(G) = −iGn(G). The Hartree potential VH (r), Eq. (15), is also conveniently calculated in G-space: VH (G) =

4π n(G)∗ . G2

(48)

Thus, in the case of norm-conserving PPs, a single FFT grid, large enough to accommodate the {Gcsoft} set, can be used for orbitals, charge density, and potential.


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The use of FFT is mathematically equivalent to a pure G-space description (we neglect here a small inconsistency in exchange-correlation potential and energy density, due to the presence of a small amount of components beyond the {Gcsoft} set). This has important consequences: working in G-space means that translational invariance is exactly conserved and that forces are analytical derivatives of the energy (apart from the effect of the small inconsistency mentioned above). Forces that are analytical derivatives of the energy ensure that the constant of motion (i.e., the sum of kinetic and potential energy of the ions in Newtonian dynamics) is conserved during the evolution.

3.1.

Double-Grid Technique

Let us focus on ultrasoft PPs. In G-space the charge density is: n(G) = n soft(G) +

I Q mn (G)φi |βnI βmI |φi .

(49)

i,nm,I

The augmentation term often requires a cutoff higher than E csoft , and as a consequence a larger set of G-vectors. Let us call {Gcdens} the set of G-vectors that are needed for the augmented part: h¯ 2 2 G < E cdens . 2m

(50)

In typical situations, using pseudized augmented charges, E cdens ranges from E csoft to ∼ 2 − 3E csoft . The same FFT grid could be used both for the augmented charge density and for KS orbitals. This however would imply using an oversized FFT grid in the most expensive part of the calculation, dramatically increasing computer time. A better solution is to introduce two FFT grids: • a coarser grid (in r-space) for the KS orbitals and the soft part of the charge density. The FFT dimensions N1 , N2 , N3 of this grid are big enough to accommodate all G-vectors in {Gcsoft}; • a denser grid (in r-space) for the total charge density and the exchangecorrelation and Hartree potentials. The FFT dimensions M1 ≥ N1 , M2 ≥ N2 , M3 ≥ N3 of this grid are big enough to accommodate all G-vectors in {Gcdens}. In this framework, the soft part of the electron density n soft , is calculated in r-space using FFTs on the coarse grid and transformed in G-space using a coarse-grid FFT on the {Gcsoft} grid. The augmented charge density is calculated in G-space on the {Gcdens} grid, using Eq. (49) as described in the next section. n(G) is used to evaluate the Hartree potential, Eq. (48). Then

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n(G) is Fourier-transformed in r-space on the dense grid, where the exchangecorrelation potential, Eq. (47), is evaluated. In real space, the two grids are not necessarily commensurate. Whenever the need arises to go from the coarse to the dense grid, or vice versa, this is done in G-space. For instance, the potential Veff , Eq. (13), is needed both on the I dense grid to calculate quantities such as the Dnm , Eq. (16), and on the coarse grid to calculate Veff φ j , Eq. (11). The connection between the two grids occurs in G-space, where Fourier filtering is performed: Veff is first transformed in G-space on the dense grid, then transferred to the coarse G-space grid by eliminating components incompatible with E csoft , and then back-transformed in r-space using a coarse-grid FFT. We remark that for each time step only a few dense-grid FFT are performed, while the number of necessary coarse-grid FFTs is much larger, proportional to the number of KS states Nks .

3.2.

Augmentation Boxes

Let us consider the augmentation functions Q nm , which appear in the calI culation of the electron density, Eq. (49), in the calculation of Dnm , Eq. (16), I and in the integrals involving ∂ Q nm /∂R I needed to compute the forces acting on the nuclei, Eq. (23). The calculation of the Q nm in G-space has a large computational cost because the cutoff for the Q nm is the large cutoff E cdens . The computational cost can be significantly reduced if we take advantage of the localization of the Q nm in the core region. We call “augmentation box” a fraction of the supercell, containing a small portion of the dense grid in real space. An augmentation box is defined only for atoms described by ultrasoft PPs. The augmentation box for atom I is centred at the point of the dense grid that is closer to the position R I . During a MD run, the centre of the I th augmentation box makes discontinuous jumps to one of the neighbouring grid points whenever the position vector R I gets closer to such grid point. In a MD run, the augmentation box must always contain completely the augmented charge belonging to the I th atom; otherwise, the augmentation box must be as small as possible. The volume of the augmentation box is much smaller than the volume of the supercell. The number of G-vectors in the reciprocal space of the augmentation box is smaller than the number of G-vectors in the dense grid by the ratio of the volumes of the augmentation box and of the supercell. As a consequence, the cost of calculations on the augmentation boxes increases linearly with the number of atoms described by ultrasoft PPs. Augmentation boxes are used (i) to construct the augmented charge density, Eq. (6), and (ii) to calculate the self-consistent contribution to the


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coefficients of the nonlocal PP, Eq. (16). In case (i), the augmented charge is conveniently calculated in G-space, following [4], and Fourier-transformed in r-space. All these calculations are done on the augmentation box grid. Then the calculated contribution at each r-point of the augmentation box grid is added to the charge density at the same point in the dense grid. In case I as follows: for every atom described (ii), it is convenient to calculate Dnm by a ultrasoft PP, take the Fourier transform of Veff (r) on the corresponding augmentation box grid and evaluate the integral of Eq. (16) in G-space.

3.3.

Parallelization

Various parallelization strategies for PW–PP calculations have been described in the literature. A strategy that ensures excellent scalability in terms of both computer time and memory consists in distributing the PW basis set and the FFT grid points in real and reciprocal space across processors. A crucial issue for the success of this approach is the FFT algorithm, which must be capable of performing three-dimensional FFT on data shared across different processors with good load balancing. The parallelization in the case of ultrasoft PPs is described in detail in Giannozzi et al. [12].

4.

Applications

Presently, systems described by supercells containing up to a few hundreds atom are within the reach of first-principles MD. A large body of techniques developed for classical MD, such as simulated annealing, finite-temperature simulations, free-energy calculations, etc. can be straightforwardly extended to first-principles MD. Typical applications include the study of aperiodic systems: liquids, atomic clusters, large molecules, including biological active sites; complex solid-state systems: defects in solids, defect diffusion, surface reconstructions; dynamical processes: chemical reactions, catalysis, and finitetemperature studies. The use of ultrasoft PPs is especially convenient in the simulation of systems containing first-row atoms (C, N, O, F) and transition metal elements, such as, e.g., biological active sites, involving Fe, Mn, Ni as catalytic centers. A good example of application of first-principles MD is the investigation of a complex organometallic reaction: the migratory insertion of carbon monoxide (CO) into zirconium–carbon bonds anchored to a calix[4]arene moiety, shown in Fig. 1 [13]. The investigated reactivity is representative of the large class of migratory insertions of carbon monoxide and alkyl-isocyanides into metal–alkyl bonds observed for most of the early d-block metals, leading to the formation of a new carbon–carbon bond [14].

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Figure 1.

Figure 2.

Geometry of calix[4]arene.

Insertion of CO into the Zr-CH3 bond of a calix[4]arene.

The CO migratory insertion is supposed to be initialized by the coordination of the nucleophilic CO species to the electron-deficient zirconium centre of [ p-But calix[4](OMe)2 (O)2 –Zr(Me)2 ], 1 in Fig. 2, to form the relatively stable adduct 2. MD simulations were started by heating up by small steps (via rescaling of atomic velocities) the structure of 2 to a temperature of 300 K. Both electronic and nuclear degrees of freedom were allowed to evolve without any constraint for 2.4 ps. The migratory CO insertion can be followed by studying the time evolution of the carbon–carbon CH3 –CO, metal–carbon Zr–CH3 and metal– oxygen Zr–O distances. Figure 3 clearly shows that the reactive CO migration takes place within ca. 0.4 ps: the fast decrease in the CH3 –CO distance from ca. 2.7 Å to ca. 1.5 Å corresponds to the formation of the new CH3–CO carbon– carbon bond. At the same time the Zr–CH3 distance follows an almost complementary trajectory with respect to the CH3 –CO distance and grows from ca. 2.4 up to ca. 3.7 Å, reflecting the methyl detachment from the metal centre upon CO insertion.


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4.5

’C-C’ ’Zr-C’ ’Zr-O’

4

Distances (Angstrom)

3.5

3

2.5

2

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1

0

0.2

0.4

0.6

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1.6

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Figure 3. Evolution of carbon–carbon CH3 –CO, metal–carbon Zr–CH3 and metal–oxygen Zr–O distances during the simulation of CO insertion into calix[4]arene.

The Zr–O distance is found to decrease from its initial value of ca. 3.5 Å in 2, to ca. 2.2 Å, corresponding to the Zr–O bond in 4, within 1.0 ps. The 0.6 ps delay between the formation of the CH3 –CO bond and the formation of the Zr–O bond suggests the initial formation of a transient species, 3 in Fig. 2, characterized by an η1 -coordination of the OC–CH3 acyl group with a formed CH3 –CO bond and still a long Zr–O bond; this η1 -acyl subsequently evolves to the corresponding η2 -bound acyl species. The short time stability of the η1 -acyl isomer (ca. 0.6 ps) suggests a negligible barrier for the conversion of the η1 into the more stable η2 -isomer, as confirmed by static DFT calculations.

Acknowledgments Algorithms and codes presented in this work have been originally developed at EPFL Lausanne by Alfredo Pasquarello and Roberto Car, and then at Princeton University by Paolo Giannozzi and Roberto Car. Several people have also contributed or are contributing to the current development and distribution under the GPL License: Kari Laasonen, Andrea Trave, Carlo Cavazzoni, and Nicola Marzari.

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References [1] A. Pasquarello, P. Giannozzi, K. Laasonen, A. Trave, N. Marzari, and R. Car, The Car–Parrinello molecular dynamics code described in this paper is freely available in the Quantum-espresso distribution, released under the GNU Public License at http://www.democritos.it/scientific.php., 2004. [2] D. Vanderbilt, “Soft Self-Consistent Pseudopotentials in a Generalized Eigenvalue Formalism,” Physical Review B, 41, 7892, 1990. [3] D.R. Hamann, M. Schlüter, and C. Chiang, “Norm-Conserving Pseudopotentials,” Physical Review Letters, 43, 1494, 1979. [4] K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt, “Car–Parrinello Molecular Dynamics with Vanderbilt Ultrasoft Pseudopotentials,” Physical Review B, 47, 10142, 1993. [5] R. Car and M. Parrinello, “Unified Approach for Molecular Dynamics and DensityFunctional Theory,” Physical Review Letters, 55, 2471, 1985. [6] G. Pastore, E. Smargiassi, and F. Buda, “Theory of Ab Initio Molecular-Dynamics Calculations,” Physical Review A, 44, 6334, 1991. [7] D. Marx and J. Hutter, “Ab-Initio Molecular Dynamics: Theory and Implementation,” In: Modern Methods and Algorithms of Quantum Chemistry, John von Neumann Institute for Computing, FZ Jülich, pp. 301–449, 2000. [8] F. Tassone, F. Mauri, and R. Car, “Acceleration Schemes for Ab Initio MolecularDynamics Simulations and Electronic-Structure Calculations,” Physical Review B, 50, 10561, 1994. [9] C. Cavazzoni and G.L. Chiarotti, “A Parallel and Modular Deformable Cell Car–Parrinello Code,” Computer Physics Communuications, 123, 56, 1999. [10] R. Car and M. Parrinello, “The Unified Approach for Molecular Dynamics and Density Functional Theory,” In: A. Polian, P. Loubeyre, and N. Boccara (eds.), Simple Molecular Systems at Very High Density, Plenum, New York, p. 455, 1989. [11] P. Pulay, “Ab Initio Calculation of Force Constants and Equilibrium Geometries,” Molecular Physics, 17, 197, 1969. [12] P. Giannozzi, F. De Angelis, and R. Car, “First-Principle Molecular Dynamics with Ultrasoft Pseudopotential: Parallel Implementation and Application to Extended Bio-Inorganic Systems,” Journal of Chemical Physics, 120, 5903–5915, 2004. [13] S. Fantacci, F. De Angelis, A. Sgamellotti, and N. Re, “Dynamical Density Functional Study of the Multistep CO Insertion into Zirconium–Carbon Bonds Anchored to a Calix[4]arene Moiety,” Organometallics, 20, 4031, 2001. [14] L.D. Durfee and I.P. Rothwell, “Chemistry of Eta-2-acyl, Eta-2-iminoacyl, and Related Functional Groups,” Chemical Reviews, 88, 1059, 1988.

1.5 ELECTRONIC STRUCTURE CALCULATIONS WITH LOCALIZED ORBITALS: THE SIESTA METHOD Emilio Artacho1 , Julian D. Gale2 , Alberto García3 , Javier Junquera4, Richard M. Martin5 , Pablo Ordejón6 , Daniel Sánchez-Portal7, and José M. Soler8 1 University of Cambridge, Cambridge, UK 2 Curtin University of Technology, Perth, Western Australia, Australia 3 Universidad del País Vasco, Bilbao, Spain 4 Rutgers University, New Jersey, USA 5 University of Illinois at Urbana, Urbana, IL, USA 6 Instituto de Materiales, CSIC, Barcelona, Spain 7 Donostia International Physics Center, Donostia, Spain 8

Universidad Autónoma de Madrid, Madrid, Spain

Practical quantum mechanical simulations of materials, which take into account explicitly the electronic degrees of freedom, are presently limited to about 1000 atoms. In contrast, the largest classical simulations, using empirical interatomic potentials, involve over 109 atoms. Much of this 106 -factor difference is due to the existence of well-developed order-N algorithms for the classical problem, in which the computer time and memory scale linearly with the number of atoms N of the simulated system. Furthermore, such algorithms are well suited for execution in parallel computers, using rather small interprocessor communications. In contrast, nearly all quantum mechanical simulations involve a computational effort which scales as O(N 3 ), that is, as the cube of the number of atoms simulated. Such an intrinsically more expensive dependence is due to the delocalized character of the electron wavefunctions. Since the electrons are fermions, every one of the ∼N occupied wavefunctions must be kept orthogonal to every other one, thus requiring ∼N 2 constraints, each involving an integral over the whole system, whose size is also proportional to N . Despite such intrinsic difficulties, the last decade has seen an intense advance in algorithms that allow quantum mechanical simulations with an 77 S. Yip (ed.), Handbook of Materials Modeling, 77–91. c 2005 Springer. Printed in the Netherlands.

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O(N ) computational effort. Such algorithms are based on avoiding the spatially extended electron eigenfunctions and using instead magnitudes, such as the one-electron density matrix, that are spatially localized, thus allowing for a spatial decomposition of the electronic problem. This strategy exploits what has been called by Walter Kohn the nearsightedness of the electron-gas [1]. Its implementation requires, or is greatly facilitated, by the use of a spatially localized basis set, such as a linear combination of atomic orbitals (LCAO). This paper gives a brief overview of such methods and describes in some detail one of them, the Spanish Initiative for Electronic Simulations with Thousands of Atoms (SIESTA).

1.

Order- N Algorithms

Despite its relatively recent development, there are already good reviews of O(N ) methods for the electronic structure problem, such as those of Ordejon [2] and Goedecker [3]. Here we will only explain briefly the basic difficulties and lines of solution, emphasizing the more practical aspects. Although some methods, such as that of Car and Parrinello, use a direct minimization approach, it is pedagogically convenient to consider the solution of the electronic problem as a two-step process. First, one needs to find the Hamiltonian (and eventually the overlap) matrix in some convenient basis. Second one has to find the solution of Schrödinger’s equation in that representation, that is, the electron wavefunctions or density matrix as a linear combination of basis functions. Since the effective electron potential, and therefore the Hamiltonian, depends on the electron density, this two-step process has to be iterated to selfconsistency. Although both steps require highly nontrivial algorithms to be performed with O(N ) effort, from a physical point of view the second one involves more fundamental problems and solutions. We will therefore give first, in this section, an overview of the second step, and leave for the next section the technical solution of the first step (the construction of the Hamiltonian), in the context of SIESTA. Although O(N ) methods have been developed for Hatree–Fock calculations as well, here we will restrict ourselves to density functional theory (DFT) because the methods are more mature and easier to understand in this context. There are numerous good introductory reviews on DFT like in Ref. [4]. A central magnitude in most O(N ) methods is the one-electron density operator ρˆ =

|ψi f (i )ψi |.

(1)

i

Its representation in real space is the density matrix ρ(r, r ) =

i

f (i ) ψi (r) ψi∗ (r ),

(2)

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where ψi (r) is the ith eigenfunction of the Kohn–Sham one-electron Hamiltonian of DFT, i is its corresponding eigenvalue, and f (i ) is its Fermi– Dirac occupation factor. Such a representation is appropriate for recent schemes that use finite difference formulae, in a real space grid of points, to solve the Kohn–Sham equations. We will assume, however, that a basis set of some kind of localized orbitals φµ (r), is used to expand the electron wavefunc = matrix takes the form tions: ψi (r) µ ciµ φµ (r). In this case the density ∗ . The density ρ(r, r ) = µν ρµν φµ (r) φν∗ (r ), where ρµν = i f (i ) ciµ ciν matrix allows to generate all the magnitudes required for a self-consistent DFT calculation. The electron density is simply its diagonal, ρ(r) = ρ(r, r), and it allows to calculate the Hartree (electrostatic) and exchange-correlation potentials. The electronic kinetic energy is given by 1 E kin = − 2

∇r2 ρ(r, r )

r=r

d3 r =

µν

ρµν Tνµ ,

(3)

where, using atomic units (e = m e = = 1),

1 φν∗ (r)∇ 2 φµ (r)d3 r. (4) 2 Notice from Eq. (2) that the electron eigenstates ψi (r) are also eigenvectors of the density matrix, whose corresponding eigenvalues are the occupation factors f (i ). However, diagonalizing ρµν is an O(N 3 ) operation, no cheaper than diagonalizing the Hamiltonian, so that magnitudes that depend on the eigenvectors, like the band structure or the density of states, are not ususally obtained in O(N ) calculations (although there are special O(N ) techniques to obtain partially some of these magnitudes [3]). The central role of ρ(r, r ) in O(N ) methods stems from the fact that it is sparse: when r and r are far away, ρ(r, r ) becomes negligibly small. To see this, it suffices to consider a uniform electron gas. In this case, the one-electron √ eigenfunctions become plane waves of the form ψk (r) = exp(ikr)/ where k is a wave vector and is the system volume. By substitution into Eq. (2), it is easy to see that ρ(r, r ), which in this case depends only on |r − r|, is simply the Fourier transform of the Fermi function in k space: f (k) = 1, if |k| ≤ k F , and f (k) = 0 otherwise, at zero temperature. Its Fourier transform ρ(|r − r|) decays as cos(k F |r − r|)/|r − r|2 . Furthermore, it turns out that the free electron gas at T = 0 is the worst possible case: at finite temperature the decay is exponential, with a decay constant proportional to the temperature. For an insulator, the decay is also exponential, even at zero temperature, with a decay constant that increases with the energy gap [3]. Therefore, the number of non-negligible values of ρ(r, r ) increases only linearly with the size of the system, with a prefactor that depends on its bonding character, and particularly on whether it is metallic or insulating. We will see that the computational effort (execution time and memory) is directly related to the number of those Tνµ = −

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non-negligible matrix elements. In practice, for metallic systems, the prefactor is so large that the crossover system size, at which O(N ) methods become computationally competitive over traditional O(N 3 ) methods, has not yet been reached. We will therefore assume that the systems that we are considering are insulators, even though some (but not all) of the methods described could in principle be applied to metals as well. Chronologically, the first quantum mechanical O(N ) method, the divide and conquer (DC) scheme of Weitao Yang et al., is also conceptually the simplest from a physical point of view (recursion and other methods based on Green’s functions were developed in the 1970s that were also linear scaling; their linear-scaling character was not the driving force behind them though, and they are not so well suited for self-consistent studies). It is based on dividing the whole system into smaller pieces, each surrounded by a buffer region, that are then treated (including the buffer) by conventional quantum mechanical methods, i.e., by diagonalizing the local Hamiltonian. Using a common value for the chemical potential (Fermi energy) allows for charge transfer among different regions. From this treatment, the density (in the first proposal) or the density matrix (in a subsequent development) of the different pieces are combined to generate that of the entire system. The matrix elements between points (or orbitals) in different spatial pieces are obtained from those between the pieces themselves and their buffer regions (the elements between two buffer points are not used). Thus, the width of the buffer regions must account fully for the decay of ρ(r, r ). Beyond this width, usually called the localization radius, the matrix elements are neglected. In practice, this implies rather large buffer regions, making the method more expensive than other, more recent, O(N ) methods. The second O(N ) method to be mentioned, the Fermi operator expansion (FOE), constructs the whole (though sparse) density matrix as an expansion of the Hamiltonian. To this end, one expands the Fermi–Dirac function (conveniently smoothed) as a polynomial, within some energy range: f () = nmax n a , for min < < max . In practice, one uses n max + 1 Chebyshev n n=0 polynomials rather than powers of for stability reasons, but this is just a technical point [3]. Then one constructs the density matrix (by performing n max multiplications of the Hamiltonian) as ρˆ =

n max

an H n ,

(5)

n=0

where the coefficients an are the same as before. To keep the O(N ) scaling of the computation, one needs to restrict the spatial range, within the required localization radius, after each matrix multiplication. To understand the effect of this operator, consider its application to an eigenvector of the Hamiltonian. Provided that the eigenvalue is within the range of the expansion, the result


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max will be ρψ ˆ = nn=0 an n ψ = f ()ψ. This is exactly the effect of the density matrix operator of Eq. (1). A closely related method is the Fermi operator projection (FOP), in which one starts from a trial set of electron wavefunctions, each constrained within a different localization region (usually around atoms) and applies the expansion (5) of the density matrix operator (without constructing it) to the trial functions, projecting them into the occupied subspace. One still needs to make them orthogonal but, since they are spatially localized by construction, the process can be performed in O(N ) operations. The resulting functions are a complete representation of the density matrix, of size Nel × Nloc , with Nel the number of electrons and Nloc the number of basis orbitals within a localization region. In contrast, the normal representation of the density matrix, used in the FOE method, has Nbasis × Nloc nonzero matrix elements, where Nbasis is the number of basis orbitals, which is substantially larger than Nel . Therefore, the FOP method is more efficient than the FOE. In the density matrix minimization (DMM) method of Li, Nunes and Vanderbilt, the entire sparse density matrix is also obtained by minimizing the total energy as a function of its matrix elements in a localized basis set of atomic orbitals [5], grid points, or some other kind of support functions [6]. Again, matrix elements separated by more than a pre-established localization radius are neglected. A complication is that in performing the minimization, one must impose the constraint that the eigenvalues of the density matrix (i.e., the occupation weights) must be between zero and one, as required by the Fermi exclusion principle (for simplicity, we will consider combined spin–orbital indexes µ and i, so that each basis orbital or electron state has a defined spin and contains a single electron). At zero temperature, the constrained energy minimization will make all the eigenvalues either zero (above the Fermi energy) or one, what amounts to making matrix ρ idempotent: ρ 2 = ρ (since all the eigenvalues of ρ 2 will be identical to those of ρ). To impose this constraint, one introduces an auxiliary matrix ρ˜µν , with the same dimensions, and defines the density matrix using the McWeeny “purification” transformation ρ = 3ρ˜ 2 − 2ρ˜ 3 . Thus, the eigenvalues of ρ and ρ˜ are related by f i = 3 f˜i2 − 2 f˜i3 . It can be easily seen that, if f˜i is between –1/2 and 3/2, then f i is within the required range 0 ≤ f i ≤ 1. And if f˜i is close to either 0 or 1, then f i is even closer to these values. This allows for an unconstrained minimization of the ˜ = min. A practical energy as a function of the auxiliary matrix: E tot (ρ(ρ)) problem is that the spatial range of ρ˜ 3 is three times larger than the localization radius of ρ. ˜ To improve efficiency, one may truncate ρ further, although this degrades its exact idempotency, introducing extra errors. If the basis set is not orthonormal, ρ˜ 3 becomes (ρ˜ S)3 and the problem worsens. Like the FOP method, the orbital minimization (OM) approach uses a set of ∼Nel localized wavefunctions, conventionally called Wannier functions.

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These wavefunctions are optimized, within their respective localization regions, by minimizing a modified total energy functional proposed by Kim, Mauri, and Galli, which has the form E = Tr[(H − µI )(2S − I )]

(6)

where Hi j and Si j are, respectively, the Hamiltonian and overlap matrix elements between the localized states i and j , Ii j ≡ δi j is the identity matrix, and µ is the chemical potential (Fermi energy). Although not immediately obvious, it has been shown that this functional form has very convenient properties. Initially, the localized orbitals need not be orthonormal, but the functional penalizes them for not being so, in such a way that they become orthogonal as a result of the unconstrained minimization. Furthermore, although more localized orbitals are used than the number of electrons, the minimization retains only Nel of them with norm equal to one, while the rest become normless. A problem with this method is that it usually requires a very large number (frequently over 1000) of iterations in the first functional minimization (for the first Hamiltonian). This is a consequence of the minimization problem becoming ill-conditioned when the localization regions are imposed on the wavefunctions. Subsequent minimizations, during the self-consistency process and geometry relaxation, require many fewer iterations (typically of the order of ten), so that the initial minimization problem is not so important in most practical calculation projects. Another practical problem is to choose the chemical potential µ, which must lie within the energy gap to ensure charge conservation. Furthermore, the self-consistency process and geometry relaxation may result in a shift of the gap, thus requiring cumbersome changes of µ during it. There are also hybrid methods. Gillan et al. use the DMM method, optimizing a density matrix expanded in a rather small basis of localized orbitals. These orbitals are in turn optimized by expanding them in terms of a much richer basis of finite elements called “blips” [6]. Bernholc et al use a similar approach, sometimes called the quasi-O(N ) method [7], in which a conventional diagonalization, rather than DMM, is used to find the eigenvectors (and the density matrix) in terms of the small basis of localized orbitals, which are then optimized in a fine real space grid. Although the diagonalization step is O(N 3 ), the small size of the localized orbital basis, and thus of the Hamiltonian, implies a small prefactor, allowing for simulations of rather large systems in practice, including metallic ones.

2.

The SIESTA method

The O(N ) methods, described in the previous section, were developed initially in the context of tight binding calculations, in which the Hamiltonian


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matrix elements, between atomic orbitals of a minimal basis set, are given by empirical formulae for any atomic positions. This allows to concentrate on the more fundamental problem of finding the electron states, given a Hamiltonian of minimum size, without caring about how to obtain selfconsistently such a Hamiltonian. This latter problem, although more prosaic and technical, involves a large number of small sub-problems, such as finding good and efficient pseudopotentials and basis sets, calculating the electron density from the electron wavefunctions, the Hartree and exchange-correlation potentials from the density, the matrix elements of the kinetic and potential operators, the atomic forces, etc. Although none of these problems poses essential difficulties, solving all of them with an O(N ) effort is a major enterprise that involves tens or hundreds of thousands of code lines. Therefore, there are not many well developed codes able to perform practical O(N ) DFT simulations. On this respect, we may cite, apart from SIESTA: the implementation of the DMM method in the GAUSSIAN code [5]; the CONQUEST code, which uses the hybrid approach mentioned in last section [6]; and the recent ONETEP code [8] using finite-cut-off representations of Dirac delta functions as basis set. Although not using strictly O(N ) methodology, we will also mention the FIREBALL code of Lewis et al. [9], which was the precursor of SIESTA in many technical aspects, as well as that of Lippert et al. [10], which also employs a very similar approach. The first major decision of any DFT implementation concerns the election of the basis set. Traditionally, most codes developed in the condensed matter community employ plane waves (PWs). They are conceptually simple and asymptotically complete. Most importantly, this completeness is very easy to approach in a systematic way, what greatly simplifies their practical use. Not depending on the atomic positions, plane waves are also spatially unbiased, what simplifies many developments and eliminates spurious effects like Pulay forces, even when the basis is far from converged. In addition, there are some very efficient techniques, particularly the fast Fourier transform (FFT), that greatly help and simplify the implementation of an efficient plane wave code. PWs have also disadvantages: being unbiased, they can equally represent any function, but they are not specially well suited to represent any one in particular. In comparison, the atomic orbitals traditionally used in quantum chemistry are very specially suited to represent the electron wavefunctions, and therefore they are much more efficient. Thus, one frequently needs tens or even hundreds of PWs per atom to achieve the same accuracy of a minimal basis of just four atomic orbitals. However, when comparing basis set efficiency, it is essential to consider the target accuracy of the calculations. LCAO basis are very efficient initially (i.e., for low accuracies). They can also achieve very high accuracies, but they are much harder to improve systematically than PWs. Therefore, in terms of both human and computational effort, LCAO basis sets become less and less convenient, compared to PW, as the required accuracy increases.

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In practice, most simulation projects involve a huge number of trial calculations, to check the importance and the convergence of many effects and parameters, to explore candidate geometries and compositions, etc. To perform efficiently this initial exploration, it is extremely useful to have a method (and a basis set in particular) that allows a uniform transition from very fast “quick and dirty” calculations to very accurate ones. And LCAO bases allow precisely that. Apart from the pros and cons of PW mentioned before, their main disadvantage for us is their intrinsic inadequacy for O(N ) calculations. This is because each plane wave extends over the whole system, making PW inadequate to expand localized wave functions. Partly because of this reason, the last decade has seen a renaissance of real space methods, in which the electron wave functions are represented directly in a grid of points [11]. Such a “basis” has many of the advantages of PW, specially its systematic completeness, while it is also perfectly adequate to represent localized wave functions. It also allows for implementing a variety of boundary conditions, apart from the periodic ones imposed by PW. In practice, considerably more real space points are required than the already numerous PW, to achieve a similar precision, thus facing important limitations, especially in computer memory. The other main alternative for bases to implement O(N ) methods is LCAO. This is the traditional workhorse basis of quantum chemistry methods, in most of which the atomic orbitals are in turn expanded as a linear combination of Gaussian orbitals. This Gaussian expansion greatly facilitates the calculation of the three- and four-center integrals required in Hartree–Fock and configuration interaction methods. However, it is not specially useful to calculate the matrix elements of the nonlinear exchange and correlation potential, needed in DFT. In this case, it is better to use numerical orbitals, given by the product of a spherical harmonic times a radial function, represented in a fine radial grid. Furthermore, in order to expand the localized electron states and density matrices, used in O(N ) methods, it is conceptually and practically useful that the basis functions are stricly localized, i.e., defined to be zero beyond a specified radius. Such orbitals were proposed by Sankey and Niklewski and implemented in the codes FIREBALL [9] and SIESTA [12, 13]. They are generated by solving, for each angular momentum, the radial Schrödinger equation for the corresponding nonlocal pseudopotential. At the atomic orbital eigenvalue, the wavefunction decays exponentially for r → ∞. Shifting the energy to a slightly higher value, the wavefunction has a node at some radius rc , and may be considered as the solution under the constraint of a hard wall at rc . Using a common “energy shift” for all atoms and angular momenta (what implies a different rc for each one) provides a balanced basis, avoiding or mitigating spurious charge transfers. This scheme has the disadvantage of generating orbitals with a discontinuous derivative at rc (kink), which has been proven to have a small effect on the energy of condensed systems.


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To generate a richer basis set, SIESTA splits these numerical atomic orbitals (NAO) as the sum of a smooth part with even shorter range, plus a remainder, treating both parts as variationally independent basis orbitals, and producing in this way a radial flexibilization of the basis set. This splitting, inspired by the “split-valence” procedure used with Gaussian-expanded orbitals in quantum chemistry, can be repeated to generate multiple-ζ bases for each valence orbital. In order to introduce also angular flexibilization, polarization orbitals with higher angular momentum can be included. To provide them, SIESTA finds the perturbation created in the valence orbitals by an applied electric field. These polarization orbitals can also be “split,” using the previously described method, to create arbitrarily rich basis sets. It is well known that the optimal atomic basis orbitals are environment dependent. The simplest example is the hydrogen molecule, in which the optimal exponential atomic orbitals decay as e−r (in atomic units) for large interatomic separations (isolated atoms) and as e−2r for zero separation (helium atom). To account for this effect, the basis orbitals can be optimized variationally (i.e., by minimizing the total energy) within an environment similar (but simpler) to that in which they will be used. The transferability will improve by increasing the number of atomic orbitals in the basis set. To eliminate the kink, present at rc , in the orbitals of Sankey and Niklewski, it is convenient to use as variational parameters those defining a soft confinement potential, which diverges at rc . As with the “energy shift” of the hard-potential orbitals, it is important to use a common “pressure” parameter, for all the atoms and angular momenta, that controls the range of the orbitals during the optimization process [14]. To handle efficiently the core electrons, SIESTA uses the norm-conserving pseudopotentials of Troullier and Martins, in the fully nonlocal form of Kleinman and Bylander: VˆPS =

PS

d r |rVlocal (r)r| + 3

lmax

|χlm Vl χlm |,

(7)

l,m

where Vlocal(r) decays as −Z val /r when r → ∞. Since these pseudopotentials have become standard in condensed matter electronic structure codes, and they have been covered in other chapters of this handbook, we will only mention that, in SIESTA, Vlocal(r) is optimized for smoothness, rather than using the semilocal pseudopotential of a given angular momentum. The Hamiltonian and ovelap matrix elements contain several terms. The simplest ones to calculate in O(N ) operations are those involving two-center integrals between overlapping orbitals, because each orbital overlaps only with a small number of other orbitals, independent of the system size. These matrix elements are the overlap elements themselves Sµν = φµ |φν , the integrals χlm |φµ involved in the second term of Eq. (7), and the kinetic matrix elements Tµν = φµ | − 12 ∇ 2 |φν . All of these are calculated in Fourier

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space, using convolution techniques, and stored as a product of spherical harmonics times numerical radial functions, interpolated in a fine radial grid [13]. To compute the matrix elements of the local potentials, we first find the electron density ρ(r), in a regular three-dimensional grid of points r, from the density matrix: ρ(r) =

µν

ρµν φµ (r)φν (r).

(8)

Notice that, for a given point r, only a few orbitals are nonzero at r and contribute to the sum, so that the evaluation of ρ(r) is an O(N ) operation, given the fact that the the number of grid points scales linearly with the volume, which in turn is proportional to N . From ρ(r) we calculate the Hartree potential VH (r) (the electrostatic potential created by ρ(r)) using FFT. This step scales as N log(N ) and is therefore not strictly O(N ). In practice it represents only a very minor part of the whole calculation, even for the largest systems considered up to now. Whenever this step becomes dominant, we may switch to other methods, like fast multipoles or multigrid algorithms, that are strictly O(N ). The exchange and correlation potential Vxc (r) is computed in the local density (LDA) or generalized gradient approximations (GGA), the latter using finite difference derivatives. We then find the total effective potential Veff (r) by adding the local pseudopotentials of all the atoms to VH (r) + Vxc (r). Since both Vlocal and VH have long range parts with opposite signs, we subtract from each of them the electrostatic potential created by a reference density, the sum of the electron densities of the free atoms. We then find the matrix elements φµ |Veff |φν by direct integration in the grid points. Like the evaluation of ρ(r), the effort of this step has O(N ) scaling, because the number of nonzero orbitals at each grid point is independent of the system size. The evaluation of the total energy, atomic forces, and stress tensor, proceeds simultaneously to that of the Hamiltonian matrix elements, using the last density matrix available during the self-consistency process. For exam ple, the kinetic and Hartree energies are given by E kin = µν ρµν Tνµ and E H = 12 µν ρµν φν |VH |φµ , respectively. The factor 1/2 prevents double counting of the electron–electron interactions. For the forces and stress we directly use the analytic derivatives of each term of the total energy. For each term, energy, forces and stresses are computed simultaneously, in the same places of the code. This ensures an exact compatibility between the computed total energy and its derivatives, including all corrections like Pulay forces. Once the Hamiltonian and overlap matrices have been calculated, a new density matrix is obtained either by: (i) solving the generalized eigenvalue problem by conventional O(N 3 ) methods of linear algebra, or (ii) using the O(N ) orbital minimization method of Kim, Mauri, and Galli, described in previous section. The first one must be used for systems that are metallic or suffer bond breakings that create partially occupied states during the


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simulation. Apart from those, systems below a threshold size actually run faster with the conventional O(N 3 ) methods. This threshold depends on the bonding nature of the system, on the size of the basis set used, on the spatial range of the basis orbitals, and on other calculation parameters, but it is typically around ∼100 atoms. Even for sizes above this threshold, it may be more efficient, specially in terms of human investment, to use plain diagonalization. This is because the O(N ) method is intrinsically more limited (specially for bond breaking) and difficult to use, with more parameters to adjust: the localization radius of the Wannier orbitals and, especially, the chemical potential. As a rule of thumb, the O(N ) method is practical for long geometry relaxations or molecular dynamics of systems with more than ∼300 atoms, or for short calculations with more than ∼500 atoms. With conventional diagonalization, an important efficiency consideration is whether the computational effort is dominated by the diagonalization itself or by the construction of the Hamiltonian. In the first case, which occurs above ∼100 atoms, the only relevant efficiency parameter is the basis set size, while other parameters, like the spatial range of the basis orbitals or the fineness of the integration grid, can be incresed at negligible cost, to improve the accuracy. In fact, it may be advantageous to increse the grid fineness even for efficiency reasons, since this will decrease the so called “eggbox effect”: a spurious ripling of the potential, due to the dependence of the total energy on the atomic positions relative to the integration grid. Though slight in the energy, the effect is larger on the atomic forces, and may increase considerably the number of iterations required to relax the geometry. We will finish this section by briefly mentioning some capabilities of SIESTA to perform a variety of calculations: • For very fast “quick and dirty” calculations, it is possible to use the non-self-consistent Harris–Foulknes functional, in which the only Hamiltonian calculated derives from a superposition of free atom densities. For diagonalization-dominated systems, with more than ∼100 atoms, and used in combination with a minimal basis set, this is essentially as fast as a tight binding calculation. • SIESTA contains algorithms for a large variety of geometry relaxation and dynamics, including the simultaneous relaxation of the lattice vectors and atomic positions, Parrinello–Rahman molecular dynamics, dynamics at constant pressure and/or temperature, etcetera. • The SIESTA program itself does not consider symmetries because it is designed for large and/or dynamical systems, which generally have low or no symmetry. However, an accompanying package contains several tools to facilitate the evaluation of phonon modes and spectra, which prepare data files with the required geometries (considering the system symmetry) and process the resulting forces to calculate the phonons.

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• SIESTA is able to apply an external electric field to systems like molecules, clusters, chains and slabs, as well as to calculate the spontaneous polarization of a solid, using the Berry phase formalism of King-Smith and Vanderbilt [4]. • It is also possible to simulate magnetic systems, using spin dependent DFT, including the ability to impose the total magnetic moment, to start with antiferromagnetic configurations, and to allow noncollinear spin solutions. • A forthcoming version will also include time-dependent DFT, using the method of Yabana and Bertsch [13].

3.

DNA: A Prototype Application

SIESTA has been applied to hundreds of different systems, including solid metals, semiconductors and insulators, liquids, molecules, surfaces, nanotubes, and biological systems [15]. Of all these, because of the reasons explained in previous section, only a minority has been studied using the O(N ) methodology to solve Schrödinger’s equation (although the Hamiltonian is always generated in O(N ) operations). A good representative of this minority is the study of the electronic structure of DNA by Artacho et al. [16]. Apart from its obvious biological interest, DNA has generated much interest recently as a candidate for controlled self assembly of molecular electronic devices. On this respect, its ability to conduct electricity is of maximum interest, but very contradictory experimental results have been obtained on this ability. Furthermore, in such devices, DNA is normally found in a dry environment, very different from its conditions in vivo, which might strongly affect its structure. Thus, the goal of the calculations was to study the structural stability and the electrical conductivity of dry DNA. A preliminary calculation used the B conformation, but later studies used the A conformation, which is known experimentally to be more stable under dry conditions. The poly(C)–poly(G) sequence (only guanines in one of the strands and only cytosines in the other one) was chosen because guanine has the smallest ionization energy (and therefore the highest apetite for electron holes, which are suspected to be the relevant carriers) and because a uniform sequence is optimal for band conductivity. The CG base pair contains 65 atoms, including those in the sugar-phosphate side chains. Since the A conformation has a helix pitch of 11 base pairs, the total number of atoms per unit cell was 715. In solution, DNA is negatively ionized, by losing a proton in each phosphate group (two per base pair). This negative charge is neutralized by positive ions in solution around the DNA chain. In dried DNA, like that deposited on surfaces, it is uncertain how the charge will be distributed, but a reasonable approximation was to restore the phosphate protons (acidic form). It must be kept in mind, however, that in reality some of


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these protons (or whatever countercations) may be missing, in which case the charge must be compensated by electron holes, like in a doped semiconductor. The calculations were done with a double-ζ basis set, with additional polarization orbitals on the hydrogen atoms involved in hydrogen bonds and on the phosphorous atoms, for a total basis set size of 4510 orbitals. To find the chemical potential, an initial selfconsistent calculation was performed using standard diagonalizations. Then, the geometry relaxation proceeded during ∼800 steps using the O(N ) method of Kim, Mauri, and Galli, with a localization radius of 4 Å for the Wannier orbitals. A final calculation, using standard diagonalization, was performed for the relaxed coordinates, to find the electron eigenfunctions and to compare the total energy and forces. The total energy with the extended eigenfunctions was only 5 meV/atom lower than with the localized Wannier functions, and the average residual force was 6 meV/Å, while it was 2 meV/Å for the linear scaling. While a geometry relaxation step takes only about one hour with the O(N ) method, it takes 20 h using standard diagonaliztion, in a single 1 GHz Intel Pentium III processor. Despite the large number of relaxation steps, the relaxed geometry was rather close to the initial one, taken from X-ray diffraction experiments. Its structural parameters are typical of the A conformation, showing that this structure is indeed stable (at least metastable) for dry DNA. The electronic structure shows clear bands, as expected for a periodic system. The highest valence band is formed by the guanine HOMO states, and has a width of only 40 meV. The lowest conduction band is formed by the cytosine LUMO states, with a width of 270 meV. Between them, there is a wide band gap of 2.0 eV, showing that nondoped poly(C)–poly(G) must be an insulator. Even for DNA doped with holes, the extremely narrow HOMO band suggests that the holes will become localized by any lattice disorder, according to Anderson’s model. To check this, we performed two calculations for “perturbed” systems. The first system has one of the base pairs inverted (GC instead of CG) as the simplest realization of sequence disorder, after which the geometry was relaxed again. As a result, the band structure of the system changed dramatically, and the extended Bloch states changed to states localized over two-three base pairs in particular sections of the 11-base-pair periodic cell. The second “perturbed” system was one of the intermediate geometries during the relaxation process, with “random” changes in the atomic coordinates, relative to the final relaxed positions. These coordinate changes lead to a total energy difference compatible with that of thermal fluctuations at 300 K. Though not as dramatic as those of the base pair inversion, the changes in the electronic band structure were also substantial, and the electron states became localized as well, indicating in this case a strong electron-phonon interaction. These results ruled out band-like conduction of holes in doped DNA, suggesting also that holes would become localized by polaronic effects (structure deformations around

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the hole). Such a suggestion was confirmed by later calculations of the hole polaron in poly(C)–poly(G) [17].

4.

Outlook

Besides the differences in scaling with system size, a large part of the advantage of classical potentials for large systems stems from the ease of parallelizing the algorithms involved in their use. In the case of quantum simulations, there are codes, like CONQUEST, which have been designed from the begining to run in massively parallel computers, and which have demonstrated their ability to run in them simulations with over ten thousand atoms. This was not the case of SIESTA, which was designed to run in modest workstations and PCs, and only later parallelized. The initial parallel versions were not very efficient, although demonstration runs with over one hundred thousand atoms were done. Recent versions have improved the parallel scaling considerably and now aim at one million atom demonstration runs. Much progress has been obtained also in a variety of acceleration techniques, from hybrid quantum mechanics–molecular mechanics to accelerated molecular dynamics. All this combined may lead very soon to unprecedented simulations of materials properties and devices with quantum mechanical methods. The major obstacle to make this possible, however, will be to find practical O(N ) methods for metals and systems with broken bonds. This is a subject of very active reseach in which much progress is expected in the coming years.

References [1] W. Kohn, “Density functional and density matrix method scaling linearly with the number of atoms,” Phys. Rev. Lett., 76, 3168–3171, 1996. [2] P. Ordejón, “Order-N tight-binding methods for electronic-structure and molecular dynamics,” Comp. Mat. Sci., 12, 157–191, 1998. [3] S. Goedecker, “Linear scaling electronic structure methods,” Rev. Mod. Phys., 71, 1085–1123, 1999. [4] R.M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, Cambridge, 2004. [5] G.E. Scuseria, “Linear scaling density functional calculations with gaussian orbitals,” J. Phys. Chem. A, 103, 4782–4790, 1999. [6] D.R. Bowler, T. Miyazaki, and M.J. Gillan, “Recent progress in linear scaling ab initio electronic structure techniques,” J. Phys. Condens. Matter, 14, 2781–2798, 2002. [7] J.L. Fattebert and J. Bernholc, “Towards grid-based O(N) density-functional theory methods: optimized nonorthogonal orbitals and multigrid acceleration,” Phys. Rev. B, 62, 1713–1722, 2000.


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[8] A.A. Mostofi, C.-K. Skylaris, P.D. Haynes, and M.C. Payne, “Total-energy calculations on a real space grid with localized functions and a plane-wave basis,” Comput. Phys. Commun., 147, 788–802, 2002. [9] J.P. Lewis, K.R. Glaesemann, G.A. Voth, J. Fritsch, A.A. Demkov, J.Ortega, and O.F. Sankey, “Further developments in the local-orbital density-functional-theory tight-binding method,” Phys. Rev. B, 64, 195103.1–10, 2001. [10] G. Lippert, J. Hutter, P. Ballone, and M. Parrinello, “A hybrid gaussian and plane wave density functional scheme,” Mol. Phys., 92, 477–487, 1997. [11] T.L. Beck, “Real-space mesh techniques in density-functional theory,” Rev. Mod. Phys., 72, 1041–1080, 2000. [12] P. Ordejón, E. Artacho, and J.M. Soler, “Selfconsistent order-N density-functional calculations for very large systems,” Phys. Rev. B, 53, R10441–R10444, 1996. [13] J.M. Soler, E. Artacho, J.D. Gale, A. García, J. Junquera, P. Ordejón, and D. SánchezPortal, “The SIESTA method for ab initio order-N materials simulation,” J. Phys. Condens. Matter, 14, 2745–2779, 2002. [14] E. Anglada, J.M. Soler, J. Junquera, and E. Artacho, “Systematic generation of finiterange atomic basis sets for linear-scaling calculations,” Phys. Rev. B, 66, 205101.1–4, 2000. [15] D. Sánchez-Portal, P. Ordejón, and E. Canadell, “Computing the properties of materials from first principles with SIESTA,” Struct. Bonding, 113, 103–170, 2004. See also http://www.uam.es/siesta. [16] E. Artacho, M. Machado, D. Sánchez-Portal, P. Ordejón, and J.M. Soler, “Electrons in dry DNA from density functional calculations,” Mol. Phys., 101, 1587–1594, 2003. [17] S.S. Alexandre, E. Artacho, J.M. Soler, and H. Chacham, “Small polarons in dry DNA,” Phys. Rev. Lett., 91, 108105–108108, 2003.

1.6 ELECTRONIC STRUCTURE METHODS: AUGMENTED WAVES, PSEUDOPOTENTIALS AND THE PROJECTOR AUGMENTED WAVE METHOD Peter E. Blöchl, Johannes Kästner, and Clemens J. Först Institute for Theoretical Physics, Clausthal University of Technology, Clausthal-Zellerfeld, Germany

The main goal of electronic structure methods is to solve the Schrödinger equation for the electrons in a molecule or solid, to evaluate the resulting total energies, forces, response functions and other quantities of interest. In this paper we describe the basic ideas behind the main electronic structure methods such as the pseudopotential and the augmented wave methods and provide selected pointers to contributions that are relevant for a beginner. We give particular emphasis to the projector augmented wave (PAW) method developed by one of us, an electronic structure method for ab initio molecular dynamics with full wavefunctions. We feel that it allows best to show the common conceptional basis of the most widespread electronic structure methods in materials science. The methods described below require as input only the charge and mass of the nuclei, the number of electrons and an initial atomic geometry. They predict binding energies accurate within a few tenths of an electron volt and bond lengths in the 1–2% range. Currently, systems with a few hundred atoms per unit cell can be handled. The dynamics of atoms can be studied up to tens of picoseconds. Quantities related to energetics, the atomic structure and to the ground-state electronic structure can be extracted. In order to lay a common ground and to define some of the symbols, let us briefly touch upon the density functional theory [1, 2]. It maps a description for interacting electrons, a nearly intractable problem, onto one of non-interacting electrons in an effective potential. Within density functional theory, the total


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energy is written as E[n (r), R R ] =

n

fn

−h 2 ¯ 2 n ∇ n 2m e 2

1 e n(r) + Z (r) n(r ) + Z (r ) + · d3r d3r 2 4π 0 |r − r | + E xc [n(r)]

(1)

occupations, n(r) = Here, |n are one-particle electron states, f n are the state ∗ f (r) (r) is the electron density and Z (r) = − n n n R Z R δ(r − R R ) is the n nuclear charge density expressed in electron charges. Z R is the atomic number of a nucleus at position R R . It is implicitly assumed that the infinite selfinteraction of the nuclei is removed. The exchange and correlation functional contains all the difficulties of the many-electron problem. The main conclusion of the density functional theory is that E xc is a functional of the density. We use Dirac’s bra and ket notation. A wavefunction n corresponds to a ket |n , the complex conjugate wave function n∗ corresponds to a bra n |, and a scalar product d3rn∗ (r)m (r) is written as n |m . Vectors in the three-dimensional coordinate space are indicated by boldfaced symbols. Note that we use R as position vector and R as atom index. In current implementations, the exchange and correlation functional E xc [n(r)] has the form

E xc [n(r)] =

d3r Fxc (n(r), |∇n(r)|),

where Fxc is a parameterized function of the density and its gradients. Such functionals are called gradient corrected. In local spin density functional theory, Fxc furthermore depends on the spin density and its derivatives. A review of the earlier developments has been given by Parr and Yang [3]. The electronic ground state is determined by minimizing the total energy functional E[n ] of Eq. (1) at a fixed ionic geometry. The one-particle wavefunctions have to be orthogonal. This constraint is implemented with the method of Lagrange multipliers. We obtain the ground state wavefunctions from the extremum condition for F[n (r), m,n ] = E[n (r)] −

[n |m − δn,m ]m,n

(2)

n,m

with respect to the wavefunctions and the Lagrange multipliers m,n . The extremum condition for the wavefunctions has the form H |n f n =

m

|m m,n

(3)

Electronic structure methods

95

2

h¯ where H = − 2m ∇2 + v eff (r) is the effective one-particle Hamilton operator. e The effective potential depends itself on the electron density via

v eff (r) =

e2 4π 0

d3r

n(r ) + Z (r ) + µxc (r), |r − r |

xc [n(r)] is the functional derivative of the exchange and correwhere µxc (r) = δ Eδn(r) lation functional. After a unitary transformation that diagonalizes the matrix of Lagrange multipliers m,n , we obtain the Kohn–Sham equations:

H |n = |n n .

(4)

The one-particle energies n are the eigenvalues of n,m 2fnf+n ffmm [4]. The remaining one-electron Schrödinger equations, namely the Kohn– Sham equations given above, still pose substantial numerical difficulties: (1) in the atomic region near the nucleus, the kinetic energy of the electrons is large, resulting in rapid oscillations of the wavefunction that require fine grids for an accurate numerical representation. On the other hand, the large kinetic energy makes the Schrödinger equation stiff, so that a change of the chemical environment has little effect on the shape of the wavefunction. Therefore, the wavefunction in the atomic region can be represented well already by a small basis set. (2) In the bonding region between the atoms the situation is opposite. The kinetic energy is small and the wavefunction is smooth. However, the wavefunction is flexible and responds strongly to the environment. This requires large and nearly complete basis sets. Combining these different requirements is nontrivial and various strategies have been developed. • The atomic point of view has been most appealing to quantum chemists. Basis functions that resemble atomic orbitals are chosen. They exploit that the wavefunction in the atomic region can be described by a few basis functions, while the chemical bond is described by the overlapping tails of these atomic orbitals. Most techniques in this class are a compromise of, on the one hand, a well-adapted basis set, where the basis functions are difficult to handle, and on the other hand numerically convenient basis functions such as Gaussians, where the inadequacies are compensated by larger basis sets. • Pseudopotentials regard an atom as a perturbation of the free electron gas. The most natural basis functions are planewaves. Plane wave basis sets are, in principle, complete and suitable for sufficiently smooth wavefunctions. The disadvantage of the comparably large basis sets required is offset by their extreme numerical simplicity. Finite plane-wave expansions are, however, absolutely inadequate to describe the strong

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oscillations of the wavefunctions near the nucleus. In the pseudopotential approach the Pauli repulsion of the core electrons is therefore described by an effective potential that expels the valence electrons from the core region. The resulting wavefunctions are smooth and can be represented well by plane-waves. The price to pay is that all information on the charge density and wavefunctions near the nucleus is lost. • Augmented wave methods compose their basis functions from atom-like wavefunctions in the atomic regions and a set of functions, called envelope functions, appropriate for the bonding in between. Space is divided accordingly into atom-centered spheres, defining the atomic regions, and an interstitial region in between. The partial solutions of the different regions, are matched at the interface between atomic and interstitial regions. The PAW method is an extension of augmented wave methods and the pseudopotential approach, which combines their traditions into a unified electronic structure method. After describing the underlying ideas of the various approaches let us briefly review the history of augmented wave methods and the pseudopotential approach. We do not discuss the atomic-orbital based methods, because our focus is the PAW method and its ancestors.

1.

Augmented Wave Methods

The augmented wave methods have been introduced in 1937 by Slater [5] and were later modified by Korringa [6], Kohn and Rostokker [7]. They approached the electronic structure as a scattered-electron problem. Consider an electron beam, represented by a plane wave, traveling through a solid. It undergoes multiple scattering at the atoms. If for some energy, the outgoing scattered waves interfere destructively, a bound state has been determined. This approach can be translated into a basis set method with energy and potential dependent basis functions. In order to make the scattered wave problem tractable, a model potential had to be chosen: The so-called muffin-tin potential approximates the true potential by a constant in the interstitial region and by a spherically symmetric potential in the atomic region. Augmented wave methods reached adulthood in the 1970s: Andersen [8] showed that the energy-dependent basis set of Slater’s APW method can be mapped onto one with energy independent basis functions, by linearizing the partial waves for the atomic regions in energy. In the original APW approach, one had to determine the zeros of the determinant of an energy dependent matrix, a nearly intractable numerical problem for complex systems. With the new energy independent basis functions, however, the problem is reduced to


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the much simpler generalized eigenvalue problem, which can be solved using efficient numerical techniques. Furthermore, the introduction of well-defined basis sets paved the way for full-potential calculations [9]. In that case the muffin-tin approximation is used solely to define the basis set |χi , while the matrix elements χi |H |χ j of the Hamiltonian are evaluated with the full potential. In the augmented wave methods one constructs the basis set for the atomic region by solving the Schrödinger equation for the spheridized effective potential

−h¯ 2 2 ∇ + v eff (r) − φ,m (, r) = 0 2m e

as function of energy. Note that a partial wave φ,m (, r) is an angular momentum eigenstate and can be expressed as a product of a radial function and a spherical harmonic. The energy-dependent partial wave is expanded in a Taylor expansion about some reference energy ν, φ,m (, r) = φν,,m (r) + ( − ν, )φ˙ ν,,m (r) + O(( − ν, )2 ), where φν,,m (r) = φ,m (ν, , r). The energy derivative of the partial wave φ˙ν (r)= ∂φ(,r) solves the equation ∂ ν,

−h¯ 2 2 ∇ + v eff (r) − ν, φ˙ ν,,m (r) = φν,,m (r). 2m e

Next, one starts from a regular basis set, such as plane waves, Gaussians or Hankel functions. These basis functions are called envelope functions |χ˜ i . Within the atomic region they are replaced by the partial waves and their energy derivatives, such that the resulting wavefunction is continuous and differentiable: χi (r) = χ˜i (r) −

R

θ R (r)χ˜ i (r) +

+ φ˙ν,R,,m (r)b R,,m,i .

θ R (r) φν,R,,m (r)a R,,m,i

R,,m

(5)

θ R (r) is a step function that is unity within the augmentation sphere centered at R R and zero elsewhere. The augmentation sphere is atom-centered and has a radius about equal to the covalent radius. This radius is called the muffintin radius, if the spheres of neighboring atoms touch. These basis functions describe only the valence states; the core states are localized within the augmentation sphere and are obtained directly by radial integration of the Schrödinger equation within the augmentation sphere.

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The coefficients a R,,m,i and b R,,m,i are obtained for each |χ˜i as follows: The envelope function is decomposed around each atomic site into spherical harmonics multiplied by radial functions: χ˜ i (r) =

u R,,m,i (|r − R R |)Y,m (r − R R ).

(6)

,m

Analytical expansions for plane waves, Hankel functions or Gaussians exist. The radial parts of the partial waves φν,R,,m and φ˙ν,R,,m are matched with value and derivative to u R,,m,i (|r|), which yields the expansion coefficients a R,,m,i and b R,,m,i . If the envelope functions are plane waves, the resulting method is called the linear augmented plane wave (LAPW) method. If the envelope functions are Hankel functions, the method is called linear muffin-tin orbital (LMTO) method. A good review of the LAPW method [8] has been given by Singh [10]. Let us now briefly mention the major developments of the LAPW method: Soler and Williams [11] introduced the idea of additive augmentation: While augmented plane waves are discontinuous at the surface of the augmentation sphere if the expansion in spherical harmonics in Eq. (5) is truncated, Soler replaced the second term in Eq. (5) by an expansion of the plane wave with the same angular momentum truncation as in the third term. This dramatically improved the convergence of the angular momentum expansion. Singh [12] introduced so-called local orbitals, which are nonzero only within a muffintin sphere, where they are superpositions of φ and φ˙ functions from different expansion energies. Local orbitals substantially increase the energy transferability. Sjöstedt et al. [13] relaxed the condition that the basis functions are differentiable at the sphere radius. In addition they introduced local orbitals, which are confined inside the sphere, and that also have a kink at the sphere boundary. Due to the large energy-cost of kinks, they will cancel, once the total energy is minimized. The increased variational degree of freedom in the basis leads to a dramatically improved plane-wave convergence [14]. The second variant of the linear methods is the LMTO method [8]. A good introduction into the LMTO method is the book by Skriver [15]. The LMTO method uses Hankel functions as envelope functions. The atomic spheres approximation (ASA) provides a particularly simple and efficient approach to the electronic structure of very large systems. In the ASA, the augmentation spheres are blown up so that their volume are equal to the total volume and the first two terms in Eq. (5) are ignored. The main deficiency of the LMTO-ASA method is the limitation to structures that can be converted into a closed packed arrangement of atomic and empty spheres. Furthermore, energy differences due to structural distortions are often qualitatively incorrect. Full potential versions of the LMTO method, that avoid these deficiencies of the ASA have been developed. The construction of tight


99

binding orbitals as superposition of muffin-tin orbitals [16] showed the underlying principles of the empirical tight-binding method and prepared the ground for electronic structure methods that scale linearly instead of with the third power of the number of atoms. The third generation LMTO [17] allows to construct true minimal basis sets, which require only one orbital per electronpair for insulators. In addition they can be made arbitrarily accurate in the valence band region, so that a matrix diagonalization becomes unnecessary. The first steps towards a full-potential implementation, that promises a good accuracy, while maintaining the simplicity of the LMTO-ASA method are currently under way. Through the minimal basis-set construction the LMTO method offers unrivaled tools for the analysis of the electronic structure and has been extensively used in hybrid methods combining density functional theory with model Hamiltonians for materials with strong electron correlations [18].

2.

Pseudopotentials

Pseudopotentials have been introduced to (1) avoid describing the core electrons explicitly and (2) to avoid the rapid oscillations of the wavefunction near the nucleus, which normally require either complicated or large basis sets. The pseudopotential approach traces back to 1940 when Herring [19] invented the orthogonalized plane-wave method. Later, Phillips and Kleinman [20] and Antoncik [21] replaced the orthogonality condition by an effective potential, which mimics the Pauli repulsion by the core electrons and thus compensates the electrostatic attraction by the nucleus. In practice, the potential was modified, for example, by cutting off the singular potential of the nucleus at a certain value. This was done with a few parameters that have been adjusted to reproduce the measured electronic band structure of the corresponding solid. Hamann et al. [22] showed in 1979 how pseudopotentials can be constructed in such a way, that their scattering properties are identical to that of an atom to first order in energy. These first-principles pseudopotentials relieved the calculations from the restrictions of empirical parameters. Highly accurate calculations have become possible especially for semiconductors and simple metals. An alternative approach towards first-principles pseudopotentials [23] preceded the one mentioned above.

2.1.

The Idea Behind Pseudopotential Construction

In order to construct a first-principles pseudopotential, one starts out with an all-electron density-functional calculation for a spherical atom. Such

100


calculations can be performed efficiently on radial grids. They yield the atomic potential and wavefunctions φ,m (r). Due to the spherical symmetry, the radial parts of the wavefunctions for different magnetic quantum numbers m are identical. For the valence wavefunctions one constructs pseudo-wavefunctions |φ˜ ,m : There are numerous ways [24–27] to construct the pseudo-wavefunctions. They must be identical to the true wave functions outside the augmentation region, which is called core-region in the context of the pseudopotential approach. Inside the augmentation region the pseudo-wavefunction should be nodeless and have the same norm as the true wavefunctions, that is φ˜ ,m |φ˜ ,m = φ,m |φ,m (compare Fig. 1). From the pseudo-wavefunction, a potential u (r) can be reconstructed by inverting the respective Schrödinger equation:

h¯ 2 2 − ∇ + u (r) − ,m φ˜,m (r) = 0 2m e ⇒ u (r) = ,m +

h¯ 2 2 ∇ φ˜,m (r). φ˜ ,m (r) 2m e 1

·

0

0

1

2

3

r [abohr] Figure 1. Illustration of the pseudopotential concept at the example of the 3s wavefunction of Si. The solid line shows the radial part of the pseudo-wavefunction φ˜,m . The dashed line corresponds to the all-electron wavefunction φ,m , which exhibits strong oscillations at small radii. The angular momentum dependent pseudopotential u (dash-dotted line) deviates from the all-electron one v eff (dotted line) inside the augmentation region. The data are generated by the fhi98PP code [28].


101

This potential u (r) (compare Fig. 1), which is also spherically symmetric, differs from one main angular momentum to the other. Next we define an effective pseudo-Hamiltonian

h¯ 2 2 e2 ps ∇ + v (r) + H˜ = − 2m e 4π 0

d3r

n(r ˜ ) + Z˜ (r ) + µxc ([n(r)], ˜ r) |r − r |

ps

and determine the pseudopotentials v such that the pseudo-Hamiltonian produces the pseudo-wavefunctions, that is ps v (r)

e2 = u (r) − 4π 0

d3r

n(r ˜ ) + Z˜ (r ) − µxc ([n(r)], ˜ r). |r − r |

(7)

This process is called “unscreening.” ˜ Z(r) mimics the charge density of the nucleus and the core electrons. It is usually an atom-centered, spherical Gaussian that is normalized to the charge of nucleus and core of that atom. In the pseudopotential approach, Z˜ R (r) does ˜ n (r) ˜ n∗ (r) not change with the potential. The pseudo density n(r) ˜ = n fn is constructed from the pseudo-wavefunctions. In this way we obtain a different potential for each angular momentum channel. In order to apply these potentials to a given wavefunction, the wavefunction must first be decomposed into angular momenta. Then each comps ponent is applied to the pseudopotential v for the corresponding angular momentum. The pseudopotential defined in this way can be expressed in a semilocal form

¯ −r)+ v (r, r ) = v(r)δ(r ps

,m

ps

Y,m (r) v (r) − v(r) ¯

δ(|r| − |r |) ∗ × Y,m (r ) . |r|2

(8)

The local potential v(r) ¯ only acts on those angular momentum components, not included in the expansion of the pseudopotential construction. Typically, it is chosen to cancel the most expensive nonlocal terms, the one corresponding to the highest physically relevant angular momentum. The pseudopotential is nonlocal as it depends on two position arguments, r and r . The expectation values are evaluated as a double integral ˜ = ˜ ps | |v

3

dr

˜ ). ˜ ∗ (r)v ps (r, r )(r d3r

102


The semilocal form of the pseudopotential given in Eq. (8) is computationally expensive. Therefore, in practice, one uses a separable form of the pseudopotential [29–31]: v ps ≈

−1

v ps |φ˜i φ˜ j |v ps |φ˜ i

i, j

i, j

φ˜ j |v ps .

(9)

Thus, the projection onto spherical harmonics used in the semilocal form of Eq. (8) is replaced by a projection onto angular momentum dependent functions |v ps φ˜ i . The indices i and j are composite indices containing the atomic-site index R, the angular momentum quantum numbers , m and an additional index α. The index α distinguishes partial waves with otherwise identical indices R, , m, as more than one partial wave per site and angular momentum is allowed. The partial waves may be constructed as eigenstates to the ps pseudopotential v for a set of energies. One can show that the identity of Eq. (9) holds by applying a wavefunction ˜ = i |φ˜ i ci to both sides. If the set of pseudo partial waves |φ˜i in Eq. (9) | is complete, the identity is exact. The advantage of the separable form is that ˜ ps | is treated as one function, so that expectation values are reduced to φv ˜ combinations of simple scalar products φ˜i v ps |. The total energy of the pseudopotential method can be written in the form E=

n

fn

h2 ¯ ˜ 2 ˜ n |v ps | ˜ n ˜ n − ∇ f n n + E self + 2m e n

˜ ) 2 n(r) ˜ + Z˜ (r) n(r ˜ ) + Z(r

1 e × + · 2 4π 0

d3r

d3r

|r − r |

+ E xc [n(r)]. ˜ (10)

The constant E self is adjusted such that the total energy of the atom is the same for an all-electron calculation and the pseudopotential calculation. For the atom, from which it has been constructed, this construction guarantees that the pseudopotential method produces the correct one-particle energies for the valence states and that the wavefunctions have the desired shape. While pseudopotentials have proven to be accurate for a large variety of systems, there is no strict guarantee that they produce the same results as an allelectron calculation, if they are used in a molecule or solid. The error sources can be divided into two classes: • Energy transferability problems: Even for the potential of the reference atom, the scattering properties are accurate only in given energy window. • Charge transferability problems: In a molecule or crystal, the potential differs from that of the isolated atom. The pseudopotential, however, is strictly valid only for the isolated atom.


103

The plane-wave basis set for the pseudo wavefunctions is defined by the shortest wave length λmin = 2π/|G max | via the so-called plane-wave cutoff h2 G2 E PW = ¯ 2mmax . It is often specified in Rydberg (1Ry = 12 H≈13.6 eV). The planee wave cutoff is the highest kinetic energy of all basis functions. The basis-set convergence can systematically be controlled by increasing the plane-wave cutoff. The charge transferability is substantially improved by including a nonlinear core correction [32] into the exchange-correlation term of Eq. (10). Hamann [33] showed how to construct pseudopotentials from unbound wavefunctions as well. Vanderbilt [31] and Laasonen et al. [34] generalized the pseudopotential method to non-norm-conserving pseudopotentials, so-called ultra-soft pseudopotentials, which dramatically improves the basis-set convergence. The formulation of ultra-soft pseudopotentials has already many similarities with the projector augmented wave method. Truncated separable pseudopotentials suffer sometimes from so-called ghost states. These are unphysical core-like states, which render the pseudopotential useless. These problems have been discussed by Gonze et al. [35] . Quantities such as hyperfine parameters that depend on the full wavefunctions near the nucleus, can be extracted approximately [36]. A good review about pseudopotential methodology has been written by Payne et al. [37] and Singh [10]. In 1985, Car and Parrinello [38] published the ab initio molecular dynamics method. Simulations of the atomic motion have become possible on the basis of state-of-the-art electronic structure methods. Besides making dynamical phenomena and finite temperature effects accessible to electronic structure calculations, the ab initio molecular dynamics method also introduced a radically new way of thinking into electronic structure methods. Diagonalization of a Hamilton matrix has been replaced by classical equations of motion for the wavefunction coefficients. If one applies friction, the system is quenched to the ground state. Without friction truly dynamical simulations of the atomic structure are performed. Using thermostats [39–42], simulations at constant temperature can be performed. The Car–Parrinello method treats electronic wavefunctions and atomic positions on an equal footing.

3.

Projector Augmented Wave Method

The Car–Parrinello method had been implemented first for the pseudopotential approach. There seemed to be unsurmountable barriers against combining the new technique with augmented wave methods. The main problem was related to the potential-dependent basis set used in augmented wave methods: the Car–Parrinello method requires a well-defined and unique total energy functional of atomic positions and basis set coefficients. Furthermore, the analytic evaluation of the first partial derivatives of the total energy with respect

104


to wavefunctions, H |n , and atomic position, the forces, must be possible. Therefore, it was one of the main goals of the PAW method to introduce energy and potential independent basis sets that are as accurate as the previously used augmented basis sets. Other requirements have been: (1) The method should at least match the efficiency of the pseudopotential approach for Car–Parrinello simulations. (2) It should become an exact theory when converged and (3) its convergence should be easily controlled. We believe that these criteria have been met, which explains why the PAW method becomes increasingly widespread today.

3.1.

Transformation Theory

At the root of the PAW method lies a transformation, that maps the true wavefunctions with their complete nodal structure onto auxiliary wavefunctions, that are numerically convenient. We aim for smooth auxiliary wavefunctions, which have a rapidly convergent plane-wave expansion. With such a transformation we can expand the auxiliary wave functions into a convenient basis set such as plane waves, and evaluate all physical properties after reconstructing the related physical (true) wavefunctions. Let us denote the physical one-particle wavefunctions as |n and the aux˜ n . Note that the tilde refers to the representation of iliary wavefunctions as | smooth auxiliary wavefunctions and n is the label for a one-particle state and contains a band index, a k-point and a spin index. The transformation from the auxiliary to the physical wavefunctions is denoted by T : ˜ n . |n = T |

(11)

Now we express the constrained density functional F of Eq. (2) in terms of our auxiliary wavefunctions ˜ n] − ˜ n , m,n ] = E[T F[T

˜ n |T † T | ˜ m − δn,m ]m,n . [

(12)

n,m

The variational principle with respect to the auxiliary wavefunctions yields ˜ n = T † T | ˜ n n . T † H T |

(13)

Again we obtain a Schrödinger-like equation (see derivation of Eq. (4)), but now the Hamilton operator has a different form, H˜ = T † H T , an overlap operator O˜ = T † T occurs, and the resulting auxiliary wavefunctions are smooth. When we evaluate physical quantities we need to evaluate expectation values of an operator A, which can be expressed in terms of either the true or the auxiliary wavefunctions: A =

n

f n n |A|n =

n

˜ n |T † AT | ˜ n . f n

(14)


105

In the representation of auxiliary wavefunctions we need to use transformed ˜ † AT . As it is, this equation only holds for the valence electrons. operators A=T The core electrons are treated differently as will be shown below. The transformation takes us conceptionally from the world of pseudopotentials to that of augmented wave methods, which deal with the full wavefunctions. We will see that our auxiliary wavefunctions, which are simply the plane-wave parts of the full wavefunctions, translate into the wavefunctions of the pseudopotential approach. In the PAW method, the auxiliary wavefunctions are used to construct the true wavefunctions and the total energy functional is evaluated from the latter. Thus it provides the missing link between augmented wave methods and the pseudopotential method, which can be derived as a well-defined approximation of the PAW method. In the original paper [4], the auxiliary wavefunctions have been termed pseudo wavefunctions and the true wavefunctions have been termed allelectron wavefunctions, in order to make the connection more evident. We avoid this notation here, because it resulted in confusion in cases, where the correspondence is not clear-cut.

3.2.

Transformation Operator

So far, we have described how we can determine the auxiliary wave functions of the ground state and how to obtain physical information from them. What is missing, is a definition of the transformation operator T . The operator T has to modify the smooth auxiliary wave function in each atomic region, so that the resulting wavefunction has the correct nodal structure. Therefore, it makes sense to write the transformation as identity plus a sum of atomic contributions S R : T =1+

SR .

(15)

R

For every atom, S R adds the difference between the true and the auxiliary wavefunction. The local terms S R are defined in terms of solutions |φi of the Schrödinger equation for the isolated atoms. This set of partial waves |φi will serve as a basis set so that, near the nucleus, all relevant valence wavefunctions can be expressed as superposition of the partial waves with yet unknown coefficients: (r) =

φi (r)ci

for |r − R R | < rc,R ,

(16)

i∈R

with i ∈ R we indicate those partial waves that belong to site R. Since the core wavefunctions do not spread out into the neighboring atoms, we will treat them differently. Currently we use the frozen-core approximation, which imports the density and the energy of the core electrons from

106


the corresponding isolated atoms. The transformation T shall produce only wavefunctions orthogonal to the core electrons, while the core electrons are treated separately. Therefore, the set of atomic partial waves |φi includes only valence states that are orthogonal to the core wavefunctions of the atom. For each of the partial waves we choose an auxiliary partial wave |φ˜i . The identity |φi = (1 + S R )|φ˜i for i ∈ R S R |φ˜i = |φi − |φ˜i

(17)

defines the local contribution S R to the transformation operator. Since 1 + S R shall change the wavefunction only locally, we require that the partial waves |φi and their auxiliary counter parts |φ˜i are pairwise identical beyond a certain radius rc,R : φi (r) = φ˜i (r)

for i ∈ R and |r − R R | > rc,R .

(18)

Note that the partial waves are not necessarily bound states and are therefore not normalizable, unless we truncate them beyond a certain radius rc,R . The PAW method is formulated such that the final results do not depend on the location where the partial waves are truncated, as long as this is not done too close to the nucleus and identical for auxiliary and all-electron partial waves. In order to be able to apply the transformation operator to an arbitrary auxiliary wavefunction, we need to be able to expand the auxiliary wavefunction locally into the auxiliary partial waves. ˜ (r) =

φ˜i (r)ci =

i∈R

˜ φ˜ i (r) p˜i |

for |r − R R | < rc,R ,

(19)

i∈R

which defines the projector functions | p˜i . The projector functions probe the local character of the auxiliary wave function in the atomic region. Examples of projector functions are shown in Fig. 2. From Eq. (19) we can derive ˜ i∈R |φi p˜i | = 1, which is valid within rc,R . It can be shown by insertion, ˜ that can be that the identity Eq. (19) holds for any auxiliary wavefunction | expanded locally into auxiliary partial waves |φ˜i , if p˜i |φ˜ j = δi, j

for i, j ∈ R.

(20)

Note that neither the projector functions nor the partial waves need to be orthogonal among themselves. The projector functions are fully determined with the above conditions and a closure relation, which is related to the unscreening of the pseudopotentials (see Eq. 90 in Ref. [4]). By combining Eqs. (17) and (19), we can apply S R to any auxiliary wavefunction: ˜ = S R |

i∈R

˜ = S R |φ˜ i p˜i |

˜ |φi − |φ˜ i p˜i |.

i∈R

(21)


107

Figure 2. Projector functions of the chlorine atom. Top: two s-type projector functions, middle: p-type, bottom: d-type.

Hence, the transformation operator is T =1+

|φi − |φ˜i p˜i |,

(22)

i

where the sum runs over all partial waves of all atoms. The true wavefunction can be expressed as ˜ + | = |

˜ = | ˜ + |φi − |φ˜i p˜i |

i

˜ R1 | R1 − |

(23)

R

with | R1 =

˜ |φi p˜i |

(24)

˜ |φ˜i p˜i |.

(25)

i∈R

˜ R1 = |

i∈R

In Fig. 3, the decomposition of Eq. (23) is shown for the example of the bonding p-σ state of the Cl2 molecule. To understand the expression Eq. (23) for the true wavefunction, let us concentrate on different regions in space. (1) Far from the atoms, the partial waves are, according to Eq. (18), pairwise identical so that the auxiliary wavefunc˜ tion is identical to the true wavefunction, that is (r) = (r). (2) Close to an atom R, however, the auxiliary wavefunction is, according to Eq. (19), identi˜ ˜ R1 (r). Hence, the true cal to its one-center expansion, that is, (r) =

108


Figure 3. Bonding p-σ orbital of the Cl2 molecule and its decomposition of the wavefunction into auxiliary wavefunction and the two one-center expansions. Top-left: True and auxiliary wave function; top-right: auxiliary wavefunction and its partial wave expansion; bottomleft: the two partial wave expansions; bottom-right: true wavefunction and its partial wave expansion.

wavefunction (r) is identical to R1 (r), which is built up from partial waves that contain the proper nodal structure. In practice, the partial wave expansions are truncated. Therefore, the identity of Eq. (19) does not hold strictly. As a result, the plane waves also contribute to the true wavefunction inside the atomic region. This has the advantage that the missing terms in a truncated partial wave expansion are partly accounted for by plane waves, which explains the rapid convergence of


109

the partial wave expansions. This idea is related to the additive augmentation of the LAPW method of Soler and Williams [11]. Frequently, the question comes up, whether the transformation Eq. (22) of the auxiliary wavefunctions indeed provides the true wavefunction. The transformation should be considered merely as a change of representation analogous to a coordinate transform. If the total energy functional is transformed consistently, its minimum will yield auxiliary wavefunctions that produce the correct wavefunctions |.

3.3.

Expectation values

Expectation values can be obtained either from the reconstructed true wavefunctions or directly from the auxiliary wave functions A =

Nc

f n n |A|n +

n

=

φnc |A|φnc

n=1

˜ n |T † AT | ˜ n + f n

n

Nc

φnc |A|φnc ,

(26)

n=1

where f n are the occupations of the valence states and Nc is the number of core states. The first sum runs over the valence states, and second over the core states |φnc . Now we can decompose the matrix element for a wavefunction into its individual contributions according to Eq. (23):

˜ + |A| =

R

˜ ˜ + = |A|

R

˜ R1 ) ( R1 −

˜ R1 |A| ˜ R1 R1 |A| R1 −

R

+

˜ R1 ) A ˜ + ( R1 −

part 1

˜ R1 |A| ˜ − ˜ R1 + ˜ − ˜ R1 |A| R1 − ˜ R1 R1 −

R

part 2 +

R/ = R

˜ R1 |A| R1 − ˜ R1 . R1 −

part 3

(27)

110


Only the first part of Eq. (27) is evaluated explicitly, while the second and third parts of Eq. (27) are neglected, because they vanish for sufficiently local operators as long as the partial wave expansion is converged: The func˜ R1 vanishes per construction beyond its augmentation region, tion R1 − because the partial waves are pairwise identical beyond that region. The func˜ − ˜ R1 vanishes inside its augmentation region, if the partial wave expantion ˜ R1 sion is sufficiently converged. In no region of space both functions R1 − ˜ − ˜ R1 are simultaneously nonzero. Similarly the functions R1 − ˜ R1 and from different sites are never non-zero in the same region in space. Hence, the second and third parts of Eq. (27) vanish for operators such as the kinetic h¯ 2 2 ∇ and the real space projection operator |rr|, which produces energy − 2m e the electron density. For truly nonlocal operators the parts 2 and 3 of Eq. (27) would have to be considered explicitly. The expression, Eq. (26), for the expectation value can therefore be written with the help of Eq. (27) as

A =

˜ n |A| ˜ n + n1 |A|n1 − ˜ n1 |A| ˜ n1 + f n

n

=

˜ n |A| ˜ n + f n

n

 

R

−

φnc |A|φnc

n=1

+

Nc

R

 

Nc

φñc |A|φñc

n=1

Di, j φ j |A|φi +

Nc,R

i, j ∈R

n∈R

Nc,R

Di, j φ˜ j |A|φ˜i +

i, j ∈R



φnc |A|φnc  

φ˜ nc |A|φñc ,

(28)

n∈R

where Di, j is the one-center density matrix defined as

Di, j =

n

˜ n | p˜ j p˜i | ˜ n = f n

˜ n f n ˜ n | p˜ j , p˜i |

(29)

n

The auxiliary core states, |φ˜ nc allow to incorporate the tails of the core wavefunction into the plane-wave part, and therefore assure, that the integrations of partial wave contributions cancel strictly beyond rc . They are identical to the true core states in the tails, but are a smooth continuation inside the atomic sphere. It is not required that the auxiliary wave functions are normalized.


111

Following this scheme, the electron density is given by n(r) = n(r) ˜ + n(r) ˜ =

n 1R (r) − n˜ 1R (r)

R ∗ ˜ n (r) ˜ n (r) fn

(30)

+ n˜ c (r)

n

n 1R (r) =

Di, j φ ∗j (r)φi (r) + n c,R (r)

i, j ∈R

n˜ 1R (r)

=

Di, j φ˜ ∗j (r)φ˜ i (r) + n˜ c,R (r),

(31)

i, j ∈R

where n c,R is the core density of the corresponding atom and n˜ c,R is the auxiliary core density, which is identical to n c,R outside the atomic region, but smooth inside. Before we continue, let us discuss a special point: The matrix element of a general operator with the auxiliary wavefunctions may be slowly converging with the plane-wave expansion, because the operator A may not be well behaved. An example for such an operator is the singular electrostatic potential of a nucleus. This problem can be alleviated by adding an “intelligent zero”: If an operator B is purely localized within an atomic region, we can use the identity between the auxiliary wavefunction and its own partial wave expansion ˜ n − ˜ n1 |B| ˜ n1 . ˜ n |B| 0 =

(32)

Now we choose an operator B so that it cancels the problematic behavior of the operator A, but is localized in a single atomic region. By adding B to the plane-wave part and the matrix elements with its one-center expansions, the plane-wave convergence can be improved without affecting the converged result. A term of this type, namely v¯ will be introduced in the next section to cancel the Coulomb singularity of the potential at the nucleus.

4.

Total Energy

Like wavefunctions and expectation values also the total energy can be divided into three parts: ˜ n , R R ] = E˜ + E[

E 1R − E˜ 1R .

(33)

R

The plane-wave part E˜ involves only smooth functions and is evaluated on equi-spaced grids in real and reciprocal space. This part is computationally

112


most demanding, and is similar to the expressions in the pseudopotential approach: E˜ =

−h 2 ¯ ˜ ñ ∇2 2m e n 2

n

+ +

e 1 · 2 4π 0

d 3r

d3r

[n(r) ˜ + Z˜ (r)][n(r ˜ ) + Z˜ (r )] |r − r |

d3r v(r) ¯ n(r) ˜ + E xc [n(r)]. ˜

(34)

Z˜ (r) is an angular-momentum dependent core-like density that will be described in detail below. The remaining parts can be evaluated on radial grids in a spherical harmonics expansion. The nodal structure of the wavefunctions can be properly described on a logarithmic radial grid that becomes very fine near the nucleus, E 1R

=

i, j ∈R

Di, j

N c,R 2 −h 2 ¯ 2 c ¯ 2 c −h φj ∇ φi + φn ∇ φn 2m e 2m e

n∈R

e2 1 [n 1 (r) + Z (r)][n 1 (r ) + Z (r )] + · d3 r d3 r 2 4π 0 |r − r | 1 + E xc [n (r)] 2 − h ¯ 1 2 Di, j φ˜ j ∇ φ˜ i E˜ R = 2m e i, j ∈R + +

e2 1 · 2 4π 0

d3 r

d3 r

(35)

[n˜ 1 (r) + Z˜ (r)][n˜ 1 (r ) + Z˜ (r )] |r − r |

d3r v(r) ¯ n˜ 1 (r) + E xc [n˜ 1 (r)].

(36)

˜ The compensation charge density Z(r) = R Z˜ R (r) is given as a sum of angular momentum dependent Gauss functions, which have an analytical plane-wave expansion. A similar term occurs also in the pseudopotential approach. In contrast to the norm-conserving pseudopotential approach, however, the compensation charge of an atom Z˜ R is nonspherical and constantly adapts to the instantaneous environment. It is constructed such that n 1R (r) + Z R (r) − n˜ 1R (r) − Z˜ R (r)

(37)

has vanishing electrostatic multipole moments for each atomic site. With this choice, the electrostatic potentials of the augmentation densities vanish outside their spheres. This is the reason that there is no electrostatic interaction of the one-center parts between different sites.


113

The compensation charge density as given here is still localized within the atomic regions. A technique similar to an Ewald summation, however, allows to replace it by a very extended charge density. Thus we can achieve, that the plane-wave convergence of the total energy is not affected by the auxiliary density. The potential v¯ = R v¯ R , which occurs in Eqs. (34) and (36), enters the total energy in the form of “intelligent zeros” described in Eq. (32) 0=

n

=

˜ n |v¯ R | ˜ n − ˜ n1 |v¯ R | ˜ n1 f n ˜ n |v¯ R | ˜ n − f n

n

Di, j φ˜i |v¯ R |φ˜ j .

(38)

i, j ∈R

The main reason for introducing this potential is to cancel the Coulomb singularity of the potential in the plane-wave part. The potential v¯ allows to influence the plane-wave convergence beneficially, without changing the converged result. v¯ must be localized within the augmentation region, where Eq. (19) holds.

5.

Approximations

Once the total energy functional provided in the previous section has been defined, everything else follows: Forces are partial derivatives with respect to atomic positions. The potential is the derivative of the nonkinetic energy contributions to the total energy with respect to the density, and the auxiliary ˜ n with respect to auxiliary wave Hamiltonian follows from derivatives H˜ | functions. The fictitious Lagrangian approach of Car and Parrinello [38] does not allow any freedom in the way these derivatives are obtained. Anything else than analytic derivatives will violate energy conservation in a dynamical simulation. Since the expressions are straightforward, even though rather involved, we will not discuss them here. All approximations are incorporated already in the total energy functional of the PAW method. What are those approximations? • First, we use the frozen-core approximation. In principle, this approximation can be overcome. • The plane-wave expansion for the auxiliary wavefunctions must be complete. The plane-wave expansion is controlled easily by increasing the plane-wave cut-off defined as E PW = 12 h¯ 2 G 2max . Typically, we use a planewave cut-off of 30 Ry. • The partial wave expansions must be converged. Typically we use one or two partial waves per angular momentum (, m) and site. It should be noted that the partial wave expansion is not variational, because it

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P.E Blöchl, J. Kästner, and C.J. Först changes the total energy functional and not the basis set for the auxiliary wavefunctions.

We do not discuss here numerical approximations such as the choice of the radial grid, since those are easily controlled.

6.

Relation to the Pseudopotentials

We mentioned earlier that the pseudopotential approach can be derived as a well-defined approximation from the PAW method: The augmentation part of the total energy E = E 1 − E˜ 1 for one atom is a functional of the one-center density matrix Di, j ∈R defined in Eq. (29). The pseudopotential approach can be recovered if we truncate a Taylor expansion of E about the atomic density matrix after the linear term. The term linear to Di, j is the energy related to the nonlocal pseudopotential. E(Di, j ) = E(Di,atj )+ = E self +

(Di, j − Di,atj )

i, j

˜ n |v ps | ˜ n − f n

∂E + O(Di, j − Di,atj )2 ∂ Di, j

d3r v(r) ¯ n(r)+ ˜ O(Di, j −Di,atj )2

n

(39) which can directly be compared to the total energy expression, Eq. (10), of the pseudopotential method. The local potential v(r) ¯ of the pseudopotential approach is identical to the corresponding potential of the projector augmented ˜ wave method. The remaining contributions in the PAW total energy, namely E, differ from the corresponding terms in Eq. (10) only in two features: our auxiliary density also contains an auxiliary core density, reflecting the nonlinear core correction of the pseudopotential approach, and the compensation density Z˜ (r) is non-spherical and depends on the wavefunction. Thus, we can look at the PAW method also as a pseudopotential method with a pseudopotential that adapts to the instantaneous electronic environment. In the PAW method, the explicit nonlinear dependence of the total energy on the one-center density matrix is properly taken into account. What are the main advantages of the PAW method compared to the pseudopotential approach? First, all errors can be systematically controlled so that there are no transferability errors. As shown by Watson and Carter [43] and Kresse and Joubert [44], most pseudopotentials fail for high-spin atoms such as Cr. While it is probably true that pseudopotentials can be constructed that cope even with this situation, a failure can not be known beforehand, so that some empiricism remains in practice: A pseudopotential constructed from an isolated atom is


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not guaranteed to be accurate for a molecule. In contrast, the converged results of the PAW method do not depend on a reference system such as an isolated atom, because PAW uses the full density and potential. Like other all-electron methods, the PAW method provides access to the full charge and spin density, which is relevant, for example, for hyperfine parameters. Hyperfine parameters are sensitive probes of the electron density near the nucleus. In many situations they are the only information available that allows to deduce atomic structure and chemical environment of an atom from experiment. The plane-wave convergence is more rapid than in norm-conserving pseudopotentials and should in principle be equivalent to that of ultra-soft pseudopotentials [31]. Compared to the ultra-soft pseudopotentials, however, the PAW method has the advantage that the total energy expression is less complex and can therefore be expected to be more efficient. The construction of pseudopotentials requires to determine a number of parameters. As they influence the results, their choice is critical. Also the PAW methods provides some flexibility in the choice of auxiliary partial waves. However, this choice does not influence the converged results.

7.

Recent Developments

Since the first implementation of the PAW method in the CP-PAW code, a number of groups have adopted the PAW method. The second implementation was done by the group of Holzwarth [45]. The resulting PWPAW code is freely available [46]. This code is also used as a basis for the PAW implementation in the AbInit project. An independent PAW code has been developed by Valiev and Weare [47]. Recently, the PAW method has been implemented into the VASP code [44]. The PAW method has also been implemented by Kromen into the ESTCoMPP code of Blügel and Schröder. Another branch of methods uses the reconstruction of the PAW method, without taking into account the full wavefunctions in the energy minimization. Following chemists’ notation, this approach could be termed “postpseudopotential PAW.” This development began with the evaluation for hyperfine parameters from a pseudopotential calculation using the PAW reconstruction operator [36] and is now used in the pseudopotential approach to calculate properties that require the correct wavefunctions such as hyperfine parameters. The implementation by Kresse and Joubert [44] has been particularly useful as they had an implementation of PAW in the same code as the ultrasoft pseudopotentials, so that they could critically compare the two approaches with each other. Their conclusion is that both methods compare well in most

116


cases, but they found that magnetic energies are seriously – by a factor 2 – in error in the pseudopotential approach, while the results of the PAW method were in line with other all-electron calculations using the linear augmented plane-wave method. As a short note, Kresse and Joubert incorrectly claim that their implementation is superior as it includes a term that is analogous to the nonlinear core correction of pseudopotentials [32]: this term however is already included in the original version in the form of the pseudized core density. Several extensions of the PAW have been done in the recent years: For applications in chemistry truly isolated systems are often of great interest. As any plane-wave based method introduces periodic images, the electrostatic interaction between these images can cause serious errors. The problem has been solved by mapping the charge density onto a point charge model, so that the electrostatic interaction could be subtracted out in a self-consistent manner [48]. In order to include the influence of the environment, the latter was simulated by simpler force fields using the molecular-mechanics–quantummechanics (QM–MM) approach [49]. In order to overcome the limitations of the density functional theory, several extensions have been performed. Bengone et al. [50] implemented the LDA+U approach into the CP-PAW code. Soon after this, Arnaud and Alouani [51] accomplished the implementation of the GW approximation into the CP-PAW code. The VASP-version of PAW [52] and the CP-PAW code have now been extended to include a noncollinear description of the magnetic moments. In a noncollinear description, the Schrödinger equation is replaced by the Pauli equation with two-component spinor wavefunctions. The PAW method has proven useful to evaluate electric field gradients [53] and magnetic hyperfine parameters with high accuracy [54]. Invaluable will be the prediction of NMR chemical shifts using the GIPAW method of Pickard and Mauri [55], which is based on their earlier work [56]. While the GIPAW is implemented in a post-pseudopotential manner, the extension to a self-consistent PAW calculation should be straightforward. An post-pseudopotential approach has also been used to evaluate core level spectra [57] and momentum matrix elements [58].

Acknowledgments We are grateful for carefully reading the manuscript to S. Boeck, J. Noffke, A. Poddey, as well as to K. Schwarz for his continuous support. This work has benefited from the collaborations within the ESF Programme on “Electronic Structure Calculations for Elucidating the Complex Atomistic Behavior of Solids and Surfaces.”


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P.E Blöchl, J. Kästner, and C.J. Först [24] G.P. Kerker, “Non-singular atomic pseudopotentials for solid state applications,” J. Phys. C, 13, L189, 1980. [25] G.B. Bachelet, D.R. Hamann, and M. Schlüter, “Pseudopotentials that work: from H to Pu,” Phys. Rev. B, 26, 4199, 1982. [26] N. Troullier and J.L. Martins, “Efficient pseudopotentials for plane-wave calculations,” Phys. Rev. B, 43, 1993, 1991. [27] J.S. Lin, A. Qteish, M.C. Payne, and V. Heine, “Optimized and transferable nonlocal separable ab initio pseudopotentials,” Phys. Rev. B, 47, 4174, 1993. [28] M. Fuchs and M. Scheffler, “Ab initio pseudopotentials for electronic structure calculations of poly-atomic systems using density-functional theory,” Comput. Phys. Commun., 119, 67, 1999. [29] L. Kleinman and D.M. Bylander, “Efficacious form for model pseudopotentials,” Phys. Rev. Lett., 48, 1425, 1982. [30] P.E. Blöchl, “Generalized separable potentials for electronic structure calculations,” Phys. Rev. B, 41, 5414, 1990. [31] D. Vanderbilt, “Soft self-consistent pseudopotentials in a generalized eigenvalue formalism,” Phys. Rev. B, 41, 17892, 1990. [32] S.G. Louie, S. Froyen, and M.L. Cohen, “Nonlinear ionic pseudopotentials in spindensity-functional calculations,” Phys. Rev. B, 26, 1738, 1982. [33] D.R. Hamann, “Generalized norm-conserving pseudopotentials,” Phys. Rev. B, 40, 2980, 1989. [34] K. Laasonen, A. Pasquarello, R. Car, C. Lee, and D. Vanderbilt, “Implementation of ultrasoft pseudopotentials in ab initio molecular dynamics,” Phys. Rev. B, 47, 110142, 1993. [35] X. Gonze, R. Stumpf, and M. Scheffler, “Analysis of separable potentials,” Phys. Rev. B, 44, 8503, 1991. [36] C.G. Van de Walle and P.E. Blöchl, “First-principles calculations of hyperfine parameters,” Phys. Rev. B, 47, 4244, 1993. [37] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, and J.D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate-gradients,” Rev. Mod. Phys., 64, 11045, 1992. [38] R. Car and M. Parrinello, “Unified approach for molecular dynamics and densityfunctional theory,” Phys. Rev. Lett., 55, 2471, 1985. [39] S. Nosé, “A unified formulation of the constant temperature molecular-dynamics methods,” Mol. Phys., 52, 255, 1984. [40] Hoover, “Canonical dynamics: equilibrium phase-space distributions,” Phys. Rev. A, 31, 1695, 1985. [41] P.E. Blöchl and M. Parrinello, “Adiabaticity in first-principles molecular dynamics,” Phys. Rev. B, 45, 9413, 1992. [42] P.E. Blöchl, “Second generation wave function thermostat for ab initio molecular dynamics,” Phys. Rev. B, 65, 1104303, 2002. [43] S.C. Watson and E.A. Carter, “Spin-dependent pseudopotentials,” Phys. Rev. B, 58, R13309, 1998. [44] G. Kresse and J. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys. Rev. B, 59, 1758, 1999. [45] N.A.W. Holzwarth, G.E. Mathews, R.B. Dunning, A.R. Tackett, and Y. Zheng, “Comparison of the projector augmented-wave, pseudopotential, and linearized augmented-plane-wave formalisms for density-functional calculations of solids,” Phys. Rev. B, 55, 2005, 1997.


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[46] A.R. Tackett, N.A.W. Holzwarth, and G.E. Matthews, “A projector augmented wave (PAW) code for electronic structure calculations. Part I: atompaw for generating atom-centered functions. A projector augmented wave (PAW) code for electronic structure calculations. Part II: pwpaw for periodic solids in a plane wave basis,” Comput. Phys. Commun., 135, 329–347, 2001. See also pp. 348–376. [47] M. Valiev and J.H. Weare, “The projector-augmented plane wave method applied to molecular bonding,” J. Phys. Chem. A, 103, 10588, 1999. [48] P.E. Blöchl, “Electrostatic decoupling of periodic images of plane-wave-expanded densities and derived atomic point charges,” J. Chem. Phys., 103, 7422, 1995. [49] T.K. Woo, P.M. Margl, P.E. Blöchl, and T. Ziegler, “A combined Car–Parrinello QM/MM implementation for ab initio molecular dynamics simulations of extended systems: application to transition metal catalysis,” J. Phys. Chem. B, 101, 7877, 1997. [50] O. Bengone, M. Alouani, P.E. Blöchl, and J. Hugel, “Implementation of the projector augmented-wave LDA+U method: application to the electronic structure of NiO,” Phys. Rev. B, 62, 16392, 2000. [51] B. Arnaud and M. Alouani, “All-electron projector-augmented-wave GW approximation: application to the electronic properties of semiconductors,” Phys. Rev. B., 62, 4464, 2000. [52] D. Hobbs, G. Kresse, and J. Hafner, “Fully unconstrained noncollinear magnetism within the projector augmented-wave method,” Phys. Rev. B, 62, 11556, 2000. [53] H.M. Petrilli, P.E. Blöchl, P. Blaha, and K. Schwarz, “Electric-field-gradient calculations using the projector augmented wave method,” Phys. Rev. B, 57, 14690, 1998. [54] P.E. Blöchl, “First-principles calculations of defects in oxygen-deficient silica exposed to hydrogen,” Phys. Rev. B, 62, 6158, 2000. [55] C.J. Pickard and F. Mauri, “All-electron magnetic response with pseudopotentials: NMR chemical shifts,” Phys. Rev. B., 63, 245101, 2001. [56] F. Mauri, B.G. Pfrommer, and S.G. Louie, “Ab initio theory of NMR chemical shifts in solids and liquids,” Phys. Rev. Lett., 77, 5300, 1996. [57] D.N. Jayawardane, C.J. Pickard, L.M. Brown, and M.C. Payne, “Cubic boron nitride: experimental and theoretical energy-loss near-edge structure,” Phys. Rev. B, 64, 115107, 2001. [58] H. Kageshima and K. Shiraishi, “Momentum-matrix-element calculation using pseudopotentials,” Phys. Rev. B, 56, 14985, 1997.

1.7 ELECTRONIC SCALE James R. Chelikowsky University of Minnesota, Minneapolis, MN, USA

1.

Real-space methods for ab initio calculations

Major computational advances in predicting the electronic and structural properties of matter come from two sources: improved performance of hardware and the creation of new algorithms, i.e., software. Improved hardware follows technical advances in computer design and electronic components. Such advances are frequently characterized by Moore’s Law, which states that computer power will double every 2 years or so. This law has held true for the past 20 or 30 years and most workers expect it to hold for the next decade, suggesting that such technical advances can be predicted. In clear contrast, the creation of new high performance algorithms defies characterization by a similar law as creativity is clearly not a predictable activity. Nonetheless, over the past half century, most advances in the theory of the electronic structure of matter have been made with new algorithms as opposed to better hardware. One may reasonably expect these advances to continue. Physical concepts such as the pseudopotentials and density functional theories coupled with numerical methods such as iterative diagonalization methods have permitted very large systems to be examined, much larger systems than could be handled solely by the increase allowed by computational hardware advances. Systems with hundreds, if not thousands, of atoms can now be examined, whereas methods of a generation ago might handle only tens of atoms. The development of real-space methods for the electronic structure over the past ten years is a notable advance in high performance algorithms for solving the electronic structure problem. Real-space methods do not require an explicit basis. The convergence of the method, assuming a uniform grid, can be tested by varying only one parameter: the grid spacing. The method can be easily be applied to neutral or charged systems, to extended or localized systems, and to diverse materials such as simple metals, semiconductors, 121 S. Yip (ed.), Handbook of Materials Modeling, 121–135. c 2005 Springer. Printed in the Netherlands.

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and transition metals. These methods are also well suited for highly parallel computing platforms as few global communications are required. Review articles on these approaches can be found in Refs. [1–3].

2.

The Electronic Structure Problem

Most contemporary descriptions of the electronic structure problem for large systems cast the problem within density functional theory [4]. The many body problem is mapped onto a one electron Schrödinger equation called the Kohn–Sham equation [5]. For an atom, this equation can be written as

−2 ∇ 2 Z e2 − + VH ( r ) + Vxc [ r , ρ( r )] 2m r

ψn ( r ) = E n ψn ( r)

(1)

where there are Z electrons in the atom, VH is the Hartree or Coulomb potential, and Vxc is the exchange-correlation potential. The Hartree and exchangecorrelation potentials can be determined from the electronic charge density. The eigenvalue and eigenfunctions, (E n , ψn ( r )), can be used to determine the total electronic energy of the atom. The density is given by ρ( r ) = −e

|ψn ( r )|2

(2)

n,occup

The summation is over all occupied states. The Hartree potential is then determined by r ) = −4π eρ( r) ∇ 2 VH (

(3)

This term can be interpreted as the electrostatic interaction of an electron with the charge density of system. The exchange-correlation potential is more problematic. Within density functional theory, one can define an exchange correlation potential as a functional of the charge density. The central tenant of the local density approximation [5] is that the total exchange-correlation energy may be written as

r ) xc (ρ( r )) d 3r E xc [ρ] = ρ(

(4)

where xc is the exchange-correlation energy density. If one has knowledge of the exchange-correlation energy density, one can extract the potential and total electronic energy of the system. As a first approximation the exchangecorrelation energy density can be extracted from a homogeneous electron gas. It is common practice to separate exchange and correlation contributions to xc : xc = x + c [4]. It is not difficult to solve the Kohn–Sham equation (Eq. 1) for an atom. The potential, and charge density, is assumed to be spherically symmetric

Electronic scale

123

and the Kohn–Sham problem reduces to solving a one-dimensional problem. The Hartree and exchange-correlation potentials can be iterated to form a selfconsistent field. Usually the process is so quick for an atom that it can be done on desktop or laptop computer in a matter of seconds. In three dimensions, as for a complex atomic cluster, liquid or crystal, the problem is highly nontrivial. One major difficulty is the range of length scales involved. For example, in the case of a multielectron atom, the most tightly bound, core electrons can be confined to within ∼0.1 Å whereas the outer valence electrons may extend over ∼1–5 Å. In addition, the nodal structure of the atomic wave functions are difficult to replicate with a simple basis, especially the cusp in a wave function at the nuclear site where the Coulomb potential diverges. One approach to this problem is to form a basis combining highly localized functions with extended functions. This approach enormously complicates the electronic structure problem as valence and core states are treated on equal footing whereas such states are not equivalent in terms of their chemical activity. Consider the physical content of the periodic table, i.e., arranging the elements into columns with similar chemical properties. The Group IV elements such as C, Si, and Ge have similar properties because they share an outer s2 p2 configuration. This chemical similarity of the valence electrons is recognized by the pseudopotential approximation [6, 7]. The pseudopotential replaces the “all electron” potential by one that reproduces only the chemically active, or valence electrons. Usually, the pseudopotential subsumes the nuclear potential with those of the core electrons to generate an “ion core potential.” As an example, consider a sodium atom whose core electron configuration is 1s2 2s2 2p6 and valence electron configuration is 3s1 . The charge on the ion core pseudopotential is +1 (the nuclear charge minus the number of core electrons). Such a pseudopotential will bind only one electrons. The length scale of the pseudopotential is now set by the valence electrons alone. This permits a great simplification of the Kohn–Sham problem in terms of choosing a basis. For the purposes of designing an ab initio pseudopotential let us consider a sodium atom. By solving for the Na atom, we know the eigenvalue, 3s , and the corresponding wave function, ψ3s (r) for the valence electron. We demand several conditions for the Na pseudopotential: (1) The potential bind only the valence electron, the 3s-electron for the case at hand. (2) The eigenvalue of the corresponding valence electron be identical to the full potential eigenvalue. The full potential is also called the all-electron potential. (3) The wave function be nodeless and identical to the “all electron” wave function outside the core region. For example, we construct a pseudo-wave function, φ3s (r) such that φ3s (r)=ψ3s (r) for r > rc where rc defines the size spanned by the ion core, i.e., the nucleus and core electrons. For Na, this means the “size” of 1s2 2s2 2p6

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states. Typically, the core is taken to be less than the distance corresponding to the maximum of the valence wave function, but greater than the distance of the outermost node. If the eigenvalue, p , and the wave function, φp (r), are known from solving the atom, it is possible to invert the Kohn–Sham equation to yield an ion core pseudopotential, i.e., a pseudopotential that when screened will yield the exact eigenvalue and wave function by construction: p

Vion(r) = p +

2 ∇ 2 φp − VH (r) − Vxc [r, ρ(r)] 2mφp

(5)

Within this construction, the pseudo-wave function, φp (r), should be identical to the all electron wave function, ψAE (r), outside the core: φp (r) = ψAE (r) for r >rc will guarantee that the pseudo-wave function will yield similar chemical properties as the all electron wave function. For r < rc , one may alter the all-electron wave function as one wishes, within certain limitations, and retain the chemical accuracy of the problem. For computational simplicity, we take the wave function in this region to be smooth and nodeless. Another very important criterion is mandated. Namely, the integral of the pseudocharge density, i.e., square of the wave function |φp (r)|2 , within the core should be equal to the integral of the all-electron charge density. Without this condition, the pseudo-wave function can differ by a scaling factor from the all-electron wave function, that is, φp (r)=C ×ψAE (r) for r > rc where the constant, C, may differ from unity. Since we expect the chemical bonding of an atom to be highly dependent on the tails of the valence wave functions, it is imperative that the normalized pseudo wave function be identical to the all-electron wave functions. The criterion by which one insures C = 1 is called norm conserving [2]. An example of a pseudopotential, in this case the Na pseudopotential, is presented in Fig. 1. The ion core pseudopotential is dependent on the angular momentum component of the wave function. This is apparent from Eq. (5) p where the Vion is “state dependent” or nonlocal. This nonlocal behavior is pronounced for first row elements, which lack p-states in the core, and for first row transition metals, which lack d-states in the core. A physical explanation for this behavior can be traced to the orthogonality requirement of the valence wave functions to the core states. This may be illustrated by considering the carbon atom. The 2s of carbon is orthogonal to the 1s state, whereas the 2p state is not required to be orthogonal to a 1p state. As such, the 2s state has a node; the 2p does not. In transforming these states to nodeless pseudo-wave functions, more kinetic energy associated with the 2s exists compared to the 2p state. The additional kinetic energy cancels the strong coulombic potential better for the 2s state than the 2p. In terms of the ion core pseudopotential, the 2s potential is weaker than the 2p state.

Electronic scale

125

2 1

s-pseudopotential

Potential (Ry)

0 ⫺1 p-pseudopotential

⫺2 d-pseudopotential

⫺3 ⫺4

all electron

⫺5

0

1

2 r (a.u.)

3

4

Figure 1. Pseudopotential compared to the all-electron potential for the sodium atom. This pseudopotential was constructed using the method of Troullier and Martins [8].

In the case of sodium, only three significant components (s, p, and d) are required for an accurate pseudopotential. Note how the d component is the strongest following the argument that no core states of similar angular momentum exist within the Na core. For more complex systems such as a rare earth metals, one might have four or more components. In Fig. 2, the 3s state for the all electron potential is illustrated. It is compared to the lowest s-state for the pseudopotential illustrated in Fig. 1 The Kohn–Sham equation can be rewritten for a pseudopotential as

−2 ∇ 2 p + Vion ( r ) + VH ( r ) + Vxc [ r , ρ( r )] 2m

ψn ( r ) = E n ψn ( r)

(6)

p

where Vion can be expressed as p

r) = Vion(

Vi,ion ( r − Ri ) p

(7)

i p

where Vi,ion is the ionic pseudopotential for the ith-atomic species located at position, Ri . The charge density in Eq. (7) corresponds to a sum over the wave functions for occupied valence states.

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J.R. Chelikowsky 0.6 Na

Wave Functions

0.4

3s

3p

0.2

0

⫺0.2 0

1

2 r (a.u.)

3

4

5

Figure 2. Pseudopotential wave functions compared to all-electron wave functions for the sodium atom. The all-electron wave functions are indicated by the dashed lines.

Since the pseudopotential and corresponding wave functions vary slowly in space, a number of simple basis sets is possible, e.g., one could use Gaussians [9] or plane waves [6, 7]. Both methods often work quite well, although each has its limitations. Owing in part to the simplicity and ease of implementation, plane wave methods have become of the method of choice for electronic structure work, especially for simple metals and semiconductors like silicon [7, 10]. Methods based on plane wave bases are often called “momentum” or “reciprocal” space approaches to the electronic structure problem. Plane wave approaches utilize a basis of “infinite extent.” The extended basis requires special techniques to describe localized systems. For example, suppose one wishes to examine a cluster of silicon atoms. A common approach is to use a “supercell method.” The cluster would be placed in a large cell, which is periodically repeated to fill up all space. The electronic structure of this system corresponds to an isolated cluster, provided sufficient “vacuum” surrounds each cluster. This method is very successful and has been used to consider localized systems such as clusters as well as extended systems such as surfaces or liquids [10]. In contrast, one can take a rather dramatic alternative view and eliminate an explicit basis altogether and solve Eq. (6) completely in real space using

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a grid. Real space or grid methods are typically used for engineering problems, e.g., one might solve for the strain field in an airplane wing using finite element methods. Such methods have not been commonly used for the electronic structure problem. There are at least two reasons for this situation. First, without the pseudopotential method, a nonlinear grid would be needed to describe the singular coulombic potential near the atomic nucleus and the corresponding cusp in the wave function. This would enormously complicate the problem and destroy the simplicity of the method. Second, the non-local nature of the pseudopotential can be easily addressed in grid methods, but until recently the formalism for this task has not been available. Real-space approaches overcome many of the complications involved with explicit basis, especially for describing nonperiodic systems such as molecules, clusters and quantum dots. Unlike localized orbitals such as Gaussians, the basis is unbiased. One need not specify whether the basis contains particular angular momentum components. Moreover, the basis is not “attached” to the atomic positions and no Pulay forces need to be considered [11]. Pulay forces arise from an incomplete basis. As atoms are moved, the basis needs to be recomputed as the convergence changes with the atomic configuration. Unlike an extended basis such as those based on plane waves, the vacuum is easily described by grid points. In contrast to plane waves, grids are efficient and easy to implement on parallel platforms. Real space algorithms avoid the use of fast Fourier transforms by performing all calculations in physical space instead of Fourier space. A benefit of avoiding Fourier transforms is that very few global communications are required. Different numerical methods can be used to implement real space methods such as finite element or finite difference methods. Both approaches have advantages and liabilities. Finite element methods can easily accommodate nonuniform grids and can reflect the variational principle as the mesh is refined [1]. This is an appropriate approach for systems in which complex boundary conditions exist. For systems where the boundary conditions are simple, e.g., outside a domain the wave function is set to zero, this is not an important consideration. Finite differencing methods are easier to implement compared to finite element methods, especially with uniform grids. Both approaches have been extensively utilized; however, owing to the ease of implementation, finite differencing methods have been applied to a wider range of materials and properties. For this reason, we will illustrate the finite differencing method. A key aspect to the success of the finite difference method is the availability of higher order finite difference expansions for the kinetic energy operator, i.e., expansions of the Laplacian [12]. Higher order finite difference methods significantly improve convergence of the eigenvalue problem when compared with standard finite difference methods. If one imposes a simple, uniform grid

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on our system where the points are described in a finite domain by (xi , y j , z k ), one may approximate the Laplacian operator at (xi , y j , z k ) by M ∂ 2ψ = Cn ψ(xi + nh, y j , z k ) + O(h 2M+2 ), ∂ x 2 n=−M

(8)

where h is the grid spacing and M is a positive integer. This approximation is accurate to O(h2M+2 ) under the assumption that ψ can be approximated accurately by a power series in h. Algorithms are available to compute the coefficients Cn for arbitrary order in h [12]. With the kinetic energy operator expanded as in Eq. (8), one can set up the Kohn–Sham equation over a grid. For simplicity, let us assume a uniform grid, but this is not a necessary requirement. ψ(xi , y j , z k ) is computed on the grid by solving the eigenvalue problem: 

M M 2  Cn1 ψn (xi + n 1 h, y j , z k ) + Cn2 ψn (xi , y j + n 2 h, z k ) − 2m n =−M n =−M 1

+

M n 3 =−M



2

Cn3 ψn (xi , y j , z k + n 3 h) + Vion(xi , y j , z k ) + VH (xi , y j , z k )

+ Vxc (xi , y j , z k ) ψn (xi , y j , z k ) = E n ψn (xi , y j , z k )

(9)

For L grid points, the size of the full matrix is L 2 . A uniformly spaced grid in a three-dimensional cube is shown in Fig. 3. Each grid point corresponds to a row in the matrix. However, many points in the cube are far from any atoms in the system and the wave function on these points may be replaced by zero. Special data structures may be used to discard these points and retain only those having a nonzero value for the wave function. The size of the Hamiltonian matrix is usually reduced by a factor of two to three with this strategy, which is quite important considering the large number of eigenvectors which must be saved. Further, since the Laplacian can be represented by a simple stencil, and since all local potentials sum up to a simple diagonal matrix, the Hamiltonian need not be stored. Nonlocality in the pseudopotential, i.e., the “state dependence” of the potential as illustrated in Fig. 1, is easily treated using a plane wave basis in Fourier space, but it may also be calculated in real space. The nonlocality appears only in the angular dependence of the potential and not in the radial coordinate. It is often advantageous to use a more advanced projection scheme, due to Kleinman and Bylander [13]. The interactions between valence electrons and pseudo-ionic cores in the Kleinman–Bylander form may be separated into a local potential and a nonlocal pseudopotential in real space [8], which differs from zero only inside the small core region around each atom.

Electronic scale

129

Figure 3. Uniform grid illustrating a typical configuration for examining the electronic structure of a localized system. The gray sphere represents the domain where the wave functions are allowed to be nonzero. The light spheres within the domain are atoms.

One can write the Kleinman–Bylander form in real space as p

r )φn ( r) = Vion(

Vloc (| ra |)φn ( r) +

a a K n,lm

1 = a Vlm

G an,lm u lm ( ra )Vl (ra ),

(10)

a, n,lm

u lm ( ra )Vl (ra )ψn ( r )d3r,

(11)

a is the normalization factor, and Vlm

= u lm ( ra )Vl (ra )u lm ( ra ) d3r,

(12)

where ra = r − Ra , and the u lm are the atomic pseudopotential wave functions of angular momentum quantum numbers (l, m) from which the l-dependent ionic pseudopotential, Vl (r), is generated. Vl (r) = Vl (r) − Vloc (r) is the difference between the l component of the ionic pseudopotential and the local ionic potential. As a specific example, in the case of Na, we might choose the local part of the potential to replicate only the l = 0 component as defined by the 3s state. The nonlocal parts of the potential would then contain only the l = 1 and l = 2 components. The choice of which angular component is chosen for the local part of the potential is somewhat arbitrary. It is often convenient to chose the local potential to correspond to the highest l-component of interest. This

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reduces the computational effort associated with higher l-components [3]. The choice of the local potential can be tested by utilizing different components for the local potential. There are several difficulties with the eigen problems generated in this application in addition to the size of the matrices. First, the number of required eigenvectors is proportional to the atoms in the system, and can grow up to thousands. Besides storage, maintaining the orthogonality of these vectors can be a formidable task. Second, the relative separation of the eigenvalues becomes increasingly poor as the matrix size increases and this has an adverse effect on the rate of convergence of the eigenvalue solvers. Preconditioning techniques attempt to alleviate this problem. A brief review of these approaches can be found in Ref. [3]. The architecture of the Hamiltonian matrix is illustrated in Fig. 4 for a diatomic molecule. Although the details of matrix structure will be a function of the geometry of the system, the essential elements remain the same. The off-diagonal elements arise from the expansion coefficients in Eq. (8) and the nonlocal potential in Eq. (10). These elements are not updated during the self-consistency cycle. The on-diagonal matrix elements consist of the local ion core pseudopotential, the Hartree potential and the exchange-correlation potential. These terms are updated each self-consistent cycle.

Figure 4. Hamiltonian matrix for a diatomic molecule in real space. Nonzero matrix elements are indicated by black dots. The diagonal matrix elements consist of the local ionic pseudopotential, Hartree potential and local density exchange-correlation potential. The off-diagonal matrix elements consistent of the coefficients in the finite difference expansion and the nonlocal matrix elements of the pseudopotential. The system contains about 4000 grid points or 16 million matrix elements.

Electronic scale

131

Figure 5. Potentials and wave functions for the oxygen dimer molecule. The total electronic potential is shown on the left along a ray connecting the two oxygen atoms. The Kohn–Sham molecular orbitals are shown on the right side of the figure. The orbitals on the left are from a real space calculation and the ones on the right from a plane wave calculation.

While the Hamiltonian matrix in real space can be large, it never needs to be explicitly saved. Also, the matrix is sparse; the sparsity is a function of M (see Eq. 8), which is the order of the higher order difference expansion. For larger values of M, the grid can be made coarse. However, this reduces the sparsity of the matrix. Conversely if we use standard finite difference methods, the matrix is sparser, but the grid size must be fine to retain the same accuracy. In practice, a value of M = 4−6 appears to work very well. There is a close relationship between the plane wave method and real-space methods. For example, one can always do a Fourier transform on a real-space method and obtain results in reciprocal space, or perform the operation in reverse to go from Fourier space to real space. In this sense, higher order finite differences can be considered an abridged Fourier transform as one does not sum over all grid points in the mesh. As a rough measure of the convergence of real space methods, one can consider a Fourier component or plane wave cut off of (π/ h)2 for a grid spacing, h. Using this criterion, a grid spacing of h = 0.5 a.u.1 would correspond to a plane wave cut-off of approximately 40 Ry. In Fig. 5, a comparison is between the plane-wave supercell method and a real-space method for the oxygen dimer. The oxygen dimer is a difficult 1 1 a.u. = 0.529 Å or one bohr unit of length.

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molecular species using pseudopotentials as the potential is rather deep and quite nonlocal as compared to second row elements such as silicon. The total local electronic potential is depicted along a ray containing the oxygen atoms [14]. Also shown are the Kohn–Sham one electron orbitals. The agreement between the two methods is quite good, certainly less than the uncertainties involved in the local density approximation. The most noticeable difference in the potential occurs at the nuclear positions. At this point, the atomic pseudopotential are quite strong and the variation in the wave function requires a fine mesh. However, it is important to note that this spatial regime is removed from the bonding region of the molecule. A survey of cluster and molecular species using both plane waves and real space method confirms that the accuracy of the two methods is comparable, but the real space method is easier to implement [14].

3.

Outlook

The focus of the electronic structure problem will likely not reside in solving for the energy bands of ordered solids. The energy band structure of crystalline matter, especially elemental solids, has largely been exhausted. This is not to say that elemental solids are no longer of interest. Certainly, interest in these materials will continue as testing grounds for new electronic structure methods. However, interest in nonperiodic systems such as amorphous solids, liquids, glasses, clusters, and nanoscale quantum dots is now a major focus of the electronic structure problem. Perhaps this is the greatest challenge for electronic structure methods, i.e., systems with many electronic and nuclear degrees of freedom and little or no symmetry. Often the structure of these materials are unknown and the materials properties may be a strong function of temperature. Real-space methods offer a new avenue for these large and complex systems. As an illustration of the potential of these methods, consider the example of quantum dots. In Fig. 6, we illustrate hydrogenated Ge clusters. These clusters are composed of bulk fragments of Ge whose dangling bonds are capped with hydrogen. The hydrogen passivates any electronically active dangling bonds. The larger clusters correspond to quantum dots, i.e., semiconductor fragments whose surface properties have been removed, but whose optical properties are dramatically altered by quantum confinement. It is well known that these systems have optical properties with much larger gaps than that of the bulk crystal. The optical spectra of such clusters are shown in Fig. 7. The largest cluster illustrated contains over 800 atoms, although even larger clusters have been examined. This size cluster would be difficult to examine with traditional methods. Although these calculations were done with a ground state method the general shape of the spectra are correct and the evolution of the

Electronic scale

Figure 6.

133

Hydrogenated germanium clusters ranging from germane (GeH4 ) to Ge147 H100 .

Ge35H36

Ge87H76

Photoabsorption (arb.un.)

Ge147H100

Ge191H148

Ge239H196

Ge293H172

Ge357H204

E

Ge525H276

1

E

2

E0 1

2

3 4 Transitionenergy (eV)

5

6

Figure 7. Photoabsorption spectra for hydrogenated germanium quantum dots. The labels E 0 , E 1 and E 2 refer to optical features.

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J.R. Chelikowsky

spectra appear bulk-like by a few hundred atoms. Surfaces, clusters, magnetic systems, complex solids have also been treated with real-space methods [1, 15]. Finally, systems approach the macroscopic limit, it is common to employ finite element or finite difference methods to describe material properties. One would like to couple these methods to those appropriate at the quantum (or nano) limit. The use of real space methods at these opposite limits would be a natural choice. Some attempts along these lines exist. For example, fracture methods often divide up a problem by treating the fracture tip with quantum mechanical methods, the surrounding area by molecular dynamics and the medium away from the tip by continuum mechanics [16].

References [1] T.L. Beck, “Real-space mesh techniques in density functional theory,” Rev. Mod. Phys., 74, 1041, 2000. [2] J.R. Chelikowsky, “The pseudopotential-density functional method applied to nanostructures,” J. Phys. D: Appl. Phys., 33, R33, 2000. [3] C.L. Bris (ed.), Handbook of Numerical Analysis (Devoted to Computational Chemistry), Volume X, Elsevier, Amsterdam, 2003. [4] S. Lundqvist and N.H. March (eds.), Theory of the Inhomogeneous Electron Gas, Plenum, New York, 1983. [5] W. Kohn and L.J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev., 140, A1133, 1965. [6] W. Pickett, “Pseudopotential methods in condensed matter applications,” Comput. Phys. Rep., 9, 115, 1989. [7] J.R. Chelikowsky and M.L. Cohen, “Ab initio pseudopotentials for semiconductors,” In: T.S. Moss and P.T. Landsberg (eds.), Handbook of Semiconductors, 2nd edn., Elsevier, Amsterdam, 1992. [8] N. Troullier and J.L. Martins, “Efficient pseudopotentials for plane-wave calculations,” Phys. Rev. B, 43, 1993, 1991. [9] J.R. Chelikowsky and S.G. Louie, “First principles linear combination of atomic orbitals method for the cohesive and structural properties of solids: application to diamond,” Phys. Rev. B, 29, 3470, 1984. [10] J.R. Chelikowsky and S.G. Louie (eds.), Quantum Theory of Materials, Kluwer, Dordrecht, 1996. [11] P. Pulay, “Ab initio calculation of force constants and equilibrium geometries,” Mol. Phys., 17, 197, 1969. [12] B. Fornberg and D.M. Sloan, “A review of pseudospectral methods for solving partial differential equations,” Acta Numerica, 94, 203, 1994. [13] L. Kleinman and D.M. Bylander, “Efficacious form for model pseudopotential,” Phys. Rev. Lett., 48, 1425, 1982. [14] J.R. Chelikowsky, N. Troullier, and Y. Saad, “The finite-difference-pseudopotential method: electronic structure calculations without a basis,” Phys. Rev. Lett., 72, 1240, 1994.

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[15] J. Bernholc, “Computational materials science: the era of applied quantum mechanics,” Phys. Today, 52, 30, 1999. [16] A. Nakano, M.E. Bachlechner, R.K. Kalia, E. Lidorkis, P. Vashishta, G.Z. Voyladjis, T.J. Campbell, S. Ogata, and F. Shimojo, “Multiscale simulation of nanosystems,” Comput. Sci. Eng., 3, 56, 2001.

1.8 AN INTRODUCTION TO ORBITAL-FREE DENSITY FUNCTIONAL THEORY Vincent L. Lignères1 and Emily A. Carter2 1 Department of Chemistry, Princeton University, Princeton, NJ 08544, USA 2

Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544, USA

Given a quantum mechanical system of N electrons and an external potential (which typically consists of the potential due to a collection of nuclei), the traditional approach to determining its ground-state energy involves the optimization of the corresponding wavefunction, a function of 3N dimensions, without considering spin variables. As the number of particles increases, the computation quickly becomes prohibitively expensive. Nevertheless, electrons are indistinguishable so one could intuitively expect that the electron density – N times the probability of finding any electron in a given region of space – might be enough to obtain all properties of interest about the system. Using the electron density as the sole variable would reduce the dimensionality of the problem from 3N to 3, thus drastically simplifying quantum mechanical calculations. This is in fact possible, and it is the goal of orbital-free density functional theory (OF-DFT). For a system of N electrons in an external potential Vext , the total energy E can be expressed as a functional of the density ρ [1], taking on the following form: E[ρ] = F[ρ] +

Vext ( r )ρ( r ) d r

(1)

Here, denotes the system volume considered, while F is the universal functional that contains all the information about how the electrons behave and interact with one another. The actual form of F is currently unknown and one has to resort to approximations in order to evaluate it. Traditionally, it is split into kinetic and potential energy contributions, the exact forms of which are also unknown. Kohn and Sham first proposed replacing the exact kinetic energy of an interacting electron system with an approximate, noninteracting, single 137 S. Yip (ed.), Handbook of Materials Modeling, 137–148. c 2005 Springer. Printed in the Netherlands.

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V. Lignères and E.A. Carter

determinantal wavefunction that gives rise to the same density [2]. This approach is general and remarkably accurate but involves the introduction of one-electron orbitals. E[ρ] = TKS [φ1 , . . . , φ N ] +

Vext( r )ρ( r )d r + J [ρ] + E xc [ρ]

(2)

TKS denotes the Kohn–Sham (KS) kinetic energy for a system of N noninteracting electrons (i.e., for the case of noninteracting electrons, a single-determinantal wavefunction is the exact solution), the φi are the corresponding one-electron orbitals, J is the classical electron–electron repulsion, and E xc is a correction term that should account for electron exchange, electron correlation, and the difference in kinetic energy between the interacting and noninteracting systems. If the φi are orthonormal, TKS has the following explicit form: TKS = −

1 2

N

φi∗ ( r )∇ 2 φi ( r ) d r

(3)

i=1

Unfortunately, the required orthogonalization of these orbitals makes the computational time scale cubically in the number of electrons. Although linearscaling KS algorithms exist, they require some degree of localization in the orbitals and, for this reason, are not applicable to metallic systems [3]. For condensed matter systems, the KS method has another bottleneck: the need to sample the Brillouin zone for the wavefunction (also called “k-point sampling”) can add several orders of magnitude in cost to the computation. Thus, a further advantage of OF-DFT is that, without a wavefunction, this very expensive computational prefactor of the number of k-points is completely absent from the calculation. At this point, many general, efficient and often accurate functionals are available to handle every term in Eq. (2) as functionals of the electron density alone, except for the kinetic energy. The development of a generally applicable, accurate, linear-scaling kinetic energy density functional (KEDF) would remove the last bottleneck in the DFT computations and enable researchers to study much larger systems than are currently accessible. In the following, we will focus our discussion on such functionals.

1.

General Overview

Historically, the first attempt at approximating the kinetic energy assumes a uniform, noninteracting electron gas [4, 5] and is known as the Thomas–Fermi (TF) model for a slowly varying electron gas. 3 (3π 2 )2/3 ρ( r )5/3d r (4) TTF = 10

Orbital-free density functional theory

139

The model, although crude, constitutes a reasonable first approximation to the kinetic energy of periodic systems. It fails for atoms and molecules, however, as it predicts no shell structure, no interatomic bonding, and the wrong behavior for ρ at the r = 0 and r = +∞ limits. We will discuss some ways to improve this model later. A deeper look at Eq. (3) reveals another approach to describing the kinetic energy as a functional of the density. Within the Hartree–Fock (HF) approximation [6], we have ρ( r) = ρ( r) =

N i=1 N

φi∗ ( r )φi ( r)

(5a)

ρi ( r)

(5b)

i=1

so that, using the hermiticity of the gradient operator, and acting on Eq. (5) we obtain r) = 2 ∇ 2 ρ(

N

φi∗ ( r )∇ 2 φi ( r ) + ∇φi∗ ( r )∇φi ( r)

(6)

i=1

Rearranging Eq. (6), integrating over , and substituting Eq. (3) into Eq. (6) yields TKS = −

1 4

∇ 2 ρ( r )d r+

1 2

N

∇φi∗ ( r )∇φi ( r ) d r

(7)

i=1

Multiplying and dividing every term of the sum by ρi naturally introduces ∇ρi TKS = −

1 4

∇ 2 ρ( r )d r+

1 8

N |∇ρi ( r )|2

i=1

ρi ( r)

d r

(8)

but does not provide a form for which the sum can be evaluated simply. Nevertheless, the first term can be rewritten as the integral of the gradient of the density around the edge of space.

∇ 2 ρ( r )d r=

∇ρ( r ) d r

(9)

For a finite system, the gradient of the density vanishes at large distances and for a periodic system the gradients on opposite sides of a periodic cell cancel each other out, so that this integral evaluates to zero in both cases. Finally, for a one-orbital system, we obtain the following exact expression for the kinetic energy [7]. 1 TVW = 8

|∇ρ( r )|2 d r ρ( r)

(10)

140


Although only exact for up to two electrons, the von Weizsäcker (VW) functional is an essential component of the true kinetic energy and provides a good first approximation in the case of quickly varying densities such as those of atoms and molecules. Unfortunately, the total energy corresponding to the ground-state electron density has the same magnitude as the exact kinetic energy. Consequently, errors made in approximating the kinetic energy have a dramatic impact on the total energy and, by extension, on the ground state electron density computed by minimization. Unlike the exchange-correlation energy functionals, which represent a much smaller component of the total energy, kinetic-energy functionals must be highly accurate in order to achieve consistently accurate energy predictions.

2.

KEDFs for Finite Systems

In the case of a finite system such as a single atom, a few molecules in the gas phase, or a cluster, the electron density varies extremely rapidly near the nuclei, making the TF functional inadequate. Although many corrections have been suggested to improve upon the TF results for atoms, these modifications only yield acceptable results when densities obtained from a different method are used, usually HF. Left to determine their own densities self-consistently, these corrections still predict no shell structure for atoms. Nevertheless, the TF functional, or some fraction of it, may still be useful as a corrective term, as we will see later. Going back to the KS expression from Eq. (8), we introduce r) = n i (

r) ρi ( ρ( r)

(11)

which, when multiplying both sides by ρ( r ) and taking the gradient, yields ∇ρi ( r ) = n i ( r )∇ρ( r ) + ρ( r )∇n i ( r)

(12)

Substituting Eq. (12) into Eq. (8) gives the following expression: TKS =

1 8

N (n i ( r )∇ρ( r ) + ρ( r )∇n i ( r ))2

n i ( r )ρ( r)

i=1

d r

(13)

The product is expanded into three sums and reorganized as TKS

1 = 8

N N |∇ρ( r )|2 n i ( r ) + 2∇ρ( r) ∇n i ( r) ρ( r ) i=1 i=1

+ ρ( r)

N |∇n i ( r )|2 i=1

n i ( r)

d r

(14)


141

From Eq. (11), it follows immediately that N

n i ( r) = 1

(15)

i=1

and so, making use of the linearity of the gradient operator in the second term of Eq. (14) N

∇n i ( r) = ∇

i=1

N

n i ( r ) = ∇(1) = 0

(16)

i=1

the expression further simplifies to

TKS =

|∇ρ( r )|2 d r+ 8ρ( r)

ρ( r)

N |∇n i ( r )|2 i=1

8n i ( r)

d r

(17)

As every quantity in the second integral is positive, we can conclude that the VW functional (the first term in Eq. 17) constitutes a lower bound on the noninteracting kinetic energy. This makes physical sense anyway, as we know that the VW kinetic energy is exact for any one-orbital system (one or two electrons, or any number of bosons). Any other orbital introduced will have to be orthogonal to the first. This introduces nodes in the wavefunction, which raises the kinetic energy of the entire system. Therefore, further improvements upon the VW model involve adding an extra term to take into account the larger kinetic energy in the regions of space in which more than one orbital is significant. Far away from the molecule, only one orbital tends to dominate the picture and the VW functional is accurate enough to account for the relatively small contribution of these regions to the total kinetic energy. Most of the deviation from the exact, noninteracting kinetic energy is located close to the nuclei, in the core region of atoms. Corrections based on adding some fraction of the TF functional to the VW have been proposed (see, for instance, Ref. [8]), but only when nonlocal functionals (those depending on more than one point in space, e.g., r and r ) are introduced is a convincing shell structure observed for atomic densities [9]. Even without such correction terms, the TF and VW functionals may still be enough to obtain an accurate description of the system in some limited cases. For instance, Wesolowski and Warshel used a simple, orbital-free KEDF to describe water molecules as a solvent for a quantum-chemically treated water molecule solute [10]. They were able to reproduce the solvation free energy of water accurately using this method. Although this result is encouraging, the ultimate goal of OF-DFT is to determine a KEDF that would be accurate even without the backup provided by the traditional quantum-mechanical method. One key to judging of the

142


quality of a given functional is to express it in terms of its kinetic-energy density.

T [ρ] =

t (ρ( r )) d r

(18)

The KS functional as it is expressed in Eq. (3) uniquely defines its kinetic-energy density. Certainly, if a given functional can reproduce the KS kinetic-energy density faithfully it must reproduce the total energy also. Any functional that differs from that one by a function that integrates to 0 over the entire system – like, for instance, the Laplacian of the density – will match the KS energy just as well but not the KS kinetic-energy density. For the VW functional, for instance, the corresponding kinetic-energy density should include a Laplacian contribution:

TVW =

tVW (ρ) d r

(19)

|∇ρ( 1 r )|2 tVW (ρ) = − ∇ 2 ρ( r) + 4 8ρ( r)

(20)

OF-DFT has experienced its most encouraging successes for periodic systems using a different class of kinetic energy functionals described below. These achievements led to attempts to use this alternative class of functionals for nonperiodic systems as well. Choly and Kaxiras recently proposed a method to approximate such functionals and adapt them for nonperiodic systems [11]. If successful, their method may further enlarge the range of applications where currently available functionals yield physically reasonable results.

3.

KEDFs for Periodic Systems

If the system exhibits translational invariance, or can be approximated using a system that does, it becomes advantageous to introduce periodic boundary conditions and thus reduce the size of the infinite system to a small number of atoms in a finite volume. A plane-wave basis set expansion most naturally describes the electron density under these conditions. As an additional advantage, quantities can be computed either in real or reciprocal space, by performing fast Fourier transforms (FFTs) on the density represented on a uniform grid. The number of functions necessary to describe the electron density in a given system is highly dependent upon the rate of fluctuation of said density. Quickly varying densities need more plane waves in real space which translate into larger reciprocal-space grids and, consequently, into finer realspace meshes. Unfortunately, in real systems, electrons tend to stay mostly


143

around atomic nuclei and only occasionally venture in the interatomic regions of space. This makes the total electron density vary extremely rapidly close to the nuclei, in the core region of space. Consequently, an extremely large number of plane waves would be necessary to describe the total electron density. One can get around this problem by realizing that the core region density is often practically invariant upon physical and chemical change. This observation is similar to the realization that only valence shell electrons are involved in chemical bonding. The valence electron density varies a lot less rapidly than the total density, so that if the core electrons could be removed, one could drastically reduce the total number of plane waves required in the basis set. Of course, the influence of the core electrons on the geometry and energy of the system must still be accounted for. This is done by introducing pseudopotentials that mimic the presence of core electrons and the nuclei. Obviously, if one is interested in any properties that require an accurate description of the electron density near the nuclei of a system, such pseudopotential-based methods will be inappropriate. Each chemical element present in the system must be represented by its own unique pseudopotential, which is typically constructed as follows. First, an all-electron calculation on an atom is performed to obtain the valence eigenvalues and wavefunctions that one seeks to reproduce within a pseudopotential calculation. Then, the oscillations of the valence wavefunction in the core region are smoothed out to create a “pseudowavefunction,” which is then used to invert the KS equations for the atom to obtain the pseudopotential that corresponds to the pseudowavefunction, subject to the constraint that the allelectron eigenvalues are reproduced. Typically, this is done for each angular momentum channel, so that one obtains a pseudopotential that has an angular dependence, usually expressed as projection operators involving the atomic pseudowavefunctions. Such a pseudopotential is referred to as “nonlocal,” because it is not simply a function of the distance from the nucleus, but also depends on the angular nature of the wavefunction it acts upon. In other words, when a nonlocal pseudopotential acts on a wavefunction, s-symmetry orbitals will be subject to a different potential than p-symmetry orbitals, etc. (as in the exact solution to the Schroedinger equation for a one-electron atom or ion). This affords a nonlocal pseudopotential enough flexibility so that it is quite accurate and transferable to a diverse set of environments. The above discussion presents a second significant challenge for OF-DFT beyond kinetic energy density functionals, since nonlocal pseudopotentials cannot be employed in OF-DFT, because no wavefunction exists to be acted upon by the orbital-based projection operators intrinsic to nonlocal pseudopotentials. In the case of an orbital-free description of the density, the pseudopotentials must be local (depending only on one point in space) and spherically symmetrical around the atomic nucleus. Thus, in OF-DFT, the challenge is to

144


construct accurate and transferable local pseudopotentials for each element. An attempt in this direction specifically for OF-DFT was made by Madden and coworkers, where the OF-DFT equation δ E xc δJ δTKS + Vext + + =µ δρ δρ δρ

(21)

is inverted to find a local pseudopotential (the second term on the left-hand side of Eq. (21)) that reproduces a crystalline density derived from a KS calculation using a nonlocal pseudopotential [12]. Here the terms on the left-hand side of Eq. (21) are the density functional variations of the same terms given in Eq. (2), except that in OF-DFT, TKS will be a functional of the density only and not of the orbitals. On the right-hand side is µ, the chemical potential. This method yielded promising results for alkali and alkaline earth metals, but was not extended beyond such elements because inherent to the method was the assumption and use of a given approximate kinetic energy density functional. Hence the pseudopotential had built into it the success and/or failure associated with any given choice of kinetic energy functional. A related approach for constructing local pseudopotentials based on embedding an ion in an electron gas was proposed by Anta and Madden; this method yielded improved results for liquid Li, for example [13]. More recently, Zhou et al. proposed that improved local pseudopotentials for condensed matter could be obtained by inverting not the OF-DFT equations but instead the KS equations so that the exact kinetic energy could be used in the inversion procedure. This was done subject to the constraint of reproducing accurate crystalline electron densities, using a modified version of the method developed by Wang and Parr for the inversion procedure [14]. Zhou et al. showed that a local pseudopotential could be constructed in this way that, e.g., for silicon, yielded bulk properties for both semiconducting and metallic phases in excellent agreement with predictions by a nonlocal pseudopotential within the KS theory. This bulk-derived local pseudopotential also exhibited improved transferability over those derived from a single atomic density. In principle, Zhou et al.’s approach is a general scheme applicable to all elements, since the exact kinetic energy is utilized [15]. With local pseudopotentials now in hand, we turn our attention back to calculating accurate valence electron densities via kinetic-energy density functionals within OF-DFT. The valence electron density in condensed matter can be viewed as fluctuating around an average value that corresponds to the total number of electrons spread homogeneously over the system. If this were exactly the case, we would have a uniform electron gas for which the kinetic energy is described exactly by the TF functional in Eq. (4) with a constant density. For an inhomogeneous density, the TF functional still constitutes an


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appropriate starting point and is the zeroth order term of the conventional gradient expansion (CGE) [16]. TKS [ρ] = TTF [ρ] + T 2 [ρ] + T 4 [ρ] + T 6 [ρ] + · · ·

(22)

Here, T 2, T 4, and T 6 correspond to the second-, fourth-, and sixth-order corrections, respectively. All odd-order corrections are zero. The second-order correction is found to be one ninth of the VW kinetic energy, while the fourthorder term is [17]: 1 T [ρ] = 540(3π 2 )2/3

4

ρ

1/3

(∇ 2 ρ)2 9∇ 2 ρ(∇ρ)2 (∇ρ)4 − + d r (23) ρ2 8ρ 3 3ρ 4

Starting with the sixth-order term, all further corrections diverge for quickly varying or exponentially decaying densities [18]. Moreover, the fourth-order correction constitutes only a minor improvement over the second-order term and its potential δT 4 [ρ]/δρ also diverges for quickly varying or exponentially decaying densities. Usually then, the CGE expansion is truncated at second order as TCGE [ρ] = TTF [ρ] + 19 TVW [ρ]

(24)

For slowly varying densities, this truncation is reasonable. For the nearly-free electron gas, linear response theory can provide an additional constraint on the kinetic-energy functional [19]. 



1 δ 2 T [ρ]

 =− = Fˆ 

2

δρ χLind ρ 0

−1

1 1 − η2

1 + η

+ ln

2 4η 1 − η

(25)

Here Fˆ denotes the Fourier transform, δ the functional derivative evaluated at a reference density ρ0 , and χLind is the Lindhard susceptibility function, the expression for which is detailed on the right-hand side, where η = q/2kF , q is the reciprocal space wave vector and kF = (3π 2 ρ0 )1/3 . Although the exact susceptibility is known in this case, the actual kinetic-energy functional is not. Its behavior at the small and large q limits can be evaluated, however. The exact linear response matches the CGE only for very slowly varying densities, which correspond to small values of q. 







δ 2 (TTF [ρ] + 19 TVW [ρ])

δ 2 T [ρ]

 ˆ = Lim F Lim Fˆ 



η→0 η→0 δρ 2 ρ δρ 2 ρ 0

(26)

0

In the limit of infinitely quickly varying densities or the large q limit (LQL), the linear response behavior is very different. 







δ 2 (− 35 TTF [ρ] + TVW [ρ])

δ 2 T [ρ]

 ˆ = Lim F Lim Fˆ 

 (27) η→+∞ η→+∞

δρ 2 ρ δρ 2 ρ 0

0

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As we saw before though, the VW kinetic energy constitutes a lower bound to the kinetic energy. Therefore, here the linear response behavior cannot be correct (we are far from the small perturbations away from the uniform gas limit required in linear response theory) and we can conclude that linear response theory inadequately describes quickly varying densities. Nevertheless, a lot of effort has been made to determine the corresponding kineticenergy functional. Bridging the gap between the small and large q to obtain the linear response kinetic-energy functional involves explicitly enforcing the correct linear response behavior. Pioneering work in this direction by Wang and Teter [20], Perrot [21], and Smargiassi and Madden [22] produced impressive results for many main group metals. A correction term is added to the TF and VW functionals to enforce the linear response. T [ρ] = TTF [ρ] + TVW [ρ] + TX [ρ]

(28)

Here TX is the correction, usually a nonlocal functional of the density that can be expressed as a double integral

TX [ρ] =

ρ α ( r)

w( r − r )ρ β ( r ) d r d r

(29)

where w is called the response kernel and is adjusted to produce the global linear response behavior, while α and β are functional-dependent parameters. More complex functionals, based either on higher-order response theories [23], for instance) or on density-dependent kernels (like those of Chacón and coworkers [24] or Wang et al. [25] can produce more general and transferable results. However, their excellent performance comes with increased computational costs and, in the case of the Chacón functional, with quadratic scaling of the computational time with system size. Nevertheless, computations using these functionals are several orders of magnitude faster than those using the KS kinetic energy. For example, Jesson and Madden performed DFT molecular dynamics simulations of solid and liquid aluminum using the Foley and Madden KEDF, on systems four times larger and for simulation times twice as long [26] as previous KS molecular dynamics studies [27] could consider. Although the melting temperature they predicted was much lower than the experimental value and previous predictions, it appears that their pseudopotential, not their KEDF, was the main source of error. It is important to emphasize that even the best of today’s functionals do not exactly match the accuracy of the KS method, exhibiting non-negligible deviations from the KS densities and energies in many cases. This should spur further developments of kinetic-energy density functionals.


4.

147

Conclusions and Outlook

Despite more than seventy years of research in this field and some tremendous progress, kinetic-energy density functionals have not yet reached a degree of sophistication that allow their use reliably and transferably for all elements in the periodic table and for all phases of matter. One could easily view the development of accurate descriptions of the kinetic energy in terms of the density alone as the last great frontier of density functional theory. Currently, OF-DFT research is moving from the development of new, approximate functionals to attempting to determine the properties of the exact one [28]. Also, it is becoming clearer that reproducing the KS energy for a given system is not a guarantee of functional accuracy. More efforts have been devoted to trying to reproduce the kinetic energy density predicted by the KS method at every point in space [29]; one can expect this type of effort to intensify in the future. If highly accurate and general forms for the kinetic-energy density functional are discovered, which retain the linear scaling efficiency of current functionals, OF-DFT will undoubtedly become the quantum-based method of choice for investigating wavefunctionindependent properties of large numbers of atoms. Aside from spectroscopic quantities, most properties of interest (e.g., vibrations, forces, dynamical evolution, structure, etc.) do not depend on knowledge of the electronic wavefunction and hence OF-DFT can be employed. For further reading about advanced technical details in kinetic-energy density functional theory, see Wang and Carter [30].

References [1] P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev., 136, B864– B871, 1964. [2] W. Kohn and L.J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev., 140, A1133–A1138, 1965. [3] S. Goedecker, “Linear scaling electronic structure models,” Rev. Mod. Phys., 71(4), 1085–1123, 1999. [4] E. Fermi, “Un metodo statistice per la determinazione di alcune proprieta dell’atomo,” Rend. Accad., Lincei 6, 602–607, 1927. [5] L.H. Thomas, “The calculation of atomic fields,” Proc. Camb. Phil. Soc., 23, 542– 548, 1927. [6] C.C.J. Roothaan, “New developments in molecular orbital theory,” Rev. Mod. Phys., 23, 69–89, 1951. [7] C.F. von Weizsäcker, “Zur Theorie der Kernmassen,” Z. Phys, 96, 431–458, 1935. [8] P.K. Acharya, L.J. Bartolotti, S.B. Sears, and R.G. Parr, “An atomic kinetic energy functional with full Weizsacker correction,” Proc. Natl. Acad. Sci. USA, 77, 6978– 6982, 1980. [9] P. García-González, J.E. Alvarellos, and E. Chacón, “Kinetic-energy density functional: atoms and shell structure,” Phys. Rev. A, 54, 1897–1905, 1996.

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V. Lignères and E.A. Carter [10] T. Wesolowski and A. Warshel, “Ab initio free-energy perturbation calculations of solvation free-energy using the frozen density-functional approach,” J. Phys. Chem., 98, 5183–5187, 1994. [11] N. Choly and E. Kaxiras, “Kinetic evergy density functionals for non-periodic systems,” Solid State Commun., 121, 281–286, 2002. [12] S. Watson, B.J. Jesson, E.A. Carter, and P. A. Madden, “Ab initio pseudopotentials for orbital-free density functionals,” Europhys. Lett., 41, 37–42, 1998. [13] J.A. Anta and P.A. Madden, “Structure and dynamics of liquid lithium: comparison of ab initio molecular dynamics predictions with scattering experiments,” J. Phys. Condens. Matter, 11, 6099–6111, 1999. [14] Y. Wang and R.G. Parr, “Construction of exact Kohn–Sham orbitals from a given electron density,” Phys. Rev. A, 47, R1591–R1593, 1993. [15] B. Zhou, Y.A. Wang, and E.A. Carter, “Transferable local pseudopotentials derived via inversion of the Kohn–Sham equations in a bulk environment,” Phys. Rev. B, 69, 125109, 2004. [16] D.A. Kirzhnits, “Quantum corrections to the Thomas–Fermi equation,” Sov. Phys. – JETP, 5, 64–71, 1957. [17] C.H. Hodges, “Quantum corrections to the Thomas–Fermi approximation – the Kirzhnits method,” Can. J. Phys., 51, 1428–1437, 1973. [18] D.R. Murphy, “The sixth-order term of the gradient expansion of the kinetic energy density functional,” Phys. Rev. A, 24, 1682–1688, 1981. [19] J. Lindhard. K. Dan. Vidensk. Selsk. Mat. Fys. Medd., 28, 8, 1954. [20] L.-W. Wang and M.P. Teter, “Kinetic-energy functional of the electron density,” Phys. Rev. B, 45, 13196–13220, 1992. [21] F. Perrot, “Hydrogen–hydrogen interaction in an electron gas,” J. Phys. Condens. Matter, 6, 431–446, 1994. [22] E. Smargiassi and P.A. Madden, “Orbital-free kinetic-energy functionals for firstprinciples molecular dynamics,” Phys. Rev. B, 49, 5220–5226, 1994. [23] M. Foley and P.A. Madden, “Further orbital-free kinetic-energy functionals for ab initio molecular dynamics,” Phys. Rev. B, 53, 10589–10598, 1996. [24] P. García-González, J.E. Alvarellos, and E. Chacón, “Nonlocal symmetrized kineticenergy density functional: application to simple surfaces,” Phys. Rev. B, 57, 4857– 4862, 1998. [25] Y.A. Wang, N. Govind, and E.A. Carter, “Orbital-free kinetic-energy density functionals with a density-dependent kernel,” Phys. Rev. B, 60, 16350–16358, 1999. [26] B.J. Jesson and P.A. Madden, “Ab initio determination of the melting point of aluminum by thermodynamic integration,” J. Chem. Phys., 113, 5924–5934, 2000. [27] G.A. de Wijs, G. Kresse, and M.J. Gillan, “First-order phase transitions by firstprinciples free-energy calculations: the melting of Al.,” Phys. Rev. B, 57, 8223–8234, 1998. ´ Nagy, “A method to get an analytical expression for the non-interacting [28] T. Gál and A. kinetic energy density functional,” J. Mol. Struct., 501–502, 167–171, 2000. [29] E. Sim, J. Larkin, and K. Burke, “Testing the kinetic energy functional: kinetic energy density as a density functional,” J. Chem. Phys., 118, 8140–8148, 2003. [30] Y.A. Wang and E.A. Carter, “Orbital-free kinetic energy density functional theory,” In: S.D. Schwartz (ed.), Theoretical Methods in Condensed Phase Chemistry, Kluwer, Dordrecht, pp. 117–184, 2000.

1.9 AB INITIO ATOMISTIC THERMODYNAMICS AND STATISTICAL MECHANICS OF SURFACE PROPERTIES AND FUNCTIONS Karsten Reuter1 , Catherine Stampfl1,2, and Matthias Scheffler1 1 Fritz-Haber-Institut der Max-Planck-Gesellschaft, Faradayweg 4-6, D-14195 Berlin, Germany 2 School of Physics, The University of Sydney, Sydney 2006, Australia

Previous and present “academic” research aiming at atomic scale understanding is mainly concerned with the study of individual molecular processes possibly underlying materials science applications. In investigations of crystal growth one would, for example, study the diffusion of adsorbed atoms at surfaces, and in the field of heterogeneous catalysis it is the reaction path of adsorbed species that is analyzed. Appealing properties of an individual process are then frequently discussed in terms of their direct importance for the envisioned material function, or reciprocally, the function of materials is often believed to be understandable by essentially one prominent elementary process only. What is often overlooked in this approach is that in macroscopic systems of technological relevance typically a large number of distinct atomic scale processes take place. Which of them are decisive for observable system properties and functions is then not only determined by the detailed individual properties of each process alone, but in many, if not most cases, also the interplay of all processes, i.e., how they act together, plays a crucial role. For a predictive materials science modeling with microscopic understanding, a description that treats the statistical interplay of a large number of microscopically well-described elementary processes must therefore be applied. Modern electronic structure theory methods such as density-functional theory (DFT) have become a standard tool for the accurate description of the individual atomic and molecular processes. In what follows we discuss the present status of emerging methodologies that attempt to achieve a (hopefully seamless) match of DFT with concepts from statistical mechanics or thermodynamics, in order to also address the interplay of the various molecular processes. The 149 S. Yip (ed.), Handbook of Materials Modeling, 149–194. c 2005 Springer. Printed in the Netherlands.

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new quality of, and the novel insights that can be gained by, such techniques is illustrated by how they allow the description of crystal surfaces in contact with realistic gas-phase environments, which is of critical importance for the manufacture and performance of advanced materials such as electronic, magnetic and optical devices, sensors, lubricants, catalysts, and hard coatings. For obtaining an understanding, and for the design, advancement or refinement of modern technology that controls many (most) aspects of our life, a large range of time and length scales needs to be described, namely, from the electronic (or microscopic/atomistic) to the macroscopic, as illustrated in Fig. 1. Obviously, this calls for a multiscale modeling, were corresponding theories (i.e., from the electronic, mesoscopic, and macroscopic regimes) and their results need to be linked appropriately. For each length and time scale regime alone, a number of methodologies are well established. It is however, the appropriate linking of the methodologies that is only now evolving. Conceptually quite challenging in this hierarchy of scales are the transitions from what is often called a micro- to a mesoscopic system description, and from a meso- to a macroscopic system description. Due to the rapidly increasing number of particles and possible processes, the former transition is methodologically primarily characterized by the rapidly increasing importance of statistics, while in the latter, the atomic substructure is finally discarded in favor of a

Statistical Mechanics or Thermodynamics

length (m) 1 10

-3

10

-6

10

-9

macroscopic regime

Density Functional Theory mesoscopic regime electronic regime

time (s) 10

-15

10

-9

10

-3

1

Figure 1. Schematic presentation of the time and length scales relevant for most material science applications. The elementary molecular processes, which rule the behavior of a system, take place in the so-called “electronic regime”. Their interplay, which frequently determines the functionalities however, only develops after meso- and macroscopic lengths or times.

Ab initio atomistic thermodynamics and statistical mechanics

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continuum modeling. In this contribution we will concentrate on the micro- to mesoscopic system transition, and correspondingly discuss some possibilities of how atomistic electronic structure theory can be linked with concepts and techniques from statistical mechanics and thermodynamics. Our aim is a materials science modeling that is based on understanding, predictive, and applicable to a wide range of realistic conditions (e.g., realistic environmental situations of varying temperatures and pressures). This then mostly excludes the use of empirical or fitted parameters – both at the electronic and at the mesoscopic level, as well as in the matching procedure itself. Electronic theories that do not rely on such parameters are often referred to as first-principles (or in latin: ab initio) techniques, and we will maintain this classification also for the linked electronic-statistical methods. Correspondingly, our discussion will mainly (nearly exclusively) focus on such ab initio studies, although mentioning some other work dealing with important (general) concepts. Furthermore, this chapter does not (or only briefly) discuss equations; instead the concepts are demonstrated (and illustrated) by selected, typical examples. Since many (possibly most) aspects of modern material science deal with surface or interface phenomena, the examples are from this area, addressing in particular surfaces of semiconductors, metals, and metal oxides. Apart from sketching the present status and achievements, we also find it important to mention the difficulties and problems (or open challenges) of the discussed approaches. This can however only be done in a qualitative and rough manner, since the problems lie mostly in the details, the explanations of which are not appropriate for such a chapter. To understand the elementary processes ruling the materials science context, microscopic theories need to address the behavior of electrons and the resulting interactions between atoms and molecules (often expressed in the terminology of chemical bonds). Electrons move and adjust to perturbations on a time scale of femtoseconds (1 fs = 10−15 s), atoms vibrate on a time scale of picoseconds (1 ps = 10−12 s), and individual molecular processes take place on a length scale of 0.1 nanometer (1 nm = 10−9 m). Because of the central importance of the electronic interactions, this time and length scale regime is also often called the “electronic regime”, and we will use this term here in particular, in order to emphasize the difference between ab initio electronic and semi-empirical microscopic theories. The former explicitly treat the electronic degrees of freedom, while the latter already coarse-grain over them and directly describe the atomic scale interactions by means of interatomic potentials. Many materials science applications depend sensitively on intricate details of bond breaking and making, which on the other hand are often not well (if at all) captured by existing semi-empiric classical potential schemes. A predictive first-principles modeling as outlined above must therefore be based on a proper description of molecular processes in the “electronic regime”, which is much harder to accomplish than just a microscopic description employing more or

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less guessed potentials. In this respect we find it also appropriate to distinguish the electronic regime from the currently frequently cited “nanophysics” (or better “nanometer-scale physics”). The latter deals with structures or objects of which at least one dimension is in the range 1–100 nm, and which due to this confinement exhibit properties that are not simply scalable from the ones of larger systems. Although already quite involved, the detailed understanding of individual molecular processes arising from electronic structure theories is, however, often still not enough. As mentioned above, in many cases the system functionalities are determined by the concerted interplay of many elementary processes, not only by the detailed individual properties of each process alone. It can, for example, very well be that an individual process exhibits very appealing properties for a desired application, yet the process may still be irrelevant in practice, because it hardly ever occurs within the “full concert” of all possible molecular processes. Evaluating this “concert” of elementary processes one obviously has to go beyond separate studies of each microscopic process. However, taking the interplay into account, naturally requires the treatment of larger system sizes, as well as an averaging over much longer time scales. The latter point is especially pronounced, since many elementary processes in materials science are activated (i.e., an energy barrier must be overcome) and thus rare. This means that the time between consecutive events can be orders of magnitude longer than the actual event time itself. Instead of the above mentioned electronic time regime, it may therefore be necessary to follow the time evolution of the system up to seconds and longer in order to arrive at meaningful conclusions concerning the effect of the statistical interplay. Apart from the system size, there is thus possibly the need to bridge some twelve orders of magnitude in time which puts new demands on theories that are to operate in the corresponding mesoscopic regime. And also at this level, the ab initio approach is much more involved than an empirical one because it is not possible to simply “lump together” several not further specified processes into one effective parameter. Each individual elementary step must be treated separately, and then combined with all the others within an appropriate framework. Methodologically, the physics in the electronic regime is best described by electronic structure theories, among which density-functional theory [1–4] has become one of the most successful and widespread approaches. Apart from detailed information about the electronic structure itself, the typical output of such DFT calculations, that is of relevance for the present discussion, is the energetics, e.g., total energies, as well as the forces acting on the nuclei for a given atomic configuration. If this energetic information is provided as function of the atomic configuration {R I }, one talks about a potential energy surface (PES) E({R I }). Obviously, a (meta)stable atomic configuration corresponds to a (local) minimum of the PES. The forces acting on the given atomic configuration are just the local gradient of the PES, and the vibrational


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modes of a (local) minimum are given by the local PES curvature around it. Although DFT mostly does not meet the frequent demand for “chemical accuracy” (1 kcal/mol ≈ 0.04 eV/atom) in the energetics, it is still often sufficiently accurate to allow for the aspired modeling with predictive character. In fact, we will see throughout this chapter that error cancellation at the statistical interplay level may give DFT-based approaches a much higher accuracy than may be expected on the basis of the PES alone. With the computed DFT forces it is possible to directly follow the motion of the atoms according to Newton’s laws [5, 6]. With the resulting ab initio molecular dynamics (MD) [7–11] only time scales up to the order of 50 ps are, however, currently accessible. Longer times may, e.g., be reached by so-called accelerated MD techniques [12], but for the desired description of a truly mesoscopic scale system which treats the statistical interplay of a large number of elementary processes over some seconds or longer, a match or combination of DFT with concepts from statistical mechanics or thermodynamics must be found. In the latter approaches, bridging of the time scale is achieved by either a suitable “coarse-graining” in time (to be specified below) or by only considering thermodynamically stable (or metastable) states. We will discuss how such a description, appropriate for a mesoscopic-scale system, can be achieved starting from electronic structure theory, as well as ensuing concepts like atomistic thermodynamics, lattice-gas Hamiltonians (LGH), equilibrium Monte Carlo simulations, or kinetic Monte Carlo simulations (kMC). Which of these approaches (or a combination) is most suitable depends on the particular type of problem. Table 1 lists the different theoretical approaches and the time and length scales that they treat. While the concepts are general, we find it instructive to illustrate their power and limitations on the basis of a particular issue that is central to the field of surface-related studies including applications as important as crystal growth and heterogeneous catalysis, namely to treat the effect of a finite gas-phase. With surfaces forming the interface to the surrounding environment, a critical dependence of their

Table 1. The time and length scales typically handled by different theoretical approaches to study chemical reactions and crystal growth Information

Time scale

Length scale

< 103 atoms Density-functional theory Microscopic – ∼ < 103 atoms Ab initio molecular dynamics Microscopic t< ∼ ∼ 50 ps < 103 atoms Semi-empirical molecular dynamics Microscopic t< ∼ ∼ 1 ns < < Kinetic Monte Carlo simulations Micro- to mesoscopic 1 ps < ∼ t ∼ 1 h ∼ 1 µm > 10 nm Ab initio atomistic thermodynamics Meso- to macroscopic Averaged ∼ > < < Rate equations Averaged 0.1 s ∼ t ∼ ∞ ∼ 10 nm < > 10 nm Continuum equations Macroscopic 1s < ∼t ∼∞ ∼

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properties on the species in this gas-phase, on their partial pressures and on the temperature can be intuitively expected [13, 14]. After all, we recall that for example in our oxygen-rich atmosphere, each atomic site of a close-packed crystal surface at room temperature is hit by of the order of 109 O2 molecules per second. That this may have profound consequences on the surface structure and composition is already highlighted by the everyday phenomena of oxide formation, and in humid oxygen-rich environments, eventually corrosion with rust and verdigris as two visible examples [15]. In fact, what is typically called a stable surface structure is nothing but the statistical average over all elementary adsorption processes from, and desorption processes to, the surrounding gas-phase. If atoms or molecules of a given species adsorb more frequently from the gas-phase than they desorb to it, the species’ concentration in the surface structure will be enriched with time, thus also increasing the total number of desorption processes. Eventually this total number of desorption processes will (averaged over time) equal the number of adsorption processes. Then the (average) surface composition and structure will remain constant, and the surface has attained its thermodynamic equilibrium with the surrounding environment. Within this context we may be interested in different aspects; for example, on the microscopic level, the first goal would be to separately study elementary processes such as adsorption and desorption in detail. With DFT one could, e.g., address the energetics of the binding of the gas-phase species to the surface in a variety of atomic configurations [16], and MD simulations could shed light on the possibly intricate gas-surface dynamics during one individual adsorption process [10, 11, 17]. Already the search for the most stable surface structure under given gas-phase conditions, however, requires the consideration of the interplay between the elementary processes (of at least adsorption and desorption) at the mesoscopic scale. If we are only interested in the equilibrated system, i.e., when the system has reached its thermodynamic ground (or a metastable) state, the natural choice would then be to combine DFT data with thermodynamic concepts. How this can be done will be exemplified in the first part of this chapter. On the other hand, the processes altering the surface geometry and composition from a known initial state to the final ground state can be very slow. And coming back to the above example of oxygen–metal interaction, corrosion is a prime example, where such a kinetic hindrance significantly slows down (and practically stops) further oxidation after an oxide film of certain thickness has formed at the surface. In such circumstances, a thermodynamic description will not be satisfactory and one would want to follow the explicit kinetics of the surface in the given gas-phase. Then the combination of DFT with concepts from statistical mechanics explicitly treating the kinetics is required, and we will illustrate some corresponding attempts in the last section entitled “First-principles kinetic Monte Carlo simulations”.


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Ab Initio Atomistic Thermodynamics

First, let us discuss the matching of electronic structure theory data with thermodynamics. Although this approach applies “only” to systems in equilibrium (or in a metastable state), we note that at least, at not too low temperatures, a surface is likely to rapidly attain thermodynamic equilibrium with the ambient atmosphere. And even if it has not yet equilibrated, at some later stage it will have and we can nevertheless learn something by knowing about this final state. Thermodynamic considerations also have the virtue of requiring comparably less microscopic information, typically only about the minima of the PES and the local curvatures around them. As such, it is often advantageous to first resort to a thermodynamic description, before embarking upon the more demanding kinetic modeling described in the last section. The goal of the thermodynamic approach is to use the data from electronic structure theory, i.e., the information on the PES, to calculate appropriate thermodynamic potential functions like the Gibbs free energy G [18–21]. Once such a quantity is known, one is immediately in the position to evaluate macroscopic system properties. Of particular relevance for the spatial aspect of our multiscale endeavor is further that within a thermodynamic description larger systems may readily be divided into smaller subsystems that are mutually in equilibrium with each other. Each of the smaller and thus potentially simpler subsystems can then first be treated separately, and the contact between the subsystems is thereafter established by relating their corresponding thermodynamic potentials. Such a “divide and conquer” type of approach can be especially efficient, if infinite, but homogeneous parts of the system like bulk or surrounding gas-phase can be separated off [22–27].

1.1.

Free Energy Plots for Surface Oxide Formation

How this quite general concept works and what it can contribute in practice may be illustrated with the case of oxide formation at late transition metal (TM) surfaces sketched in Fig. 2 [28, 29]. These materials have widespread technological use, for example, in the area of oxidation catalysis [30]. Although they are likely to form oxidic structures (i.e., ordered oxygen–metal compounds) in technologically-relevant high oxygen pressure environments, it is difficult to address this issue at the atomic scale with the corresponding experimental techniques of surface science because they often require Ultra-High Vacuum (UHV) [31]. Instead of direct, so-called in situ measurements, the surfaces are usually first exposed to a defined oxygen dosage, and the produced oxygen-enriched surface structures are then cooled down and analyzed in UHV. Due to the low temperatures, it is hoped that the surfaces do not attain their equilibrium structure in UHV during the time of the measurement, and

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Figure 2. Cartoon sideviews illustrating the effect of an increasingly oxygen-rich atmosphere on a metal surface. Whereas in perfect vacuum (left) the clean surface prevails, finite O2 pressures in the environment lead to an oxygen-enrichment in the solid and its surface. Apart from some bulk dissolved oxygen, frequently observed stages in this oxidation process comprise (from left to right) on-surface adsorbed O, the formation of thin (surface) oxide films, and eventually the transformation to an ordered bulk oxide compound. Note, that all stages can be strongly kinetically-inhibited. It is, e.g., not clear whether the observation of a thin surface oxide film means that this is the stable surface composition and structure at the given gas-phase pressure and temperature, or whether the system has simply not yet attained its real equilibrium structure (possibly in form of the full bulk oxide). Such limitations can be due to quite different microscopic reasons: adsorption from or desorption to the gas-phase could be slow/hindered, or (bulk) oxide growth may be inhibited because metal diffusion through the oxide to its surface or oxygen diffusion from the surface to the oxide/metal interface is very slow.

thus provide information about the corresponding surface structure at higher oxygen pressures. This is, however, not fully certain, and it is also not guaranteed that the surface has reached its equilibrium structure during the time of oxygen exposure. Typically, a large variety of potentially kinetically-limited surface structures can be produced this way. Even though it can be academically very interesting to study all of them in detail, one would still like to have some guidance as to which of them would ultimately correspond to an equilibrium structure under which environmental conditions. Furthermore, the knowledge of a corresponding, so-called surface phase diagram as a function of, in this case, the temperature T and oxygen pressure pO2 can also provide useful information to the now surging in situ techniques, as to which phase to expect. The task for an ab initio atomistic thermodynamic approach would therefore be to screen a number of known (or possibly relevant) oxygen-containing surface structures, and evaluate which of them turns out to be the most stable one under which (T, pO2 ) conditions [24–27]. Most stable translated into the thermodynamic language meaning that the corresponding structure minimizes an appropriate thermodynamic function, which would in this case be the Gibbs free energy of adsorption G [32, 33]. In other words, one has to compute G as a function of the environmental variables for each structural model,


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and the one with the lowest G is identified as most stable. What needs to be computed are all thermodynamic potentials entering into the thermodynamic function to be minimized. In the present case of the Gibbs free energy of adsorption these are for example the Gibbs free energies of bulk and surface structural models, as well as the chemical potential of the O2 gas phase. The latter may, at the accuracy level necessary for the surface phase stability issue, well be approximated by an ideal gas. The calculation of the chemical potential µO (T, pO2 ) is then straightforward and can be found in standard statistical mechanics text books, (e.g., [34]). Required input from a microscopic theory like DFT are properties like bond lengths and vibrational frequencies of the gas-phase species. Alternatively, the chemical potential may be directly obtained from thermochemical tables [35]. Compared to this, the evaluation of the Gibbs free energies of the solid bulk and surface is more involved. While in principle contributions from total energy, vibrational free energy or configurational entropy have to be calculated [24–26], a key point to notice here is that not the absolute Gibbs free energies enter into the computation of G, but only the difference of the Gibbs free energies of bulk and surface. This often implies some error cancellation in the DFT total energies. It also leads to quite some (partial) cancellation in the free energy contributions like the vibrational energy. In a physical picture, it is thus not the effect of the absolute vibrations that matters for our considerations, but only the changes of vibrational modes at the surface as compared to the bulk. Under such circumstances it may result that the difference between the bulk and surface Gibbs free energies is already well approximated by the difference of their leading total energy terms, i.e., the direct output of the DFT calculations [24]. Although this is of course appealing from a computational point of view, and one would always want to formulate the thermodynamic equations in a way that they contain such differences, we stress that it is not a general result and needs to be carefully checked for every specific system. Once the Gibbs free energies of adsorption G(T, pO2 ) are calculated for each surface structural model, they can be plotted as a function of the environmental conditions. In fact, under the imposed equilibrium the two-dimensional dependence on T and pO2 can be summarized into a one-dimensional dependence on the gas-phase chemical potential µO (T, pO2 ) [24]. This is done in Fig. 3(a) for the Pd(100) surface including, apart from the clean surface, a number of previously characterized oxygen-containing surface structures. These are two structures with ordered on-surface√O adsorbate layers of differ√ ent density ( p(2 × 2) and c(2 × 2)), a so-called ( 5 × 5)R27◦ surface oxide containing one layer of PdO on top of Pd(100), and finally the infinitely thick PdO bulk oxide [37]. If we start at very low oxygen chemical potential, corresponding to a low oxygen concentration in the gas-phase, we expectedly find the clean Pd(100) surface to yield the lowest G line, which in fact is used here as the reference zero. Upon increasing µO in the gas-phase,

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the Gibbs free energies of adsorption of the other oxygen-containing surfaces decrease gradually, however, as it becomes more favorable to stabilize such structures with more and more oxygen atoms being present in the gas-phase. The more oxygen the structural models contain, the steeper the slope of their G curves becomes, and above a critical µO we eventually find the surface oxide to be more stable than the clean surface. Since the PdO bulk oxide contains a macroscopic (or at least mesoscopic) number of oxygen atoms, the slope of its G line exhibits an infinite slope and cuts the other lines vertically at µO ≈ − 0.8 eV. For any higher oxygen chemical potential in the gas-phase, the bulk PdO phase will then always result as most stable.


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With the clean surface, the surface and the bulk oxide, the thermodynamic analysis yields therefore three equilibrium phases for Pd(100) depending on the chemical potential of the O2 environment. Exploiting ideal gas laws, this one-dimensional dependence can be translated into the physically more intuitive dependence on temperature and oxygen pressure. For two fixed temperatures, this is also indicated by the resulting pressure scales at the top axis of Fig. 3(a). Alternatively, the stability range of the three phases can be directly plotted in (T, pO2 )-space, as shown Fig. 3(b). A most intriguing result is that the thermodynamic stability range of the recently identified surface oxide extends well beyond the one of the common PdO bulk oxide, i.e., the surface oxide could well be present under environmental conditions where the PdO bulk oxide is known to be unstable. This result is somewhat unexpected, in two ways: First, it had hitherto been believed that it is the slow growth kinetics (not the thermodynamics) that exclusively controls the thickness of oxide films at surfaces. Second, the possibility of only few atomic layer thick (surface) oxides with structures not necessarily related to the known bulk oxides was traditionally not perceived. √ √ The additional stabilization of the ( 5 × 5)R27◦ surface oxide is attributed to the strong coupling of the ultrathin film to the Pd(100) substrate [37]. Similar findings have recently been obtained at the Pd(111) [28, 38] and Ag(111) [33, 39] surfaces. Interestingly, the low stability of the bulk oxide phases of these more noble TMs had hitherto often been used as argument against the relevance of oxide formation in technological environments like in oxidation catalysis [30]. It remains to be seen whether the surface oxide phases and their extended stability range, which have recently been intensively discussed, will change this common perception.

1.2.

Free Energy Plots of Semiconductor Surfaces

Already in the introduction we had mentioned that the concepts discussed here are general and applicable to a wide range of problems. To illustrate this, we supplement the discussion by an example from the field of semiconductors, where the concepts of ab initio atomistic thermodynamics had in fact been developed first [18–21, 40]. Semiconductor surfaces exhibit complex reconstructions, i.e., surface structures that differ significantly in their atomic composition and geometry from the one of the bulk-truncated structure [13]. Knowledge of the correct surface atomic structure is, on the other hand, a prerequisite to understand and control the surface or interface electronic properties, as well as the detailed growth characteristics. While the number of possible configurations with complex surface unit-cell reconstructions is already large, searching for possible structural models becomes even more involved for surfaces of compound semiconductors. In order to minimize the number

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of dangling bonds, the surface may exchange atoms with the surrounding gasphase, which in molecular beam epitaxy (MBE) growth is composed of the substrate species at elevated temperatures and varying partial pressures. As a consequence of the interaction with this gas-phase, the surface stoichiometry may be altered and surface atoms be displaced to assume a more favorable bonding geometry. The resulting surface structure depends thus on the environment, and atomistic thermodynamics may again be employed to compare the stability of existing (or newly suggested) structural models as a function of the conditions in the surrounding gas-phase. The thermodynamic quantity that is minimized by the most stable structure is in this case the surface free energy, which in turn depends on the Gibbs free energies of the bulk and surface of the compound, as well as on the chemical potentials in the gasphase. The procedure of evaluating these quantities goes exactly along the lines described above, where in addition, one frequently assumes the surface fringe not only to be in thermodynamic equilibrium with the surrounding gasphase, but also with the underlying compound bulk [24]. With this additional constraint, the dependence of the surface structure and composition on the environment can, even for the two component gas-phase in MBE, be discussed as a function of the chemical potential of only one of the compound species alone. Figure 4 shows as an example the dependence on the As content in the gas-phase for a number of surface structural models of the GaAs(001)

Figure 4. Surface energies for GaAs(001) terminations as a function of the As chemical potential, µAs . The thermodynamically allowed range of µAs is bounded by the formation of Ga droplets at the surface (As-poor limit at −0.58 eV) and the condensation of arsenic at the surface (As-rich limit at 0.00 eV). The ζ (4 × 2) geometry is significantly lower in energy than the previously proposed β2(4 × 2) model for the c(8 × 2) surface reconstruction observed under As-poor growth conditions (from Ref. [41]).


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surface. A reasonable lower limit for this content is given, when there is so little As2 in the gas-phase that it becomes thermodynamically more favorable for the arsenic to leave the compound. The resulting GaAs decomposition and formation of Ga droplets at the surface denotes the lower limit of As chemical potentials considered (As-poor limit), while the condensation of arsenic on the surface forms an appropriate upper bound (As-rich limit). Depending on the As to Ga stoichiometry at the surface, the surface free energies of the individual models have either a positive slope (As-poor terminations), a negative slope (As-rich terminations) or remain constant (stoichiometric termination). While the detailed atomic geometries behind the considered models in Fig. 4 are not relevant here, most of them may roughly be characterized as different ways of forming dimers at the surface in order to reduce the number of dangling orbitals [42]. In fact, it is this general “rule” of dangling bond minimization by dimer formation that has hitherto mainly served as inspiration in the creation of new structural models for the (001) surfaces of III–V zinc-blende semiconductors, thereby leading to some prejudice in the type of structures considered. In contrast, at first the theoretically proposed so-called ζ(4 × 2) structure is actuated by the filling of all As dangling orbitals and emptying of all Ga dangling orbitals, as well as a favorable electrostatic (Ewald) interaction between the surface atoms [41]. The virtue of the atomistic thermodynamic approach is now that such a new structural model can be directly compared in its stability against all existing ones. And indeed, the ζ(4 × 2) phase was found to be more stable than all previously proposed reconstructions at low As pressure. Returning to the methodological discussion, the results shown in Figs. 3 and 4 nicely summarize the contribution that can be made by such analysis. While ab initio atomistic thermodynamics has a much wider applicability (see Sections 1.3–1.5), the approach followed for obtaining Figs. 3 and 4 has some limitations. Most prominently, one has to be aware that the reliability is restricted to the number of considered configurations, or in other words that only the stability of those structures plugged in can be compared. Had, for example, the surface oxide structure not been considered in Fig. 3, the p(2×2) adlayer structure would have yielded the lowest Gibbs free energy of adsorption in a range of µO intermediate to the stability ranges of the clean surface and the bulk oxide, changing the resulting surface phase diagram √ √ accordingly. Alternatively, it is at present not completely clear, whether the ( 5× 5)R27◦ structure is really the only surface oxide on Pd(100). If another yet unknown surface oxide exists and exhibits a sufficiently low G for some oxygen chemical potential, it will similarly affect the surface phase diagram, as would another novel and hitherto unconsidered surface reconstruction with sufficiently low surface free energy in the GaAs example. As such, appropriate care should be in place when addressing systems where only limited information about surface structures is available. With this in mind, even in such systems the

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atomistic thermodynamics approach can still be a particularly valuable tool though, since it allows, for example, to rapidly compare the stability of newly devised structural models against existing ones. In this way, it gives tutorial insight into what structural motives may be particularly important. This may even yield ideas about other structures that one should test, as well, and the theoretical identification of the ζ(4 × 2) structure in Fig. 4 by Lee et al. [41] is a prominent example. In the Section 1.4 we will discuss an approach that is able to overcome this limitation. This comes unfortunately at a significantly higher computational demand, so that it has up to now only be used to study simple adsorption layers on surfaces. This will then also provide more detailed insight into the transitions between stable phases. In Figs. 3 and 4, the transitions are simply drawn abrupt, and no reference is made to the finite phase coexistence regions that should occur at finite temperatures, i.e., regions in which with changing pressure or temperature one phase gradually becomes populated and the other one depopulated. That this is not the case in the discussed examples is not a general deficiency of the approach, but has to do with that the configurational entropy contribution to the Gibbs free energy of the surface phases has been deliberately neglected in the two corresponding studies. This is justified, since for the well-ordered surface structural models considered, this contribution is indeed small and will affect only a small region close to the phase boundaries. The width of this affected phase coexistence region can even be estimated [26], but if more detailed insight into this very region is desired, or if disorder becomes more important e.g., at more elevated temperatures, then an explicit calculation of the configurational entropy contribution will become necessary. For this, equilibrium MC simulations as described below are the method of choice, but before we turn to them there is yet another twist to free energy plots that deserves mentioning.

1.3.

“Constrained Equilibrium”

Although a thermodynamic approach can strictly describe only the situation where the surface is in equilibrium with the surrounding gas-phase (or in a metastable state), the idea is that it can still give some insight when the system is close to thermodynamic equilibrium, or even when it is only close to thermodynamic equilibrium with some of the present gas-phase species [25]. For such situations it can be useful to consider “constrained equilibria,” and one would expect to get some ideas as to where in (T, p)-space thermodynamic phases may still exist, but also to identify those regions where kinetics may control the material function.


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We will discuss heterogeneous catalysis as a prominent example. Here, a constant stream of reactants is fed over the catalyst surface and the formed products are rapidly carried away. If we take the CO oxidation reaction to further specify our example, the surface would be exposed to an environment composed of O2 and CO molecules, while the produced CO2 desorbs from the catalyst surface at the technologically employed temperatures and is then transported away. Neglecting the presence of the CO2 , one could therefore model the effect of an O2 /CO gas-phase on the surface, in order to get some first ideas of the structure and composition of the catalyst under steady-state operation conditions. Under the assumption that the adsorption and desorption processes of the reactants occur much faster than the CO2 formation reaction, the latter would not significantly disturb the average surface population, i.e., the surface could be close to maintaining its equilibrium with the reactant gas-phase. If at all, this equilibrium holds, however, only with each gasphase species separately. Were the latter fully equilibrated among each other, too, only the products would be present under all environmental conditions of interest. It is in fact particularly the high free energy barrier for the direct gas-phase reaction that prevents such an equilibration on a reasonable time scale, and necessitates the use of a catalyst in the first place. The situation that is correspondingly modeled in an atomistic thermodynamics approach to heterogeneous catalysis is thus a surface in “constrained equilibrium” with independent reservoirs representing all reactant gas-phase species, namely O2 and CO in the present example [25]. It should immediately be stressed though, that such a setup should only be viewed as a thought construct to get a first idea about the catalyst surface structure in a high-pressure environment. Whereas we could write before that the surface will sooner or later necessarily equilibrate with the gas-phase in the case of a pure O2 atmosphere, this must no longer be the case for a “constrained equilibrium”. The on-going catalytic reaction at the surface consumes adsorbed reactant species, i.e., it continuously drives the surface populations away from their equilibrium value, and even more so in the interesting regions of high catalytic activity. That the “constrained equilibrium” concept can still yield valuable insight is nicely exemplified for the CO oxidation over a “Ru” catalyst [43]. For ruthenium, the afore described tendency to oxidize under oxygen-rich environmental conditions is much more pronounced than for the above discussed nobler metals Pd and Ag [28]. While for the latter the relevance of (surface) oxide formation under the conditions of technological oxidation catalysis is still under discussion [28, 33, 39, 44], it is by now established that a film of bulklike oxide forms on the Ru(0001) model catalyst during high-pressure CO oxidation, and that this RuO2 (110) is the active surface for the reaction [45]. When evaluating its surface structure in “constrained equilibrium” with an O2 and CO environment, four different “surface phases” result depending on the gas-phase conditions that are now described by the chemical potentials of both

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reactants, cf. Fig. 5. The “phases” differ from each other in the occupation of two prominent adsorption site types exhibited by this surface, called bridge (br) and coordinatively unsaturated (cus) sites. At very low µCO , i.e., a very low CO concentration in the gas-phase, either only the bridge, or bridge and cus sites are occupied by oxygen depending on the O2 pressure. Under increased CO concentration in the gas-phase, both the corresponding Obr /− and the Obr /Ocus phase have to compete with CO that would also like to adsorb at the cus sites. And eventually the Obr /COcus phase develops. Finally, under very reducing gas-phase conditions with a lot of CO and essentially no oxygen, a completely CO covered surface results (CObr /COcus). Under these conditions the RuO2 (110) surface can at best be metastable, however, as above the white-dotted line in Fig. 5 the RuO2 bulk oxide is already unstable against CO-induced decomposition. With the already described difficulty of operating the atomic-resolution experimental techniques of surface science at high pressures, the possibility of reliably bridging the so-called pressure gap is of key interest in heterogeneous catalysis research [30, 43, 46]. The hope is that the atomic-scale understanding gained in experiments with some suitably chosen low pressure conditions would also be representative of the technological ambient pressure situation. Surface phase diagrams like the one shown in Fig. 5 could give some valuable guidance in this endeavor. If the (T, pO2 , pCO ) conditions of the low pressure experiment are chosen such that they lie within the stability region of the same surface phase as at high-pressures, the same surface structure and composition will be present and scalable results may be expected. If, however, temperature and pressure are varied in such a way, that one crosses from one stability region to another one, different surfaces are exposed and there is no reason to hope for comparable functionality. This would, e.g., also hold for a naive bridging of the pressure gap by simply maintaining a constant partial pressure ratio. In fact, the comparability holds not only within the regions of the stable phases themselves, but with the same argument also for the phase coexistence regions along the phase boundaries. The extent of these configurational entropy induced phase coexistence regions has been indicated in Fig. 5 by white regions. Although as already discussed, the above mentioned approach gives no insight into the detailed surface structure under these conditions, pronounced fluctuations due to an enhanced dynamics of the involved elementary processes can generally be expected due to the vicinity of a phase transition. Since catalytic activity is based on the same dynamics, these regions are therefore likely candidates for efficient catalyst functionality [25]. And indeed, very high and comparable reaction rates have recently been noticed for different environmental conditions that all lie close to the white region between the Obr /Ocus and Obr /COcus phases. It must be stressed, however, that exactly in this region of high catalytic activity one would similarly expect the


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Figure 5. Top panel: Top view of the RuO2 (110) surface explaining the location of the two prominent adsorption sites (coordinatively unsaturated, cus, and bridge, br). Also shown are perspective views of the four stable phases present in the phase diagram shown below (Ru = light large spheres, O = dark medium spheres, C = white small spheres). Bottom panel: Surface phase diagram for RuO2 (110) in “constrained equilibrium” with an oxygen and CO environment. Depending on the gas-phase chemical potentials (µO , µCO ), br and cus sites are either occupied by O or CO, or empty (–), yielding a total of four different surface phases. For T = 300 and 600 K, this dependence is also given in the corresponding pressure scales. Regions that are expected to be particularly strongly affected by phase coexistence or kinetics are marked by white hatching (see text). Note that conditions representative for technological CO oxidation catalysis (ambient pressures, 300–600 K) fall exactly into one of these ranges (after Refs. [25, 26]).

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breakdown of the “constrained equilibrium” assumption of a negligible effect of the on-going reaction on the average surface structure and stoichiometry. At least everywhere in the corresponding hatched regions in Fig. 5 such kinetic effects will lead to significant deviations from the surface phases obtained within the approach described above, even at “infinite” times after steady-state has been reached. Atomistic thermodynamics may therefore be employed to identify interesting regions in phase space. Their surface coverage and structure, i.e., the very dynamic behavior, must then however be modeled by statistical mechanics explicitly accounting for the kinetics, and the corresponding kMC simulations will be discussed towards the end of the chapter.

1.4.

Ab Initio Lattice-gas Hamiltonian

The predictive power of the approach discussed in the previous sections extends only to the structures that are directly considered, i.e., it cannot predict the existence of unanticipated geometries or stoichiometries. To overcome this limitation, and to include a more general and systematic way of treating phase coexistence and order–disorder transitions, a proper sampling of configuration space must be achieved, instead of considering only a set of plausible structural models. Modern statistical mechanical methods like Monte Carlo (MC) simulations are particularly designed to efficiently fulfill this purpose [6, 47]. The straightforward matching with electronic structure theories would thus be to determine with DFT the energetics of all system configurations generated in the course of the statistical simulation. Unfortunately, this direct linking is currently, and also in the foreseeable future, computationally unfeasible. The exceedingly large configuration spaces of most materials science problems require a prohibitively large number of free energy evaluations (which can easily go beyond 106 for moderately complex systems), including also disordered configurations. With the direct matching impossible, an efficient alternative is to map the real system somehow onto a simpler, typically discretized model system, the Hamiltonian of which is sufficiently fast to evaluate. This then enables us to evaluate the extensive number of free energies required by the statistical mechanics. Obvious uncertainties of this approach are how appropriate the model system represents the real system, and how its parameters can be determined from the first-principles calculations. The advantage, on the other hand, is that such a detour via an appropriate (“coarse-grained”) model system often provides deeper insight and understanding of the ruling mechanisms. If the considered problem can be described by a lattice defining the possible sites for the species in the system, a prominent example for such a mapping approach is given by the concept of a LGH (or in other languages, an “Isingtype model” [48] or a “cluster-expansion” [49, 50]). Here, any system state


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is defined by the occupation of the sites in the lattice and the total energy of any configuration is expanded into a sum of discrete interactions between these lattice sites. For a one component system with only one site type, the LGH would then for example read (with obvious generalizations to multicomponent, multi-site systems): H=F

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Figure 6. (a) Illustration of some types of lateral interactions for the case of a twodimensional adsorbate layer (small dark spheres) that can occupy the two distinct threefold pair hollow sites of a (111) close-packed surface. Vm (n = 1, 2, 3) are two-body (or pair) interactions at first, second, and third nearest neighbor distances of like hollow sites (i.e., fcc–fcc or hcp–hcp). Vmtrio (n = 1, 2, 3) are the three possible three-body (or trio) interactions between pair(h,f)

three atoms in like nearest neighbor hollow sites, and Vm (n = 1, 2, 3) represent pair interactions between atoms that occupy unlike hollow sites (i.e., one in fcc and the other in hcp or vice versa). (b) Example of an adsorbate arrangement from which an expression can be obtained for use in solving for interaction parameters. The (3 × 3) periodic surface unit-cell is indicated by the large darker spheres. The arrows indicate interactions between the adatoms. Apart from the obvious first nearest-neighbor interactions (short arrows), also third nearestneighbor two-body interactions (long arrows) exist, due to the periodic images outside of the unit cell.

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adsorbate layer that can occupy the two distinct threefold hollow sites of a (111) close-packed surface. In particular, the pair interactions up to third nearest neighbor between like and unlike hollow sites are shown, as well as three possible trio interactions between adsorbates in like sites. It is apparent that such a LGH is very general. The Hamiltonian can be equally well evaluated for any lattice occupation, be it dense or sparse, periodic or disordered. And in all cases it merely comprises performing an algebraic sum over a finite number of terms, i.e., it is computationally very fast. The disadvantage is, on the other hand, that for more complex systems with multiple sites and several species, the number of interaction terms in the expansion increases rapidly. Which of these (far-reaching or multi-body) interaction terms need to be considered, i.e., where the sum in Eq. (1) may be truncated, and how the interaction energies in these terms may be determined, is the really sensitive part of such a LGH approach that must be carefully checked. The methodology in itself is not new, and traditionally the interatomic interactions have often been assumed to be just pairwise additive (i.e., higherorder terms beyond pair interactions were neglected); the interaction energies were then obtained by simply fitting to experimental data (see, e.g., [51–53]). This procedure obviously results in “effective parameters” with an unclear microscopic basis, “hiding” or “masking” the effect and possible importance of three-body (trio) and higher-order interactions. This has the consequence that while the Hamiltonian may be able to reproduce certain specific experimental data to which the parameters were fitted, it is questionable and unlikely that it will be general and transferable to calculations of other properties of the system. Indeed, the decisive contribution to the observed behavior of adparticles by higher-order, many-atom interactions has in the meanwhile been pointed out by a number of studies (see, e.g., [54–58]). As an alternative to this empirical procedure, the lateral interactions between the particles in the lattice can be deduced from detailed DFT calculations, and it is this approach in combination with the statistical mechanics methods that is of interest for this chapter. The straightforward way to do this is to directly compute these interactions as differences of calculations, with different occupations at the corresponding lattice sites. For the example of a pair interaction between two adsorbates at a surface, this would translate into two DFT calculations where only either one of the adsorbates sits at its lattice site, and one calculation where both are present simultaneously. Unfortunately, this type of approach is hard to combine with the periodic boundary conditions that are typically required to describe the electronic structure of solids and surfaces [16]. In order to avoid interactions with the periodic images of the considered lattice species, huge (actually often prohibitively large) supercells would be required. A more efficient and intelligent way of addressing the problem is instead to specifically exploit the interaction with the periodic images. For this, different configurations in various (feasible)


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supercells are computed with DFT, and the obtained energies expressed in terms of the corresponding interatomic interactions. Figure 6 illustrates this for the case of two adsorbed atoms in a laterally periodic surface unit-cell. Due to this periodicity, each atom has images in the neighboring cells. Because of these images, each of the atoms in the unit-cell experiences not only the obvious pair interaction at the first neighbor distance, but also a pair interaction at the third neighbor distance (neglecting higher pairwise or multi-body interactions for the moment). The computed DFT binding energy for this conpair pair (3×3),i = 2E + 2V1 + 2V3 . Doing figuration i can therefore be written as E DFT this for a set of different configurations thus generates a system of linear equations that can be solved for the interaction energies either by direct inversion (or by fitting techniques, if more configurations than interaction parameters were determined). The crucial aspect in this procedure is the number and type of interactions to include in the LGH expansion, and the number and type of configurations that are computed to determine them. We note that there is no a priori way to know at how many, and what type of, interactions to terminate the expansion. While there are some attempts to automatize this procedure [59–61], it is probably fair to say that the actual implementation remains to date a delicate task. Some guidelines to judge on the convergence of the constructed Hamiltonian include its ability to predict the energies of a number of DFT-computed configurations that were not employed in the fit, or that it reproduces the correct lowest-energy configurations at T = 0 K (so-called “ground-state line”) [50].

1.5.

Equilibrium Monte Carlo Simulations

Once an accurate LGH has been constructed, one has at hand a very fast and flexible tool to provide the energies of arbitrary system configurations. This may in turn be used for MC simulations to obtain a good sampling of the available configuration space, i.e., to determine the partition function of the system. An important aspect of modern MC techniques is that this sampling is done very efficiently by concentrating on those parts of the configuration space that contribute significantly to the latter. The Metropolis algorithm [62], as a famous example of such so-called importance sampling schemes, proceeds therefore by generating at random new system configurations. If the new configuration exhibits a lower energy than the previous one, it is automatically “accepted” to a gradually built-up sequence of configurations. And even if the configuration has a higher energy, it still has an appropriately Boltzmann weighted probability to make it to the considered set. Otherwise it is “rejected” and the last configuration copied anew to the sequence. This way, the algorithm preferentially samples low energy configurations, which contribute most to the partition function. The acceptance criteria of the Metropolis, and of other

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importance sampling schemes, furthermore fulfill detailed balance. This means that the forward probability of accepting a new configuration j from state i is related to the backward probability of accepting configuration i from state j by the free energy difference of both configurations. Taking averages of system observables over the thus generated configurations yields then their correct thermodynamic average for the considered ensemble. Technical issues regard finally how new trial configurations are generated, or how long and in what system size the simulation must be run in order to obtain good statistical averages [6, 47]. The kind of insights that can be gained by such a first-principles LGH + MC approach is nicely exemplified by the problem of on-surface adsorption at a close-packed surface, when the latter is in equilibrium with a surrounding gas-phase. If this environment consists of oxygen, this would, e.g., contribute to the understanding of one of the early oxidation stages sketched in Fig. 2. What would be of interest is for instance to know how much oxygen is adsorbed at the surface given a certain temperature and pressure in the gas-phase, and whether the adsorbate forms ordered or disordered phases. As outlined above, the approach proceeds by first determining a LGH from a number of DFT-computed ordered adsorbate configurations. This is followed by grand-canonical MC simulations, in which new trial system configurations are generated by randomly adding or removing adsorbates from the lattice positions and where the energies of these configurations are provided by the LGH. Evaluating appropriate order parameters that check on prevailing lateral periodicities in the generated sequence of configurations, one may finally plot the phase diagram, i.e., what phase exists under which (T, p)-conditions (or equivalently (T, µ)-conditions) in the gas-phase. The result of one of the first studies of this kind is shown in Fig. 7 for the system O/Ru(0001). The employed LGH comprised two types of adsorption sites, namely the hcp and fcc hollows, lateral pair interactions up to third neighbor and three types of trio interactions between like and unlike sites, thus amounting to a total of fifteen independent interaction parameters. At low temperature, the simulations yield a number of ordered phases corresponding to different periodicities and oxygen coverages. Two of these ordered phases had already been reported experimentally at the time the work was carried out. The prediction of two new (higher coverage) periodic structures, namely a 3/4 and a 1 monolayer phase, has in the meanwhile been confirmed by various experimental studies. This example thus demonstrates the predictive nature of the first-principles approach, and the stimulating and synergetic interplay between theory and experiment. It is also worth pointing out that these new phases and their coexistence in certain coverage regions were not obtained in early MC calculations of this system based on an empirical LGH, which was determined by simply fitting a minimal number of pair interactions to the then available experimental phase diagram [51]. We also like to


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Figure 7. Phase diagram for O/Ru(0001) as obtained using the ab initio LGH approach in combination with MC calculations. The triangles indicate first order transitions and the circles second order The identified ordered structures are labeled as: (2×2)-O (A), (2×1)√ transitions. √ O (B), ( 3 × 3)R30◦ (C), (2 × 2)-3O (D), and disordered lattice-gas (l.g.) (from Ref. [63]).

stress the superior transferability of the first-principles interaction parameters. As an example we name simulations of temperature programmed desorption (TPD) spectra, which can among other possibilities be obtained by combining the LGH with a transfer-matrix approach and kinetic rate equations [61]. Figure 8 shows the result obtained with exactly the same LGH that also underlies the phase diagram of Fig. 7. Although empirical fits of TPD spectra may give better agreement between calculated and experimental results, we note that the agreement visible in Fig. 8 is in fact quite good. The advantage, on the other hand, is that no empirical parameters were used in the LGH, which allows to unambiguously trace back the TPD features to lateral interactions with well-defined microscopic meaning. The results summarized in Fig. 7 also serve quite well to illustrate the already mentioned differences between the initially described free energy plots and the LGH + MC method. In the first approach, the stability of a fixed set of configurations is compared in order to arrive at the phase diagram. Consider, for example, that we would have restricted our free energy analysis of the O/Ru(0001) system to only the O(2 × 2) and O(2 × 1) adlayer structures that were the two experimentally known ordered phases before 1995. The stability region of the prior phase, bounded at lower chemical potentials by the clean surface and at higher chemical potentials by O(2 × 1) phase, then comes

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out just as much as in Fig. 7. This stability range will be independent of temperature, however, there is no order–disorder transition at higher temperature due to the neglect of configurational entropy. More importantly, since the two higher-coverage phases would not have been explicitly considered, the stability of the O(2 × 1) phase would falsely extend over the whole higher chemical potential range. One would have to include these two configurations into the analysis to obtain the right result shown in Fig. 7, whereas the LGH + MC method yields them automatically. While this emphasizes the deeper insight and increased predictive power that is achieved by the proper sampling of configuration space in the LGH + MC technique, one must also recognize that the computational cost of the latter is significantly higher. It is, in particular, straightforward to directly compare the stability of qualitatively different geometries like the on-surface adsorption and the surface oxide phases in Fig. 3 in a free energy plot (or the various surface reconstructions entering Fig. 4). Setting up an LGH that would equally describe both systems, on the other hand, is far from trival. Even if it were feasible to find a generalized lattice that would be able to treat all system states, disentangling and determining the manifold of interaction energies in such a lattice will be very involved. The required discretization of the real system, i.e., the mapping onto a lattice, is therefore to date the major limitation of the LGH + MC technique – be it applied to two-dimensional pure surface systems or


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even worse to three-dimensional problems addressing a surface fringe of finite width. Still, it is also precisely this mapping and the resulting very fast analysis of the properties of the LGH that allows for an extensive and reliable sampling of the configuration space of complex systems that is hitherto unparalleled by other approaches. Having highlighted the importance of this sampling for the determination of unanticipated new ordered phases at lower temperatures, the final example in this section illustrates specifically the decisive role it also plays for the simulation and understanding of order-disorder transitions at elevated temperatures. A particularly intriguing transition of this kind is observed for Na on Al(001). The interest in such alkali metal adsorption systems has been intense, especially since in the early 1990s it was found (first for Na on Al(111) and then on Al(100)) that the alkali metal atoms may kick-out surface Al atoms and adsorb substitutionally [65–67]. This was in sharp contrast to the “experimental evidence” and the generally accepted understanding of the time, which was that alkali-metal atoms adsorb in the highest coordinated on-surface hollow site, and cause little disturbance to a close-packed metal surface. For the specific system Na on Al(001) at a coverage of 0.2 monolayer, recent low energy electron diffraction experiments observed furthermore a reversible phase transition √ √ in the◦temperature range 220 K–300 K. Below this range, an ordered ( 5 × 5)R27 structure forms, where the Na atoms occupy surface substitutional sites, while above it, the Na atoms, still in the substitutional sites, form a disordered arrangement in the surface. Using the ab initio LGH + MC approach the ordered phase and the disorder transition could be successfully reproduced [67]. Pair interactions up to the sixth nearest neighbor and two different trio interactions, as well as one quarto interaction were included in the LGH expansion. We note that determining these interaction parameters requires care, and careful cross-validation. To specifically identify the crucial role played by configurational entropy in the temperature induced order–disorder transition, a specific MC algorithm proposed by Wang and Landau [68] was employed. In contrast to the above outlined Metropolis algorithm, this scheme affords an explicit calculation of the density of configuration states, g(E), i.e., the number of system configurations with a certain energy E. This quantity provides in turn all major thermodynamic functions, e.g., the canonical distributionat a given temperature, g(E)e−E/ kB T , the free energy, F(T ) = − kB T ln( E g(E)e−E/kB T ) = −kB T ln(Z ), where Z is the partition function, the internal energy, U (T ) = [ E Eg(E)e−E/kB T ]/Z , and the entropy S = (U − F)/T . Figure 9 shows the calculated density of configuration states g(E), together with the internal and free energy derived from it. In the latter two quantities, the abrupt change corresponding to the first-order phase transition obtained at 301 K can be nicely discerned. This is also visible as a double peak in the logarithm of the canonical distribution (Fig. 9(a), inset) and as a singularity

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Figure 9. (a) Calculated density of configuration states, g(E), for Na on Al(100) at a coverage of 0.2 monolayers. Inset: Logarithm of the canonical distribution P(E, T ) = g(E)e E/ kB T , at the critical temperature. (b) Free energy F(T ) and internal energy U (T ) as a function of temperature, derived from g(E). The cusp in F(T ) and discontinuity in U (T ) at 301 K reflect the occurrence of the disorder–order phase transition, experimentally observed in the range 220–300 K (from Ref.[67]).


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in the specific heat at the critical temperature (not shown) [67]. It can be seen that the free energy decreases notably with increasing temperature. The reason for this is clearly the entropic contribution (difference in the free and internal energies), the magnitude of which suddenly increases at the transition temperature and continues to increase steadily thereafter. Taking this configurational entropy into account is therefore (and obviously) the crucial aspect in the simulation and understanding of this order–disorder phase transition, and only the LGH+MC approach with its proper sampling of configuration space can provide it. What the approach does not yield, on the other hand, is how the phase transition actually takes place microscopically, i.e., how the substitutional Na atoms move their positions by necessarily displacing surface Al atoms, and on what time scale (with what kinetic hindrance) this all happens. For this, one necessarily needs to go beyond a thermodynamic description, and explicitly follow the kinetics of the system over time, which will be the topic of the following section.

2.

First-Principles Kinetic Monte Carlo Simulations

Up to now we had discussed how equilibrium MC simulations can be used to explicitly evaluate the partition function, in order to arrive at surface phase diagrams as function of temperature and partial pressures of the surrounding gas-phase. For this, statistical averages over a sequence of appropriately sampled configurations were taken, and it is appealing to also connect some time evolution to this sequence of generated configurations (MC steps). In fact, certain nonequilibrium problems can already be tackled on the basis of this uncalibrated “MC time” [47]. The reason why this does not work in general is twofold. First, equilibrium MC is designed to achieve an optimum sampling of configurational space. As such, also MC moves that are unphysical like a particle hop from an occupied site to an unoccupied one, hundreds of lattice spacings away may be allowed, if they help to obtain an efficient sampling of the relevant configurations. The remedy for this obstacle is straightforward, though, as one only needs to restrict the possible MC moves to “physical” elementary processes. The second reason is more involved, as it has to do with the probabilities with which the individual events are executed. In equilibrium MC the forward and backward acceptance probabilities of time-reversed processes like hops back and forth between two sites only have to fulfill the detailed balance criterion, and this is not enough to establish a proper relationship between MC time and “real time” [69]. In kinetic Monte Carlo simulations (kMC) a proper relationship between MC time and real time is achieved by interpreting the MC process as providing a numerical solution to the Markovian master equation describing the

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dynamic system evolution [70–74]. The simulation itself still looks superficially similar to equilibrium MC in that a sequence of configurations is generated using random numbers. At each configuration, however, all possible elementary processes and the rates with which they occur are evaluated. Appropriately weighted by these different rates one of the possible processes is then executed randomly to achieve the new system configuration, as sketched in Fig. 10. This way, the kMC algorithm effectively simulates stochastic processes, and a direct and unambiguous relationship between kMC time and real time can be established [74]. Not only does this open the door to a treatment of the kinetics of nonequilibrium problems, but also it does so very efficiently, since the time evolution is actually coarse-grained to the really decisive rare events, passing over the irrelevant short-time dynamics. Time scales of the order of seconds or longer for mesoscopically-sized systems are therefore readily accessible by kMC simulations [12].

Figure 10. Flow diagram illustrating the basic steps in a kMC simulation. (i) Loop over all lattice sites of the system and determine the atomic processes that are possible for the current system configuration. Calculate or look up the corresponding rates. (ii) Generate two random numbers, (iii) advance the system according to the process selected by the first random number (this could, e.g., be moving an atom from one lattice site to a neighboring one, if the corresponding diffusion process was selected). (iv) Increment the clock according to the rates and the second random number, as prescribed by an ensemble of Poisson processes, and (v) start all over or stop, if a sufficiently long time span has been simulated.


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Insights from MD, MC, and kMC

To further clarify the different insights provided by molecular dynamics, equilibrium and kinetic Monte Carlo simulations, consider the simple, but typical rare event type model system shown in Fig. 11. An isolated adsorbate vibrates at finite temperature T with a frequency on the picosecond time scale and diffuses about every microsecond between two neighboring sites of different stability. In terms of a PES, this situation is described by two stable minima of different depths separated by a sizable barrier. Starting with the particle in any of the two sites, a MD simulation would follow the thermal motion of the adsorbate in detail. In order to do this accurately, timesteps in the femtosecond range are required. Before the first diffusion event can be observed at all, of the order of 109 time steps have therefore to be calculated first, in which the particle does nothing but just vibrate around the stable minimum. Computationally this is unfeasible for any but the simplest model systems, and even if it were feasible it would obviously not be an efficient tool to study the long-term time evolution of this system. For Monte Carlo simulations on the other hand, the system first has to be mapped onto a lattice. This is unproblematic for the present model and results

Figure 11. Schematic potential energy surface (PES) representing the thermal diffusion of an isolated adsorbate between two stable lattice sites A and B of different stability. A MD simulation would explicitly follow the dynamics of the vibrations around a minimum, and is thus inefficient to address the rare diffusion events happening on a much longer time scale. Equilibrium Monte Carlo simulations provide information about the average thermal occupation of the two sites, , based on the depth of the two PES minima (E A and E B ). Kinetic Monte Carlo simulations follow the “coarse-grained” time evolution of the system, N(t), employing the rates for the diffusion events between the minima (rA→B , rB→A ). For this, PES information not only about the minima, but also about the barrier height at the transition state (TS) between initial and final state is required (E A , E B ).

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in two possible system states with the particle being in one or the other minimum. Equilibrium Monte Carlo provides then only time-averaged information about the equilibrated system. For this, a sequence of configurations with the system in either of the two system states is generated, and considering the higher stability of one of the minima, appropriately more configurations with the system in this state are sampled. When taking the average, one arrives at the obvious result that the particle is with a certain higher (Boltzmann-weighted) probability in the lower minimum than in the higher one. Real information on the long-term time-evolution of the system, i.e., focusing on the rare diffusion events, is finally provided by kMC simulations. For this, first the two rates of the diffusion events from one system state to the other and vice versa have to be known. We will describe below that they can be obtained from knowledge of the barrier between the two states and the vibrational properties of the particle in the minima and at the barrier, i.e., from the local curvatures. A lot more information on the PES is therefore required for a kMC simulation than for equilibrium MC, which only needs input about the PES minima. Once the rates are known, a kMC simulation starting from any arbitrary system configuration will first evaluate all possible processes and their rates and then execute one of them with appropriate probability. In the present example, this list of events is trivial, since with the particle in either minimum only the diffusion to the other minimum is possible. When the event is executed, on average the time (rate)−1 has passed and the clock is advanced accordingly. Note that as described initially, the rare diffusion events happen on a time scale of nano- to microseconds, i.e., with only one executed event the system time will be directly incremented by this amount. In other words, the time is coarse-grained to the rare event time, and all the short-time dynamics (corresponding in the present case to the picosecond vibrations around the minimum) are efficiently contained in the process rate itself. Since the barrier seen by the particle when in the shallower minimum is lower than when in the deeper one, cf. Fig. 11, the rate to jump into the deeper minimum will correspondingly be higher than the one for the backwards jump. Generating the sequence of configurations, each time more time will therefore have passed after a diffusion event from deep to shallow compared to the reverse process. When taking a long-time average, describing then the equilibrated system, one therefore arrives necessarily at the result that the particle is on average longer in the lower minimum than in the higher one. This is identical to the result provided by equilibrium Monte Carlo, and if only this information is required, the latter technique would most often be the much more efficient way to obtain it. KMC, on the other hand, has the additional advantage of shedding light on the detailed time-evolution itself, and can in particular also follow the explicit kinetics of systems that are not (or not yet) in thermal equilibrium.


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From the discussion of this simple model system, it is clear that the key ingredients of a kMC simulation are the analysis and identification of all possibly relevant elementary processes and the determination of the associated rates. Once this is known, the coarse graining in time achieved in kMC immediately allows to follow the time evolution and the statistical occurrence and interplay of the molecular processes of mesoscopically sized systems up to seconds or longer. As such it is currently the most efficient approach to study long time and larger length scales, while still providing atomistic information. In its original development, kMC was exclusively applied to simplified model systems, employing a few processes with guessed or fitted rates (see, e.g., Ref. [69]). The new aspect brought into play by so-called first-principles kMC simulations [75, 76] is that these rates and the processes are directly provided from electronic structure theory calculations, i.e., that the parameters fed into the kMC simulation have a clear microscopic meaning.

2.2.

Getting the Processes and Their Rates

For the rare event type molecular processes mostly encountered in the surface science context, an efficient and reliable way to obtain the individual process rates is transition-state theory (TST) [77–79]. The two basic quantities entering this theory are an effective attempt frequency, ◦ , and the minimum energy barrier E that needs to be overcome for the event to take place, i.e., to bring the system from the initial to the final state. The atomic configuration corresponding to E is accordingly called the transition state (TS). Within the harmonic approximation, the effective attempt frequency is proportional to the ratio of normal vibrational modes at the initial and transition state. Just like the barrier E, ◦ is thus also related to properties of the PES, and as such directly amenable to a calculation with electronic structure theory methods like DFT [80]. In the end, the crucial additional PES information required in kMC compared to equilibrium MC is therefore the location of the transition state in form of the PES saddle point along a reaction path of the process. Particularly for high-dimensional PES this is not at all a trivial problem, and the development of efficient and reliable transition-state-search algorithms is a very active area of current research [81, 82]. For many surface related elementary processes (e.g., diffusion, adsorption, desorption or reaction events) the dimensionality is fortunately not excessive, or can be mapped onto a couple of prominent reaction coordinates as exemplified in Fig. 12. The identification of the TS and the ensuing calculation of the rate for individual identified elementary processes with TST are then computationally involved, but just feasible. This still leaves as a fundamental problem, how the relevant elementary processes for any given system configuration can be identified in the first place.

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Figure 12. Calculated DFT-PES of a CO oxidation reaction process at the RuO2 (110) model catalyst surface. The high-dimensional PES is projected onto two reaction coordinates, representing two lateral coordinates of the adsorbed Ocus and COcus (cf. Fig. 5). The energy zero corresponds to the initial state at (0.00 Å, 3.12 Å), and the transition state is at the saddle point of the PES, yielding a barrier of 0.89 eV. Details of the corresponding transition state geometry are shown in the inset. Ru = light, large spheres, O = dark, medium spheres, and C = small, white spheres (only the atoms lying in the reaction plane itself are drawn as three-dimensional spheres) (from Ref. [26]).

Most TS-search algorithms require not only the automatically provided information of the actual system state, but also knowledge of the final state after the process has taken place [81]. In other words, quite some insight into the physics of the elementary process is needed in order to determine its rate and include it in the list of possible processes in the kMC simulation. How difficult and nonobvious this can be even for the simplest kind of processes is nicely exemplified by the diffusion of an isolated metal atom over a close-packed surface [82]. Such a process is of fundamental importance for the epitaxial growth of metal films, which is a necessary prerequisite in many applications like catalysis, magneto-optic storage media or interconnects in microelectronics. Intuitively, one would expect the surface diffusion to proceed by simple hops from one lattice site to a neighboring lattice site, as illustrated in Fig. 13(a) for an fcc (100) surface. Having said that, it is in the meanwhile well established that on a number of substrates diffusion does not operate preferentially by such hopping processes, but by atomic exchange as explained in Fig. 13(b). Here, the adatom replaces a surface atom, and the latter then assumes the adsorption site. Even much more complicated, correlated exchange diffusion processes involving a larger number of surface atoms are currently discussed for some materials. And the complexity increases of course further, when diffusion along island edges, across steps and around defects needs to be treated in detail [82].


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Figure 13. Schematic top view of a fcc(100) surface, explaining diffusion processes of an isolated metal adatom (white circle). (a) Diffusion by hopping to a neighboring lattice site, (b) diffusion by exchange with a surface atom.

While it is therefore straightforward to say that one wants to include, e.g., diffusion in a kMC simulation, it can in practice be very involved to identify the individual processes actually contributing to it. Some attempts to automatize the search for the elementary processes possible for a given system configuration are currently undertaken, but in the first-principles kMC studies performed up to date (and in the foreseeable future), the process lists are simply generated by physical insight. This obviously bears the risk of overlooking a potentially relevant molecular process, and on this note this just evolving method has to be seen. Contrary to traditional kMC studies, where an unknown number of real molecular processes is often lumped together into a handful effective processes with optimized rates, first-principles kMC has the advantage, however, that the omission of a relevant elementary process will definitely show up in the simulation results. As such, first experience [15] tells that a much larger number of molecular processes needs to be accounted for in a corresponding modeling “with microscopic understanding” compared to traditional empirical kMC. In other words, that the statistical interplay determining the observable function of materials takes places between quite a number of different elementary processes, and is therefore often way too complex to be understood by just studying in detail the one or other elementary process alone.

2.3.

Applications to Semiconductor Growth and Catalysis

The new quality of and the novel insights that can be gained by mesoscopic first-principles kMC simulations was first demonstrated in the area of nucleation

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and growth in metal and semiconductor epitaxy [75, 76, 83–87]. As one example from this field we return to the GaAs(001) surface already discussed in the context of the free energy plots. As apparent from Fig. 4, the so-called β2(2 × 4) reconstruction represents the most stable phase under moderately As-rich conditions, which are typically employed in the MBE growth of this material. Aiming at an atomic-scale understanding of this technologically most relevant process, first-principles LGH + kMC simulations were performed, including the deposition of As2 and Ga from the gas phase, as well as diffusion on this complex β2(2 × 4) semiconductor surface. In order to reach a trustworthy modeling, the consideration of more than 30 different elementary processes was found to be necessary, underlining our general message that complex materials properties cannot be understood by analyzing isolated molecular processes alone. Snapshots of characteristic stages during a typical simulation at realistic deposition fluxes and temperature are given in Fig. 14. They show a small part (namely 1/60) of the total mesoscopic simulation area, focusing on one “trench” of the β2(2 × 4) reconstruction. At the chosen conditions, island nucleation is observed in these reconstructed surface trenches, which is followed by growth along the trench, thereby extending into a new layer. Monitoring the density of the nucleated islands in huge simulation cells (160 × 320 surface lattice constants), a saturation indicating the beginning of steady-state growth is only reached after simulation times of the order of seconds for quite a range of temperatures. Obviously, neither such system sizes, nor time scales would have been accessible by direct electronic structure theory calculations combined, e.g., with MD simulations. In the ensuing steady-state growth, attachment of a deposited Ga atom to an existing island typically takes place before the adatom could take part in a new nucleation event. This leads to a very small nucleation rate that is counterbalanced by a simultaneous decrease in the number of islands due to coalescence. The resulting constant island density during steady-state growth is plotted in Fig. 15 for a range of technologically relevant temperatures. At the lower end around 500–600 K, this density decreases, as is consistent with the frequently employed standard nucleation theory. Under these conditions, the island morphology is predominantly determined by Ga surface diffusion alone, i.e., it may be understood on the basis of one molecular process class. Around 600 K the island density becomes almost constant, however, and even increases again above around 800 K. The determined magnitude is then orders of magnitude away from the prediction of classical nucleation theory, cf. Fig. 15, but in very good agreement with existing experimental data. The reason for this unusual behavior is that the adsorption of As2 molecules at reactive surface sites becomes reversible at these elevated temperatures. The initially formed Ga–As–As–Ga2 complexes required for nucleation, cf. Fig. 14(b), become unstable against As2 desorption, and a decreasing fraction of them can stabilize into larger aggregates. Due to the contribution of the decaying complexes, an

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Figure 14. Snapshots of characteristic stages during a first-principles kMC simulation of GaAs homoepitaxy. Ga and As substrate atoms appear in medium and dark grey, Ga adatoms in white. (a) Ga adatoms preferentially wander around in the trenches. (b) Under the growth conditions used here, an As2 molecule adsorbing on a Ga adatom in the trench initiates island formation. (c) Growth proceeds into a new atomic layer via Ga adatoms forming Ga dimers. (d) Eventually, a new layer of arsenic starts to grow, and the island extends itself towards the foreground, while more material attaches along the trench. The complete movie can be retrieved via the EPAPS homepage (http://www.aip.org/pubservs/epaps.html), document No. E-PRLTAO-87-031152 (from Ref. [86]).

effectively higher density of mobile Ga adatoms results at the surface, which in turn yields a higher nucleation rate of new islands. The temperature window around 700–800 K, which is frequently used by MBE crystal growers, may therefore be understood as permitting a compromise between high Ga adatom mobility and stability of As complexes that leads to a low island density and correspondingly smooth films. Exactly under the technologically most relevant conditions, surface properties that decisively influence the growth behavior (and therewith the targeted functionality) result therefore from the concerted interdependence of distinct molecular processes, i.e., in this case diffusion, adsorption and desorption. To further show that this interdependence is to our opinion more the rule than an exception in materials science applications, we return in the remainder of

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Figure 15. Saturation island density corresponding to steady-state MBE of GaAs as a function of the inverse growth temperature. The dashed line shows the prediction of classical nucleation theory for diffusion-limited attachment and a critical nucleus size equal to 1. The significant deviation at higher temperatures is caused by arsenic losses due to desorption, which is not considered in classical nucleation theory (from Ref. [87]).

this section to the field of heterogeneous catalysis. Here, the conversion of reactants into products by means of surface chemical reactions (A + B → C) adds another qualitatively different class of processes to the statistical interplay. In the context of the thermodynamic free energy plots we had already discussed that these on-going catalytic reactions at the surface continuously consume the adsorbed reactants, driving the surface populations away from their equilibrium value. If this has a significant effect, presumably, e.g., in regions of very high catalytic activity, the average surface coverage and structure does even under steady-state operation never reach its equilibrium with the surrounding reactant gas phase, and must thence be modeled by explicitly accounting for the surface kinetics [88–90]. In terms of kMC, this means that in addition to the diffusion, adsorption and desorption of the reactants and products, also reaction events have to be considered. For the case of CO oxidation, as one of the central reactions taking place in our car catalytic converters, this translates into the conversion of adsorbed O and CO into CO2 . Even for the afore discussed, moderately complex model catalyst RuO2 (110), again close to 30 elementary processes result, comprising both adsorption to and desorption from the two prominent site-types at the surface (br and cus, cf. Fig. 5), as well as diffusion between any nearest neighbor site-combination (br→br, br→cus, cus→br, cus→cus). Finally, reaction events account for the catalytic activity and are possible


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whenever O and CO are simultaneously adsorbed in any nearest neighbor sitecombination. For given temperature and reactant pressures, the corresponding kMC simulations are then first run until steady-state conditions are reached, and the average surface populations are thereafter evaluated over sufficiently long times. We note that even for elevated temperatures, both time periods may again largely exceed the time span accessible by current MD techniques as exemplified in Fig. 16. The obtained steady-state average surface populations at T = 600 K are shown in Fig. 17 as a function of the gas-phase partial pressures. Comparing with the surface phase diagram of Fig. 5 from ab initio atomistic thermodynamics, i.e., neglecting the effect of the on-going catalytic reactions at the surface, similarities, but also the expected significant differences under some environmental conditions can be discerned. The differences affect most prominently the presence of oxygen at the br sites, where it is much more strongly bound than CO. For the thermodynamic approach only the ratio of adsorption to desorption matters, and due to the ensuing very low desorption rate, Obr is correspondingly stabilized even when there is much more CO in the gas-phase than O2 (left upper part of Fig. 5). The surface reactions, on the other hand, provide a very efficient means of 100 Site occupation number (%)

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Figure 16. Time evolution of the site occupation by O and CO of the two prominent adsorption sites of the RuO2 (110) model catalyst surface shown in Fig. 5. The temperature and pressure conditions chosen (T = 600 K, pCO = 20 atm, pO2 = 1 atm) correspond to an optimum catalytic performance. Under these conditions kinetics builds up a steady-state surface population in which O and CO compete for either site type at the surface, as reflected by the strong fluctuations in the site occupations. Note the extended time scale, also for the “induction period” until the steady-state populations are reached when starting from a purely oxygen covered surface. A movie displaying these changes in the surface population can be retrieved via the EPAPS homepage (http://www.aip.org/pubservs/spaps.html), document No. E-PRLTAO93-006438 (from Ref. [90]).

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Figure 17. Left panel: Steady state surface structures of RuO2 (110) in an O2 /CO environment obtained by first-principles kMC calculations at T = 600 K. In all non-white areas, the average site occupation is dominated (> 90 %) by one species, and the site nomenclature is the same as in Fig. 5, where the same surface structure was addressed within the ab initio atomistic thermodynamics approach. Right panel: Map of the corresponding catalytic CO oxidation activity measured as so-called turn-over frequencies (TOFs), i.e., CO2 conversion per cm2 and second: White areas have a TOF < 1011 cm−2 s−1 , and each increasing gray level represents one order of magnitude higher activity. The highest catalytic activity (black region, TOF > 1017 cm−2 s−1 ) is narrowly concentrated around the phase coexistence region that was already suggested by the thermodynamic treatment (from Ref. [90]).

removing this Obr species that is not accounted for in the thermodynamic treatment. As net result, under most CO-rich conditions in the gas phase, oxygen is faster consumed by the reaction than it can be replenished from the gas phase. The kMC simulations covering this effect yield then a much lower surface concentration of Obr , and in turn show a much larger stability range of surface structures with CObr at the surface (blue and hatched blue regions). It is particularly interesting to notice, that this yields a stability region of a surface structure consisting of only adsorbed CO at br sites that does not exist in the thermodynamic phase diagram at all, cf. Fig. 5. The corresponding CObr /− “phase” (hatched blue region) is thus a stable structure with defined average surface population that is entirely stabilized by the kinetics of this open catalytic system. These differences were conceptually anticipated in the thermodynamic phase diagram, and qualitatively delineated by the hatched regions in Fig. 5. Due to the vicinity to a phase transition and the ensuing enhanced dynamics at the surface, these regions were also considered as potential candidates for highly efficient catalytic activity. This is in fact confirmed by the first-principles kMC simulations as shown in the right panel of Fig. 17. Since the detailed statistics of all elementary processes is explicitly accounted for in the latter type simulations, it is straightforward to also evaluate the average occurrence of


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the reaction events over long time periods as a measure of the catalytic activity. The obtained so-called turnover frequencies (TOF, in units of formed CO2 per cm2 per second) are indeed narrowly peaked around the phase coexistence line, where the kinetics builds up a surface population in which O and CO compete for either site type at the surface. This competition is in fact nicely reflected by the large fluctuations in the surface populations apparent in Fig. 16. The partial pressures and temperatures corresponding to this high activity “phase”, and even the absolute TOF values under these conditions, agree extremely well with detailed experimental studies measuring the steady-state activity in the temperature range from 300–600 K and both at high pressures and in UHV. Interestingly, under the conditions of highest catalytic performance it is not the reaction with the highest rate (lowest barrier) that dominates the activity. Although the particular elementary process itself exhibits very suitable properties for catalysis, it occurs too rarely in the full concert of all possible events to decisively affect the observable macroscopic functionality. This emphasizes again the importance of the statistical interplay and the novel level of understanding that can only be provided by first-principles based mesoscopic studies.

3.

Outlook

As highlighted by the few examples from surface physics, many materials’ properties and functions arise out of the interplay of a large number of distinct molecular processes. Theoretical approaches aiming at an atomic-scale understanding and predictive modeling of such phenomena have therefore to achieve both an accurate description of the individual elementary processes at the electronic regime and a proper treatment of how they act together on the mesoscopic level. We have sketched the current status and future direction of some emerging methods which correspondingly try to combine electronic structure theory with concepts from statistical mechanics and thermodynamics. The results already achieved with these techniques give a clear indication of the new quality and novelty of insights that can be gained by such descriptions. On the other hand, it is also apparent that we are only at the beginning of a successful bridging of the micro- to mesoscopic transition in the multiscale materials modeling endeavor. Some of the major conceptual challenges we see at present that need to be tackled when applying these schemes to more complex systems have been touched in this chapter. They may be summarized under the keywords accuracy, mapping and efficiency, and as outlook we briefly comment further on them. Accuracy: The reliability of the statistical treatment depends predominantly on the accuracy of the description of the individual molecular processes that are input to it. For the mesoscopic methods themselves it makes in fact no

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difference, whether the underlying PES comes from a semi-empirical potential or from first-principles calculations, but the predictive power of the obtained results (and the physical meaning of the parameters) will obviously be significantly different. In this respect, we only mention two somehow diverging aspects. For the interplay of several (possibly competing) molecular processes, an “exact” description of the energetics of each individual process, e.g., in form of a rate for kMC simulations may be less important than the relative ordering among the processes as, e.g., provided by the correct trend in their energetics. In this case, the frequently requested chemical accuracy in the description of single processes could be a misleading concept, and modest errors in the PES would tend to cancel (or compensate each other) in the statistical mechanics part. Here, we stress the words modest errors, however, which, e.g., largely precludes semi-empiric potentials. Particularly for systems where bond breaking and making is relevant, the latter do not have the required accuracy. On the other hand, for the particular case of DFT as the current workhorse of electronic structure theories it appears that the present uncertainties due to the approximate treatment of electronic exchange and correlation are less problematic than hitherto often assumed (still caution, and systematic tests are necessary). On the other hand, in other cases where for example one process strongly dominates the concerted interplay, such an error cancellation in the statistical mechanics part will certainly not occur. Then, a more accurate description of this process will be required than can be provided by the exchangecorrelation functionals in DFT that are available today. Improved descriptions based on wave-function methods and on local corrections to DFT exist or are being developed, but come so far at a high computational cost. Assessing what kind of accuracy is required for which process under which system state, possibly achieved by evolutionary schemes based on gradually improving PES descriptions, will therefore play a central role in making atomistic statistical mechanics methods computationally feasible for increasingly complex systems. Mapping: The configuration space of most materials science problems is exceedingly large. In order to arrive at meaningful statistics, even the most efficient sampling of such spaces still requires (at present and in the foreseeable future) a number of PES evaluations that is prohibitively large to be directly provided by first-principles calculations. This problem is mostly circumvented by mapping the actual system onto a coarse-grained lattice model, in which the real Hamiltonian is approximated by discretized expansions, e.g., in certain interactions (LGH) or elementary processes (kMC). The expansions are then first parametrized by the first-principles calculations, while the statistical mechanics problem is thereafter solved exploiting the fast evaluations of the model Hamiltonians. Since in practice these expansions can only comprise a finite number of terms, the mapping procedure intrinsically bears the problem of overlooking a relevant interaction or process. Such an omission can


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obviously jeopardize the validity of the complete statistical simulation, and there are at present no fool-proof or practical, let alone automatized schemes as to which terms to include in the expansion, neither how to judge on the convergence of the latter. In particular when going to more complex systems the present “hand-made” expansions that are mostly based on educated guesses will become increasingly cumbersome. Eventually, the complexity of the system may become so large, that even the mapping onto a discretized lattice itself will be problematic. Overcoming these limitations may be achieved by adaptive, self-refining approaches, and will certainly be of paramount importance to ensure the general applicability of the atomistic statistical techniques. Efficiency: Even if an accurate mapping onto a model Hamiltonian is achieved, the sampling of the huge configuration spaces will still put increasing demands on the statistical mechanics treatment. In the examples discussed above, the actual evaluation of the system partition function, e.g., by MC simulations is a small add-on compared to the computational cost of the underlying DFT calculations. With increasing system complexity, different problems and an increasing number of processes this may change eventually, requiring the use of more efficient sampling schemes. A major challenge for increasing efficiency is for example the treatment of kinetics, in particular when processes operate at largely different time scales. The computational cost of a certain time span in kMC simulations is dictated by the fastest process in the system, while the slowest process governs what total time period needs actually to be covered. If both process scales differ largely, kMC becomes expensive. A remedy may, e.g., be provided by assuming the fast process to be always equilibrated at the time scale of the slow one, and correspondingly an appropriate mixing of equilibrium MC with kMC simulations may significantly increase the efficiency (as typically done in nowadays TPD simulations). Alternatively, the fast process could not be explicitly considered anymore on the atomistic level, and only its effect incorporated into the remaining processes. Obviously, with such a grouping of processes one approaches already the meso- to macroscopic transition, gradually giving up the atomistic description in favor of a more coarse-grained or even continuum modeling. The crucial point to note here is that such a transition is done in a controlled and hierarchical manner, i.e., necessarily as the outcome and understanding from the analysis of the statistical interplay at the mesoscopic level. This is therefore in marked contrast to, e.g., the frequently employed rate equation approach in heterogeneous catalysis modeling, where macroscopic differential equations are directly fed with effective microscopic parameters. If the latter are simply fitted to reproduce some experimental data, at best a qualitative description can be achieved anyway. If really microscopically meaningful parameters are to be used, one does not know which of the many in principle possible elementary processes to consider. Simple-minded “intuitive” approaches like,

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e.g., parametrizing the reaction equation with the data from the reaction process with the highest rate may be questionable in view of the results described above. This process may never occur in the full concert of the other processes, or it may only contribute under particular environmental conditions, or be significantly enhanced or suppressed due to an intricate interplay with another process. All this can only be filtered out by the statistical mechanics at the mesoscopic level, and can therefore not be grasped by the traditional rate equation approach omitting this intermediate time and length scale regime. The two key features of the atomistic statistical schemes reviewed here are in summary that they treat the statistical interplay of the possible molecular processes, and that these processes have a well-defined microscopic meaning, i.e., they are described by parameters that are provided by first-principles calculations. This distinguishes these techniques from approaches where molecular process parameters are either directly put into macroscopic equations neglecting the interplay, or where only effective processes with fitted or empirical parameters are employed in the statistical simulations. In the latter case, the individual processes lose their well-defined microscopic meaning and typically represent an unspecified lump sum of not further resolved processes. Both the clear cut microscopic meaning of the individual processes and their interplay are, however, decisive for the transferability and predictive nature of the obtained results. Furthermore, it is also precisely these two ingredients that ensure the possibility of reverse-mapping, i.e., the unambiguous tracing back of the microscopic origin of (appealing) materials’ properties identified at the meso- or macroscopic modeling level. We are convinced that primarily the latter point will be crucial when trying to overcome the present trial and error based system engineering in materials sciences in the near future. An advancement based on understanding requires theories that straddle various traditional disciplines. The approaches discussed here employ methods from various areas of electronic structure theory (physics as well as chemistry), statistical mechanics, mathematics, materials science, and computer science. This high interdisciplinarity makes the field challenging, but is also part of the reason why it is exciting, timely, and full with future perspectives.

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1.10 DENSITY-FUNCTIONAL PERTURBATION THEORY Paolo Giannozzi1 and Stefano Baroni2 1 DEMOCRITOS-INFM, Scuola Normale Superiore, Pisa, Italy 2

DEMOCRITOS-INFM, SISSA-ISAS, Trieste, Italy

The calculation of vibrational properties of materials from their electronic structure is an important goal for materials modeling. A wide variety of physical properties of materials depend on their lattice-dynamical behavior: specific heats, thermal expansion, and heat conduction; phenomena related to the electron–phonon interaction such as the resistivity of metals, superconductivity, and the temperature dependence of optical spectra, are just a few of them. Moreover, vibrational spectroscopy is a very important tool for the characterization of materials. Vibrational frequencies are routinely and accurately measured mainly using infrared and Raman spectroscopy, as well as inelastic neutron scattering. The resulting vibrational spectra are a sensitive probe of the local bonding and chemical structure. Accurate calculations of frequencies and displacement patterns can thus yield a wealth of information on the atomic and electronic structure of materials. In the Born–Oppenheimer (adiabatic) approximation, the nuclear motion is determined by the nuclear Hamiltonian H: H=−

h¯ 2 I

∂2 + E({R}), 2M I ∂R2I

(1)

where R I is the coordinate of the I th nucleus, M I its mass, {R} indicates the set of all the nuclear coordinates, and E({R}) is the ground-state energy of the Hamiltonian, H{R} , of a system of N interacting electrons moving in the field of fixed nuclei with coordinates {R}: H{R} = −

1 h¯ 2 ∂ 2 e2 + v I (ri − R I ) + E N ({R}), + 2 2m i ∂ri 2 i=/ j |ri − r j | i,I

(2) 195 S. Yip (ed.), Handbook of Materials Modeling, 195–214. c 2005 Springer. Printed in the Netherlands.

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where ri is the coordinate of the ith electron, m is the electron mass, −e is the electron charge, E N ({R}) is the nuclear electrostatic energy: E N ({R}) =

e2 Z I Z J , 2 I =/ J |R I − R J |

(3)

Z I being the charge of the I th nucleus, and v I is the electron–nucleus Coulomb interaction: v I (r) = −Z I e2 /r. In a pseudopotential scheme each nucleus is thought to be lumped together with its own core electrons in a frozen ion which interacts with the valence electrons through a smooth pseudopotential, v I (r). The equilibrium geometry of the system is determined by the condition that the forces acting on all nuclei vanish. The forces F I can be calculated by applying the Hellmann–Feynman theorem to the Born–Oppenheimer Hamiltonian H{R} :

∂ H{R} ∂ E({R}) {R} , = − {R} FI ≡ − ∂R I ∂R I

(4)

where {R} (r1 , . . . , r N ) is the ground-state wavefunction of the electronic Hamiltonian, H{R} . Eq. (4) can be rewritten as: FI = −

n(r)

∂v I (r − R I ) ∂ E N ({R}) dr − , ∂R I ∂R I

(5)

where n(r) is the electron charge density for the nuclear configuration {R}:

n(r) = N

|{R} (r, r2 , . . . , r N )|2 dr2 · · · dr N .

(6)

For a system near its equilibrium geometry, the harmonic approximation applies and the nuclear Hamiltonian of Eq. (1) reduces the Hamiltonian of a system of independent harmonic oscillators, called normal modes. Normal mode frequencies, ω, and displacement patterns, U Iα for the αth Cartesian component of the I th atom, are determined by the secular equation:

αβ

β

C IJ − M I ω2 δ IJ δαβ U J = 0,

(7)

J,β αβ

where C IJ is the matrix of interatomic force constants (IFCs): αβ

C IJ ≡

∂ 2 E({R}) β

∂ R αI ∂ R J

=−

∂ FIα

β.

∂ RJ

(8)

Various dynamical models, based on empirical or semiempirical inter-atomic potentials, can be used to calculate the IFCs. In most cases, the parameters of the model are obtained from a fit to some known experimental data, such as a set of frequencies. Although simple and often effective, such approaches tend

Density-functional perturbation theory

197

to have a limited predictive power beyond the range of cases included in the fitting procedure. It is often desirable to resort to first-principles methods, such as density-functional theory, that have a far better predictive power even in the absence of any experimental input.

1.

Density-Functional Theory

Within the framework of density-functional theory (DFT), the energy E({R}) can be seen as the minimum of a functional of the charge density n(r): e2 E({R}) = T0 [n(r)] + 2 +

n(r)n(r ) dr dr + E xc [n(r)] |r − r |

V{R} (r)n(r)dr + E N ({R}),

(9)

with the constrain that the integral of n(r) equals the number of electrons in the system, N . InEq. (9), V{R} indicates the external potential acting on the electrons, V{R} = I v I (r − R I ), T0 [n(r)] is the kinetic energy of a system of noninteracting electrons having n(r) as ground-state density, N/2 h¯ 2 ∂ 2 ψn (r) T0 [n(r)] = −2 ψn∗ (r) dr 2m n=1 ∂r2

n(r) = 2

N/2

|ψn (r)|2 ,

(10) (11)

n=1

and E xc is the so-called exchange-correlation energy. For notational simplicity, the system is supposed here to be a nonmagnetic insulator, so that each of the N/2 lowest-lying orbital states accommodates two electrons of opposite spin. The Kohn-Sham (KS) orbitals are the solutions of the KS equation:

HSCF ψn (r) ≡

h¯ 2 ∂ 2 − + VSCF (r) ψn (r) = n ψn (r), 2m ∂r2

(12)

where HSCF is the Hamiltonian for an electron under an effective potential VSCF : n(r ) 2 VSCF (r) = V{R} (r) + e (13) dr + v xc (r), |r − r | and v xc – the exchange-correlation potential – is the functional derivative of the exchange-correlation energy: v xc (r) ≡ δ E xc /δn(r). The form of E xc is unknown: the entire procedure is useful only if reliable approximate expressions for E xc are available. It turns out that even the simplest of such expressions, the local-density approximation (LDA), is surprisingly good in many

198


cases, at least for the determination of electronic and structural ground-state properties. Well-established methods for the solution of KS equations, Eq. (12), in both finite (molecules, clusters) and infinite (crystals) systems, are described in the literature. The use of more sophisticated and more performing functionals than LDA (such as generalized gradient approximation, or GGA) is now widespread. An important consequence of the variational character of DFT is that the Hellmann–Feynman form for forces, Eq. (5), is still valid in a DFT framework. In fact, the DFT expression for forces contains a term coming from explicit derivation of the energy functional E({R}) with respect to atomic positions, plus a term coming from implicit dependence via the derivative of the charge density: =− FDFT I

n(r)

∂ V{R}(r) ∂ E N ({R}) dr − − ∂R I ∂R I

δ E({R}) ∂n(r) dr. (14) δn(r) ∂R I

The last term in Eq. (14) vanishes exactly for the ground-state charge density: the minimum condition implies in fact that the functional derivative of E({R}) equals a constant – the Lagrange multiplier that enforces the constrain on the total number of electrons – and the integral of the derivative of the electron = FI density is zero because of charge conservation. As a consequence, FDFT I as in Eq. (5). Forces in DFT can thus be calculated from the knowledge of the electron charge-density. IFCs can be calculated as finite differences of Hellmann–Feynman forces for small finite displacements of atoms around the equilibrium positions. For finite systems (molecules, clusters) this technique is straightforward, but it may also be used in solid-state physics (frozen phonon technique). An alternative technique is the direct calculation of IFCs using density-functional perturbation theory (DFPT) [1–3].

2.

Density-Functional Perturbation Theory

An explicit expression for the IFCs can be obtained by differentiating the forces with respect to nuclear coordinates, as in Eq. (8): ∂ 2 E({R}) = ∂R I ∂R J

∂n(r) ∂ V{R} (r) ∂ 2 V{R} (r) ∂ 2 E N ({R}) dr + δ IJ n(r) dr + . ∂R J ∂R I ∂R I ∂R J ∂R I ∂R J (15)

The calculation of the IFCs thus requires the knowledge of the ground-state charge density, n(r), as well as of its linear response to a distortion of the nuclear geometry, ∂n(r)/∂R I .


199

The charge-density linear response can be evaluated by linearizing Eqs. (11)–(13), with respect to derivatives of KS orbitals, density, and potential, respectively. Linearization of Eq. (11) leads to: ∂ψn (r) ∂n(r) = 4 Re ψn∗ (r) . ∂R I ∂R I n=1 N/2

(16)

Whenever the unperturbed Hamiltonian is time-reversal invariant, eigenfunctions are either real, or they occur in conjugate pairs, so that the prescription to keep only the real part in the above formula can be dropped. The derivatives of the KS orbitals, ∂ψn (r)/∂R I , are obtained from linearization of Eqs. (12) and (13):

(HSCF − n )

∂ψn (r) ∂n ∂ VSCF (r) =− − ∂R I ∂R I ∂R I

ψn (r),

(17)

where ∂ VSCF (r) ∂ V{R} (r) = + e2 ∂R I ∂R I

1 ∂n(r ) dr + |r − r | ∂R I

δv xc (r) ∂n(r ) dr δn(r ) ∂R I (18)

is the first-order derivative of the self-consistent potential, and

∂ VSCF ∂n ψn = ψn ∂R I ∂R I

(19)

is the first-order derivative of the KS eigenvalue, n . The form of the righthand side of Eq. (17) ensures that ∂ψn (r)/∂R I can be chosen so as to have a vanishing component along ψn (r) and thus the singularity of the linear system in Eq. (17) can be ignored. Equations (16)–(18) form a set of self-consistent linear equations. The linear system, Eq. (17), can be solved for each of the N/2 derivatives ∂ψn (r)/∂R I separately, the charge-density response calculated from Eq. (16), and the potential response ∂ VSCF /∂R I is updated from Eq. (18), until self-consistency is achieved. Only the knowledge of the occupied states of the system is needed to construct the right-hand side of the equation, and efficient iterative algorithms – such as conjugate gradient or minimal residual methods – can be used for the solution of the linear system. In the atomic physics literature, an equation analogous to Eq. (17) is known as the Sternheimer equation, and its self-consistent version was used to calculate atomic polarizabilities. Similar methods are known in the quantum chemistry literature, under the name of coupled Hartree–Fock method for the Hartree–Fock approximation [4, 5].

200


The connection with standard first-order perturbation (linear-response) theory can be established by expressing Eq. (17) as a sum over the spectrum of the unperturbed Hamiltonian: 1 ∂ψn (r) = ψm (r) ∂R I n − m m= /n

∂ VSCF ψn , ψm ∂R

(20)

I

running over all the states of the system, occupied and empty. Using Eq. (20), the electron charge-density linear response, Eq. (16), can be recast into the form: 1 ∂ψn (r) =4 ψn∗ (r)ψm (r) ∂R I n − m /n n=1 m= N/2

∂ VSCF ψn . ψm ∂R

(21)

I

This equations shows that the contributions to the electron-density response coming from products of occupied states cancel each other. As a consequence, in Eq. (17) the derivatives ∂ψn (r)/∂R I can be assumed to be orthogonal to all states of the occupied manifold. An alternative and equivalent point of view is obtained by inserting Eq. (16) into Eq. (18) and the resulting equation into Eq. (17). The set of N/2 selfconsistent linear systems is thus recast into a single huge linear system for all the N/2 derivatives ∂ψn (r)/∂R I

∂ψn (r) ∂ψm + K nm ∂R I ∂R I m=1 N/2

(HSCF − n )

(r) = −

∂ V{R} (r) ψn (r), ∂R I

(22)

under the orthogonality constraints: ∂ψn ψn = 0. ∂R

(23)

I

The nonlocal operator K nm is defined as:

∂ψm K nm ∂R I

(r) = 4

ψn (r)

δv xc (r) e2 + |r − r | δn(r )

ψm∗ (r )

∂ψm (r ) dr . ∂R I (24)

The same expression can be derived from a variational principle. The energy functional, Eq. (9), is written in terms of the perturbing potential and of the perturbed KS orbitals: V (u I ) V{R} (r) + u I

∂ V{R} (r) , ∂R I

ψn(u I ) ψn (r) + u I

∂ψn (r) , ∂R I

(25)

and expanded up to second order in the strength u I of the perturbation. The first-order term gives the Hellmann–Feynman forces. The second-order one is a quadratic functional in the ∂ψn (r)/∂R I s whose minimization yields


201

Eq. (22). This approach forms the basis of variational DFPT [6, 7], in which all the IFCs are expressed as minima of suitable functionals. The big linear system of Eq. (22) can be directly solved with iterative methods, yielding a solution that is perfectly equivalent to the self-consistent solution of the smaller linear systems of Eq. (17). The choice between the two approaches is thus a matter of computational strategy.

3.

Phonon Modes in Crystals In perfect crystalline solids, the position of the I th atom can be written as: R I = Rl + τs = l1 a1 + l2 a2 + l3 a3 + τs

(26)

where Rl is the position of the lth unit cell in the Bravais lattice and τs is the equilibrium position of the sth atom in the unit cell. Rl can be expressed as a sum of the three primitive translation vectors a1 , a2 , a3 , with integer coefficients l1 , l2 , l3 . The electronic states are classified by a wave-vector k and a band index ν: ψn (r) ≡ ψν,k (r),

ψν,k (r + Rl ) = eik·Rl ψν,k (r)

∀l,

(27)

where k is in the first Brillouin zone, i.e.: the unit cell of the reciprocal lattice, defined as the set of all vectors {G} such that Gl · Rm = 2π n, with n an integer number. Normal modes in crystals (phonons) are also classified by a wave-vector q and a mode index ν. Phonon frequencies, ω(q), and displacement patterns, Usα (q), are determined by the secular equation:

C˜ stαβ (q) − Ms ω2 (q)δst δαβ Utβ (q) = 0.

(28)

t,β αβ

The dynamical matrix, C˜ st (q), is the Fourier transform of real-space IFCs: C˜ stαβ (q) =

e−iq·Rl Cstαβ (Rl ).

(29)

l

The latter are defined as Cstαβ (l, m) ≡

∂2 E β

∂u αs (l)∂u t (m)

= Cstαβ (Rl − Rm ),

(30)

where us (l) is the deviation from the equilibrium position of atom s in the lth unit cell: R I = Rl + τs + us (l).

(31)

Because of translational invariance, the real-space IFCs, Eq. (30), depend on l and m only through the difference Rl − Rm . The derivatives are evaluated

202


at us (l) = 0 for all the atoms. The direct calculation of such derivatives in an infinite periodic system is however not possible, since the displacement of a single atom would break the translational symmetry of the system. The elements of the dynamical matrix, Eq. (29), can be written as second derivatives of the energy with respect to a lattice distortion of wave-vector q: 1 ∂2 E , C˜ stαβ (q) = β Nc ∂u ∗α s (q)∂u t (q)

(32)

where Nc is the number of unit cells in the crystal, and us (q) is the amplitude of the lattice distortion: us (l) = us (q)eiq·Rl .

(33)

In the frozen-phonon approach, the calculation of the dynamical matrix at a generic point of the Brillouin zone presents the additional difficulty that a crystal with a small distortion, Eq. (33), “frozen-in,” loses the original periodicity, unless q = 0. As a consequence, an enlarged unit cell, called supercell, is required for the calculation of IFCs at any q =/ 0. The suitable supercell for a perturbation of wave-vector q must be big enough to accommodate q as one of the reciprocal-lattice vectors. Since the computational effort needed to determine the forces (i.e., the electronic states) grows approximately as the cube of the supercell size, the frozen-phonon method is in practice limited to lattice distortions that do not increase the unit cell size by more than a small factor, or to lattice-periodical (q = 0) phonons. The dynamical matrix, Eq. (32), can be decomposed into an electronic and an ionic contribution: (34) C˜ stαβ (q) = el C˜ stαβ (q) +ion C˜ stαβ (q), where: 1 el ˜ αβ Cst (q) = Nc

+ δst

∂n(r) ∂u αs (q)

n(r)

∗

∂ V{R} (r) β

∂u t (q)

dr

∂ 2 V{R}(r) β

∂u ∗α s (q = 0)∂u t (q = 0)

dr .

(35)

The ionic contribution – the last term in Eq. (15) – comes from the derivatives of the nuclear electrostatic energy, Eq. (3), and does not depend on the electronic structure. The second term in Eq. (34) depends only on the charge density of the unperturbed system and it is easy to evaluate. The first term in Eq. (34) depends on the charge-density linear response to the lattice distortion of Eq. (33), corresponding to a perturbing potential characterized by a single wave-vector q: ∂v s (r − Rl − τs ) ∂ V{R} (r) =− eiq·Rl . (36) ∂us (q) ∂r l


203

An advantage of DFPT with respect to the frozen-phonon technique is that the linear response to a monochromatic perturbation is also monochromatic with the same wave-vector q. This is a consequence of the linearity of DFPT equations with respect to the perturbing potential, especially evident in Eq. (22). The calculation of the dynamical matrix can thus be performed for any q−vector without introducing supercells: the dependence on q factors out and all the calculations can be performed on lattice-periodic functions. Real-space IFCs can then be obtained via discrete (fast) Fourier transforms. To this end, dynamical matrices are first calculated on a uniform grid of q-vectors in the Brillouin zone: b1 b2 b3 + l2 + l3 , (37) ql1 ,l2 ,l3 = l1 N1 N2 N3 where b1 , b2 , b3 are the primitive translation vectors of the reciprocal lattice, l1 , l2 , l3 are integers running from 0 to N1 − 1, N2 − 1, N3 − 1, respectively. αβ A discrete Fourier transform produces the IFCs in real space: C˜ st (ql1 ,l2 ,l3 ) → αβ Cst (Rl1 ,l2 ,l3 ), where the real-space grid contains all R−vectors inside a supercell, whose primitive translation vectors are N1 a1 , N2 a2 , N3 a3 : Rl1 ,l2 ,l3 = l1 a1 + l2 a2 + l3 a3 .

(38)

Once this has been done, the IFCs thus obtained can be used to calculate inexpensively via (inverse) Fourier transform dynamical matrices at any q vector not included in the original reciprocal-space mesh. This procedure is known as Fourier interpolation. The number of dynamical matrix calculations to be performed, N1 N2 N3 , is related to the range of the IFCs in real space: the realspace grid must be big enough to yield negligible values for the IFCs at the boundary vectors. In simple crystals, this goal is typically achieved for relatively small values of N1 , N2 , N3 [8, 9]. For instance, the phonon dispersions of Si and Ge shown in Fig. 1 were obtained with N1 = N2 = N3 = 4.

4.

Phonons and Macroscopic Electric Fields

Phonons in the long-wavelength limit (q → 0) may be associated with a macroscopic polarization, and thus a homogeneous electric field, due to the long-range character of the Coulomb forces. The splitting between longitudinal optic (LO) and transverse optic (TO) modes at q = 0 for simple polar semiconductors (e.g., GaAs), and the absence of LO–TO splitting in nonpolar semiconductors (e.g., Si), is a textbook example of the consequences of such phenomenon. Macroscopic electrostatics in extended systems is a tricky subject from the standpoint of microscopic ab initio theory. In fact, on the one hand, the macroscopic polarization of an extended system depends on surface effects; on the

204

P. Giannozzi and S. Baroni 600 Frequency [cm-1]

Si

400

200

0

Frequency [cm-1]

Ge

⌫

K

X

⌫

L

X

W

L

Dos

⌫

K

X

⌫

L

X

W

L

Dos

400 300 200 100 0

Figure 1. Calculated phonon dispersions and density of states for crystalline Si and Ge. Experimental data are denoted by diamonds. Reproduced from Ref. [8].

other hand, the potential which generates a homogeneous electric field is both nonperiodic and not bounded from below: an unpleasant situation when doing calculations using Born–von Kármán periodic boundary conditions. In the last decade, the whole field has been revolutionized by the advent of the so called modern theory of electric polarization [10, 11]. From the point of view of lattice dynamics, a more traditional approach based on perturbation theory is however appropriate because all the pathologies of macroscopic electrostatics disappear in the linear regime, and the polarization response to a homogeneous electric field and/or to a periodic lattice distortion – which is all one needs in order to calculate long-wavelength phonon modes – is perfectly well-defined. In the long-wavelength limit, the most general expression of the energy as a quadratic function of atomic displacements, us (q = 0) for atom s, and of a macroscopic electric field, E, is:

E({u}, E) =

1 ˜ st · ut − E · ∞ · E − e us · an C us · Z s · E, 2 st αβ 8π s

(39)


205

where is the volume of the unit cell; ∞ is the electronic (i.e., clamped nuclei) dielectric tensor of the crystal; Z s is the tensor of Born effective charges ˜ is the q =0 dynamical matrix of the system, calculated [12] for atom s; and an C at vanishing macroscopic electric field. Because of Maxwell’s equations, the polarization induced by a longitudinal phonon in the q → 0 limit generates a macroscopic electric field which exerts a force on the atoms, thus affecting the phonon frequency. This, in a nutshell, is the physical origin of the LO–TO splitting in polar materials. Minimizing Eq. (39) with respect to the electric field amplitude at fixed lattice distortion yields an expression for the energy which depends on atomic displacements only, defining an effective dynamical matrix which contains an additional (“nonanalytic”) contribution: C˜ stαβ =an C˜ stαβ +na C˜ stαβ ,

(40)

where na

C˜ stαβ

4π e2 =

γ

νβ Z γα 4π e2 (q · Z s )α (q · Z t )β ν Z t qν s qγ = γν q · ∞ · q γ ,ν qγ ∞ qν

(41)

displays a nonanalytic behavior in the limit q → 0. As a consequence, the resulting IFCs are long-range in real space, with a dependence on the interatomic distance, which is typical of the dipole–dipole interaction. Because of this long-range behavior, the Fourier technique described above must be modified: a suitably chosen function of q, whose q → 0 limit is the same as in Eq. (41), is subtracted from the dynamical matrix in q-space. This procedure makes residual IFCs short-range and suitable for Fourier transform on a relatively small grid of points. The nonanalytic term previously subtracted out in q-space is then readded in real space. An example of application of such procedure is shown in Fig. 2, for phonon dispersions of some III–VI semiconductors. The link between the phenomenological parameters Z and ∞ of Eq. (39) and their microscopic expression is provided by conventional electrostatics. From Eq. (39) we obtain the expression for the electric induction D: D≡−

4π ∂ E 4π e = Z s · us + ∞ E, ∂E s

(42)

from which the macroscopic polarization, P, is obtained via D = E + 4π P. One finds the known result relating Z to the polarization induced by atomic displacements, at zero electric field:

Z αβ s

∂Pα = ; β e ∂u s (q = 0) E=0

(43)

206

P. Giannozzi and S. Baroni 400

Frequency [cm-1]

GaAs 300 200 100 0

Frequency [cm-1]

AlAs

⌫

K X

⌫

L

X

W

L

Dos

⌫

K

X

⌫

L

X

W

L

Dos

⌫

K X

⌫

L

X

W

L

Dos

⌫

K

⌫

L

X

W

L

Dos

500

250

0

Frequency [cm-1]

GaSb

300

200

100

0

Frequency [cm-1]

AlSb

400 300 200 100 0

X

Figure 2. Calculated phonon dispersions and density of states for several III-V zincblende semiconductors. Experimental data are denoted by diamonds. Reproduced from Ref. [8].


207

while the electronic dielectric-constant tensor ∞ is the derivative of the polarization with respect to the macroscopic electric field at clamped nuclei:

αβ = δαβ ∞

∂Pα + 4π . ∂Eβ u (q=0)=0

(44)

s

DFPT provides an easy way to calculate Z and ∞ from first principles [8, 9]. The polarization linearly induced by an atomic displacement is given by the sum of an electronic plus an ionic term: ∂Pα

e =− β Nc ∂u s (q = 0)

r

e ∂n(r) dr + Z s δαβ . ∂u s (q = 0)

(45)

This expression is ill-defined for an infinite crystal with Born–von Kármán periodic boundary conditions, because r is not a lattice-periodic operator. We remark, however, that we actually only need off-diagonal matrix elements / n (see the discussion of Eqs. 20 and 21). These can be ψm |r|ψn with m = rewritten as matrix elements of a lattice-periodic operator, using the following trick: ψm |r|ψn =

ψm |[HSCF , r]|ψn

, m − n

∀ m =/ n.

(46)

The quantity |ψ¯ nα = rα |ψn is the solution of a linear system, analogous to Eq. (17): (HSCF − n )|ψ¯ nα = Pc [HSCF , rα ]|ψn ,

(47)

N/2

where Pc = 1 − n=1 |ψn ψn | projects out the component over the occupiedstate manifold. If the self-consistent potential acting on the electrons is local, the above commutator is simply proportional to the momentum operator: [HSCF , r] = −

h¯ 2 ∂ . m ∂r

(48)

Otherwise, the commutator will contain an explicit contribution from the nonlocal part of the potential [13]. The final expression for the effective charges reads:

Z αβ s

N/2 4 ¯ α ∂ψn = Zs + . ψn ∂u β (q = 0) Nc n=1

(49)

The calculation of ∞ requires the response of a crystal to an applied electric field E. The latter is described by a potential, V (r) = eE · r, that is neither lattice-periodic nor bounded from below. In the linear-response regime,

208


however, we can use the same trick as in Eq. (46) and replace all the occurrences of r|ψn with |ψ¯ nα calculated as in Eq. (47). The simplest way to calculate ∞ is to keep the electric field E fixed and to iterate on the potential: ∂ VSCF (r) ∂ V (r) = + ∂E ∂E

e2 δv xc (r) + |r − r | δn(r )

∂n(r ) dr . ∂E

(50)

One finally obtains:

αβ ∞

= δαβ

N/2 ∂ψ 16π e n − ψ¯ nα ∂Eβ Nc n=1

.

(51)

Effective charges can also be calculated from the response to an electric field. In fact, they are also proportional to the force acting on an atom upon application of an electric field. Mathematically, this is simply a consequence of the fact that the effective charge can be seen as the second derivative of the energy with respect to an ion displacement and an applied electric field, and its value is obviously independent of the order of differentiation. Alternative approaches – not using perturbation theory – to the calculation of effective charges and of dielectric tensors have been recently developed. Effective charges can be calculated as finite differences of the macroscopic polarization induced by atomic displacements, which in turn can be expressed in terms of a topological quantity – depending on the phase of ground-state orbitals – called the Berry’s phase [10, 11]. When used at the same level of accuracy, the linear-response and Berry’s phase approaches yield the same results. The calculation of the dielectric tensor using the same technique is possible by performing finite electric-field calculations (the electrical equivalent of the frozen-phonon approach). Recently, practical finite-field calculations have become possible [14, 15], using an expression of the position operator that is suitable for periodic systems.

5.

Applications

The calculation of vibrational properties in the frozen-phonon approach can be performed using any methods that provide accurate forces on atoms. Localized basis-set implementations suffers from the problem of Pulay forces: the last term of Eq. (14) does not vanish if the basis set is incomplete. In order to obtain accurate forces, the Pulay term must be taken into account. The plane-wave (PW) basis set is instead free from such problem: the last term in Eq. (14) vanishes exactly even if the PW basis set is incomplete.


209

Practical implementations of DFPT equations is straightforward with PW’s and norm-conserving pseudopotentials (PPs). In a PW-PP calculation, only valence electrons are explicitly accounted for, while the electron-ionic cores interactions are described by suitable atomic PPs. Norm-conserving PPs contain a nonlocal term of the form: NL (r, r ) = V{R}

Dnm βn∗ (r − Rl − τs )βm (r − Rl − τs ).

(52)

sl n,m

The nonlocal character of the PP requires some generalizations of the formulas described in the previous section, which are straightforward. More extensive modifications are necessary for “ultrasoft” PPs [16], which are appropriate to effectively deal with systems containing transition metal or other atoms that would otherwise require a very large PW basis set when using normconserving PPs. Implementations for other kinds of basis sets, such as LMTO, FLAPW, mixed basis sets (localized atomic-like functions plus PWs) exist as well. Presently, phonon spectra can be calculated for materials described by unit cells or supercells containing up to several tens atoms. Calculations in simple semiconductors (Fig. 1 and 2) and metals (Fig. 3) are routinely performed with modest computer hardware. Systems that are well described by some flavor of DFT in terms of structural properties have a comparable accuracy in their phonon frequencies (with typical error in the order of a few percent points) and phonon-related quantities. The real interest of phonon calculations in simple systems, however, stems from the possibility to calculate real-space IFCs also in cases for which experimental data would not be sufficient to set up a reliable dynamical model (as, for instance, in AlAs, Fig. 2). The availability of IFCs in real space and thus of the complete phonon spectra allows for the accurate evaluation of thermal properties (such as thermal expansion coefficients in the quasi-harmonic approximation) and of electron–phonon coupling coefficients in metals. Calculations in more complex materials are computationally more demanding, but still feasible for a number of nontrivial systems [2]: semiconductor superlattices and heterostructures, ferroelectrics, semiconductor surfaces [18], metal surfaces, high-Tc superconductors are just a few examples of systems successfully treated in the recent literature. A detailed knowledge of phonon spectra is crucial for the explanation of phonon-related phenomena such as structural phase transitions (under pressure or with temperature) driven by “soft phonons,” pressure-induced amorphization, Kohn anomalies. Some examples of such phonon-related phenomenology are shown in Fig. 4–6. Figure 4 shows the onset of a phonon anomaly at an incommensurate q-vector under pressure in ice XI, believed to be connected to the observed amorphization under pressure. Figure 5 displays a Kohn anomaly and the related lattice instability in the phonon spectra of ferromagnetic shape-memory alloy

210


Fe

ω [cm-1]

300

200

100

H

⌫

P

H

N

P

⌫

N

Ni

ω [cm-1]

300

200

100

⌫

X

W

X

K

⌫

L

Figure 3. Calculated phonon dispersions, with spin-polarized GGA (solid lines) and LDA (dotted lines), for Ni in the face-centered cubic structure and Fe in the body-centered cubic structure. Experimental data are denoted by diamonds. Reproduced from Ref. [17].

Ni2 MnGa. Figure 6 shows a similar anomaly in the phonon spectra of the hydrogenated W(110) surface. DFT-based methods can also be employed to determine Raman and infrared cross sections – very helpful quantities when analyzing experimental data. Infrared cross sections are proportional to the square of the polarization induced by a phonon mode. For the νth zone-center (q = 0) mode,

Density-functional perturbation theory (a) 500

0 kbar

(b)

211

15 kbar

(c)

35 kbar

400

kz

ω(cm⫺1)

Z A

200

Σ

100

B

T E

Λ ∆ Γ

V C

ky

kx

0 Γ

Figure 4.

Σ

C

Y

∆

Γ

Γ

Σ

C

Y

∆

Γ

Γ

Σ

C

Y

∆

Γ

Phonon dispersions in ice XI at 0, 15, and 35 kbar. Reproduced from Ref. [19].

Γ

K

X

125 LA

frequency (cm1)

100

75 TA1 50

25 TA2 0 theory 370oK

25

250oK 50 0

0.2

0.4 0.6 q=ζ[110] 2π/a

0.8

1

Figure 5. Calculated phonon dispersion of Ni2 MnGa in the fcc Heusler structure, along the − K − Z line in the [110] direction. Experimental data taken at 250 and 370 K are shown for comparison. Reproduced from Ref. [20].

characterized by a normalized vibrational eigenvector Usβ , the oscillator strength f is given by 2 αβ β Z s Us . f = α sβ

(53)

212

P. Giannozzi and S. Baroni clean

hydrogenated

frequency (cm⫺1)

200

[110] N

100

S

[001] H

Γ

Γ

[112]

H

N

S

Γ

H

N

S

Figure 6. Phonon dispersions of the clean (left panel) and hydrogenated (right panel) W(110). Full dots indicate electron energy-loss data, open diamonds helium-atom scattering data. Reproduced from Ref. [21].

The calculation of Raman cross sections is difficult in resonance conditions, since the knowledge of excited-state Born–Oppenheimer surfaces is required. Off-resonance Raman cross sections are however simply related to the change of the dielectric constant induced by a phonon mode. If the frequency of the incident light, ωi , is much smaller than the energy band gap, the contribution of the νth vibrational mode to the intensity of the light diffused in Stokes Raman scattering is: I (ν) ∝

(ωi − ων )4 αβ r (ν), ων

(54)

where α and β are the polarizations of the incoming and outgoing light beams, ων is the frequency of the νth mode, and the Raman tensor r αβ (ν) is defined as:

r

αβ

∂χ αβ 2 (ν) = , ∂eν

(55)

where χ = (∞ − 1)/4π is the electric polarizability of the system, eν is the coordinate along the vibrational eigenvector Usβ for mode ν, and indicates an average over all the modes degenerate with the νth one. The Raman tensor can be calculated as a finite difference of the dielectric tensor with a phonon frozen-in, or directly from higher-order perturbation theory [22].


6.

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Outlook

The field of lattice-dynamical calculations based on DFT, in particular in conjunction with perturbation theory, is ripe enough to allow a systematic application to systems and materials of increasing complexity. Among the most promising fields of application, we mention the characterization of materials through the prediction of the relation existing between their atomistic structure and experimentally detectable spectroscopic properties; the study of the structural (in)stability of materials at extreme pressure conditions; the prediction of the thermal dependence of different materials properties using the quasi-harmonic approximation; the prediction of superconductive properties via the calculation of electron–phonon coupling coefficients. We conclude mentioning that sophisticated open-source codes for lattice dynamical calculations [23] are freely available for download from the web.

References [1] S. Baroni, P. Giannozzi, and A. Testa, “Green’s-function approach to linear response in solids,” Phys. Rev. Lett., 58, 1861, 1987. [2] S. Baroni, S. de Gironcoli, A. Dal Corso, and P. Giannozzi, etc. “Phonons and related crystal properties from density-functional perturbation theory,” Rev. Mod. Phys., 73, 515–562, 2001. [3] X. Gonze, “Adiabatic density-functional perturbation theory,” Phys. Rev. A, 52, 1096, 1995. [4] J. Gerratt and I.M. Mills, J. Chem. Phys., 49, 1719, 1968. [5] R.D. Amos, In: K.P. Lawley (ed.), Ab initio Methods in Quantum Chemistry – I, Wiley, New York, p. 99, 1987. [6] X. Gonze, “Perturbation expansion of variational principles at arbitrary order,” Phys. Rev. A, 52, 1086, 1995. [7] X. Gonze, “First-principles responses of solids to atomic displacements and homogeneous electric fields: Implementation of a conjugate-gradient algorithm,” Phys. Rev. B, 55, 10337, 1997. [8] P. Giannozzi, S. de Gironcoli, P. Pavone, and S. Baroni, “Ab initio calculation of phonon dispersions in semiconductors,” Phys. Rev. B, 43, 7231, 1991. [9] X. Gonze and C. Lee, “Dynamical matrices, Born effective charges, dielectric permittivity tensors, and interatomic force constants from density-functional perturbation theory,” Phys. Rev. B, 55, 10355, 1997. [10] D. Vanderbilt and R.D. King-Smith, “Electric polarization as a bulk quantity and its relation to surface charge,” Phys. Rev. B, 48, 4442, 1993. [11] R. Resta, “Macroscopic polarization in crystalline dielectrics: the geometrical phase approach,” Rev. Mod. Phys., 66, 899, 1994. [12] M. Born and K. Huang, Dynamical Theory of Crystal Lattices., Oxford University Press, Oxford, 1954. [13] S. Baroni and R. Resta, “Ab initio calculation of the macroscopic dielectric constant in silicon,” Phys. Rev. B, 33, 7017, 1986.

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P. Giannozzi and S. Baroni [14] P. Umari and A. Pasquarello, “Ab initio molecular dynamics in a finite homogeneous electric field,” Phys. Rev. Lett., 89, 157602, 2002. [15] I. Souza, J. Íñiguez, and D. Vanderbilt, “First-principles approach to insulators in finite electric fields,” Phys. Rev. Lett., 89, 117602, 2002. [16] D. Vanderbilt, “Soft self-consistent pseudopotentials in a generalized eigenvalue formalism,” Phys. Rev. B, 41, 7892, 1990. [17] A. Dal Corso and S. de Gironcoli, “Density-functional perturbation theory for lattice dynamics with ultrasoft pseudo-potentials,” Phys. Rev. B, 62, 273, 2000. [18] J. Fritsch and U. Schröder, “Density-functional calculation of semiconductor surface phonons,” Phys. Rep., 309, 209–331, 1999. [19] K. Umemoto, R.M. Wentzcovitch, S. Baroni, and S. de Gironcoli, “Anomalous pressure-induced transition(s) in ice XI,” Phys. Rev. Lett., 92, 105502, 2004. [20] C. Bungaro, K.M. Rabe, and A. Dal Corso, “First-principle study of lattice instabilities in ferromagnetic Ni2 MnGa,” Phys. Rev. B, 68, 134104, 2003. [21] C. Bungaro, S. de Gironcoli, and S. Baroni, “Theory of the anomalous Rayleigh dispersion at H/W(110) surfaces,” Phys. Rev. Lett., 77, 2491, 1996. [22] M. Lazzeri and F. Mauri, “High-order density-matrix perturbation theory,” Phys. Rev. B, 68, 161101, 2003. [23] PWscf package: www.pwscf.org. ABINIT: www.abinit.org.

1.11 QUASIPARTICLE AND OPTICAL PROPERTIES OF SOLIDS AND NANOSTRUCTURES: THE GW-BSE APPROACH Steven G. Louie1 and Angel Rubio2 1

Department of Physics, University of California at Berkeley and Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA 2 ´ ´ Departamento Fisica de Materiales and Unidad de Fisica de Materiales ´ Vasco and Centro Mixto CSIC-UPV, Universidad del Pais Donosita Internacional Phycis Center (DIPC)

We present a review of recent progress in the first-principles study of the spectroscopic properties of solids and nanostructures employing a many-body Green’s function approach based on the GW approximation to the electron self-energy. The approach has been widely used to investigate the excitedstate properties of condensed matter as probed by photoemission, tunneling, optical, and related techniques. In this article, we first give a brief overview of the theoretical foundations of the approach, then present a sample of applications to systems ranging from extended solids to surfaces to nanostructures and discuss some possible ideas for further developments.

1.

Background

A large part of research in condensed matter science is related to the characterization of the electronic properties of interacting many-electron systems. In particular, an accurate description of the electronic structure and its response to external probes is essential for understanding the behavior of systems ranging from atoms, molecules, and nanostructures to complex materials. Moreover, many characterization tools in physics, chemistry and materials science as well as electro/optical devices are spectroscopic in nature, based on the interaction 215 S. Yip (ed.), Handbook of Materials Modeling, 215–240. c 2005 Springer. Printed in the Netherlands.

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of photons, electrons, or other quanta with matter exciting the system to higher energy states. Yet, many fundamental questions concerning the conceptual and quantitative descriptions of excited states of condensed matter and their interactions with external probes are still open. Hence there is a strong need for theoretical approaches which can provide an accurate description of the excitedstate electronic structure of a system and its response to external probes. In what follows we discuss some recent progress along a very fruitful direction in the first-principles studies of the electronic excited-state properties of materials, employing a many-electron Green’s function approach based on the so-called GW approximation [1–3]. Solving for the electronic structure of an interacting electron system (in terms of the many-particle Schrödinger equation) has an intrinsic high complexity: while the problem is completely well defined in terms of the total number of particles N and the external potential V(r), its solution depends on 3N coordinates. This makes the direct search for either exact or approximate solutions to the many-body problem a task of rapidly increasing complexity. Fortunately, in the study of either ground- or excited-state properties, we seldom need the full solution to the Schrödinger equation. When one is interested in structural properties, the ground-state total energy is sufficient. In other cases, we want to study how the system responds to some external probe. Then knowledge of a few excited-state properties must be added. For instance, in a direct photoemission experiment, a photon impinges on the system and an electron is removed. In an inverse photoemission process, an electron is absorbed and a photon is ejected. In both cases we just have to deal with the gain or loss of energy of the N electron system when a single particle is added or removed, i.e., with the one-particle excitation spectrum. If the electron was not removed after the absorption of the photon, the system evolves from its ground state to a neutral excited state, and the process may be described by correlated electron–hole excitation amplitudes. At the simplest level of treating the many-electron problem, the Hartree– Fock theory (HF) is obtained by considering the ground-state wavefunction to be a single Slater determinant of single-particle orbitals. In this way the N-body problem is reduced to N one-body problems with a self-consistent requirement due to the dependence of the HF effective potential on the wavefunction. By the variational theorem, the HF total energy is a variational upper bound of the ground-state energy for a particular symmetry. The HF-eigenvalues may also be used as rough estimates of the one-electron excitation energies. The validity of this procedure hinges on the assumption that the single-particle orbitals in the N and (N-1) system are the same (Koopman’s theorem), i.e., neglecting the electronic relaxation of the system. A better procedure to estimate excitation energies is to perform self-consistent calculations for the N and (N-1) systems and subtract the total energies (this is called the “-SCF method” for excitation energies which has also been used in other theoretical frameworks such as the

Quasiparticle and optical properties of solids and nanostructures

217

density-functional theory). For infinitely extended system, this scheme gives the same result as Koopman’s theorem and more refined methods are needed to address the problem of one-particle (quasiparticle) excitation energies in solids. The HF theory in general is far from accurate because typically the wavefunction of a system cannot be written as a single determinant for the ground state and Koopman’s theorem is a poor approximation. On the other hand, within density-functional-theory (DFT), the ground-state energy of an interacting system can be exactly written as a functional of the ground-state electronic density [4]. When comparing to conventional quantum chemistry methods, this approach is particularly appealing since solving the ground-state energy does not rely on the complete knowledge of the N-electron wavefunction but only on the electronic density, reducing the problem to that of a self-consistent field calculation. However, although the theory is exact, the energy functional contains an unknown quantity called the exchange-correlation energy, E xc [n], that has to be approximated in practical implementations. For ground-state properties, in particular those of solids and larger molecular systems, present-day DFT results are comparable or even surpassing in quality to those from standard ab initio quantum chemistry techniques. Its use has continued to increase due to a better scaling in computational effort with the number of atoms in the system. As in HF theory, the Kohn–Sham eigenvalues of the DFT cannot be directly interpreted as the quasiparticle excitation energies. Such interpretation has led to the well-known bandgap problem for semiconductors and insulators: the Kohn–Sham gap is typically 30–50% less than the observed band gap. Indeed, the original formulation of the DFT is not applicable to excited states nor to problems involving time-dependent external fields, thus excluding the calculation of optical response, quasiparticle excitation spectrum, photochemistry, etc. Theorems have, however, been proved subsequently for time-dependent density functional theory (TDDFT) which extends the applicability of the approach to excited-state phenomena [5, 6]. The main result of TDDFT is a set of time-dependent Kohn–Sham equations that include all the many-body effects through a time-dependent exchange-correlation potential. As for static DFT, this potential is unknown and has to be approximated in any practical application. TDDFT has been applied with success to the calculations of quantities such as the electron polarizabilities for the optical spectra of finite systems. However, TDDFT encounters problems in studying spectroscopic properties of extended systems [7] and severely underestimates the high-lying excitation energies in molecules when simple exchange and correlation functionals are employed. These failures are related to our ignorance of the exact exchangecorrelation potential in DFT. The actual functional relation between density, n(r), and the exchange-correlation potential, Vxc (r), is highly non-analytical and non-local. A very active field of current research is in the search of robust, new exchange-correlation functionals for real material applications.

218


Alternatively, a theoretically well-grounded and rigorous approach for the excited-state properties of condensed matter is the interacting Green’s function approach. The n-particle Green’s function describes the propagation of the n-particle amplitude in an interacting electron system. It provides a proper framework for accurately computing the N-particle excitation properties. For example, knowledge of the one-particle and two-particle Green’s functions yields information, respectively, on the quasiparticle excitations and optical response of a system. The use of this approach for practical study of the spectroscopic properties of real materials is the focus of the present review. In the remainder of the article, we first present a brief overview of the theoretical framework for many-body perturbation theory and discuss the firstprinciples calculation of properties related to the one- and two-particle Green’s functions within the GW approximation to the electron self-energy operator. Then, we present some selected examples of applications to solids and reduced dimensional systems. Finally, some conclusions and perspectives are given.

2.

Many-body Perturbation Theory and Green’s Functions

A very successful and fruitful development for computing electron excitations has been a first-principles self-energy approach [1–3, 8] in which the quasiparticle’s (excited electron or hole) energy is determined directly by calculating the contribution of the dynamical polarization of the surrounding electrons. In many-body theory, this is obtained by evaluating the evolution of the amplitude of the added particle via the single-particle Green’s function, G(xt, x t ) = −iN |T {ψ(xt)ψ † (x t )}|N ,∗ from which one obtains the dispersion relation and lifetime of the quasiparticle excited state. There are no adjustable parameters in the theory and, from the equation of motion of the single-particle Green’s function, the quasiparticle energies E nk and wavefunctions ψnk are determined by solving a Schrödinger-like equation: (T + Vext + VH )ψk (r) +

dr(r,r ; E nk )ψnk (r ) = E nk ψnk (r),

(1)

where T is the kinetic energy operator, Vext is the external potential due to the ions, VH is the Hartree potential of the electrons, and is the self-energy operator where all the many-body exchange and correlation effects are included. The self-energy operator describes an effective potential on the quasiparticle * This corresponds to the Green’s function at zero temperature where |N > is the many-electron ground state, ψ(xt) is the field operator in the Heisenberg picture, x stands for the spatial coordinates r plus the spin coordinate, and T is the time ordered operator. In this context, ψ † (xt)|N> represents an (N + 1)-electron state in which an electron has been added at time t onto position r.


219

resulting from the interaction with all the other electrons in the system. In general is non-local, energy dependent and non-Hermitian, with the imaginary part giving the lifetime of the excited state. Similarly, from the two-particle Green’s function, we can obtain the correlated electron–hole amplitude and excitation spectrum, and hence the optical properties. For details of the Green’s function formalism and many-body techniques applied to condensed matter, we refer the reader to several comprehensive papers in the literature [2, 3, 7–10]. Here we shall just present some of the main equations used for the quasiparticle and optical spectra calculations. (To simplify the presentation, we use in the following atomic units, e = h¯ = m = 1.) In standard textbook, the unperturbed system is often taken to be the noninteracting system of electrons under the potential Vion(r) + VH (r). However, for rapid convergence in a perturbation series, it is better to start from a different non-interacting or mean-field scenario, like the Kohn–Sham DFT system, which already includes an attempt to describe exchange and correlations in the actual system. Also, in a many-electron system, the Coulomb interaction between two electrons is readily screened by a dynamic rearrangement of the other electrons, reducing its strength. It is more natural to describe the electron–electron interaction in terms of a screened Coulomb potential W and formulate the self energy as a perturbation series in terms of W. In this approach [1–3], the electron self-energy can then be obtained from a self-consistent set of Dyson-like equations: P(12) = −i

d(34)G(13)G(41+ ) (34, 2)

W (12) = v(12) + (12) = i

d(34)W (13)P(34)v(42)

d(34)G(14+ )W (13)(42, 3)

G(12) = G 0 (12) +

d(34)G 0 (13)[(34) − δ(34)Vxc (4)]G(42)

(12, 3) = δ(12)δ(13) +

(2) (3) (4) (5)

d(4567)[δ(12)/δG(45)] × G(46)G(75)(67, 3)

(6)

where 1 ≡ (x1 , t1 ) and 1+ ≡ (x1 , t1 + η)(η >0 infinitesimal). v stands for the bare Coulomb interaction, P is the irreducible polarization, W is the dynamical screened Coulomb interaction, and is the so-called vertex function. Here G 0 is the single-particle DFT Green’s function, G 0 (x, x ; ω) = n ψn (x)ψn∗ (x)/[ω−εn −iηsgn(µn )], with η a positive infinitesimal and ψn and εn the corresponding DFT wavefunctions and eigenenergies. This way of writing down the equations is in fact appealing since it highlights the important physical ingredients: the polarization (which contains the response of the system to the additional particle or hole) is built up by the creation of particle–hole pairs

220


(described by the two-particle Green’s functions). The vertex function contains the information that the hole and the electron interact. This set of equations defines an iterative approach that allows us to gather information about quasiparticle excitations and dynamics. The iterative approach of course has to be approximated. We now describe some of the approximations used in the literature to address quasiparticle excitations and their subsequent extension to optical spectroscopy and exciton states.

3.

Quasiparticle Excitations: the GW Approach

In practical first-principles implementations, the GW approximation [1] is employed in which the self-energy operator is taken to be the first order term in a series expansion in terms of the screened Coulomb interaction W and the dressed Green function G of the electron P(12) = −i G(12)G(21) (12) = i G(12+ )W (12)

(7) (8)

(in frequency space: (r, r ; ω) = i/2π dω e−iω η G(r, r , ω − ω )W (r, r , ω )). Vertex corrections are not included in this approximation. This corresponds to the simplest approximation for (123), assuming it to be diagonal in space and time coordinates, i.e., (123) = δ(12)δ(13). This has to be complemented with Eq. (5) above. Thus, even at the GW level, we have a many-body self-consistent problem. Most ab initio GW applications do this self-consistent loop by (1) taking the DFT results as the mean field and (2) varying the energy of the quasiparticle but keeping fixed its wavefunction (equal to the DFT wavefunction). This corresponds to the G 0 W0 scheme for the calculation of quasiparticle energy as a first-order perturbation to the Kohn–Sham energy εnk : E nk ≈ εnk + nk|(E nk ) − Vxc |nk,

(9)

where Vxc is the exchange-correlation potential within DFT and |nk > is the corresponding wavefunction. This “G 0 W0 ” approximation reproduces to within 0.1 eV the experimental band gaps for many semiconductors and insulators and their surfaces, thus circumventing the well-known bandgap problem [2, 3]. Also it gives much better HOMO–LUMO gaps and ionization energies in localized systems, and results for the lifetimes of hot electrons in metals and image states at surfaces [7]. For some systems, the quasiparticle wavefunction can differ significantly from the DFT wavefunction; one then needs to solve the quasiparticle equation, Eq. (1), directly.


4.

221

Optical Response: the Bethe–Salpeter Equation

From Eqs. (2)–(6) for the GW self energy, we have a non-vanishing functional derivative δ/δG. One obtains a second-order correction to the bare vertex (1) (123) = δ(12)δ(13): (2)

(123) = δ(12)δ(13) +

d(4567)[δ (1) (12)/δG 0 (45)]G 0 (46) × G 0 (75) (1) (673).

(10)

This can be viewed as the linear response of the self-energy to a change in the total potential of the system. The vertex correction accounts for exchangecorrelation effects between an electron and the other electrons in the screening density cloud. In particular it includes the electron–hole interaction (excitonic effects) in the dielectric response∗ . Indeed, the functional derivative of G is responsible for the attractive direct term in the electron–hole interaction that goes into the effective two-particle equation, the Bethe–Salpeter equation, which determines the spectrum and wavefunctions of the correlated electron– hole neutral excitations created, for example, in optical experiments. Taking as first-order self energy (1) = G 0 W0 , it is easy to derive a Bethe–Salpeter equation, which correctly yields features like bound excitons and changes in absorption strength in the optical absorption spectra. Within this scheme [7, 10], the effective two-particle Hamiltonian takes (when static screening is used in W) a particularly simple, energy-independent form

[(εn1 − εn2 )δn1n3 δn2n4 + u (n1n2)(n3n4) − W(n1n2)(n3n4)]AS(n3n4)

n3n4

= S AS (n1n2)

(11)

where AS is the electron–hole amplitude and the matrix elements are taken with respect to the quasiparticle wavefunctions n 1 , . . . , n 4 as follows: u (n1n2)(n3n4) = n 1 n 2 |u|n 3 n 4 and W(n1n2)(n3n4) = n 1 n 3 |W |n 2 n 4 , with u equal to the Coulomb potential v except for the long-range component q = 0 that is set to zero (that is, u(q)=4π/q 2 but with u(0) = 0). The solution of Eq. (11) allows one to construct the optical absorption spectrum from the imaginary part of the macroscopic dielectric function ε M : Im[εM (ω)] = 16π e2 /ω2

|ê· < 0|i/h¯ [H, r]|S > |2 δ(ω − S )

(12)

S

* Vertex corrections and self-consistency tend to cancel to a large extent for the 3D homogeneous electron

gas. This cancellation of vertex corrections with self-consistency seems to be a quite general feature. However, there is no formal justification for it and further work along the direction of including consistently dynamical effects and vertex corrections should be explored (Aryasetiawan and Gunnarsson, 1998; and references therein).

222


where eˆ is the normalized polarization vector of the light and i/h¯ [H ,r] is the single-particle velocity operator. The sum runs over all the excited states |S> of the system (with excitation energy S ) and |0 > is the ground state. One of the main effects of the electron–hole interaction is the coupling of different electron–hole configurations (denoted by |he >) which modifies the usual interband transition matrix elements that appear in Eq. (12) to: electrons (h,e) = holes AS < h|i/h¯ [H, r]|e >. h e In this context, the Bethe–Salpeter approach to the calculation of two-particle excited states is a natural extension of the GW approach for the calculation of one-particle excited states, within a same theoretical framework and set of approximations (the GW-BSE scheme). As we shall see below, GW-BSE calculations have helped elucidate the optical spectra for a wide range of systems from nanostructures to bulk semiconductors to surfaces and 1D polymers and nanotubes.

5.

Applications to Bulk Materials and Surfaces

Since the mid 1980s, the GW approach has been employed with success to the study of quasiparticle excitations in bulk semiconductors and insulators [2, 3, 9, 11, 12]. In Fig. 1, the calculated GW band gaps of a number of insulating materials are plotted against the measured quasiparticle gaps [11]. A perfect agreement between theory and experiment would place the data points on the diagonal line. As seen from the figure, the Kohn–Sham gaps in the local density approximation (LDA) significantly underestimate the experimental values, giving rise to the bandgap problem. Some of the Kohn–Sham gaps are even negative. However, the GW results (which provide an appropriate description of particle-like excitations in an interacting systems) are in excellent agreement with experiments for a range of materials – from the small gap semiconductors such as InSb, to moderate size gap materials such as GaN and solid C60 , and to the large gap insulators such as LiF. In addition, the GW quasiparticle band structures for semiconductors and conventional metals in general compare very well with data from photoemission and inverse photoemission measurements. Figure 2 depicts the calculated quasiparticle band structure of germanium [11] and copper [13] as compared to photoemission data for the occupied states and inverse photoemission data for the unoccupied states. For Ge, the agreement is within the error bars of experiments. In fact, the conduction band energies of Ge were theoretically predicted before the inverse photoemission measurement. The results for Cu agree with photoemission data to within 30 meV for the highest d-band, correcting 90% of the LDA error. The energies of the other d-bands throughout the Brillouin zone are reproduced within 300 meV, and the maximum error (about 600 meV) is found for the bottom valence band at the


223

Theoretical Band Gap (eV)

15

10 Quasiparticle theory

5 Many-body corrections

LDA

0 0

5 10 Experimental Band Gap (eV)

15

Figure 1. Comparison of the GW bandgap with experiment for a wide range of semiconductors and insulators. The Kohn–Sham eigenvalue gaps calculated within the local density approximation (LDA) are also included for comparison. (after Ref. [11]).

Figure 2. Calculated GW quasiparticle band structure of Ge (left panel) and Cu (right panel) as compared with experiments (open and full symbols). In the case of Cu we also provide the DFT-LDA band structure as dashed lines. (after Ref. [11, 13]).

224


Figure 3. Computed GW quasiparticle bandstructure for the Si(111) 2 × 1 surface compared with experimental results (dots). On the left we show a model of the surface reconstruction (after Ref. [15]).

point, where only 50% of the LDA error is corrected. This level of agreement for the d-bands cannot be obtained without including self-energy contributions∗ . Similar results have been obtained for other materials and even for some nonconventional insulating systems such as the transition metal oxides and metal hydrides. The GW approach has also been used to investigate the quasiparticle excitation spectrum of surfaces, interfaces and clusters. Figure 3 gives the example of the Si(111)2 × 1 surface [14, 15]. This surface has a very interesting geometric and electronic structure. At low temperature, to minimize the surface energy, the surface undergoes a 2 × 1 reconstruction with the surface atoms forming buckled π -bonded chains. The ensuing structure has an occupied and an unoccupied quasi-1D surface-state band, which are dispersive only along the π -bonded chains and give rise to a quasiparticle surface-state bandgap of 0.7 eV that is very different from the bulk Si bandgap of 1.2 eV. The calculated quasiparticle surface-state bands are compared to photoemission and inversed photoemission data in Fig. 3. As seen in the figure, both the calculated surface-state band dispersion and bandgap are in good agreement with experiment, and these results are also in accord with results from scanning tunneling spectroscopy (STS) which physically also probes quasiparticle excitations. But, a long-standing puzzle in the literature has been that the measured surface-state gap of this system from

* On the other hand, the total bandwidth is still larger than the measured one. This overestimate of the GW bandwidth for metals with respect to the experimental one seems to be a rather general feature, which is not yet properly understood.


225

optical experiments differs significantly (by nearly 0.3 eV) from the quasiparticle gap, indicative of perhaps very strong electron-hole interaction on this surface. We shall take up this issue later when we discuss optical response. Owing to interactions with other excitations, quasiparticle excitations in a material are not exact eigenstates of the system and thus possess a finite lifetime. The relaxation lifetimes of excited electrons in solids can be attributed to a variety of inelastic and elastic scattering mechanisms, such as electron–electron (e–e), electron–phonon (e–p), and electron–imperfection interactions. The theoretical framework to investigate the inelastic lifetime of the quasiparticle (due to electron–electron interaction as manifested in the imaginary part of ) has been based for many years on the electron gas model of Fermi liquids, characterized by the electron-density parameter rs . In this simple model for either electrons or holes with energy E very near the Fermi level, the inelastic lifetime is found to be, in the high-density limit (rs 475 K). A simple improvement over the harmonic approximation, called the quasiharmonic approximation, is obtained by employing volume-dependent force constant tensors. This approach maintains all the computational advantages of the harmonic approximation while permitting the modeling of thermal expansion. The volume dependence of the phonon frequencies induced by the volume dependence of the force constants is traditionally described by the Grüneisen parameter γkb = −∂ ln νb (k)/∂ ln V . However, for the purpose of

354

A. van de Walle and M. Asta

Figure 2. Temperature-dependence of the free energy of the θ and θ phases of the Al2 Cu compound. Insets show the crystal structures of each phase and the corresponding phonon density of states. Dashed lines indicate region of metastability and the θ phase is seen to become stable above about 475 K. (Adapted from Ref. [5] with the permission of the authors.)

modeling thermal expansion, it is more convenient to directly parametrize the volume-dependence of the free energy itself. This dependence has two sources: the change in entropy due to the change in the phonon frequencies and the elastic energy change due to the expansion of the lattice: F(T, V ) = E 0 (V ) + Fvib (T, V )

(11)

where E 0 (V ) is the energy of a motionless lattice whose unit cell is constrained to remain at volume V, while Fvib (T, V ) is the vibrational free energy of a harmonic system constrained to remain with a unit cell volume V at temperature T . The equilibrium volume V ∗ (T ) at temperature T is obtained by minimizing F(T, V ) with respect to V . The resulting free energy F(T ) at temperature T is then given by F(T, V ∗ (T )). The quasiharmonic approximation has been shown to provide a reliable description of thermal expansion of numerous elements up to their melting points, as illustrated in Fig. 3. First-principles calculations can be used to provide the necessary input parameters for the above formalism. The so-called direct force method proceeds by calculating, from first principles, the forces experienced by the atoms in response to various imposed displacements and by determining the value of the force constant tensors that match these forces through a least-squares fit.

First-principles modeling of phase equilibria 2.0 Na

0.0 ⫺1.0 ⫺2.0

∆1/1(%)

∆1/1(%)

1.0

355

A1

1.0 0.0

⫺1.0 0

100 200 300 400 Temperature (K)

0

200 400 600 800 1000 Temperature (K)

Figure 3. Thermal expansion of selected metals calculated within the quasiharmonic approximation. (Reproduced from Ref. [6] with the permission of the authors.)

Note that the simultaneous displacements of the periodic images of each displaced atom due to the periodic boundary conditions used in most ab initio methods typically requires the use of a supercell geometry, in order to be able to sample all the displacements needed to determine the force constants. While the number of force constants to be determined is in principle infinite, in practice, it can be reduced to a manageable finite number by noting that the force constant tensor associated with two atoms that lie farther than a few nearest neighbor shells can be accurately neglected for many systems. Alternatively, linear response theory (Rabe, Chapter 1) can be used to calculate the dynamical matrix D(k) directly using second-order perturbation theory, thus circumventing the need for supercell calculations. Linear response theory is also particularly useful when a system is characterized by non-negligible long-range force-constants, as in the presence of Fermi-surface instabilities or long-ranged electrostatic contributions. The above discussion has centered around the application of harmonic (or quasiharmonic) approximations to the statistical modeling of vibrational contributions to free energies of solids. While harmonic theory is known to be highly accurate for a wide class of materials, important cases exist where this approximation breaks down due to large anharmonic effects. Examples include the modeling of ferroelectric and martensitic phase transformations where the high-temperature phases are often dynamically unstable at zero temperature, i.e., their phonon spectra are characterized by unstable modes. In such cases, effective Hamiltonian methods have been developed to model structural phase transitions from first principles (Rabe, Chapter 1). Alternatively, direct application of ab initio molecular-dynamics offers a general framework for modeling thermodynamic properties of anharmonic solids [1, 2].

356

3.


Thermodynamics of Compositionally Disordered Solids

We now relax the main assumption made in the previous section, by allowing atoms to exit the neighborhood of their local equilibrium position. This is accomplished by considering every possible way to arrange the atoms on a given lattice. As illustrated in Fig. 1(b), the state of order of an alloy can be described by occupation variables σi specifying the chemical identity of the atom associated with lattice site i. In the case of a binary alloy, the occupations are traditionally chosen to take the values +1 or −1, depending on the chemical identity of the atom. Returning to Eq. (2), all the thermodynamic information of a system is contained in its partition function Z and in the case of a crystalline alloy system, the sum over all possible states of the system can be conveniently factored as follows: Z=

σ v∈σ e∈v

exp[−β E(σ, v, e)]

(12)

where β = (k B T )−1 and where • σ denotes a configuration (i.e., the vector of all occupation variables); • v denotes the displacement of each atom away from its local equilibrium position; • e is a particular electronic state when the nuclei are constrained to be in a state described by σ and v; and • E(σ, v, e) is the energy of the alloy in a state characterized by σ , v and e. Each summation defines an increasingly coarser level of hierarchy in the set of microscopic states. For instance, the sum over v includes all displacements such that the atoms remain close to the undistorded configuration σ . Equation (12) implies that the free energy of the system can be written as

F(T ) = −kB T ln

σ

exp[−β F(σ, T )]

(13)

where F(σ, T ) is nothing but the free energy of an alloy with a fixed atomic configuration, as obtained in the previous section

F(σ, T ) = −kB T ln

v∈σ e∈v

exp[−β E(σ, v, e)]

(14)

The so-called “coarse graining” of the partition function illustrated by Eq. (13) enables, in principle, an exact mapping of a real alloy onto a simple lattice model characterized by the occupation variables σ and a temperaturedependent Hamiltonian F(σ, T ) [7, 8].

First-principles modeling of phase equilibria

357

Although we have reduced the problem of modeling the thermodynamic properties of configurationally disordered solids to a more tractable calculation for a lattice model, the above formalism would still require the calculation of the free energy for every possible configuration σ , which is computationally intractable. Fortunately, the configurational dependence of the free energy can often be parametrized using a convenient expansion known as a cluster expansion [7, 9]. This expansion takes the form of a polynomial in the occupation variables F(σ, T ) = J∅ +

Ji σi +

i

i, j

Ji j σi σ j +

Ji j k σi σ j σk + · · ·

i, j,k

where the so-called effective cluster interactions (ECI) J∅ , Ji , Ji j , . . . , need to be determined. The cluster expansion can be recast into a form which exploits the symmetry of the lattice by regrouping the terms as follows F (σ, T ) =

α

m a Ja

σi

i∈α

where α is a cluster (i.e., a set of lattice sites) and where the summation is taken over all clusters that are symmetrically distinct while the average . . .

is taken over all clusters α that are symmetrically equivalent to α. The multiplicity m α weight each term by the number of symmetrically equivalent clusters in a given reference volume (e.g., a unit cell). While the cluster expansion is presented here in the context of binary alloys, an extension to multicomponent alloys (where σi can take more than two different values) is straightforward [9]. It can be shown that when all clusters α are considered in the sum, the cluster expansion is able to represent any function of configuration σ by an appropriate selection of the values of Jα . However, the real advantage of the cluster expansion is that, for many systems, it is found to converge rapidly. An accuracy that is sufficient for phase diagram calculations can often be achieved by keeping only clusters α that are relatively compact (e.g., short-range pairs or small triplets, as illustrated in the left panel of Fig. 4). The unknown parameters of the cluster expansion (the ECI Jα ) can then determined by fitting them to F(σ, T ) for a relatively small number of configurations σ obtained from first-principles computations. Once the ECI have been determined, the free energy of the alloy for any given configuration can be quickly calculated, making it possible to explore a large number of configurations without recalculating the free energy of each of them from first principles. In some applications the development of a converged cluster expansion can be complicated by the presence of long-ranged interatomic interactions mediated by electronic-structure (Fermi-surface), electrostatic and/or elastic effects. Long-ranged interactions lead to an increase in the number of ECIs

358


Figure 4. Typical choice of clusters (left) and structures (right) used for the construction of a cluster expansion on the hcp lattice. Big circles, small circles and crosses represent consecutive close-packed planes of the hcp lattice. Concentric circles represent two sites, one above the other in the [0001] direction. The unit cell of the structures (right) along the (0001) plane is indicated by lines while the third lattice vector, along [0001], is identical to the one of the hcp primitive cell. (Adapted, with the permission of the authors, from Ref. [10], a first-principles study of the metastable hcp phase diagram of the Ag–Al system.)

that must be computed, and a concomitant increase in the number of configurations that must be sampled to derive them. For metals it has been demonstrated how long-ranged electronic interactions can be derived from perturbation theory using coherent-potential approximations to the electronic structure of a configurationally disordered solid as a reference state [11]. Effective approaches to modeling long-ranged elastically mediated interactions have also been formulated [12]. Such elastic effects are known to be particularly important in describing the thermodynamics of mixtures of species with very large differences in atomic “size”. The cluster expansion tremendously simplifies the search for the lowest energy configuration at each composition of the alloy system. Determining these ground states is important because they determine the general topology of the alloy phase diagram. Each ground state is typically associated with one of the stable phases of the alloy system. There are three main approaches to identify the ground states of an alloy system. With the enumeration method, all the configurations whose unit cell contains less than a given number of atoms are enumerated and their energy


359

is quickly calculated using the value of F(σ, 0) predicted from the cluster expansion. The energy of each structure can then be plotted as a function of its composition (see Fig. 5) and the points touching the lower portion of the convex hull of all points indicate the ground states. While this method is approximate, as it ignores ground states with unit cell larger than the given threshold, it is simple to implement and has been found to be quite reliable, thanks to the fact that most ground states indeed have a small unit cell. Simulated annealing offers another way to find the ground states. It proceeds by generating random configurations via MC simulations using the Metropolis algorithm (G. Gilmer, Chapter 2) that mimic the ensemble sampled in thermal equilibrium at a given temperature. As the temperature is lowered, the simulation should converge to the ground state. Thermal fluctuations are used as an effective means of preventing the system from getting trapped in local minima of energy. While the constraints on the unit cell size are considerably relaxed relative to the enumeration method, the main disadvantage of this method is that, whenever the simulation cell size is not an exact multiple of the ground state unit cell, artificial defects will be introduced in the simulation that need to be manually identified and removed. Also, the risk of obtaining local rather than global minima of energy is not negligible and must be controlled by adjusting the rate of decay of the simulation temperature.

Figure 5. Ground state search using the enumeration method in the Scx -Vacancy1−x S system. Diamonds represent the formation energies of about 3×106 structures, predicted from a cluster expansion fitted to LDA energies. The ground states, indicated by open circles, are the structures whose formation energy touches the convex hull (solid line) of all points. (Reproduced from Ref. [13], with the permission of the authors.)

360


Finally, there exists an exact, although computational demanding, algorithm to identify the ground states [14]. This approach relies on the fact that σ the cluster expansion is linear in the correlations σα ≡ i∈a i . Moreover, it can be shown that the set of correlations σα that correspond to “real” structures can be defined by a set of linear inequalities. These inequalities are the result of lattice-specific geometric constraints and there exists systematic methods to generate them [14]. As an example of such constraints, consider the fact that it is impossible to construct a binary configuration on a triangular lattice where the nearest neighbor pair correlations take the value −1 (i.e., where all nearest neighbors are between different atomic species). Since both the objective function and the constraints are linear in the correlations, linear programming techniques can be used to determine the ground states. The main difficulties associated with this method is the fact that the resulting linear programming problem involves a number of dimensions and a number of inequalities that grows exponentially fast with the range of interactions included in the cluster expansion. Once the ground states have been identified, thermodynamic properties at finite temperature must be obtained. Historically, the infinite summation defining the alloy partition function has been approximated through various mean-field methods [7, 14]. However, the difficulties associated with extending such methods to systems with medium to long-ranged interactions, and the increase in available computational power enabling MC simulations to be directly applied, have led to reduced reliance upon these techniques more recently. MC simulations readily provide thermodynamic quantities such as energy or composition by making use of the fact that averages over an infinite ensemble of microscopic states can be accurately approximated by averages over a finite number of states generated by “importance” sampling. Moreover, quantities such as the free energy, which cannot be written as ensemble averages, can nevertheless be obtained via thermodynamic integration (Frenkel, Chapter 2; de Koning, Chapter 2) using standard thermodynamic relationships to rewrite the free energy in terms of integrals of quantities that can be obtained via ensemble averages. For instance, since energy E(T ) and free energy F(T ) are related through E(T ) = ∂(F (T )/T )/∂(1/T ) we have T

F(T0 ) F(T ) − =− T T0

T0

E(T ) dT T2

(15)

and free energy differences can therefore be obtained from MC simulations providing E (T ). Figures 6 and 7 show two phase diagrams obtained by combining first principles calculations, the cluster expansion formalism and MC simulations, an approach which offers the advantage of handling, in a


361

Figure 6. Calculated composition–temperature phase diagram for a metastable hcp Ag–Al alloy. Note that the cluster expansion formalism enables a unified treatment of both solid solutions and ordered compounds. (Reproduced from Ref. [10], with the permission of the authors.)

Figure 7. Calculated composition–temperature solid-state phase diagram for a rocksalt-type CaO–MgO alloy. The inclusion of lattice vibrations via the coarse-graining formalism is seen to substantially improve in agreement with experimental observations (filled circles). (Reproduced from Ref. [15], with the permission of the authors.)

362


unified framework, both ordered phases (with potential thermal defects) and disordered phases (with potential short-range order).

4.

Liquids and Melting Transitions

While first-principles thermodynamic methods have found the widest application in studies of solids, recent progress has been realized also in the development and application of methods for ab initio calculations of solid–liquid phase boundaries. This section provides a brief overview of such methods, based upon the application of thermodynamic integration methods within the framework of ab initio molecular dynamics simulations. Consider the ab initio calculation of the melting point for an elemental system, as was first demonstrated by Sugino and Car [1] in an application to elemental Si. The approach is based on the use of thermodynamic-integration methods to compute temperature-dependent free energies for bulk solid and liquid phases. Let U1 (r1 , r2 , . . . , r N ) denote the DFT potential energy for a collection of ions at positions (r1 , . . . , r N ), while U0 (r1 , r2 , . . . , r N ) corresponds to the energy of the same collection of ions described by a reference classical-potential model. We suppose that the free energy of the reference system, F0 , has been accurately calculated, either analytically (as in the case of an Einstein crystal) or using the atomistic simulation methods reviewed by Kofke and Frenkel in Chapter 2. We proceed to calculate the difference F1 − F0 between the DFT free energy (F1 ) and F0 employing the statistical-mechanical relation: F1 − F0 =

1 0

dUλ dλ dλ

1

= λ

dλ U1 − U0 λ

(16)

0

where the brackets · · · λ denote an average over the ensemble generated by the potential energy Uλ = λU1 + (1 − λ)U0 . In practice, · · · λ can be calculated from a time average over an MD trajectory generated with forces derived from the hybrid energy Uλ . The integral in Eq. (16) is evaluated from results computed for a discrete set of λ values, or from a time average over a simulation where λ is slowly “switched” on from zero to one. Practical applications of this approach rely on the careful choice of the reference system to provide energies that are sufficiently “close” to DFT to allow the ensemble averages in Eq. (16) to be precisely calculated from relatively short MD simulations. It should be emphasized that the approach outlined in this paragraph, when applied to the solid phase, provides a framework for accurately calculating anharmonic contributions to the vibrational free energy. Figure 8 shows results derived from the above procedure by Sugino and Car [1] in an application to elemental Si (using the Stillinger–Weber potential as a reference system). Temperature-dependent chemical potentials for solid and

Chemical potential (eV/atom)


363

0.0 ⫺0.2 ⫺0.4 ⫺0.6 ⫺0.8 0.0

Solid Liquid

0.4

0.8 1.2 1.6 Temperature (⫻ 1000 K)

2.0

Figure 8. Calculated chemical potential of solid and liquid silicon. Full lines correspond to theory and dashed lines to experiments. (Reproduced from Ref. [1], with the permission of the authors.)

liquid phases (referenced to the zero-temperature free energy of the crystal) are plotted with symbols and are compared to experimental data represented by the dashed lines. It can be seen that the temperature-dependence of the solid and liquid free energies (i.e., the slopes of the curves in Fig. 8) are accurately predicted. Relative to the solid, the liquid chemical potentials are approximately 0.1 eV/atom lower than experiment, leading to a calculated melting temperature that is approximately 300 K lower than the measured value. Comparable and even somewhat higher accuracies have been demonstrated in more recent applications of this approach to the calculation of melting temperatures in elemental metal systems (see, e.g., the references cited in [2]). The above formalism has been extended as a basis for calculating solid and liquid chemical potentials in binary mixtures [2]. In this application, thermodynamic integration for the liquid phase is used to compute the change in free energy accompanying the continuous interconversion of atoms from solute to solvent species. Such calculations form the basis for extracting solute and solvent atom chemical potentials. For the solid phase the vibrational free energy of formation of substitutional impurities is extracted either within the harmonic approximation (along the lines described above) and/or from thermodynamic integration to derive anharmonic contributions. In applications to Fe-based systems relevant to studies of the Earth’s core, the approach has been used to compute the equilibrium partitioning of solute atoms between

364


solid and liquid phases in binary mixtures at pressures that are beyond the range of direct experimental measurements.

5.

Outlook

The techniques described in this article provide a framework for computing the thermodynamic properties of elements and alloys from first principles, i.e., requiring, in principle, only the atomic numbers of the elemental constituents as input. In the most favorable cases, these methods have been demonstrated to yield finite-temperature thermodynamic properties with an accuracy that is limited only by the approximations inherent in electronic DFT. For a growing number of metallic alloy systems, such accuracy can be comparable to that achievable in direct measurements of thermodynamic properties. In such cases, ab initio methods have found applications as a framework for augmenting the experimental databases that form the basis of “computationalthermodynamics” modeling in the design of alloy microstructure. Firstprinciples methods offer the advantage of being able to provide estimates of thermodynamic properties in situations where direct experimental measurements are difficult due to constraints imposed by sluggish kinetics, metastability or extreme conditions (e.g., high pressures or temperatures). In the development of new materials, first-principles methods can be employed as a framework for rapidly assessing the thermodynamic stability of hypothetical structures before they are synthesized. With the continuing increase in computational power and improvements in the accuracy of first-principles electronicstructure methods, it is anticipated that ab initio techniques will find growing applications in predictive studies of phase stability for a wide range of materials systems.

References [1] O. Sugino and R. Car, “Ab initio molecular dynamics study of first-order phase transitions: melting of silicon,” Phys. Rev. Lett., 74, 1823, 1995. [2] D. Alfè, M.J. Gillan, and G.D. Price, “Ab initio chemical potentials of solid and liquid solutions and the chemistry of the Earth’s core,” J. Chem. Phys., 116, 7127, 2002. [3] N.D. Mermin, “Thermal properties of the inhomogeneous electron gas,” Phys. Rev., 137, A1441, 1965. [4] A.A. Maradudin, E.W. Montroll, and G.H. Weiss, Theory of Lattice Dynamics in the Harmonic Approximation, 2nd edn., Academic Press, New York, 1971. [5] C. Wolverton and V. Ozoli¸nsˇ, “Entropically favored ordering: the metallurgy of Al2 Cu revisited,” Phys. Rev. Lett., 86, 5518, 2001. [6] A.A. Quong and A.Y. Lui, “First-principles calculations of the thermal expansion of metals,” Phys. Rev. B, 56, 7767, 1997.


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[7] D. de Fontaine, “Cluster approach to order-disorder transformation in alloys,” Solid State Phys., 47, 33, 1994. [8] A. van de Walle and G. Ceder, “The effect of lattice vibrations on substitutional alloy thermodynamics,” Rev. Mod. Phys., 74, 11, 2002. [9] J.M. Sanchez, F. Ducastelle, and D. Gratias, “Generalized cluster description of multicomponent systems,” Physica, 128A, 334, 1984. [10] N.A. Zarkevich and D.D. Johnson, “Predicted hcp Ag–Al metastable phase diagram, equilibrium ground states, and precipitate structure,” Phys. Rev. B, 67, 064104, 2003. [11] G.M. Stocks, D.M.C. Nicholson, W.A. Shelton, B.L. Gyorffy, F.J. Pinski, D.D. Johnson, J.B. Staunton, P.E.A. Turchi, and M. Sluiter, “First Principles Theory of Disordered Alloys and Alloy Phase Stability,” In: P.E. Turchi and A. Gonis (eds.), NATO ASI on Statics and Dynamics of Alloy Phase Transformation, vol. 319, Plenum Press, New York, p. 305, 1994. [12] C. Wolverton and A. Zunger, “An ising-like description of structurally-relaxed ordered and disordered alloys,” Phys. Rev. Lett., 75, 3162, 1995. [13] G.L. Hart and A. Zunger, “Origins of nonstoichiometry and vacancy ordering in Sc1−x x S,” Phys. Rev. Lett., 87, 275508, 2001. [14] F. Ducastelle, Order and Phase Stability in Alloys., Elsevier Science, New York, 1991. [15] P.D. Tepesch, A.F. Kohan, and G.D. Garbulsky, et al., “A model to compute phase diagrams in oxides with empirical or first-principles energy methods and application to the solubility limits in the CaO–MgO system,” J. Am. Ceram., 49, 2033, 1996.

1.17 DIFFUSION AND CONFIGURATIONAL DISORDER IN MULTICOMPONENT SOLIDS A. Van der Ven and G. Ceder Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA

1.

Introduction

Atomic diffusion in solids is a kinetic property that affects the rates of important nonequilibrium phenomena in materials. The kinetics of atomic redistribution in response to concentration gradients determine not only the speed, but often also the mechanism by which phase transformations in multicomponent solids occur. In electrode materials for batteries and fuel cells high mobilities of specific ions ranging from lithium or sodium to oxygen or hydrogen are essential. In many instances, diffusion occurs in nondilute regimes in which different migrating atoms interact with each other. For example, lithium intercalation compounds such as Lix CoO2 and Lix C6 which serve as electrodes in lithium-ion batteries, can undergo large variations in lithium concentrations, ranging from very dilute concentrations to complete filling of all interstitial sites available for Li in the host. In nondilute regimes, diffusing atoms interact with each other, both electronically and elastically. A complete theory of nondilute diffusion in multi-component solids needs to account for the dependence of the energy and migration barriers on the configuration of diffusing ions. In this chapter, we present the formalism to describe and model diffusion in multicomponent solids. With tools from alloy theory to describe configurational thermodynamics [1–3], it is now possible to rigorously calculate diffusion coefficients in nondilute alloys from first-principles. The approach relies on the use of the alloy cluster expansion which has proven to be an invaluable statistical mechanical tool that links first-principles energies to the thermodynamic and kinetic properties of solids with configurational disorder. Although diffusion is a nonequilibrium phenomenon, diffusion coefficients 367 S. Yip (ed.), Handbook of Materials Modeling, 367–394. c 2005 Springer. Printed in the Netherlands.

368

A. Van der Ven and G. Ceder

can nevertheless be calculated by considering fluctuations at equilibrium using Green–Kubo relations [4]. We first elaborate on the atomistic mechanisms of diffusion in solids with interacting diffusing species. This is followed with a discussion of the relevant Green–Kubo expressions for diffusion coefficients. We then introduce the cluster expansion formalism to describe the configurational energy of a multi-component solid. We conclude with several examples of first-principles calculations of diffusion coefficients in multi-component solids.

2.

Migration in Solids with Configurational Disorder

Multi-component crystalline solids under most thermodynamic boundary conditions are characterized by a certain degree of configurational disorder. The most extreme example of configurational disorder occurs in a solid solution in which on average the arrangements of the different components of the solid approximate randomness. But even ordered compounds exhibit some degree of disorder due to thermal excitations or slight off-stoichiometry of the bulk composition. Atoms diffusing over crystal sites of a disordered solid sample a variety of different local environments along their trajectory. Diffusion in most crystals can be characterized as a Markov process whereby atoms after each hop completely thermalize before migrating to the next site along its trajectory. Hence each hop is independent of all previous hops. With reasonable accuracy, the rate with which individual atomic hops occur, can be described with transition state theory according to = ν ∗ exp

−E b kB T

(1)

where ν ∗ is a vibrational prefactor (having units of Hz) and E b is an activation barrier. Within the harmonic approximation, the vibrational prefactor is a ratio between the vibrational eigenmodes of the solid at the initial state of the hop to the vibrational eigenmodes when the migrating atom is at the activated state [5]. In the presence of configurational disorder, the activation barrier and frequency prefactor depend on the local arrangement of atoms around the migrating atom. Modeling of diffusion in a multicomponent system therefore requires a knowledge of the dependence of E b and ν ∗ on configuration. Especially, the configuration dependence of E b is of importance as the hop frequency, , depends on it exponentially. We restrict ourselves here to migration that occurs by individual atomic hops to adjacent vacant sites. Hence we do not consider diffusion that occurs through either a ring or intersticialicy mechanism. We also make a distinction between diffusion of interstitial species and substitutional species.

Diffusion and configurational disorder in multicomponent solids

2.1.

369

Interstitial Diffusion

Interstitial diffusion occurs in many important materials. A common example is the diffusion of carbon atoms over the interstitial sites of bcc or fcc iron (i.e., steel). Many phase transformations in steel involve the redistribution of carbon atoms between growing precipitate phases and the consumed matrix phase. A defining characteristic of interstitial diffusion is the existence of an externally imposed lattice of sites over which atoms can diffuse. In steel, the crystallized iron atoms create the interstitial sites for carbon. A similar situation exists in Lix CoO2 in which a crystalline CoO2 host creates an array of intersitial sites that can be occupied by lithium. While in Lix CoO2 , the lithium concentration x can be varied from 0 to 1, in steel FeC y , the carbon concentration y is typically very low. Individual carbon atoms interfere minimally with each other as they wander over the interstitial sites of iron. In Lix CoO2 , however, as the lithium concentration is typically large, migrating lithium atoms interact strongly with each other and influence each other’s diffusive trajectories. Another type of system that we place in the category of interstitial diffusion is adatom diffusion on the surface of a crystalline substrate. Often a crystalline surface creates an array of well defined sites on which adsorbed atoms reside, such as the fcc sites on a (111) terminated surface of an fcc crystal. Diffusion then involves the migration of adsorbed atoms over these surface sites. The presence of many diffusing atoms creates a state of configurational disorder over the interstitial sites that evolves over time as a result of the activated hops of individual atoms. Not only does the activation barrier of a migrating atom depend on the local arrangement of the surrounding interstitial atoms, but also the migration mechanism can depend on that arrangement. This is the case in Lix CoO2 , a layered compound consisting of close packed oxygen planes stacked with an ABCABC sequence. Between the oxygen layers are alternating layers of Li and Co which occupy octahedral sites of the oxygen sublattice. Within each lithium plane, the lithium ions occupy a two dimensional triangular lattice. As lithium is removed from LiCoO2 , vacancies are created in the lithium planes. First-principles density functional theory calculations (LDA) have shown that two migration mechanisms for lithium exchange with an adjacent vacancy exist depending on the arrangement of surrounding lithium atoms [3]. This is illustrated in Fig. 1. If the two sites adjacent to the end points of the hop (sites (a) and (b) in Fig. 1a) are simultaneously occupied by lithium ions, then the migration mechanism follows a direct path, passing through a dumbel of oxygen atoms. The calculated activation barrier for this mechanism is high, approaching 0.8 eV. This mechanism occurs when lithium migrates to an isolated vacancy. If, however, one or both of the sites adjacent to the end points of the hop are vacant (Fig. 1b), then the migrating lithium follows a curved path which passes through an adjacent tetrahedral

370


O Li

a

a b

b O Co O Single vacancy hop (a)

Divacancy hop 35 (b)

Figure 1. Two lithium migration mechanims in Lix CoO2 depending on the arrangement of lithium ions around the migrating ion. (a) When both sites a and b are occupied by Li, the migrating lithium performs a direct hop whereby it has to squeeze through a dumbel of oxygen ions. This mechanism occurs when the migrating lithium ion hops into an isolated vacancy (square). (b) When either site a or site b are vacant, the migrating lithium ion performs a curved hop whereby it passes through a tetrahedrally coordinated site. This mechanism occurs when the migrating atom hops into a divacancy.

site, out of the plane formed by the Li sites. For this mechanism, the activation barrier is low, taking values in the vicinity of 0.3–0.4 eV. This mechanism occurs when lithium migrates into a divacancy. Comparison of the activation barriers for the two mechanisms clearly shows that lithium diffusion mediated with divacancies is more rapid than with single vacancies. Nevertheless, we can already anticipate that the availability of divacancies will depend on the overall lithium concentration. The complexity of diffusion in a disordered solid is evident in Fig. 2 which schematically illustrates a typical disordered arrangement of lithium atoms within a lithium plane of Lix CoO2 . Hop 1, for example, must occur with a large activation barrier as the lithium is migrating to an isolated vacancy. In hop 2, lithium migrates to a vacant site that belongs to a divacancy and hence follows a curved path passing through an adjacent tetrahedral site characterized by a low activation barrier. In hop 3, lithium migrates to a vacant site belonging to two divacancies simultaneously, and hence has two low energy paths available. Similar complexities can be expected for adatom diffusion on crystalline substrates.

2.2.

Substitutional Diffusion

Substitutional diffusion is qualitatively different from interstitial diffusion in that an externally imposed lattice of sites for the diffusing atoms is absent.


371

Cobalt Lithium

C

Oxygen

B c

A C B

3

A

2 a

LixCoO2

1 Lithium plane

Figure 2. A typical disordered lithium-vacancy arrangement within the lithium planes of Lix CoO2 . In a given lithium-vacancy arrangement, several different migration mechanisms can occur.

Instead, the diffusing atoms themselves form the network of crystal sites. This describes the situation for most metallic and semiconductor alloys. Vacancies with which to exchange with do exist in these crystalline alloys, however, the concentrations are often very dilute. Examples where substitutional diffusion is relevant are alloys such as Si–Ge, Al–Ti and Al–Li, in which the different species reside on the same crystal structure, and migrate by exchanging with vacancies. As with intersitial compounds, widely varying degrees of local order or disorder exist, affecting migration barriers. Al(1−x)Lix for example is metastable on the fcc crystal structure for low x and forms an ordered L12 compound at x = 0.25. Diffusion within a solid solution is different than in the ordered compound as the local arrangement of Li and Al are different. Figure 3 illustrates a diffusive hop of an Al atom to a neighboring vacancy within the ordered L12 Al3 Li phase. The energy along the migration path as calculated with LDA is also

372


1000

Energy (meV)

800

600

400

200

0 0

0.5

1

1.5

2

2.5

3

Migration path (Angstrom) Figure 3. The energy along the migration path of an Al atom hopping into a vacancy (square) on the lithium sublattice of L12 Al3 Li. Lighter atoms are Al, darker atoms are Li.

illustrated in Fig. 3. Clearly, the vacancy prefers the Li sublattice as the energy of the solid increases as the vacancy migrates from the Li sublattice to the Al sublattice by exchanging with an Al atom.

3.

Green–Kubo Expressions for Diffusion

While diffusion is complex at the atomic length scale, of central importance at the macroscopic length scale is the rate with which gradients in concentration dissipate. These rates can be described by diffusion coefficients that relate atomic fluxes to gradients in concentration. Green–Kubo methods make it possible to link kinetic coefficients to microscopic fluctuations of appropriate quantities at equilibrium. In this section we present the relevant Green–Kubo equations that allow us to calculate diffusion coefficients in multi-component solids from first-principles. We again make a distinction between interstitial and substitutional diffusers.


3.1.

373


3.1.1. Single component diffusion For a single component occuping interstitial sites of a host, such as carbon in iron, or Li in Lix CoO2 , irreversible thermodynamics [4] stipulates that a net flux J in particles occurs when a gradient in the chemical potential µ of the interstitial specie exists according to J = −L∇µ

(2)

where L is a kinetic coefficient that depends on the mobility of the diffusing atoms. Often it is more practical to express the flux in terms of a concentration gradient instead of a chemical potential gradient as the former is more accessible experimentally J = −D∇C.

(3)

D in Eq. (3) is the diffusion coefficient and the concentration C refers to the number of interstitial particles per unit volume. While the true driving force for diffusion is a gradient in chemical potential, it is nevertheless possible to work with Eq. (3) provided the diffusion coefficient is expressed as

D=L

dµ . dC

(4)

Hence the diffusion coefficient consists of a kinetic factor L and a thermodynamic factor dµ/dC. The validity of irreversible thermodynamics is restricted to systems that are not too far removed from equilibrium. To quantify this, it is useful to mentally divide the solid into small subregions that are microscopically large enough for thermodynamic variables to be meaningful yet macroscopically small enough that the same thermodynamic quantities can be considered constant within each subregion. Hence, although the solid itself is removed from equilibrium, each subregion is locally at equilbrium. This is called the local equilibrium approximation, and it is within this approximation that the linear kinetic equation Eq. (2) is considered valid. Within the local equilibrium approximation, the kinetic parameters D and L can be derived by a consideration of relevant fluctuations at thermodynamic equilibrium. Crucial in this derivation, is the assumption made by Onsager in his proof of the reciprocity relations of kinetic parameters, that the regression of a fluctuation of a particular extensive property around its equilibrium value occurs on average according to the same linear phenomenological laws as those governing the regression of artificially induced fluxes of the same extensive quantity [4]. This regression hypothesis is a consequence of the fluctuation–dissipation theorem of nonequilibrium statistical mechanics [6].

374


Several techniques, collectively referred to as Green–Kubo methods, exist to link microscopic fluctuations to macroscopic kinetic quantities [7–9]. Neglecting crystallographic anisotropy, the Green–Kubo expression for the kinetic factor for diffusion can be written as [10–12]

L=

ζ

Rζ (t)

2

(2d)t Mv s kB T

(5)

where Rζ (t) is the vector connecting the end points of the trajectory of particle ζ after a time t, M refers to the total number of interstitial sites available, v s is the volume per interstitial site, kB is the Boltzmann constant, T is the temperature and d refers to the dimension of the interstitial network. The brackets indicate an ensemble average performed at equilibrium. Often, the diffusion coefficient is also written in an equivalent form as [10] D = DJ F where

DJ =

ζ Rζ (t) (2d)t N

and

d F=

(6)

µ kB T

2

(7)

d ln(x)

.

(8)

N refers to the number of diffusing atoms and x = N/M to the fraction of filled interstitial sites. F is often called a thermodynamic factor and DJ is sometimes called the jump-diffusion or self-diffusion coefficient. A common approximation is to neglect cross correlations between different diffusing species and to replace DJ with the tracer diffusion coefficient defined as

D∗ =

Rζ2 (t) (2d)t

.

(9)

The difference between DJ and D ∗ is that the former depends on the square of the displacement of all the particles while the latter depends on the average of the square of the displacement of individual diffusing atoms. DJ is a measure of collective fluctuations of the center of mass of all the diffusing particles. Figure 4 compares DJ and D ∗ calculated with kinetic Monte Carlo simulations for the Lix CoO2 system. Notice in Fig. 4 that DJ is systematically larger than D ∗ for all lithium concentrations x, only approaching D ∗ for dilute lithium concentrations.


375

⫺6

13

D ( 10 ) (cm2/s) ν∗

⫺7 ⫺8 ⫺9

⫺10 ⫺11 0

0.2

0.6 0.4 Li concentration

0.8

1

Figure 4. A comparison of the self diffusion coefficient DJ (crosses), and the tracer diffusion coefficient D ∗ (squares), for lithium diffusion in Lix CoO2 calculated at 400 K.

For interstitial components, the chemical potential of the diffusing atoms is defined as dG dg = (10) dN dx where G is the free energy of the crystal containing the interstitial species and g is the free energy normalized per interstitial site. While the thermodynamic factor is related to the chemical potential according Eq. (8) it is often convenient to determine F by considering fluctuations in the number of interstitial atoms within the grand canonical ensemble (constant µ, T and M). µ=

F=

N N 2 − N 2

(11)

Diffusion involves redistribution of particles from subregions of the solid with a high concentration of interstitial atoms to other subregions with a low concentration. The thermodynamic factor describes the thermodynamic response to concentration fluctuations within sub-regions.

3.1.2. Two component system A similar formalism emerges when two different species reside and diffuse over the same interstitial sites of a host. This is the case for example for carbon and nitrogen diffusion in iron or lithium and sodium diffusion over the

376


interstitial sites of a transition metal oxide host. Referring to the two diffusing species as A and B, the flux equations become JA = −L AA ∇µA − L AB ∇µB JB = −L BA ∇µA − L BB ∇µB

(12)

where L i j (i, j = A or B) are kinetic coefficients similar to L of Eq. (2). As with Eq. (2), gradients in chemical potential are often not readily accessible experimentally and Eq. (12) can be written as JA = −DAA ∇CA − DAB ∇CB JB = −DBA ∇CA − DBB ∇CB .

(13)

where the matrix of diffusion coefficients

DAA DB A

D AB DB B

=

L AA LBA

L AB L BB



   

∂µ A ∂C A ∂µ B ∂C A

∂µ A ∂C B ∂µ B ∂C B

    

(14)

can again be factorized into a product of a kinetic term (the 2×2 L matrix) and a thermodynamic factor (the 2 × 2 matrix of partial derivative of the chemical potentials). The Green–Kubo expressions relating the macroscopic diffusion coefficients to atomic fluctuations are [13, 14]

ζ

Lij =

Rζi (t)

ξ

Rξ (t) j

.

(2d)tv s MkB T

(15)

where Rζi is the vector linking the end points of the trajectory of atom ζ of specie i after time t. Another factorization of D is practical when studying diffusion with a lattice model description of the interactions between the different constituents residing on the crystal network ˜ D = L˜ Θ

(16)

where

L˜ i j =

ζ

Rζi (t)

(2d)t M

ξ

Rξ (t) j

.

(17)

Diffusion and configurational disorder in multicomponent solids and ˜ ij =

∂

µi kB T

377

∂x j

.

(18)

are respectively matrices of kinetic coefficients and thermodynamic factors. As with the single component intersitial systems the chemical potentials for a binary component interstitial system are defined as µi =

∂G ∂g = ∂ Ni ∂ x i

(19)

˜ can also be written in terms where i refers to either A or B. The components of of variances of the number of particles residing on the M site crystal network at constant chemical potentials, that is in terms of measures of fluctuations M ˜ = Θ Q

N B2 − N B 2 − (N B N A − N A N B )

− (N A N B − N A N B ) N A2 − N A 2

(20) where

Q=

NA2 − NA 2

NB2 − NB 2 − (NA NB − NA NB )2

These fluctuations in NA and NB are to be calculated in the grand canonical ensemble at the chemical potentials µA and µB corresponding to the concentrations at which the diffusion coefficient is desired.

3.2.


The starting point for treating substitutional diffusion in a binary alloy are the Green–Kubo relations of Eqs. (14)–(18). However, several modifications and qualifications are necessary. These arise from the fact that alloys are characterized by a dilute concentration of vacancies and that the crystallographic sites on which the diffusing atoms reside are not created externally by a host, but are rather formed by the diffusing atoms themselves. The consequences of this for diffusion is that the chemical potentials appearing in the thermodynamic factor are not the conventional chemical potentials for the individual species A and B of a substitutional alloy, but are rather differences in chemical potentials between that of each diffusing specie and the vacancy chemical potential. Hence the chemical potentials of Eqs. (12), (14) and (18) need to be replaced by µ˜ i in which µ˜ i = µi − µV

(21)

378


where µV is the vacancy chemical potential in the solid. The reason for this modification arises from the fact that the chemical potential appearing in the Green–Kubo expression for the diffusion coefficient matrix Eq. (14) and defined in Eq. (19) corresponds to the change in free energy as component i is added by holding the number of crystalline sites constant, meaning that i is added at the expense of vacancies. This differs from the conventional chemical potentials of alloys which are defined as the change in free energy of the solid as component i is added by extending the crystalline network of the solid. µ˜ i refers to the chemical potential for a fixed crystalline network, while µi and µV correspond to chemical potentials for a solid in which the crystalline network is enlarged as more species are added. The use of µ˜ i instead of µi in the thermodynamic factor of the Green–Kubo expressions for the diffusion coefficients of crystalline solids also follows from irreversible thermodynamics [15, 16] as well as thermodynamic considerations of crystalline solids [17]. It can also be understood on physical grounds. By dividing the crystalline solid up into subregions, diffusion can be viewed as the flow of particles from one subregion to the next. Because of the constraint imposed by the crystalline network, the only way for excess atoms from one sub-region to be accommodated by a neighboring subregion is through the exchange of vacancies. One subregion gains vacancies the other loses them. The change in free energy in each subregion due to diffusion occurs by adding or removing atoms at the expense of vacancies. Another important modification to the treatment of binary interstitial diffusion is the identification of interdiffusion. Interdiffusion in its most explicit form refers to the dissipation of concentration gradients by the intermixing of A and B atoms. It is this phenomenon of intermixing that enters into continuum descriptions of diffusion couples and phase transformations involving atomic redistribution. Kehr et al. [18] demonstrated that in the limit of dilute vacancy concentrations, the full 2 × 2 diffusion matrix can be diagonalized producing an eigenvalue λ+ corresponding to density relaxations due to inhomegeneities in vacancies and an eigenvalue λ− corresponding to interdiffusion. The diagonalization of the D matrix is accompanied by a coordinate transformation of the fluxes and the concentration gradients. In matrix notation, J = −D∇C

(22)

where J and ∇C are column vectors containing as elements JA , JB and ∇CA , ∇CB , respectively. Diagonalization of D leads to D = EλE−1

(23)


379

where λ is a diagonal matrix with components λ+ (the larger eigenvalue) and λ− (the smaller eigenvalue) in the notation of Kehr et al. [18], i.e.,

λ=

λ+ 0 0 λ−

The flux equation (22) can then be rewritten as E−1 J = −λE−1 ∇C.

(24)

The eigenvalue λ− , which describes the rate with which gradients in the concentration of A and B atoms dissipate by an intermixing mode is the most rigorous formulation of what is commonly referred to as an interdiffusion coefficient.

4.

Cluster Expansion

The Green–Kubo expressions for diffusion coefficients are proportional to the ensemble averages of the square of the collective distance travelled by the diffusing particles of the solid. Trajectories of interacting diffusing particles can be obtained with kinetic Monte Carlo simulations in which particles migrate on a crystalline network with migration rates given by Eq. (1). The migration rates of a specific atom, however, depend on the local arrangement of the other diffusing atoms through the configuration dependence of the activation barrier and frequency prefactor. Ideally, the activation barrier for each local environment could be calculated from first-principles. Nevertheless, this is computationally impossible, as the number of configurations are exceedingly large, and firstprinciples activation barrier calculations have a high computational cost. It is here that the cluster expansion formalism [1–3] becomes invaluable as a tool to extrapolate energy values calculated for a few configurations to determine the energy for any arrangement of atoms in a crystalline solid. In this section, we describe the cluster expansion formalism and how it can be applied to characterize the configuration dependence of the activation barrier for diffusion. We first focus on describing the configurational energy of atoms residing at their equilibrium sites, i.e., of the configurational energy of the end points of any hop.

4.1.

General Formalism

We restrict ourselves to binary problems though the cluster expansion formalism is valid for systems with any number of species [1, 2]. While it is clear that two component alloys without crystalline defects such as vacancies are

380


binary problems, atoms residing on the interstitial sites of a host can be treated as a binary system as well, with the interstitial atoms constituting one of the components and the vacancies the other. In crystals, atoms can be assigned to well defined sites, even when relaxations from ideal crystallographic positions occur. There is always a one to one correspondence between each atom and a crystallographic site. If there are M crystallographic sites, then there are 2 M possible arrangements of two species over those sites. To characterize a particular configuration, it is useful to introduce occupation variables σi that are +1 (−1) if an A (B which could be an atom different from A or a vacancy) resides at site i. The vector σ =(σ1 , σ2 , . . . , σi , . . . , σ M ) then uniquely specifies a configuration. The use of σ , however, is cumbersome and a more versatile way of uniquely characterizing configurations can be achieved with polynomials φα of occupation variables defined as [1] σ) = φα (

σi

(25)

i∈α

where i are sites belonging to a cluster α of crystal sites. Typical examples of clusters are a nearest neighbor pair cluster, a next nearest neighbor pair cluster, a triplet cluster etc. Examples of clusters on a two dimensional triangular lattice are illustrated in Fig. 5. There are 2 M different clusters of sites and σ ). therefore 2 M cluster functions φα ( σ ) form a complete It can be shown [1] that the set of cluster functions φα ( and orthonormal basis in configuration space with respect to the scalar product 1 f ( σ )g( σ) (26) f, g = M 2 σ where f and g are any scalar functions of configuration. The sum in Eq. (26) extends over all possible configurations of A and B atoms over the M sites of the crystal. Because of their completeness and orthonormality over the space of configurations, it is possible to expand any function of configuration f ( σ) σ ). In particular, the conas a linear combination of the cluster functions φα ( figurational energy (with atoms relaxed around the crystallographic positions of the crystal) can be written as E( σ ) = Eo +

α

Vα φα ( σ)

(27)

where the sum extends over all clusters α over the M sites. The coefficients σ ) with the Vα are constants and formally follow from the scalar product of E( σ) cluster function φα ( 1 σ ), φα ( σ ) = M E( σ )φα ( σ ). (28) Vα = E( 2 σ σ) E o is the coefficient of the empty cluster φo = 1 and is the average of E( over all configurations. Equation (27) is referred to as a cluster expansion of


381

b a γ

α

β

Figure 5. Examples of clusters for a two dimensional triangular lattice.

the configurational energy and the coefficients of the expansion Vα are called effective cluster interactions (ECI). Equation (27) can be viewed as a generalized Ising model Hamiltonian containing not only nearest neighbor pair interactions, but also all other pair and multibody interactions extending beyond the nearest neighbors. Through Eq. (28), a formal link is made between the interaction parameters of the generalized Ising model and the configuration dependent ground state energies of the solid in each configuration σ . Clearly, the cluster expansion for the configurational energy, Eq. (27), is only useful if it converges rapidly, i.e., there exists a maximal cluster αmax such that all ECI corresponding to clusters larger than αmax can be neglected. In this case, the cluster expansion can be truncated to yield E( σ ) = Eo +

α max α

Vα φα ( σ)

(29)

382


A priori mathematical criteria for the convergence of the configurational energy cluster expansion do not exist. Experience indicates that convergence depends on the particular system being considered. In general, though, it can be expected that the lower order clusters extending over a limited range within the crystal will have the largest contribution in the cluster expansion.

4.2.

Symmetry and the Cluster Expansion

Simplifications to the cluster expansion (27) or (29) can be made by taking the symmetry of the crystal into account [2]. Clusters are said to be equivalent by symmetry if they can be mapped onto each other with at least one space group symmetry operation. For example, clusters α and β of Fig. 5 are equivalent since a clockwise rotation of α by 60◦ followed by a translation by the vector 2b maps α onto β. The ECI corresponding to clusters that are equivalent by symmetry have the same numerical value. In the case of α and β of Fig. 5, Vα = Vβ . All clusters that are equivalent by symmetry are said to belong to an orbit α where α is a representative cluster of the orbit. For any arrangement σ we can define averages over cluster functions φα ( σ ) as φα ( σ ) =

1 φβ ( σ) | α | β∈ α

(30)

where the sum extends over all clusters β belonging to the orbit α and | α | represents the number of clusters that are symmetrically equivalent to α. The φα ( σ ) are commonly referred to as correlation functions. Using the definition of the correlation functions and the fact that symmetrically equivalent clusters have the same ECI, we can rewrite the configurational energy normalized by the number of primitive unit cells Np (i.e., number of Bravais lattice points of the crystal which is not necessarily equal to the number of crystal sites M), as e( σ) =

E( σ) = Vo + m α Vα φα ( σ ) Np α

(31)

where m α is the multiplicity of the cluster α, defined as the number of clusters per Bravais lattice point symmetrically equivalent with α (i.e., m α = | α |/Np ) and Vo = E o /Np . The sum in (31) is only performed over the symmetrically non-equivalent clusters.

4.3.

Determination of the ECI

According to Eq. (28), the ECI for the energy cluster expansion are determined by the first-principles ground state energies for all the different


383

configurations σ . Explicitly calculating the ECI according to the scalar product Eq. (28) is intractable. Techniques, such as direct configurational averaging (DCA), though, have been devised to approximate the scalar product (28) [2, 19, 20]. In recent years, the preferred method of obtaining ECI has been with an inversion method [21–29]. In this approach, energies E( σ I ) for a set of P periodic configurations σ I with I = 1, . . . , P are calculated from firstprinciples and a truncated form of (31) is inverted such that it reproduces the E( σ I ) within a tolerable error when Eq. (31) is evaluated for configuration σ I . The simplest inversion scheme uses a least squares fit. More sophisticated algorithms involving linear programming techniques [30], cross-validation optimization [32] or the inclusion of k-space terms to account for long-range elastic strain have been developed [33, 34].

4.4.

Local Cluster Expansion

The traditional cluster expansion formalism described so far is applicable to the configurational energy of the solid which is an extensive quantity. We will refer to these expansions as extended cluster expansions. Activation barriers, however, are equal to the difference between the energy of the solid when the migrating atom is at the activated state and that when the migrating atom is at the initial equilibrium site. Hence, the configuration dependence of the activation barrier of an atom needs to be described by a cluster expansion with no translational symmetry and as such it converges to a fixed value as the system size grows. Not only is the activation barrier a function of configuration, but it also depends on the direction of the hop. This is schematically illustrated in Fig. 6 in which the end points of the hop have a different configurational energy. Describing the configuration dependence of the activation barrier independent of the direction of the hop is straightforward if a kinetically resolved activation barrier is introduced [3], defined as E KRA = E act −

n 1 Ej n j =1

(32)

∆Eb

∆EKRA

∆Eb

Figure 6. The activation barrier for migration depends on the direction of the hop when the energies of the end points of the hop are different.

384


where E act is the energy of the solid with the migrating atom at the activated state and E j are the energies of the solid with the migrating atom at the end points j of the hop. In most solids, there are n=2 end points to a hop, however, it is possible that more end points exist. All terms in Eq. (32) depend on the arrangement of atoms surrounding the end points of the hop and the activated state. The dependence of E KRA on configuration can, be described with a cluster expansion that has a point group symmetry compatible with the symmetry of the crystal as well as that of the activated state. For this reason, the cluster expansion of E KRA is called a local cluster expansion [3]. The kinetically resolved activation barrier is not the true activation barrier that enters in the transition state theory expression for the hop frequency, Eq. (1). It is merely a useful quantity that characterizes the configuration dependence of the activated state independent of the direction of the hop. The true activation barrier can be calculated from E KRA using 



n 1  E j  − Ei E b = E KRA + n j =1

(33)

where E i is the energy of the crystal with the migrating atom at the initial site of the hop. All quantities on the right hand side of Eq. (33) can be described with either a local cluster expansion (for E KRA ) or an extended cluster expansion (for the configurational energy of the solid).

5.

Practical Implementation

Calculating diffusion coefficients from first-principles in multicomponent solids involves three steps. First, a variety of ab initio energies for different atomic arrangements need to be calculated with an accurate first-principles method. This includes energies for a wide range of atomic arrangements over the sites of the crystal, as well as energies for migrating atoms placed at activated states surrounded by different arrangements. The latter calculations are typically performed with an atom at the activated state in large supercells. A useful technique to find the activated state between two equilibrium end points is the nudged elastic band method [31] which determines the lowest energy path between two equilibrium states. Calculating the vibrational prefactor requires a calculation of the phonon density of states for different atomic arrangements both with the migrating atom at its equilibrium site and at the activated state. While sophisticated techniques have been devised to characterize the configurational dependence of the vibrational free energy of a solid [35], for diffusion studies, a convenient simplification is the local harmonic approximation [36].


385

In the second step, the first-principles energy values for different atomic arrangements are used to determine the coefficients of both a local cluster expansion (for the kinetically resolved activation barriers) and a traditional extended cluster expansion (for the energy of the crystal with all atoms at non-activated crystallographic sites) with either a least squares fit or with one of the more sophisticated methods alluded to above. The cluster expansions enable the calculation of the energy and activation barrier for any arrangement of atoms on the crystal. They serve as a convenient and robust tool to extrapolate accurate first-principles energies calculated for a few configurations to the energy of any configuration. Hence the migration rates of Eq. (1) can be calculated for any arrangement of atoms. The final step is the combination of the cluster expansions with kinetic Monte Carlo simulations to calculate the quantities entering the Green–Kubo expressions for the diffusion coefficients. Kinetic Monte Carlo simulations have been discussed extensively elsewhere [3, 37, 38]. Applied to diffusion in crystals, kinetic Monte Carlo algorithms are used to simulate the stochastic migrations of many atoms, hopping to neighboring sites with frequencies given by Eq. (1). A kinetic Monte Carlo simulation starts from a representative arrangement of atoms (typically obtained with a standard Monte Carlo method for lattice models). As atoms migrate, their trajectories and the time are kept track of, enabling the calculation of the quantities between the brackets in the Green–Kubo expressions. Since the Green–Kubo expressions involve ensemble averages, many kinetic Monte Carlo runs which start from different representative initial conditions are necessary. Depending on the desired accuracy, averages need to be performed over the trajectories departing from between 100 and 10 000 different initial conditions.

6.

Examples

Two examples of first-principles calculations of diffusion coefficients in multi-component solids are reviewed in this section. The first is for lithium diffusion in Lix CoO2 and is an example of nondilute interstitial diffusion. The second example, diffusion in the fcc based Al–Li alloy, corresponds to a substitutional system.

6.1.


Lix CoO2 consists of a host structure made up of a CoO2 frame work. Layers of interstitial sites that can be occupied by lithium ions reside between O–Co–O slabs. The interstitial sites are octahedrally coordinated by oxygen and they form two dimensional triangular lattices. As described in Section 2.1,

386


two migration mechanisms exist for lithium: a single vacancy mechanism whereby lithium squeezes through a dumbell of oxygen atoms into an isolated vacancy and a divacancy mechanism in which lithium migrates through an adjacent tetrahedral site into a vacant site that is part of a divacancy [3]. The two migration mechanisms are illustrated in Fig. 1. Not only does the local arrangement of lithium ions around a hopping ion determine the migration mechanism, it also affects the value of the activation barrier for a particular migration mechanism. Figure 7 illustrates kinetically resolved activation barriers calculated from first- principles (LDA) for a variety of different lithium-vacancy arrangements around the migrating ion at different bulk lithium concentrations [3]. Note that for a given bulk composition, many possible lithium-vacancy arrangements around an atom in the activated state exist. The kinetically resolved activation barriers illustrated in Fig. 7 correspond to only a small subset of the these many configurations. The local cluster expansion is used to extrapolate from this set to all the configurations needed in a kinetic Monte Carlo simulation. Figure 7 shows that the activation barrier for the divacancy migration mechanism can vary by more that 200 meV with lithium concentration. The increase in activation barrier upon lithium removal from the host can be traced to the contraction of the host along the c-axis as the lithium concentration is reduced [3].

Activation Barrier (meV)

1000

800

600

400

200

0

0

0.2

0.4

0.6

0.8

1

Li concentration Figure 7. A sample of first-principles (LDA) kinetically resolved activation barriers E KRA for the divacancy hop mechanism (circles) and the single vacancy mechanism (squares).


387

This contraction disporportionately penalizes the activated state over the end point states of the divacancy hop mechanism. Another contribution to the variation in activation barrier with composition derives from the fact that the activated state is very close in proximity to a Co ion, which becomes progressively more oxidized (i.e., its eff ective charge becomes more positive) as the overall lithium concentration is reduced [3, 29]. This leads to an increase in the electrostatic repulsion between the activated Li and the Co as x is reduced. Extended and local cluster expansions can be constructed to describe both the configurational energy of Lix CoO2 and the configuration dependence of the kinetically resolved activation barriers. An extended cluster expansion for the first-principles configurational energy of Lix CoO2 has been described in detail in Ref. [29]. This cluster expansion when combined with Monte Carlo simulations accurately predicts phase stability in Lix CoO2 . In particular, two ordered lithium-vacancy phases are predicted at x = 1/2 and x = 1/3. Both phases are observed experimentally [39, 40]. A local cluster expansion for the kinetically resolved activation barriers has been described in Ref [3]. Figure 8 illustrates calculated diffusion coefficients at 300 K determined by applying kinetic Monte Carlo simulations to the cluster expansions of Lix CoO2 [3]. While the configuration dependence of the activation barriers were rigorously accounted for with the cluster expansions, no attempt in these calculations was made to describe the migration rate prefactor ν ∗ from first- principles. Instead, a value of 1013 Hz was used for all compositions and environments. Figure 8(a) shows both DJ and the chemical diffusion coefficient D, while Fig. 8(b) illustrates the thermodynamic factor F, which was determined by calculating fluctuations in the number of lithium particles in grand canonical Monte Carlo simulations [3] (see Section 3.1). Notice that the calculated diffusion coefficient varies by several orders of magnitude with composition, showing that the assumption of a concentration independent diffusion coefficient in this system is unjustified. The thermodynamic factor F is a measure for the deviation from ideality. In the dilute limit (x → 0), interactions between lithium ions are negligible and the configurational thermodynamics approximates that of an ideal solution. In this limit the thermodynamic factor is 1. As x increases from 0, and the solid departs from ideal behavior, the thermodynamic factor increases substantially. The local minima in DJ and D at x = 1/2 and x = 1/3 are a result of lithium ordering at those compositions. Lithium-vacancy ordering effectively locks in lithium ions into energetically favorable sublattice positions which reduces ionic mobility. The thermodynamic factor on the other hand exhibits peaks at x = 1/2 and x = 1/3 as the configurational thermodynamics of an ordered phase deviates strongly from ideal behavior. The peak signifies the fact that in an ordered phase, a small gradient in composition leads to an enormous gradient in chemical potential, and hence a large thermodynamic driving force for diffusion. This partly compensates the reduction in DJ .

388


⫺7

D

13

D (10 ) (cm2/s) ν∗

⫺8 ⫺9 ⫺10

DJ

⫺11 ⫺12 ⫺13 ⫺14

0

0.2

0.4

0.6

0.8

1

Li concentration 100000

Thermodynamic factor

10000

1000

100

100

1 1

0.2

0.4

0.6

0.8

1

Li concentration

Figure 8. (a) Calculated self diffusion coefficient DJ and chemical diffusion coefficient D for Li x CoO2 at 300 K. (b) The thermodynamic factor of Lix CoO2 at 300 K.


389

A similar computational approach can be followed to determine for example the diffusion coefficient for oxygen diffusion on a platinum (111) surface. If in addition to oxygen, sulfur atoms are also adsorbed on the platinum surface, Green–Kubo relations for binary interstitial diffusion would be needed. Furthermore, ternary cluster expansions are then necessary to describe the configuration dependence of the energy and kinetically resolved activation barrier as there are then three species: oxygen, sulfur and vacancies.

6.2.


To illustrate diffusion in a binary substitutional solid, we consider the fcc Al–Li alloy. While Al1−x Lix is predominantly stable in the bcc based crystal structure, it is metastable in fcc up to x = 0.25. In fact, it is the metastable form of fcc Al1−x Lix that strengthens the important candidate alloy for aerospace applications. A first step in determining the diffusion coefficients in this system is an accurate first-principles characterization of the alloy thermodynamics. This can be done with a binary cluster expansion for the configurational energy [26]. The expansion coefficients of the cluster expansion were fit to the first-principles energies (LDA) of more than 70 different periodic lithium-aluminum arrangements on the fcc lattice [41]. Figure 9(a) illustrates the calculated metastable fcc based phase diagram of Al1−x Lix obtained by applying Monte Carlo simulations to the cluster expansion [41]. The phase diagram shows that a solid solution phase is stable at low lithium concentration and at high temperature. At x = 0.25, the L12 ordered phase is stable. In this ordered phase the Li atoms occupy the corner points of the conventional cubic fcc unit cell. Diffusion in most metals is dominated by a vacancy mechanism. Hence it is not sufficient to simply characterize the thermodynamics of the strictly binary Al–Li alloy. Real alloys always have a dilute concentration of vacancies that wander through the crystal and in the process redistribute the atoms of the solid. The vacancies themselves have a thermodynamic preference for particular local environments over others which in turn affects the mobility of the vacancies. Treating vacancies in addition to Al and Li makes the problem a ternary one and in principles would require a ternary cluster expansion. Nevertheless, since vacancies are present in dilute concentrations, a ternary cluster expansion can be avoided by using a local cluster expansion to describe the configuration dependence of the vacancy formation energy [41]. In effect, the local cluster expansion serves as a perturbation to the binary cluster expansion to describe the interaction of a dilute concentration of a third component, in this case the vacancy. A local cluster expansion for the vacancy formation energy in fcc Al–Li was constructed by fitting to first-principles (LDA) vacancy formation energies in 23 different Al–Li arrangements [41]. Combining the vacancy

390

A. Van der Ven and G. Ceder 800 (a) solid solution

Temperature (K)

700

600

L12

500

Vacancy concentration

(b)

1e-06

1e-07 (c)

Li concentration around vacancy

0.8 0.6 2nd NN 0.4 1st NN

0.2 0 0

0.1

0.2

0.3

x in LixAl(1-x) Figure 9. (a) First-principles calculated phase diagram of fcc based Al(1−x) Lix alloy. (b) Calculated equilibrium vacancy concentration as a function of bulk alloy composition at 600 K. (c) Average lithium concentration in the first two nearest neighbor shells around a vacancy. The dashed line corresponds to the average bulk lithium concentration.


391

formation local cluster expansion with the binary cluster expansion for Al–Li in Monte Carlo simulations enables a calculation of the equilibrium vacancy concentration as a function of alloy composition and temperature. Figure 9(b) illustrates the result for Al–Li at 600 K [41]. While the vacancy concentration is more or less constant in the solid solution phase, it can vary by an order of magnitude over a small concentration range in the ordered L12 phase at 600 K. Another relevant thermodynamic property that is of importance for diffusion is the equilibrium short range order around a vacancy in fcc Al–Li. Monte Carlo simulations using the cluster expansions predict that the vacancy repels lithium ions, preferring a nearest neighbor shell rich in aluminum. Illustrated in Fig. 9(c) is the lithium concentration in shells with varying distance around a vacancy. The lithium concentration in the first nearest neighbor shell is less than the bulk alloy composition, while it is slightly higher than the bulk composition in the second nearest neighbor shell. This indicates that the vacancy repels Li and attracts Al. In the ordered phase, stable at 600 K between x = 0.23 and 0.3, the degree of order around the vacancy is even more pronounced as illustrated in Fig. 9(c). Between x = 0.23 and 0.3, the vacancy is predominantly surrounded by Al in its first and third nearest neighbor shells and by Li in its second and fourth nearest neighbor shells. This corresponds to a situation in which the vacancy occupies the lithium sublattice of the L12 ordered phase. Clearly the thermodynamic preference of the vacancies for a specific local environment will have an impact on their mobility through the crystal. While thermodynamic equilibrium determines the degree of order within the alloy and which environments the vacancies are attracted to, atomic migration mediated by a vacancy mechanism involves passing through activated states, which requires passing over an energy barrier that also depends on the local degree of order. Contrary to what is predicted for Lix CoO2 , the kinetically resolved activation barriers in fcc Al1−x Lix are not very sensitive to configuration and bulk composition [42]. For each type of atom (Al or Li), the variations in kinetically resolved activation barriers are within the numerical errors of the first-principles method (50 meV for a plane wave pseudopotential method using 107 atom supercells). This is likely the result of a negligible variation in volume of fcc Al1−x Lix with composition. But while the migration barriers do not depend significantly on configuration, they are very different depending on which atom performs the hop. The first-principles calculated migration barrier for Al hops are systematically between 150 to 200 meV larger than for Li hops [42]. The thermodynamic tendency of the vacancy to repel lithium atoms deprives Li of diffusion mediating defects. Kinetically, though, Li has a lower activation barrier relative to Al for migration into an adjacent vacancy. Hence a trade-off exists between thermodynamics and kinetics. While Li exchanges more readily with a neighboring vacancy, thermodynamically it has less access to those vacancies. Quantitatively determining the effect of this trade-off requires explicit

392


Interdiffusion coefficient (cm2/s)

10⫺11

10⫺12 Two phase coexistence

10⫺13

10⫺14 0

0.05

0.1 0.15 x in Al(1-x)Lix

0.2

0.25

0.3

Figure 10. Calculated interdiffusion coefficient (the λ− eigenvalue of the 2 × 2 D matrix) for fcc Al(1−x) Lix alloy at 600 K.

evaluation of diffusion coefficients. This can be done by applying kinetic Monte Carlo simulations to cluster expansions that describe the configurational energy and kinetically resolved activation barriers for Al, Li and dilute vacancies on the fcc lattice. Figure 10 illustrates the calculated interdiffusion coefficient at 600 K obtained by diagonalizing the D matrix of Eq. (14) [42]. The coefficient for interdiffusion describes the rate with which the Al and Li atoms intermix in the presence of a concentration gradient in the two species. The calculated interdiffusion coefficient is more or less constant in the solid solution phase, but drops by more than an order of magnitude in the L12 ordered phase. The thermodynamic preference of the vacancies for the lithium sublattice sites of L12 dramatically constricts the trajectory of the vacancies, leading to a drop in overall mobility of Li and Al.

7.

Conclusion

In this chapter, we have presented the statistical mechanical formalism that relates phenomenological diffusion coefficients for multicomponent solids to microscopic fluctuations of the solid at equilibrium. We have focussed on


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diffusion that is mediated by a vacancy mechanism and have distinguished between interstitial systems and substitional systems. An important property of multicomponent solids is the existence of configurational disorder among the constituent species. This adds a level of complexity in calculating diffusion coefficients from first- principles since the activation barriers vary along an atom’s trajectory as a result of variations in the local degree of atomic order. In this respect, the cluster expansion is an invaluable tool to describe the dependence of the energy, in particular of the activation barrier, on atomic configuration. While the formalism of calculating diffusion coefficients from firstprinciples in multicomponent solids has been established, many opportunities exist to apply it to a wide variety of multicomponent crystalline solids, including metals, ceramics and semiconductors. Faster computers and improvements to electronic structure methods that go beyond density functional theory will lead to more accurate first-principles approximations to activation barriers and vibrational prefactors. It is only a matter of time before first-principles diffusion coefficients for multicomponent solids are routinely used in continuum simulations of diffusional phase transformations and electrochemical devices such as batteries and fuel cells.

Acknowledgments We acknowledge support from the AFOSR, grant F49620-99-1-0272 and the Department of Energy, Office of Basic Energy Sciences under Contract No. DE-FG02-96ER45571. Additional support came from NSF (ACI-9619020) through computing resources provided by NPACI at the San Diego Supercomputer Center.

References [1] J.M. Sanchez, F. Ducastelle, and D. Gratias, Physica A, 128, 334, 1984. [2] D. de Fontaine, In: H. Ehrenreich and D. Turnbull (eds.), Solid State Physics., Academic Press, New York, pp. 33, 1994. [3] A. Van der Ven, G. Ceder, M. Asta, and P.D. Tepesch, Phys. Rev. B, 64, 184307, 2001. [4] S.R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, Dover Publications, Mineola, NY, 1984. [5] G.H. Vineyard, J. Phys. Chem. Solids, 3, 121, 1957. [6] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, 1987. [7] R. Zwanzig, Annu. Rev. Phys. Chem., 16, 67, 1965. [8] R. Zwanzig, J. Chem. Phys., 40, 2527, 1964. [9] Y. Zhou and G.H. Miller, J. Phys. Chem., 100, 5516, 1996. [10] R. Gomer, Rep. Prog. Phys., 53, 917, 1990.

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M. Tringides and R. Gomer, Surf. Sci., 145, 121, 1984. C. Uebing and R. Gomer, J. Chem. Phys., 95, 7626, 1991. A.R. Allnatt, J. Chem. Phys., 43, 1855, 1965. A.R. Allnatt, J. Phys. C: Solid State Phys., 15, 5605, 1982. R.E. Howard and A.B. Lidiard, Rep. Prog. Phys., 27, 161, 1964. A.R. Allnatt and A.B. Lidiard, Rep. Prog. Phys., 50, 373, 1987. J.W. Cahn and F.C. Larche, Scripta Met., 17, 927, 1983. K.W. Kehr, K. Binder, and S.M. Reulein, Phys. Rev. B, 39, 4891, 1989. C. Wolverton, G. Ceder, D. de Fontaine, and H. Dreysse, Phys. Rev. B, 45, 13105, 1992. C. Wolverton and A. Zunger, Phys. Rev. B, 50, 10548, 1994. J.W.D. Connolly and A.R. Williams, Phys. Rev. B, 27, 5169, 1983. J.M. Sanchez, J.P. Stark, and V.L. Moruzzi, Phys. Rev. B, 44, 5411, 1991. Z.W. Lu, S.H. Wei, A. Zunger, S. Frotapessoa, and L.G. Ferreira, Phys. Rev. B, 44, 512, 1991. M. Asta, D. de Fontaine, M. Vanschilfgaarde, M. Sluiter, and M. Methfessel, Phys. Rev. B, 46, 5055, 1992. M. Asta, R. McCormack, and D. de Fontaine, Phys. Rev. B, 48, 748, 1993. M.H.F. Sluiter, Y. Watanabe, D. de Fontaine, and Y. Kazazoe, Phys. Rev. B, 53, 6136, 1996. P.D. Tepesch, et al., J. Am. Cer. Soc., 79, 2033, 1996. V. Ozolins, C. Wolverton, and A. Zunger, Phys. Rev. B, 57, 6427, 1998. A. Van der Ven, M.K. Aydinol, G. Ceder, G. Kresse, and J. Hafner, Phys. Rev. B, 58, 2975, 1998. G.D. Garbulsky and G. Ceder, Phys. Rev. B, 51, 67, 1995. G. Mills, H. Jonsson, and G.K. Schenter, Surf. Sci., 324, 305, 1995. A. van de Walle and G. Ceder, J. Phase Eqilib., 23, 348, 2002. D.B. Laks, L.G. Ferreira, S. Froyen, and A. Zunger, Phys. Rev. B, 46, 12587, 1992. C. Wolverton, Philos. Mag. Lett., 79, 683, 1999. A. van de Walle and G. Ceder, Rev. Mod. Phys., 74, 11, 2002. R. LeSar, R. Najafabadi, and D.J. Srolovitz, Phys. Rev. Lett., 63, 624, 1989. A.B. Bortz, M.H. Kalos, and J.L. Lebowitz, J. Comput. Phys., 17, 10, 1975. F.M. Bulnes, V.D. Pereyra, and J.L. Riccardo, Phys. Rev. E, 58, 86, 1998. J.N. Reimers and J.R. Dahn, J. Electrochem. Soc., 139, 2091, 1992. Y. Shao-Horn, S. Levasseur, F. Weill, and C. Delmas, J. Electrochem. Soc., 150, A366, 2003. A. Van der Ven and G. Ceder, Phys. Rev. B., 2005 (in press). A. Van der Ven and G. Ceder, Phys. Rev. Lett., 2005 (in press).

1.18 DATA MINING IN MATERIALS DEVELOPMENT Dane Morgan and Gerbrand Ceder Massachusetts Institute of Technology, Cambridge MA, USA

1.

Introduction

Data Mining (DM) has become a powerful tool in a wide range of areas, from e-commerce, to finance, to bioinformatics, and increasingly, in materials science [1, 2]. Miners think about problems with a somewhat different focus than traditional scientists, and DM techniques offer the possibility of making quantitative predictions in many areas where traditional approaches have had limited success. Scientists generally try to make predictions through constitutive relations, derived mathematically from basic laws of physics, such as the diffusion equation or the ideal gas law. However, in many areas, including materials development, the problems are so complex that constitutive relations either cannot be derived, or are too approximate or intractable for practical quantitative use. The philosophy of a DM approach is to assume that useful constitutive relations exist, and to attempt to derive them primarily from data, rather than from basic laws of physics. As an example, consider what will likely stand forever as the greatest application of DM in the hard sciences, the periodic table. In 1869 Mendeleev organized the elements based on their properties, without any guiding theory, into the first modern periodic table [3]. With the advent of quantum theory it became possible to predict the structure of the periodic table and DM was no longer strictly necessary, but the results had already been known and used for many years. Even today, the easy organization of data made possible by the classifications in the periodic table make it an everyday tool for research scientists. Mendeleev established a simple ordering based on a relatively small amount of data, and so could do it on paper. However, today’s data sets can be many orders of magnitude larger, and an impressive array of computational algorithms have been developed to automate the task of identifying relationships within data. 395 S. Yip (ed.), Handbook of Materials Modeling, 395–421. c 2005 Springer. Printed in the Netherlands.

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DM is becoming an increasingly valuable tool in the general area of materials development, and there are good reasons why this area is particularly fruitful for DM applications. There is an enormous range of possible new materials, and it is often difficult to physically model the relationships between constituents, and processing, and final properties. For this reason, materials are primarily still developed by what one might call informed trial-and-error, where researchers are guided by experience and heuristic rules to a somewhat restricted space of constituents and processing conditions, but then try as many combinations as possible to find materials with desired properties. This is essentially human DM, where one’s brain, rather than the computer, is being used find correlations, make predictions, and design optimal strategies. Transferring DM tasks from human to computer offers the potential to enhance accuracy, handle more data, and allow wider dissemination of accrued knowledge. Other key drivers for growing DM use in materials development are ease of access to large databases of materials properties, new data being generated in large quantities by high-throughput experiments and quantitative computational models, and improved algorithms, computer speed, and software packages leading to more effective and easy to use DM methods. Note that DM is also used in other areas of materials science beside materials development, e.g., design and manufacturing [4, 5], but this work will not be discussed here. The interdisciplinary nature of DM creates a special challenge, since a typical materials scientist’s education does not provide an introduction to DM techniques, and the computer scientists and statisticians usually involved in developing DM methods are equally unlikely to be versed in materials science. The goal of this paper is to help foster communication between the disciplines and show examples of how they can be joined productively. We introduce DM concepts in a fairly general framework, discuss a few of the more common methods, and describe how DM is being used to tackle some materials development problems, including predicting physiochemical properties of compounds, modeling electrical and mechanical properties, developing more effective catalysts, and predicting crystal structure. The breadth of methods and applications makes a comprehensive discussion impossible, but hopefully this brief introduction will be enough to allow the interested reader to follow up on specific areas of interest.

2.

Key Methods of Data Mining

Data Mining (DM) is a vast and rapidly changing topic, with many different techniques appearing in many different fields. Broad reviews of the issues, methods, and applications are given in Refs. [1, 2] and somewhat less comprehensively but more in depth in Refs. [6, 7]. There is some disagreement about exactly what constitutes DM, as opposed to, e.g., knowledge discovery or

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statistical analysis. We will not worry much about such distinctions, and give DM the rather all encompassing definition of using your data to obtain information. This essentially defines every discovery task as some kind of DM, but there is really a continuum. The more data one has, and the less physical modeling one includes, then the more time one will spend on data management, models, and investigation, and the more DM the task will be. If one has eight data points of force and acceleration, and one performs a linear regression to fit mass, it is silly to consider it DM. There is very little time spent on the data, and one is essentially just fitting an unknown parameter in the known physical law F = ma. However, if one is trying to predict what song can be a commercial hit based on a database of song characteristics and sales data, then the primacy of data, and the absence of any guiding theory, make it clearly a DM problem [8]. DM in materials development generally focuses on prediction. Relationships are established between desired dependent properties (e.g., melting temperature or catalytic activity) and independent properties that are easily controlled and measured (e.g., precursor concentrations or annealing temperatures). Once such a relationship is established, dependent properties can be quickly predicted from independent ones, without having to perform costly and time consuming experiments. It is then possible to optimize over a large space of possible independent properties to obtain the desired dependent property. In general, we will define X as the independent properties or variables, Y as the dependent properties or variables, F as the derived relationship between X and Y , and YPred as the predicted values of Y based on F and X . The goal of a DM effort is usually to determine F such that YPred represents Y as effectively as possible. There are several key areas that need to be considered in a DM application such as the one described above: data management and preparation, prediction methods, assessment, optimization, and software.

2.1.

Data Preparation and Management

Data preparation and management will not be discussed in detail since the issues are very dependent on the specific data being used. However, the tasks associated with cleaning and managing the data can often take up the bulk of a DM project, and should not be underestimated. Data must be stored so that it can be accessed efficiently, interfaced with equipment, updated, etc. Solutions can range from simple flat files to sophisticated database software. Issues often exist with the type and quality of the data, and it is frequently necessary to make significant transformations to bring the data into a universally comparable format, and to regroup data into appropriate new variables. There is sometimes erroneous or just missing data, which may need to be dealt with

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in some manner before or during the DM process. Finally, data must be adequately comprehensive to be amenable to DM. It may be necessary to obtain further data in key areas, perhaps guided by the DM results in an iterative procedure. These issues are described in many data mining books, e.g., Ref. [7].

2.2.

Prediction Methods

Prediction methods form the heart of DM tools relevant for materials development. Although there are many DM approaches that can be used for prediction, here we focus only on three of the most popular, linear regression, neural networks, and classification methods. Linear regression is often one of the first approaches to try in a DM project, unless one has reasons to expect nonlinear behavior. It is assumed that the relationship F is a linear function, and the unknown parameters are determined by multivariate linear regression to minimize the squared error between YPred and Y (these methods are discussed in many textbooks, e.g., Refs. [9, 10]. Linear regression is generally performed by matrix manipulations and is very robust and rapid. There are many variations on strict regression, e.g., adding weights or transforming variables with logarithms. Some of the most useful regression tools are those for reducing the number of independent variables (X ), sometimes called dimensional reduction. It is frequently the case that there are many possible independent variables, but not all of them will be truly independent or important. Furthermore, the original data categories may not be optimal, and linear combinations of the variables, called latent variables, might be more effective. For example, alloy properties affected by strain will depend on the differences in atomic sizes, rather than the size of each constituent element separately. It is often difficult to have enough data to properly fit coefficients for a large number of variables (e.g., uniformly gridding a space of n variables with m points for each variable requires n m data points, which rapidly becomes unmanageable. This is sometime called the “curse of dimensionality” and is a much more significant problem in nonlinear fitting methods, such as the neural networks described below). Having too many variables that are not well constrained can lead to overfitting and poor predictive ability of the function F. Ideally, the DM method will help the user define and include the most effective latent variables for prediction. One common method for defining latent variables is Principal Component Analysis (PCA), which yields latent variables that are orthogonal and ordered by decreasing variance [11]. Assuming that variance correlates well with the importance of the latent variable to the dependent variables, then the principal components are ordered in a sensible fashion and can be truncated at some point. Orthogonality assures that latent variables are independent and


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will represent different variations. A limitation of this approach is that no information about Y is used in picking the variables. Some improvement can often be obtained by using Partial Least Squares (PLS) regression [9, 12–14], which is similar in spirit to PCA, but constructs orthogonal latent variables that maximize the covariance between X and Y . PLS latent variables capture a lot of the variation of X , but are also well correlated with Y , and so are likely to provide effective predictions. However one defines the latent variables, it is important to test their effectiveness, and there are a number of methods to identify statistically significant variables in a regression (e.g., ANOVA) [7, 9]. Another popular method is to make use of cross validation, which is discussed below, to exclude variables that are not predictive. Neural Network (NN) methods [15] are more general than linear approaches and have become a popular prediction tool for many areas. NNs loosely model the functioning of the brain, and consist of a network of neurons that can take inputs, sum them with weights, operate on the sum with a transfer function, and then emit an output. The NN is generally viewed as having layers, the first takes input from outside the NN, and the last outputs the final results to the user, while layers in between are called hidden and communicate only with other layers. For the problems considered here, the NN plays the role of the relationship F between X and Y . The weights of the neurons are unknown and must be determined by training based on known input X and output Y , where the goal is generally to minimize |YPred − Y |. The training process is analogous to a linear regression, except that the unknown weights are much more difficult to determine and many different training methods exist. Similar problems occur with excessive numbers of independent variables, and some dimensional reduction, e.g., by PCA, may be necessary. The strength of NNs is that they are very flexible, and with enough training can in principle represent any function, making them more powerful than linear methods. However, this increased power comes at a price of increased complexity. NNs have many choices that must be made correctly for optimal performance, including the number of layers, the number of neurons in each layer, the type of transfer function for each neuron, and the method of training the neural network. In general, training a NN is orders of magnitude slower than a linear regression, and convergence to the optimal parameters is by no means assured. NNs also have the drawback that it is less obvious how the X and Y variables are related than in a linear regression, making intuitive understanding more challenging. The problems of inadequate training and overfitting data are quite serious with NN’s. Some NN’s make use of “Bayesian regularization” [16–19], which includes uncertainty in the NN weights and provides some protection against overfitting. Another common solution is combining predictions from a number of differently trained NN’s (prediction by “committee”) (this approach is used

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in, e.g., Refs. [20, 21]). Another interesting approach, which can only be used in cases where one if faced with many similar problems, is to retrain NNs on related problems, making use of the information already gained in their previous training (this is done in, e.g., Ref. [22]). Classification maps data into predefined classes rather than continuous variables, where the classes are defined based on the dependent properties Y . For example, if Y is conductivity, one could classify materials into metals and insulators, and try to predict to which class a material should belong based on X , rather than performing a full regression of Y on X to predict the continuous conductivity values. Another example is predicting crystal structure, where each different structure type can be considered a class, and the goal is to be able to predict class (assign a structure type) based on the independent data X . In classification DM the relation F maps X onto categories YPred , rather than continuous values. There are a range of different classification methods, as described in most standard textbooks (we found Ref. [6] particularly lucid on these issues). The only classification scheme that will be discussed here is the K -nearest neighbor method, which is one of the simplest. This approach requires that one can define a distance between any two samples, dij = distance between X i and X j . Classification for a new X i is performed by calculating its K nearest neighbors in the existing data set, and then assigning X i to the class that contains the most items from the K neighbors. The spirit of this approach underlies structure maps for crystal structure prediction, discussed in more detail below. Other classification approaches use Bayesian probabilistic methods, decision trees, NNs, etc. but will not be described here [1, 6, 7]. There are some issues with defining a metric of success for classifications. Since YPred and Y represent class occupancies, there is not necessarily any way to measure a distance between them. One way to view the results is what is rather wonderfully called a confusion matrix, where matrix element m ij gives the number of times a sample belonging in class Ci was assigned to C j . In order to define a metric for success it is important to realize that when assigning samples to a class there are two parameters that characterize the accuracy, the fraction of samples correctly placed into the class (true positives), and the fraction of samples incorrectly placed into the class (false positives). These can vary independently and their importance can be very dependent on the problem (for example, in classifying blood as safe, it is important to get as many true positives as possible, but absolutely essential not to allow any false positives, since that would allow unsafe blood into the blood supply). Therefore, the metric for success in classification must be chosen with some care. Note that clustering, which is similar to classification, is differentiated by the fact that clustering groups data without the data clusters being predefined. This is sometimes called “unsupervised” learning and will not be discussed further here, but can be found in most DM references.


2.3.

401

Assessment

Cross-validation (CV) [23, 24] is a technique to assess the predictive ability of a fit and reduce the danger of overfitting. In a CV test with N data points, N − n data points are fit and used to predict the n points excluded from the fit. The predicted error of the excluded points is the CV score. This process can be averaged over many possible subsets of the data, which is called “leave n out CV”. The key concept behind CV is that the CV score is based on data not used in the fit. For this reason, the CV score will decrease as the model becomes more predictive, but will start to increase if the model under- or overfits the data. This in contrast to predicted errors in data that is included in the fit, which will always decrease with more fitting degrees of freedom. For example, consider a linear regression on a set of latent variables. The root mean square (RMS) error in the fit data will be a monotonically decreasing function of the number of latent variables used in the regression. However, the CV score will generally decrease for the initial principal components, and then start to increase again as the number of principal components gets large. The initial decrease in the CV score occurs because statistically meaningful variables are being added and the regression model is becoming more accurate. The increasing CV score signals that too many variables are being used, the regression is fitting noise, and that the model is overfit. By minimizing the CV score it is therefore possible to select an optimal set of latent variables for prediction. This idea is illustrated schematically in Fig. 1.

CV Error

Optimal

RMS

Number of latent variables Figure 1. A schematic comparison of the error calculated with data included in the fit (normal RMS fitting error – solid line) and excluded from the fit (CV score – dashed line).

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Test data is another important assessment tool, and simply refers to a set of data that is excluded from working data at the beginning of the project and then used to validate the model at the end of model building. To some extent, the CV method does this already, but in the common case where the model is altered to optimize the CV score, it will overestimate the true predictive accuracy of the model [23]. It is only by testing on an entirely new data set, which the model has not previously encountered, that a reliable estimate of the predictive capacity of the model can be established. Sometimes there is not enough data to create an effective test data set, but it is certainly advisable to do so if at all possible.

2.3.1. Optimization Optimization methods [25, 26] are not usually considered DM, but they are an essential tool of many DM projects. For example, once a predictive model has been established, one frequently wants to optimize the inputs to give a desired output. This usually cannot be done with local optimization schemes (e.g., conjugate gradient methods) due to a rough optimization surface with many local minima. It is therefore frequently necessary to use an optimization method capable of finding at least close to the global minimum in a landscape with many local minima. A detailed discussion of these methods is beyond the scope of this article, but common approaches include simulated annealing Monte Carlo, genetic algorithms, and branch and bound strategies. Genetic algorithms seem to be the most popular in the DM applications discussed here, and work by “evolving” toward an optimal sample population through operations such as mixing, changing, and removing samples.

2.3.2. Software Many DM algorithms are fairly simple, and can be programmed relatively quickly. Often the underlying numerical operations involve no more than standard matrix operations, and access to widely available basic linear algebra subroutines (BLAS) is adequate. However, DM is generally very explorative, and it is common to try many different approaches. Coding everything from scratch becomes prohibitive, and will lock the user into the few things they can readily implement. Fortunately, there are a large number of both free and commercial DM tools available for users. Some tools, like the Neural Net Toolbox in Matlab, are implemented in languages likely to be familiar to the materials scientist, and are readily accessible. An impressive list of possible tools is given in Appendix A of Refs. [6, 7]. It should also be remembered that for the academic user many companies will have special rates, so it is worth exploring commercial software.


3.

403

Applications

There are far too many studies using DM methods to offer a comprehensive revue. Therefore, we focus on a few key areas where DM techniques are highlighted and seem to be playing an increasingly important role.

3.1.

Quantitative Structure–Property Relationships (QSPR)

Quantitative Structure–Property Relationships (QSPR), and the closely related techniques of Quantitative Structure–Activity Relationships (QSAR), are based on the fundamental tenet that many molecular properties, from boiling point to biological activity, can be derived from basic descriptors of molecular structure. For some examples, see the general review of using NNs to predict physiochemical properties in Ref. [27] QSPR/QSAR are generally considered methods of chemistry, but are closely related to the activities of a DM material scientist. QSPR/QSAR is a large field and here we consider only one particularly illustrative example, the work of Chalk et al., predicting boiling points for molecules [20]. The boiling point for any given compound is not a particularly hard measurement, but the ability to quickly predict boiling points for many compounds, particularly ones that only exist as computer models, can be useful for screening in, e.g., drug design. Computing the boiling point of a compound directly from physical principles requires a very accurate model of the energetics and significant computation. Therefore, researchers have generally turned to DM applications in this area. Chalk et al. have a database of 6629 molecular structures and boiling points. The dependent variables Y are taken as the boiling points. A set of descriptors, X 0 , are developed based on structural and electronic characteristics (derived from semiempirical atomistic models). A technique called formal inference-based recursive modeling (FIRM) is then used to asses the relevance of each variable (this technique will not be described here but allows the influence of a variable to be tested). A set of 18 descriptors are settled on as likely to be significant and they are used for the independent variables X . A test data set of 629 molecules that span the whole range of boiling temperatures is removed. The remaining 6000 molecules are then used to find the optimal model function F to map X to Y . F is represented by a NN, and after some initial testing one is chosen with 18 first layer nodes, 10 nodes in the hidden second layer, and a single node in the third layer. The transfer functions are all sigmoids (sig(x) = 1/(1 + exp(−x))) and trained with a back-propagation algorithm. In order to control for overfitting the data is broken up into 10 disjoint subsets and a “leave

404


600 out” cross validation is performed. This trains 10 distinct NNs on 5400 molecules each. The NN training is stopped when the CV score reaches a minimum. The prediction function F is taken to be a committee, and uses the mean result of the values predicted by all 10 NNs. The final test for F is done by comparing the predicted and true boiling points for the 629 molecule test set, giving errors with a standard deviation of only 19 K (the predicted vs. true melting temperatures for the test set are shown in Fig. 2). The predictive capacity is good enough that for many of the largest prediction errors it was possible to go back to the experimental data and show that the input data itself was in error. One could now imagine using a genetic algorithm and the predicting function F to search the space of molecular structures to find, e.g., a very high melting temperature molecule, although no such work was performed by the authors. It is worth noting that computation plays an important role in providing the basic input data in the study. All of the structural and electrostatic descriptors were generated by semi-empirical atomistic models. Using computational methods can be an efficient way to generate large amounts of descriptor information, greatly reducing the amount of experimental work required.

Figure 2. Predicted vs. true boiling points for 629 compounds. Prediction is done by neural networks fit to 6000 boiling points that did not include the 629 shown here. (After [20], reproduced with permission).


3.2.

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Processing–Structure–Property Relationships

Processing–Structure–Property (PSP) relationships refer to the challenging materials problem of connecting the processing parameters of a material to its structure and properties. Processing conditions might include such things as initial composition of reactants and annealing schedule, while structural aspects might be crystal structure or grain size, and final properties are such characteristics as yield stress and corrosion resistance. PSP relationships are very important because they allow processing parameters to be adjusted to create optimal materials. PSP relationships tend to involve many different phenomena, with widely varying length and time scales, making direct modeling extremely challenging. However, analogous to QSPR’s reliance on the fact that properties must be a function of the structure of the molecules involved, in PSP relationships we know that properties must follow from structure in some manner, and that structure is somehow determined by processing. The assurance that PSP relationships exist, combined with the challenge of directly modeling them, makes this a good area for DM applications. One of the most active groups in this area has been Bhadeshia and co-workers. Bhadeshia’s review in 1999 [21] covers a lot of the material’s work that had been done up to that time in neural network (NN) modeling, and he and co-workers have continued to apply NN techniques in PSP applications to such areas as creep modeling [28, 29], mechanical weld properties [30, 31], and phase fractions in steel [32]. In general, these studies follow the DM framework used in QSPR above. Many of the data and codes used by Bhadeshia et al., as well as many others, can be found online as part of the Materials Algorithm Project [33]. Malinov and co-workers have also done extensive work with DM tools in PSP relationships, and have developed a code suite, complete with graphical user interface, to make use of their models [34]. Their work has focused primarily on Ti alloys [35–37] and nitrocarburized steels [38, 39]. The NN software they developed uses a cross validation (CV)-like strategy to assess the effectiveness of different NN architectures, training methods, and trainings, so that the best network can be obtained by optimization, rather than intuitive choice. It is a general trend in DM applications to try to automatically optimize as many choices as possible, since this gives the best results with the least user intervention. Many apparent DM choices, such as which latent variables or NN architectures to use, can in fact be determined by performing a large number of tests. Implementing this type of automation is generally limited by the user’s willingness to code the required tests, the time it takes to perform the optimization, and the amount of data required for sufficient testing. Also, one should ideally have a test set that is entirely excluded from all the optimization processes for final testing.

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A particularly interesting application by Malinov et al. is the prediction of time–temperature-transformation (TTT) diagrams for Ti alloys [34, 35, 37]. TTT diagrams give the time to reach a specified fraction of phase transformation at each temperature, and for a given phase fraction they are a curve in time–temperature space. They can be modeled to some extent directly with Johnson–Mehl–Avrami theory, but Malinov et al. chose to use a NN model so as to be able to predict for many systems and composition variations. The details discussed here are all from Ref. [35]. The data set was 189 TTT diagrams for Ti alloys, and the independent variables were taken to be the compositions of the 8 most common alloying elements and oxygen. Some additional elements that were not prevalent enough in the data set for accurate treatment had to be removed or mapped onto a Mo equivalent. It should be noted that the authors are careful to identify the ranges of the concentrations of alloying elements present in the test set. This is very important, since given the limited data, it is not clear that this NN would give accurate predictions outside the concentration ranges used in training. The dependent variables represented more of a problem, since TTT diagrams are curves, not single values. Malinov et al. solved this problem by representing the TTT diagram as a 23-tuple. Two entries gave the position of the TTT graph nose, its time and temperature. Ten entries gave the upper portion of the curve, where each entry was the fractional change in time for a fixed change in temperature, and ten more the lower portion. Finally, one entry was reserved for the martensite start temperature. These considerations, for both the independent and dependent variables, demonstrate some of the data processing that can be required for successful DM. The final predictions are quite accurate for test sets, and allowed exploration of the dependence of TTT curves on alloy composition. A number of TTT diagram predictions for (at that time) unmeasured materials were given, and some of these have since been measured, demonstrating reasonably good predictive ability for the NN model (see Fig. 3) [37]. A set of studies using DM techniques to model Al alloys recently came out of Southampton University [40–44]. The work by Starink et al. [44] summarizes studies on strength, electrical conductivity, and toughness. These studies are particularly interesting since they directly compare different DM methods as well as more physically based modeling, based on known constitutive relations. Starink et al. make use of linear regression and Bayesian NN models like those discussed above, but also apply neurofuzzy methods and support vector machines. We will not discuss these further except to point out that the latter is a relatively new development that seems to have some improved ability to give accurate predictions over the more common NN methods, and will likely grow in importance [45–47]. For the cases of direct comparison, Starink et al. find that physically based modeling performs slightly better. However, these examples involve very small data sets (around 30 samples),


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Figure 3. Comparison of predicted and measured TTT diagrams for different Ti alloys. These predictions were made and published before the experimental measurements were taken. (After Ref. [37], reproduced with permission.)

so one expects there to be significant undertraining in DM methods. Also of interest is the over three-fold decrease in predictive error for conductivity when going from linear to nonlinear DM methods, demonstrating why nonlinear NN methods have become the dominant tool for many applications. Starink et al. make some use of the concept of hybrid physical and DM approaches. This is a very natural idea, but worth mentioning explicitly. The spirit of DM is often one of using as little physical knowledge as possible, and allowing the data to guide the results. However, by introducing a certain amount of physical knowledge, a DM effort can be greatly improved. As summarized by Starink et al., this can be done through initially choosing independent variables based on known physics, using functional forms that are physically motivated in the DM, and using DM to fit remaining errors after a physical model has been used.

3.3.

Catalysis

A particularly exciting area of DM applications at present is in catalysis. A lot of recent activity in this field has been driven by the advent of highthroughput experiments, where the ability to rapidly create large data sets has created a new need for data mining concepts to interpret and guide experiment. Some reviews in this area can be found in Refs. [48–50].

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Some authors have taken approaches similar to those used in QSPR/QSAR applications and the PSP modeling described above – finding a NN model to connect the properties of interest to tractable descriptors, and then exploring that model to understand dependencies or optimize properties [22, 50–56]. The input independent variables are generally the compositions of possible alloying materials in the catalyst, and the output is some measure of the catalytic activity. Note that it is quite possible to have multiple final nodes in the network to output multiple measures of interest, such as conversion of the reactants and percentages of different products [51, 52]. It is also possible to look at catalytic behavior for a fixed catalyst under different reactor conditions, where the reactor conditions become the independent variables [22]. Once a NN has been trained, the best catalyst can be found through optimization of the function defined by the NN. This is generally done with a genetic algorithm [51, 54, 56], but other methods have also been explored [55]. Baerns et al. have done influential work in using a genetic algorithm to design new catalysts, but have skipped the step of fitting a model altogether, directly running experiments on each new generation of catalysts suggested by the genetic algorithm [57–59]. For example, Baerns et al. studied oxidative dehydrogenation of propane to propene using metal oxide catalysts with up to eight metal constituents, and found a general trend toward better catalytic activity with each generation, as shown in Fig. 4. Although optimizing the direct experimental data limits the number of samples that can be examined (Baerns et al. generally look at only a few hundred) the results have been very encouraging, e.g., leading to an effective multicomponent catalyst for low-temperature oxidation of low-concentration propane [58]. Further success

Figure 4. The best (open bar) and mean (solid bar) yield of propene at each generation of catalysts created by genetic algorithm. (After [57], reproduced with permission.)


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was obtained in studying oxidative dehydrogenation of propane to propene by following up on materials suggested by the combinatorial genetic algorithm search with further noncombinatorial “fundamental” studies [57]. Baerns et al.’s work demonstrates that the best results are sometimes obtained by combining DM and more traditional approaches. Further improvements in high-throughput methods will make direct iterative optimization of the experiments increasingly effective, but a fitted model will likely always be able to explore more samples and provide more rigorous optimization. The choice to use a fitted model is then a balance between the advantage of being able to optimize more accurately and the disadvantage of having a less accurate function to optimize. Umegaki et al. suggest that, in direct comparisons, a combined NN and genetic algorithm approach is more effective than direct optimization of experimental results, but this is a complex issue and will be problem dependent [56]. Despite many encouraging successes, DM in catalysis still faces a number of challenges. As pointed out by Hutchings and Scurrell [49] extending the independent variables to include more preparation and processing variables might significantly broaden the search for optimal materials. In addition, issues related to lifetime, stability, and other aspects of long-term performance are often difficult to predict and need to be addressed. Finally, Klanner et al. point out that there are different challenges for optimizing a library over a well known space of possible compositions and designing a discovery program for development in areas where there is essentially no precedent [50]. In the case of development of truly new materials, the problem of using a QSPR/QSAR approach in catalysis design is complicated because of the inherent difficulties of characterizing heterogeneous solids to build diverse initial libraries. Structure is a good metric for measuring diversity of molecular behavior, and therefore allows relatively easy assembly of diverse libraries for exploration. However, the very nonlinear behavior of solid catalysts, where activity is often dependent on such subtle details as surface defects, means that at this point there is no metric for measuring, a priori, the diversity of solid catalysts. Klanner et al. therefore suggest that development work will have to take place through building a large initial set of descriptors, based on synthesis data and properties of the constituent elements, and then use dimensional reduction to get a manageable number. Finally, no effort has been made here to make comparisons of DM to direct kinetic equation modeling in catalysis design. Some comments with regards to theses methods, and how they can be integrated with DM approaches, are given in Ref. [60]. It should be noted that the above issue of assembling diverse libraries, along with using genetic algorithms for intelligent searching, can be viewed as parts of the general problem of optimized experimental design. This is not a new area, but has become increasingly important due to the advent of high-throughput methods. It also encompasses such well developed fields as

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statistical Design of Experiments. This is a fruitful area for statistical and DM methods, and many of the relevant issues have already been mentioned, but we will not discuss it further here. The interested reader can consult the review by Harmon and references therein [48]. Another DM area that has been receiving increased attention due to high-throughput experiments is correlating the results of cheap and fast experimental measurements with properties of interest. This becomes particularly important when it is necessary to characterize large numbers of samples quickly, and careful measurement of the desired properties is not practical. For a discussion of this issue in high-throughput polymer research see Refs. [61, 62] and a number of rapid screening tools and detection schemes used in high-throughput catalysis development are described in Ref. [63].

3.4.

Crystal Structure

The prediction of crystal structure is a classic materials problem that has been an area of ongoing research for many years. Now that modeling efforts have made computational materials design a real possibility in many areas, the problem of predicting crystal structure has become more practically pressing, since it is usually a prerequisitie for any extensive materials modeling. Crystal structure prediction is an area well suited for DM efforts, since there is no generally reliable and tractable method to predict structure, and there is a lot of structural data collected in crystallographic databases (e.g., ICSD [64], Pauling files [65], CRYSTMET [66], ICDD [67]). Some of the most successful methods for crystal structure prediction are what are known as structure maps, reviewed at length in Refs. [68, 69]. Structure maps exist primarily for binary and ternary compounds, and the best known examples are probably the Pettifor maps [70]. To understand how Pettifor maps work, consider the map designed for AB binary alloys. Each possible element is assigned a number, called the Mendeleev number. Then each alloy AB can be plotted on a Cartesian axis by assigning it the position (x, y), where x is the Mendeleev number for element A and y is the Mendeleev number for element B. At position (x, y) one places a symbol representing the structure type for alloy AB. When enough data is plotted the like symbols tend to cluster – in other words, alloys with the same structure type tend to be located near each other on the map. This can be clearly seen in the Pettifor map in Fig. 5. The probable structure type for a new alloy can simply be found by locating where the new alloy should reside in the map and examining the nearby structure types. Structure maps were not originally introduced as an example of DM, but can be understood within that framework. One can extend the idea of using Mendeleev number to a general “vector map,” which maps each alloy to a


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Figure 5. An AB binary alloy Pettifor map. Notice that like structure types show a clear tendency to cluster near one another. Provided by John Rodgers using the CRYSTMET database [66].

multicomponent vector. The vector components might be any set of descriptors for the alloy, such as Mendeleev numbers, melting temperatures, or differences in electronegativities. Once the alloys have been mapped to representative vectors they are amenable to different DM schemes. Since crystal structures are discrete categories, not continuous values, some sort of classification DM is going to be required. Structure maps work by defining a simple Euclidean metric on the alloy vectors and making the assumption that alloys with the same structure types will be close together. When a new alloy is encountered its crystal structure

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is predicted by examining the neighborhood of the new alloy in the structure map. Structure types that appear frequently in a small neighborhood of the new alloy are good candidates for the alloy’s structure type. This is a geometric classification scheme, along the lines of K -nearest-neighbors described above. There is no unique way to define the vectors that create the structure map, and many different physical quantities, such as electronegativities and effective radii, have been proposed for constructing structure maps. Ref. [64] lists at least 53 different atomic parameters that could be used as descriptors to define a structure map. The most accurate Pettifor maps are built by mapping alloys to vectors using a specially devised chemical scale [71]. The chemical scale was motivated by many physical concerns, but is fundamentally an empirical way to map alloys to vectors, chosen to optimize the clustering of alloys with the same crystal structures. A number of new ideas are suggested by viewing crystal structure prediction from a DM framework. First, it is clear that many standard assessment techniques have only recently begun to be incorporated. It was not until about 20 years after the first Pettifor maps that an effort was made to formalize their clustering algorithm and assess their accuracy using cross validation techniques (the accuracy was found to very good, in some cases giving correct predictions for non-unique structures 95% of the time) [72]. Also, the question of how to assess errors can be fruitfully thought of in terms of false positives (predicting a crystal structure that is wrong) and false negatives (failing to predict the crystal structure that is right). For many situations, e.g., predicting structures to be checked by ab initio methods or used as input for Rietveld refinements, a false positive is not a large problem, since the error will likely be discarded at a later stage, but a false negative is critical, since it means the correct answer will not be found with further investigation. This leads to the idea of using maps to suggest a candidate structure list, rather than a single candidate structure [72]. Using a list creates many false positives, but greatly reduces the chance of false negatives. A DM perspective on structure prediction encourages one to think of moving beyond present structure map methods. For example, different metrics, other classification algorithms, or mining on more complex alloy descriptors, might yield more accurate results. Some work along these lines has already occurred, including machine learning based structure maps [73] and NN and clustering predictions of compound formation [74]. A similarly spirited application used partial least squares to predict higher level structural features of zeolites in terms of simpler structural descriptors [75], and is part of a more general center focused on DM in materials science [76]. The structure maps have at least two severe limitations. As described above, they predict structure type given that the alloy has a structure at a given stoichiometry, but do not consider the question of whether or not an alloy will have an ordered phase at that stoichiometry. This is not a problem when a structure


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is known to exist and one wants to identify it, but in many cases that information is not available. There are some successful methods for identifying alloys as compound forming versus having no-compounds, e.g., Meidema’s rules [77] or Villar’s maps for ternary compounds [68], but the problem of identifying when an alloy will show ordering at a given composition has not been thoroughly investigated in the context of structure maps. However, it is certainly possible that further DM work could be of value solving this problem, and some potentially useful methods are discussed below. Another serious limitation on structure maps is that classification DM is only effective when an adequate number of samples of each class are available. There are already thousands of structure types, the number is still increasing, and only a small percentage of possible multicomponent alloy systems have been explored [68]. Therefore, it seems unlikely that sufficiently many examples of all the structure type classes will ever be available for totally general application of structure maps. Infrequent structure types are less robustly predicted with structure maps, and totally new structure types cannot be predicted at all. The problem of limited sampling can be alleviated by restricting the area of focus, e.g., considering only the most common structure types, which are likely to be well sampled, or only a subset of alloys, where all the relevant structure types can be discovered. However, the very significant challenge of sampling all the relevant structure types creates a need for other methods. One promising idea is to abandon the use of structure types as the most effective way to classify structures and replace it with a scheme easier to sample. An idea along these lines is to classify alloys by the local environments around each atom [68, 78]. Local environments may in fact be a more relevant method of classification than structure type for understanding physical properties, and there seem to be far fewer local environments than different structure types. This is analogous to classifying proteins by their different folds, which are essential to function and come in limited variety [79]. Computational methods, using different Hamiltonians, offer an increasingly practical route toward crystal structure prediction. Given an accurate Hamiltonian for an alloy, the stable crystal structures can be calculated by minimizing the total energy. These techniques can also predict entirely new structures never seen experimentally, since the prediction is done on the computer. Unfortunately, the structural energy landscape has many local minima, and it cannot be explored quickly or easily. Researchers in this area therefore are forced to make a tradeoff between the speed and accuracy of the energy methods, and the range of possible structures that are explored. For example, Jansen has used simple pair potentials to explore the energy landscape, and then applied more accurate ab initio methods for likely structural candidates [80]. This is a common approach, to optimize with simplified expressions and then use slower and more accurate ab initio energy methods on only the more promising areas. A similar approach was taken to predict a range of

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inorganic structures from a genetic algorithm [81]. If one restricts the possible structures, then direct optimization of ab initio energies can be performed. For example, low cohesive energy structures for 32 possible alloying elements were found on a four atom, face centered cubic unit cell by optimizing ab initio energies using a genetic algorithm [82]. Although these approaches are quite promising, optimizing the energy over the space of all possible atomic arrangements is generally not practical. It is necessary to find some approach to guide the calculations to regions of structure space that are likely to have the lowest energy structures and can be explored effectively. A practical and common method to guide calculations is sometimes colloquially referred as the “round up the usual suspects” approach, borrowing a quotation from Captain Louis Renault in the end of Casablanca. This approach simply involves calculating structures one thinks are likely to be ground states and is another example of human DM, where the scientist is drawing on their own experience to guide the calculations toward the correct structure. As mentioned in the introduction, formalizing human DM on the computer offers many advantages in accuracy, verification, portability, and efficiency. An improvement can be made by limiting the human component to suggesting a few likely parent lattices, and then fitting simplified Hamiltonians on each parent lattice to predict stable structures. This approach, called cluster expansion, has been well validated in many systems [83, 84] and has been successful in predicting some structures that had not been previously identified experimentally [85, 86]. However, choosing the correct parent lattice and performing the fitting required for cluster expansion is at present still difficult to automate, although efforts along these lines are being made [87]. Ideally, the process of guiding computational crystal structure prediction would be entirely automated by DM methods. A step in this direction has been taken by Curtarolo et al. who have demonstrated how one might combine experimental data, high-throughput computation, and DM methods to guide new calculations toward likely stable crystal structures [88]. Experimental information is used to get a list of commonly occurring structure types, and then these are calculated using automated scripts for a large number of systems. Mined correlations between structural energies are then used to guide calculations on new systems toward stable regions, reducing the number of calculations required to predict crystal structures. This approach can, in theory, be expanded to totally new structure types, since these can be generated on the computer, and work in this direction is under development.

4.

Conclusions

We have seen here a number of different examples of DM applications in different areas, and it is valuable to step back and note some overall


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features. In general, DM applications in materials development still need to prove themselves, and relatively few new discoveries have been made using them. Many of the results in this field consist primarily of exploring new models to demonstrate that such modeling is possible, that accurate predictions can be made, and that useful understanding of dependencies on key variables can be obtained. This will inevitably cause some skepticism about the final utility of the methods, but it is appropriate for a field which is still relatively young and finding its place. A similar evolution has been taken by, e.g., ab initio quantum mechanical techniques. It is only recently that these methods have moved out the stage where the accuracy of the model was the key issue to the stage where the bulk of papers focus on the materials results, not the techniques. All the drivers for using DM methods identified in the introduction, more data, databases, and DM tools, will only become increasingly forceful with continuing advances in experiment, computation, algorithms, and information technology. For these reasons, we believe that DM approaches are going to be increasingly important tools for the modern materials developer. A number of the above examples showed the necessity of combining DM methods with more traditional physical approaches. Whether it is microstructural modeling in the area of processing–structure–property prediction or kinetic equation modeling in catalysis design, physical modeling is by no means standing still, and its utility will continue to expand. In the few cases where authors make direct comparisons, it is not clear that DM applications have been more effective [44, 89]. It is already true that DM approaches, although more data focused, are deeply intertwined with traditional physical modeling. A researchers knowledge of the physics of the problem strongly influences such things as choices of descriptors (e.g., exponentiating parameters where thermal activation is expected), choices in the predictive model (e.g., using linear models when linear relationships are expected), and many unwritten small decisions about how the DM is done. DM and physical modeling, despite an apparent conflict, are really best used collaboratively, and effective materials researchers will need to combine both tools to have maximal impact. Another important feature to note is the difference between DM in materials science and the more established areas of drug design and QSPR/QSAR. Although the overall framework is very similar, establishing effective descriptors for independent variables seems to be harder in materials applications. Bulk materials, more common in traditional materials science applications, often have atomic-, nano-, and micro-structural features that are hard to characterize and quantify with effective descriptors. In their absence, further progress on many problems will require additional descriptors relating to processing choices.

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Finally, we would like to stress the natural synergy between DM and other kinds of computational modeling. High-throughput computation can help provide the wealth of data needed for robust data mining, as was illustrated above in the use of computationally optimized structures for boiling point modeling [20] and crystal structure prediction [80–82, 88]. Impressive examples of high-throughput ab initio computation providing large amounts of accurate materials data can be found in Refs. [90–92]. High-throughput computation not only increases the effectiveness of DM methods, but extends the reach of computational modeling, since DM methods can help span the challenging range of length and time scales involved in materials phenomena. The growing power of DM and other computational methods will only increase their interdependence in the future. Finally, on a more personal note, we have found that one of the most valuable contributions of DM to our research has been to expand how we think about problems. DM encourages one to ask how one can make optimal use of data and to look deeply for patterns that might provide valuable information. DM makes one think on a large scale, thereby encouraging the automation of experiment, computation, and data analysis for high-throughput production. DM also encourages a culture of careful testing for any kind of fitting, through cross validation and statistical methods. Finally, DM is inherently inderdisciplinary, encouraging materials scientists to learn more about analogous problems and techniques from across the hard and soft sciences, thereby enriching us all as researchers.

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1.19 FINITE ELEMENTS IN AB INITIO ELECTRONIC-STRUCTURE CALULATIONS J.E. Pask and P.A. Sterne Lawrence Livermore National Laboratory, Livermore, CA, USA

Over the course of the past two decades, the density functional theory (DFT) (see e.g., [1]) of Hohenberg, Kohn, and Sham has proven to be an accurate and reliable basis for the understanding and prediction of a wide range of materials properties from first principles (ab initio), with no experimental input or empirical parameters. However, the solution of the Kohn–Sham equations of DFT is a formidable task and this has limited the range of physical systems which can be investigated by such rigorous, quantum mechanical means. In order to extend the interpretive and predictive power of such quantum mechanical theories further into the domain of “real materials”, involving nonstoichiometric deviations, defects, grain boundaries, surfaces, interfaces, and the like; robust and efficient methods for the solution of the associated quantum mechanical equations are critical. The finite-element (FE) method (see e.g., [2]) is a general method for the solution of partial differential and integral equations which has found wide application in diverse fields ranging from particle physics to civil engineering. Here, we discuss its application to large-scale ab initio electronic-structure calculations. Like the traditional planewave (PW) method (see e.g., [3]), the FE method is a variational expansion approach, in which solutions are represented as a linear combination of basis functions. However, whereas the PW method employs a Fourier basis, with every basis function overlapping every other, the FE method employs a basis of strictly local piecewise polynomials, each overlapping only its immediate neighbors. Because the FE basis consists of polynomials, the method is completely general and systematically improvable, like the PW method. Because the basis is strictly local, however, the method offers some significant advantages. First, because the basis functions are localized, they can be concentrated where needed in real space to increase the efficiency 423 S. Yip (ed.), Handbook of Materials Modeling, 423–437. c 2005 Springer. Printed in the Netherlands.

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of the representation. Second, a variety of boundary conditions can be accommodated, including Dirichlet boundary conditions for molecules or clusters, Bloch boundary conditions for crystals, or a mixture of these for surfaces. Finally, and most significantly for large-scale calculations, the strict locality of the basis facilitates implementation on massively parallel computational architectures by minimizing the need for nonlocal communications. The advantages of such a local, real-space approach in large-scale calculations have been amply demonstrated in the context of finite-difference (FD) methods (see, e.g., [4]). However, FD methods are not variational expansion methods, and this leads to disadvantages such as limited accuracy in integrations and nonvariational convergence. By retaining the use of a basis while remaining strictly local in real space, FE methods combine significant advantages of both PW and FD approaches.

1.

Finite Element Bases

The construction and key properties of FE bases are perhaps best conveyed in the simplest case: a one-dimensional (1D), piecewise-linear basis. Figure 1 shows the steps involved in the construction of such a basis on a domain = (0, 1). The domain is partitioned into subdomains called elements (Fig. 1a). In this case, the domain is partitioned into three elements 1 –3 ; in practice, there are typically many more, so that each element encompasses only a small fraction of the domain. For simplicity, we have chosen a uniform partition, but this need not be the case in general. (Indeed, it is precisely the flexibility to partition the domain as desired which allows for the substantial efficiency of the basis in highly inhomogeneous problems.) A parent basis φˆ i ˆ = (−1, 1) (Fig. 1b). In this case, the is then defined on the parent element parent basis functions are φˆ 1 (ξ ) = (1 − ξ )/2 and φˆ 2 (ξ ) = (1 + ξ )/2. Since the parent basis consists of two (independent) linear polynomials, it is complete to linear order, i.e., a linear combination can represent any linear polynomial exactly. Furthermore, it is defined such that each function takes on the value 1 at exactly one point, called its node, and vanishes at all (one, in this case) other nodes. Local basis functions φi(e) are then generated by transformations ξ (e) (x) ˆ to each element e of the parent basis functions φˆ i from the parent element (Fig. 1c). In present case, for example, φ1(1) (x) ≡ φˆ1 (ξ (1)(x)) = 1 − 3x and φ2(1) (x) ≡ φˆ2 (ξ (1)(x)) = 3x, where ξ (1)(x) = 6x − 1. Finally, the piecewisepolynomial basis functions φi of the method are generated by piecing together the local basis functions (Fig. 1d). In the present case, for example, φ2 (x) =

 (1)   φ2 (x),  

φ1(2) (x),

0,

x ∈ [0, 1/3] x ∈ [1/3, 2/3] otherwise.

Finite elements in ab initio electronic-structure calculations

425

Figure 1. 1D piecewise-linear FE bases. (a) Domain and elements. (b) Parent element and parent basis functions. (c) Local basis functions generated by transformations of parent basis functions to each element. (d) General piecewise-linear basis, generated by piecing together local basis functions across interelement boundaries. (e) Dirichlet basis, generated by omitting boundary functions. (f) Periodic basis, generated by piecing together boundary functions.

The above 1D piecewise-linear FE basis possesses the key properties of all such bases, whether of higher dimension or higher polynomial order. First, the basis functions are strictly local, i.e., nonzero over only a small fraction of the domain. This leads to sparse matrices and scalability, as in FD approaches,

426


while retaining the use of a basis, as in PW approaches. Second, within each element, the basis functions are simple, low-order polynomials, which leads to computational efficiency, generality, and systematic improvability, as in FD and PW approaches. Third, the basis functions are C 0 in nature, i.e., continuous but not necessarily smooth. As we shall discuss, this necessitates extra care in the solution of second-order problems, with periodic boundary conditions in particular. Finally, the basis functions have the key property φi (x j ) = δi j i.e., each basis function takes on a value of 1 at its associated node and vanishes at all other nodes. By virtue of this property, an FE expansion f (x) = c φ i i i (x) has the property f (x j ) = c j , so that the expansion coefficients have a direct, real-space meaning. This eliminates the need for computationally intensive transforms, such as Fourier transforms in PW approaches, and facilitates preconditioning in iterative solutions, such as multigrid in FD approaches (see, e.g., [4]). Figure 1(d) shows a general FE basis, capable of representing any piecewise linear function (having the same polynomial subintervals) exactly. To solve a problem subject to vanishing Dirichlet boundary conditions, as occurs in molecular or cluster calculations, one can restrict the basis as in Fig. 1(e), i.e., omit boundary functions. To solve a problem subject to periodic boundary conditions, as occurs in solid-state electronic-structure calculations, one can restrict the basis as in Fig. 1(f), i.e., piece together local basis functions across the domain boundary in addition to piecing together across interelement boundaries. Regarding this periodic basis, however, it should be noted that an arbitrary linear combination f (x) = i ci φi (x) necessarily satisfies f (0) = f (1),

(1)

but does not necessarily satisfy f (0) = f (1).

(2)

Thus, unlike PW or other such smooth bases, while the value condition (1) is enforced by the use of such an FE basis, the derivative condition (2) is not. And so for problems requiring the enforcement of both, as in solid-state electronic-structure, the derivative condition must be enforced by other means [5]. We address this further in the next section. Higher-order FE bases are constructed by defining more independent parent basis functions, which requires that some basis functions be of higher order than linear. And, as in the linear case, what is typically done is to define all functions to be of the same order so that, for example, to define a 1D quadratic basis, one would define three quadratic parent basis functions; for a 1D cubic basis, four cubic parent basis functions, etc. With higher-order basis functions,


427

however, come new possibilities. For example, with cubic basis functions there are sufficient degrees of freedom to specify both value and slope at end points, thus allowing for the possibility of both value and slope continuity across interelement boundaries, and so allowing for the possibility of a C 1 (continuous value and slope) rather than C 0 basis. For sufficiently smooth problems, such higher order continuity can yield greater accuracy per degree of freedom and such bases have been used in the electronic-structure context [6, 7]. However, while straightforward in one dimension, in higher dimensions this requires matching both values and derivatives (including cross terms) across entire curves or surfaces, which becomes increasingly difficult to accomplish and leads to additional constraints on the transformations, and thus meshes, which can be employed [8]. Higher-dimensional FE bases are constructed along the same lines as the 1D case: partition the domain into elements, define local basis functions within each element via transformations of parent basis functions, and piece together the resulting local basis functions to form the piecewise-polynomial FE basis. In higher dimensions, however, there arises a significant additional choice: that of shape. The most common 2D element shapes are triangles and quadrilaterals. In 3D, tetrahedra, hexahedra (e.g., parallelepipeds), and wedges are among the most common. A variety of shapes have been employed in atomic and molecular calculations (see, e.g., [9]). In solid-state electronic-structure calculations, the domain can be reduced to a parallelepiped and C 0 [5] as well as C 1 [7] parallelepiped elements have been employed.

2.

Solution of the Schrödinger and Poisson Equations

The solution of the Kohn–Sham equations can be accomplished by a number of approaches, including direct minimization of the energy functional [10], solution of the associated Lagrangian equations [11], and self-consistent (SC) solution of associated Schrödinger and Poisson equations (see, e.g., [3]). A finite-element based energy minimization approach has been described by Tsuchida and Tsukada [7] in the context of molecular and -point crystalline calculations. Here, we shall describe a finite-element based SC approach. In this section, we discuss the solution of the Schrödinger and Poisson equations; in the next, we discuss self-consistency. The solution of such equations subject to Dirichlet boundary conditions, as appropriate for molecular or cluster calculations, is discussed extensively in the standard texts and literature (see, e.g., [2, 9]). Here, we shall discuss their solution subject to boundary conditions appropriate for a periodic (crystalline) solid.

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In a perfect crystal, the electronic potential is periodic, i.e., V (x + R) = V (x)

(3)

for all lattice vectors R, and the solutions of the Schrödinger equation satisfy Bloch’s theorem ψ(x + R) = eik·R ψ(x)

(4)

for all lattice vectors R and wavevectors k [12]. Thus the values of V (x) and ψ(x) throughout the crystal are completely determined by their values in a single unit cell, and so the solutions of the Poisson and Schrödinger equations in the crystal can be reduced to their solutions in a single unit cell, subject to boundary conditions consistent with Eqs. (3) and (4), respectively. We consider first the Schrödinger problem: 1 − ∇ 2 ψ + V ψ = εψ 2

(5)

in a unit cell, subject to boundary conditions consistent with Bloch’s theorem, where V is an arbitrary periodic potential (atomic units are used throughout, unless otherwise specified). Since V is periodic, ψ can be written in the form ψ(x) = u(x)eik·x ,

(6)

where u is a complex, cell-periodic function satisfying u(x) = u(x + R) for all lattice vectors R [12]. Assuming the form (6), the Schrödinger equation (5) becomes 1 1 − ∇ 2 u − ik · ∇u + k 2 u + VL u + e−ik·x VNL eik·x u = εu, 2 2

(7)

where, allowing for the possibility of nonlocality, VL and VNL are the local and nonlocal parts of V . From the periodicity condition (4), the required boundary conditions on the unit cell are then [12] u(x) = u(x + Rl ),

x ∈ l

(8)

and nˆ · ∇u(x) = nˆ · ∇u(x + Rl ),

x ∈ l ,

(9)

where l and Rl are the surfaces of the boundary and associated lattice vectors R shown in Fig. 2, and nˆ is the outward unit normal at x. The required Bloch-periodic problem can thus be reduced to the periodic problem (7)–(9). However, since the domain has been reduced to the unit cell, nonlocal operators require further consideration. In particular, if as is typically the case for ab initio pseudopotentials, the domain of definition is all space (i.e., the


429

R3

R2

R1

Figure 2. Parallelepiped unit cell (domain) , boundary , surfaces 1 –3 , and associated lattice vectors R1 –R3 .

full crystal), they must be transformed to the relevant finite subdomain (i.e., the unit cell) [13]. For a separable potential of the usual form VNL (x, x ) =

a a vlm (x − τa − Rn )h la vlm (x − τa − Rn ),

(10)

n,a,l,m

where n runs over all lattice vectors and a runs over atoms in the unit cell, the nonlocal term e−ik·x VNL eik·x u in Eq. (7) is

e−ik·x

a vlm (x − τa − Rn )h la

n,a,l,m

a dx vlm (x − τa − Rn )eik·x u(x ),

R3

where the integral is over all space. Upon transformation to the unit cell , this becomes

e−ik·x

a eik·Rn vlm (x − τa − Rn )h la

a,l,m n

×

dx

n

a e−ik·Rn vlm (x − τa − Rn )eik·x u(x ).

Having reduced the required problem to a periodic problem on a finite domain, solutions may be obtained using a periodic FE basis. However, if the

430


basis is C 0 , as is typically the case, rather than C 1 or smoother, some additional consideration is required. First, the direct application of the Laplacian to such a basis is problematic. Second, being periodic in value but not in derivative (as discussed in the preceding section), the basis does not satisfy the required boundary conditions. Both issues can be resolved by reformulating the original differential formulation in weak (integral) form. Such a weak formulation can be constructed which contains no derivatives higher than first order, and which requires only value-periodicity (i.e., Eq. (8)) of the basis, thus resolving both issues. Such a weak formulation of the required problem (7)–(9) is [5]: Find scalars ε and functions u ∈ V such that 1 2

1 dx ∇v ∗ · ∇u + dxv ∗ −ik · ∇u + k 2 u + VL u + e−ik·x VNL eik·x u 2

= ε dxv ∗ u

∀v ∈ V,

where V = {v : v(x) = v(x + Rl ), x ∈ l }, and the x dependence of u and v has been suppressed for compactness. Having reformulated the problem in weak form,solutions may be obtained using a C 0 FE basis. Letting u = j c j φ j and v = j d j φ j , where φ j are real periodic finite element basis functions and c j and d j are complex coefficients, leads to a generalized Hermitian eigenproblem determining the approximate eigenvalues ε and eigenfunctions u of the weak formulation and thus of the required problem [5]:

Hi j c j = ε

j

Si j c j ,

(11)

j

where

Hi j =

dx

1 1 ∇φi · ∇φ j − ik · φi ∇φ j + k 2 φi φ j + VL φi φ j 2 2

+ φi e−ik·x VNL eik·x φ j and

Si j =

(12)

dx φi φ j ,

(13)

and again the x dependence of φi and φ j has been suppressed for compactness. For a separable potential of the form (10), the nonlocal term in (12) becomes [13]

dx φi (x)e−ik·x VNL eik·x φ j (x) =

a,l,m

ai a flm hl

aj ∗

flm ,


431

where ai = flm

dx φi (x)e−ik·x

a eik·Rn vlm (x − τa − Rn ).

n

As in the PW method, the above matrix elements can be evaluated to any desired accuracy, so that the basis need only be large enough to provide a sufficient representation of the required solution, though other functions such as the nonlocal potential may be more rapidly varying. As in the FD method, the above matrices are sparse and structured due to the strict locality of the basis. Figure 3 shows a series of FE results for a Si pseudopotential [14]. Since the method allows for the direct treatment of any Bravais lattice, results are shown for a two-atom fcc primitive cell. The figure shows the sequence of band structures obtained for 3 × 3 × 3, 4 × 4 × 4, and 6 × 6 × 6 uniform meshes vs. exact values at selected k points (where “exact values” were obtained from a well converged PW calculation). The variational nature of the method is clearly manifested: the error is strictly positive and the entire band structure converges rapidly and uniformly from above as the number of basis functions is increased. Further analysis [5] shows that the convergence of the eigenvalues is in fact sextic, i.e., the error is of order h 6 , where h is the mesh spacing, consistent with asymptotic convergence theorems for the cubic-complete case [8]. The Poisson solution proceeds along the same lines as the Schrödinger solution. In this case, the required problem is −∇ 2 VC (x) = f (x),

x∈

(14)

subject to boundary conditions VC (x) = VC (x + Rl ),

x ∈ l

(15)

and nˆ · ∇ VC (x) = nˆ · ∇ VC (x + Rl ),

x ∈ l ,

(16)

where the source term f (x) = −4πρ(x), VC (x) is the potential energy of an electron in the charge density ρ(x), and the domain , bounding surfaces l , and lattice vectors Rl are again as in Fig. 2. Reformulation of (14)–(16) in weak form and subsequent discretization in a real periodic FE basis φ j leads to a symmetric linear system determining the approximate solution VC (x) = j c j φ j (x) of the weak formulation and thus of the required problem [5]: j

L i j c j = fi ,

(17)

432


Si 15

Energy (eV)

10

3⫻3⫻3 4⫻4⫻4 6⫻6⫻6

5

FE Exact 0

L

Γ

X

Figure 3. Exact and finite-element (FE) band structures for a series of meshes, for a Si primitive cell. The convergence is rapid and variational: the entire band structure converges from above, with an error of O(h 6 ), where h is the mesh spacing.

where

Lij =

dx ∇φi (x) · ∇φ j (x)

(18)

and

fi =

dx φi (x) f (x).

(19)

As in the FD method, the above matrices are sparse and structured due to the strict locality of the basis, requiring only O(n) storage and O(n) operations


433

for solution by iterative methods, whereas O(n log n) operations are required in a PW basis, where n is the number of basis functions.

3.

Self-Consistency

The above Schrödinger and Poisson solutions can be employed in a fixed point iteration to obtain the self-consistent solution of the Kohn–Sham equations. In the context of a periodic solid, the process is generally as follows (see, e.g., Ref. [3]): an initial electronic charge density ρein is constructed (e.g., by overlapping atomic charge densities). An effective potential Veff is constructed based upon ρein (see below). The eigenstates ψi of Veff are computed by solving the associated Schrödinger equation subject to Bloch boundary conditions. From these eigenstates, or “orbitals”, a new electronic charge density ρe is then constructed according to ρe = −

f i |ψi |2 ,

i

where the sum is over occupied orbitals with occupations f i . If ρe is sufficiently close to ρein , then self-consistency has been reached; otherwise, a new ρein is constructed based on ρe and the process is repeated until self-consistency is achieved. The resulting density minimizes the total energy and is the DFT approximation of the physical density, from which other observables may be derived. The effective potential can be constructed as the sum of ionic (or nuclear, in an all-electron context), Hartree, and exchange-correlation parts: Veff = ViL + ViNL + VH + VXC ,

(20)

where, allowing for the possibility of nonlocality, ViL and ViNL are the local and nonlocal parts of the ionic term. For definiteness, we shall assume that the atomic cores are represented by nonlocal pseudopotentials. ViNL is then determined by the choice of pseudopotential. VXC is a functional of the electronic density determined by the choice of exchange-correlation functional. ViL is the Coulomb potential associated with the ions (sum of local ionic pseudopotentials). VH is the Coulomb potential associated with electrons (the Hartree potential). In the limit of an infinite crystal, ViL and VH are divergent due to the long range 1/r nature of the Coulomb interaction, and so their computation requires careful consideration. A common approach is to add and subtract analytic neutralizing densities and associated potentials, solve the resulting neutralized problems, and add analytic corrections (see, e.g., Ref. [3] in a reciprocal space context, [15] in real space). Alternatively [13], it may be L associated with each atom noted that the local parts of the ionic potentials Vi,a

434


can be replaced by corresponding localized ionic charge densities ρi,a since the potentials fall off as −Z /r (or rapidly approach this behavior) for r > rc , where Z is the number of valence electrons, r is the distance from the ion center, and rc is on the order of half the nearest neighbor distance. The total Coulomb potential VC = ViL + VH in the unit cell may then be computed at once by solving the Poisson equation ∇ 2 VC = 4πρ subject to periodic boundary conditions, where ρ = ρi + ρe is the sum of electronic and ionic charge densities in the unit cell, and the ionic charge densities ρi,a associated with each atom a are related to their respective local ionic L potentials Vi,a by Poisson’s equation L ρi,a = ∇ 2 Vi,a /4π.

Since the ionic charge densities are localized, their summation in the unit cell is readily accomplished, whereas the summation of ionic potentials is not, due to their long range 1/r tails. With VC determined, Veff can then be constructed as in Eq. (20), and the self-consistent iteration can proceed.

4.

Total Energy

Like Veff , the computation of the total energy in a crystal requires careful consideration due to the long range nature of the Coulomb interaction and resulting divergent terms. In this case, the electron–electron and ion–ion terms are divergent and positive, while the electron–ion term is divergent and negative. As in the computation of Veff , a common approach involves the addition and subtraction of analytic neutralizing densities (see, e.g., Refs. [3, 15]). Alternatively, it may be noted that the replacement of the local parts of the ionic potentials by corresponding localized charge densities, as discussed above, yields a net neutral charge density ρ = ρi + ρe , and all convergent terms in the total energy. For sufficiently localized ρi,a , a quadratically convergent expression for the total energy in terms of Kohn–Sham eigenvalues εi is then [13] E tot =

i

1 − 2

f i εi +

dx ρe (x)

VLin (x)

1 dx ρi (x)VC (x) + 2 a

1 − VC (x) − εXC [ρe (x)] 2

L dx ρi,a (x)Vi,a (x),

(21)

R3

where VLin is the local part of Veff constructed from the input charge density ρein , VC is the Coulomb potential associated with ρe , i.e., ∇ 2 VC = 4π(ρi + ρe ), εXC


435

is the exchange-correlation energy density, i runs over occupied states with occupations f i , and a runs over atoms in the unit cell. Figure 4 shows the convergence of FE results to well converged PW results as the number of elements in each direction of the wavefunction mesh is increased in a self-consistent GaAs calculation at an arbitrary k point, using the same pseudopotentials [16] and exchange-correlation functional. As in the PW method, higher resolution is employed in the calculation of the charge density and potential (twice that employed in the calculation of the of the wavefunctions, in the present case). The rapid, variational convergence of the FE approximations to the exact self-consistent solution is clearly manifested: the error is strictly positive and monotonically decreasing, with an asymptotic slope of ∼−6 on a log–log scale, indicating an error of O(h 6 ), where h is the mesh spacing, consistent with the cubic completeness of the basis. This is in contrast to FD approaches where, lacking a variational foundation, the error can be of either sign and may oscillate.

5.

Outlook

Because FE bases are simultaneously polynomial and strictly local in nature, FE methods retain significant advantages of FD methods without sacrificing the use of a basis, and in this sense, combine advantages of both PW

GaAs self-consistent total energy and eigenvalues

EFE⫺EEXACT (Ha)

10⫺2

10⫺2

10⫺3

10⫺3

10⫺4

10⫺4 Etot E1 E2 E3

10⫺5 10⫺6 8

10⫺5 10⫺6 12

16

20

24

28

32

Elements in each direction Figure 4. Convergence of self-consistent FE total energy and eigenvalues with respect to number of elements, for a GaAs primitive cell. As for a fixed potential, the convergence is rapid and variational: the error is strictly positive and monotonically decreasing, with an error of O(h 6 ), where h is the mesh spacing.

436


and FD based approaches for ab initio electronic structure calculations. In particular, while variational and systematically improvable, the method produces sparse matrices and requires no computation- or communication-intensive transforms; and so is well suited to large, accurate calculations on massively parallel architectures. However, FE methods produce generalized rather than standard eigenproblems, require more memory than FD based approaches, and are more difficult to implement. Because of the relative merits of each approach, and because FE based approaches are yet at a relatively early stage of development, it is not clear which approach will prove superior in the largescale ab initio electronic structure context in the years to come [4]. Early nonself-consistent applications to ab initio positron distribution and lifetime calculations involving over 4000 atoms [5] are promising indications, however, and the development and optimization of FE based approaches for a range of large-scale applications remains a very active area of research.

Acknowledgment This work was performed under the auspices of the U.S. Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

References [1] R.O. Jones and O. Gunnarsson, “The density functional formalism, its applications and prospects,” Rev. Mod. Phys., 61, 689–746, 1989. [2] O.C. Zienkiewicz and R.L. Taylor, The Finite Element Method, McGraw-Hill, New York, 4th edn., 1988. [3] W.E. Pickett, “Pseudopotential methods in condensed matter applications,” Comput. Phys. Rep., 9, 115–198, 1989. [4] T.L. Beck, “Real-space mesh techniques in density-functional theory,” Rev. Mod. Phys., 72, 1041–1080, 2000. [5] J.E. Pask, B.M. Klein, P.A. Sterne, and C.Y. Fong, “Finite-element methods in electronic-structure theory,” Comput. Phys. Commun., 135, 1–34, 2001. [6] S.R. White, J.W. Wilkins, and M.P. Teter, “Finite-element method for electronic structure,” Phys. Rev. B, 39, 5819–5833, 1989. [7] E. Tsuchida and M. Tsukada, “Large-scale electronic-structure calculations based on the adaptive finite-element method,” J. Phys. Soc. Japan, 67, 3844–3858, 1998. [8] G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Prentice-Hall, Englewood Cliffs, NJ, 1973. [9] L.R. Ram-Mohan, Finite Element and Boundary Element Applications in Quantum Mechanics, Oxford University Press, New York, 2002. [10] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, and J.D. Joannopoulos, “Iterative minimization techniques for ab initio total-energy calculations: molecular dynamics and conjugate gradients,” Rev. Mod. Phys., 64, 1045–1097, 1992.


437

[11] T.A. Arias, “Multiresolution analysis of electronic structure: semicardinal and wavelet bases,” Rev. Mod. Phys., 71, 267–311, 1992. [12] N.W. Ashcroft and N.D. Mermin, Solid State Physics, Holt, Rinehart and Winston, New York, 1976. [13] J.E. Pask and P.A. Sterne, “Finite-element methods in ab initio electronic-structure calculations,” Modell. Simul. Mater. Sci. Eng., to appear, 2004. [14] M.L. Cohen and T.K. Bergstresser, “Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures,” Phys. Rev., 141, 789–796, 1966. [15] J.L. Fattebert and M.B. Nardelli, “Finite difference methods in ab initio electronic structure and quantum transport calculations of nanostructures,” In: P.G. Ciarlet, (ed.), Handbook of Numerical Analysis, vol. X: Computational Chemistry, Elsevier, Amsterdam, 2003. [16] C. Hartwigsen, S. Goedecker, and J. Hutter, “Relativistic separable dual-space gaussian pseudopotentials from H to Rn,” Phys. Rev. B, 58, 3641–3662, 1998.

1.20 AB INITIO STUDY OF MECHANICAL DEFORMATION Shigenobu Ogata Osaka University, Osaka, Japan

The Mechanical properties of materials under finite deformation are very interesting and are important topics for material scientists, physicists, and mechanical and materials engineers. Many insightful experimental tests of the mechanical properties of such deformed materials have afforded an increased understanding of their behavior. Recently, since nanotechnologies have started to occupy the scientific spotlight, we must accept the challenge of studying these properties in small nano-scaled specimens and in perfect crystals under ideal conditions. While state-of-the-art experimental techniques have the capacity to make measurements in extreme situations, they are still expensive and require specialized knowledge. However, the considerable improvement in calculation methods and the striking development of computational capacity bring such problems within the range of atomic-scale numerical simulations. In particular, within the past decade, ab initio simulations, which can often give qualitatively reliable results without any experimental data as input, have become readily available. In this section, we discuss methods for studying the mechanical properties of materials using ab initio simulations. At present, we have many ab initio methods that have the potential to perform such mechanical tests. Here, however, we employ planewave methods based on density functional theory (DFT) and pseudopotential approximations because they are widely used in solid state physics. Details of the theory and of more sophisticated, state-of-the-art techniques can be found in the other section of this volume and in a review article [1]. Concrete examples of parameters settings appearing in this section presuppose that the reader is using the VASP (Vienna Ab initio Simulation Package) code [2, 3] and the ultrasoft pseudopotential. Other codes based on the same theory, such as ABINIT, CASTEP, and so on, should basically accept the same parameter settings as on VASP. 439 S. Yip (ed.), Handbook of Materials Modeling, 439–448. c 2005 Springer. Printed in the Netherlands.

440

1.

S. Ogata

Applying Deformation to Supercell

In the planewave methods, we usually use a parallelepiped-shaped supercell that has a periodic boundary condition in all directions and includes one or more atoms. The supercell can be defined by three, linearly independent basis vectors, h1 = (h 11 , h 12 , h 13 ), h2 = (h 21 , h 22 , h 23 ), h3 = (h 31, h 32 , h 33 ). In investigating the phenomena connected with a local atomic displacement, for example, a slip of the adjacent atomic planes in a crystal, an atomic position in the supercell can be directly moved within the system of fixed basis vectors. However, when we need a uniform deformation of the system under consideration, we can accomplish this by changing the basis vectors directly as we would do, for example, in simulating a phase transition or crystal twinning, and in calculations of the elastic constants and ideal strength of a perfect crystal. Let a deformation gradient tensor F represent the uniform deformation of the system. The F can be defined as Fi j =

dxi , dX j

where x and X are, respectively, the positions of a material particle in a deformed and in a reference state. By using the F, each basis vector is mapped to a new basis vector h via h k = Fkj h j . For example, for a simple shear deformation, F can be written as, 



1 0 γ F =  0 1 0 , 0 0 1 where γ represents the magnitude of the shear corresponding to the engineering shear strain. In some cases, for ease of understanding, different coordinate systems for F and for the basis vectors are taken. In this case, F is transformed into the coordinate system for a basis vector by an orthogonal tensor Q ( Q Q T = I). F = Q F Q T, h k = Fkj h j .

2.

Simulation Setting

In DFT calculations, the pseudopotential (if the code is not full-potential code) and the exchange correlation potentials should be carefully selected.

Ab initio study of mechanical deformation

441

Since these problems are not particular to deformation analysis, the reader who needs a more detailed discussion can find it elsewhere. Only a short commentary is given here. When we use the pseudopotential in a separable form [4], we need to pay attention to a possible ghost band [5], because almost all DFT codes use the separable form to save computational time and memory resources. Usually the pseudopotentials in the package codes were very carefully determined to avoid a ghost band in an equilibrium state. However, even when a pseudopotential does not generate a ghost band in the equilibrium state, such a band may still appear in a deformed state. Therefore, it is strongly recommended that a pseudopotential result should be confirmed by comparing it with the result of a full-potential calculation where possible. For the exchange correlation potential, we can normally use functions derived from the local density approximation (LDA), generalized gradient approximation (GGA), and LDA+U. In many cases, the former two methods are equally accurate. The LDA tends to underestimate lattice constants, and overestimate elastic constants and strength, and the GGA to overestimate elastic constants and strength, and underestimate lattice constants. The LDA+U sometimes offers a significantly improved accuracy [6]. The above discussions of the pseudopotential and exchange-correlation potential pertain to error sources resulting from theoretical approximations. However, as well as attending to errors from this source, we should also take care of numerical errors. Numerical errors in the planewave DFT calculation usually derive from the finite size of the k-point set and the finite number of planewaves which are uniquely determined by the supercell shape and the planewave cut-off energy. With regard to other problems, a good estimation of the stress tensor to MPa accuracy requires a finer k-point sampling than does that for an energy estimation with meV accuracy. Figure 1 shows the

-3.6

3.5 3 2.5 Stress GPa

Total energy eV

-3.62

-3.64

-3.66

2 1.5 1

-3.68 0.5 -3.7

0 0

10000 20000 30000 40000 50000 60000 70000 80000 Number of k-points

(a) Total energy vs. number of k-points

0

10000 20000 30000 40000 50000 60000 70000 80000 Number of k-points

(b) Shear stress vs. number of k-points

Figure 1. Total energy and stress vs. number of k-points curves for an aluminum primitive ¯ direction. cell under 20% shear in the {111}112

442

S. Ogata

convergence of the energy and stress as the number of k-points is increased. The model supercell is a primitive cell with an fcc structure which contains ¯ just one aluminum atom. An engineering shear strain of 0.2 to the {111}112 direction has already been applied to the primitive cell. Only the shear stress component corresponding to the shearing direction is shown. Clearly, the stress converges very slowly even though the energy converges relatively quickly. Figure 2 shows the stress–strain curves of the Al primitive cell under a {111} ¯ shear deformation using two sets of k-points, the normal 15 × 15 × 15 112 and a fine 43 × 43 × 43 Monkhorst–Pack Brillouin zone sampling [7]. This sampling scheme is explained later. The curve for 15×15×15 is significantly wavy even though the total free energy of the primitive cell agrees to the order of meV with the energy of the 43 × 43 × 43 case. Apparently, a small set of k-points does not produce a smooth stress–strain curve. This is not a small problem for the study of mechanical properties of materials, because, in the above case, the ideal strength, that is, the maximum stress of the stress– strain curve, is overestimated by 20%, a level which is usually corresponds to 2 ∼ 20 GPa. Although there are many k-points sampling schemes, in recent practice, the Monkhorst–Pack sampling scheme is typically used for testing mechanical properties. Since more efficient schemes [8], in which a smaller number

3.5

Shear Stress GPa

3 2.5 2

43x43x43 k-points

1.5 1 0.5

15x15x15 k-points

0 0

0.05

0.1 0.15 0.2 0.25 Engineering Shear Strain

0.3

0.35

Figure 2. Shear stress vs. strain curves calculated with different numbers of k-point sets. ¯ direction is applied. A shear deformation in the {111}112


443

of k-points can be used without loss of accuracy, are constructed based on crystal symmetries, a deformation which would break the crystal symmetries would remove their advantage. Therefore, the Monkhorst–Pack scheme is often favored because of its simplicity. In it, the sampling points are defined in the following manner: k(n, m, l) = nb1 + mb2 + l b3 , 2r − q − 1 ; r = 1, 2, 3, . . . , q n, m, l = 2q where bi are the reciprocal lattice vectors of the supercell and n, m, and l are the mesh sizes for each reciprocal lattice vector direction. Therefore, the total number of sampled k-points is n×m ×l. If we find that, under the symmetries of the supercell, some of the k-points are equivalent we consider only the nonequivalent k-points to save computational time. The planewave cut-off energy should also be carefully determined. We should use a large enough planewave cut-off energy to achieve a convergence of energy and stress to the required degree of accuracy. Since the atomic configuration affects the cut-off energy, it is better that we estimate that energy for the particular atomic configuration under consideration. However, in mechanical deformation analysis, it is difficult to fix the cut-off energy before starting the simulation because the deformation path cannot be predicted at the simulation’s starting point. In such a case, we have to add a safety margin of 10–20 % to the cut-off energy estimated from a known atomic configuration, for example, that of an equivalent structure. In principle, a complete basis set is necessary to express an arbitrary function by a linear combination of the basis functions. As discussed above, the planewave basis set is used to express the wave functions of electrons in ordinary DFT calculations using the pseudopotential. Because a FFT algorithm can be easily used to calculate the Hamiltonian, we can save computational time. To achieve completeness, a infinite number of the planewaves is necessary; however, to perform a practical numerical calculation, we must somehow reduce the infinite number to a finite one. Fortunately, we can ignore planewaves which have a higher energy than a cut-off value, termed the planewave cut-off energy, because the wave functions of electrons in real system do not have a component of extremely high frequencies. To estimate the cut-off energy, we can perform a series of calculations with an increasing cut-off energy for a single system. By this means, we can find a cut-off energy which is large enough to ensure that the total energy and the stress convergence of the supercell of interest fall within the required accuracy. Usually, the incompleteness of a finite number of planewave basis sets produces an unphysical stress, that is, a Puley stress. However, by using a large enough number of planewaves, we can avoid this problem. Therefore, both the stress convergence check and the energy convergence check are important in

444

S. Ogata 3.5

Shear Stress GPa

3

Ecut=90 eV

2.5 2 Ecut=129 eV

1.5 1 0.5 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Engineering Shear Strain Figure 3. Shear stress vs. strain curves calculated with different cut-off energies. A shear ¯ direction is applied. deformation in the {111}112

deformation. Figure 3 shows the stress–strain curves obtained by the use of different planewave cut-off energies. The model and simulation procedure are the same as those we have utilized in the above k-point check. Clearly, even though the error due to a small cut-off energy is small in a near equilibrium structure, it becomes larger at in a highly strained structure.

3.

Mechanical Deformation of Al and Cu

Many ab initio studies of mechanical deformation, such as tensile and shear deformation studies for metals and ceramics, have been done in the past two decades. An excellent summary of the history of ab initio mechanical testing ˇ [9]. can be found in a review paper written by Sob Here, we discuss as examples both a fully relaxed and an unrelaxed uniform shear deformation analysis [10], that is, an analysis of a pure shear and a simple shear, for aluminum and copper. The shear mode is the most important deformation mode in our consideration of the strength of a perfect crystalline solid. The shear deformation analysis usually involves more computational cost than the tensile analysis; because the shear deformation breaks many of the crystal symmetries, many nonequivalent k-points should be treated in the calculation.


445

The following analysis has been performed using the VASP code. The exchange-correlation density functional potential adopted is the Perdew–Wang generalized gradient approximation (GGA) [11]; the ultrasoft pseudopotentials [12] are used. Brillouin zone k-point sampling is performed using the Monkhorst–Pack algorithm, and the integration follows the Methfessel–Paxton scheme [13] with the smearing width chosen so that the entropic free energy (a “-T S” term) is less than 0.5 meV/atom. A six atom fcc supercell which has three {111} layer is used, and 18×25×11 k-points for Al and 12×17×7 k-points for Cu are adopted. The k-point convergence is checked as shown in Table 1. The carefully determined cut-off energies of the planewaves for the Al and Cu supercells are 162 and 292 eV, respectively. Incremental affine shear strains of 1% as described above are imposed on each crystal along the experimentally determined common slip systems to obtain the corresponding energies and stresses. In each step, the stress components, excluding the resolved shear stress along the slip system, are kept to a value less than 0.1 GPa during the simulation. In Table 2, the equilibrium lattice constants a0 obtained from the energy minimization are listed and compared with the experimental data. The calculated relaxed and unrelaxed shear moduli G r , G u for the common slip systems are compared with computed analytical values based on the experimental elastic constants. A value of γ = 0.5% is used to interpolate the resolved shear stress (σ ) versus the engineering shear strain (γ ) curves and to calculate the resolved shear moduli. In the relaxed analysis, the stress components are relaxed to within a convergence tolerance of 0.05 GPa. Table 1. Calculated ideal pure shear σr and simple shear strengths σu using different k-point sets No. of k-points 12 × 17 × 7 18 × 25 × 11 21 × 28 × 12 27 × 38 × 16

Al

Cu

σ u (GPa)

σ r (GPa)

σ u (GPa)

σ r (GPa)

3.67 3.73 – 3.71

2.76 2.84 – 2.84

3.42 3.44 3.45 –

2.16 2.15 2.15 –

Table 2. Equilibrium lattice constant (a0 ), relaxed (G r ) ¯ shear moduli of Al and Cu and unrelaxed (G u ) {111}112 Al (calc.) Al (expt.) Cu (calc.) Cu (expt.)

a0 (Å)

G r (GPa)

G u (GPa)

4.04 4.03 3.64 3.62

25.4 27.4 31.0 33.3

25.4 27.6 40.9 44.4

446

S. Ogata 3

Stress (GPa)

2.5 2 1.5 1 0.5 0

0

0.1

0.2

0.3

0.4

0.5

x/bp Figure 4. Shear stress vs. displacement curves for Al and Cu of the fully relaxed shear ¯ direction. deformation in the {111}112

At equilibrium, the Cu is considerably stiffer, with simple and pure shear moduli greater by 65 and 25%, respectively, than those of the Al. However, the Al ends up with a 32% larger ideal pure shear strength σmr than the Cu, because it has a longer range of strain before softening (see Fig. 4): γm = 0.200 in the Al, γm = 0.137 in the Cu. Figure 5 shows the changes of the iso-surfaces of the valence charge density during the shear deformation (h ≡ Vcell ρv , Vcell and ρv are the supercell volume and valence charge density, respectively). At the octahedral interstice in Al, the pocket of charge density has cubic symmetry and is angular in shape, with a volume comparable to the pocket centered on every ion. In contrast, in Cu, there is no such interstitial charge pocket, the charge density being nearly spherical about each ion. The Al has an inhomogeneous charge distribution in the interstitial region and bond directionality, while the Cu has relatively homogeneous charge distributions and little bond directionality. The charge density analysis gives a clear view of the electron activity under shear deformation, and sometime informs us about the origin of the mechanical behavior of the solids.

4.

Outlook

Currently, we can perform ab initio mechanical deformation analyses for many types of materials and for primitive and nano systems. However, in the

Ab initio study of mechanical deformation (a)

c

(b)

x=x1=0.196

a

c a

c

x=x2=0.436

b

b

b

c a

a

x=0.000

x=x2=0.494

b

b

a

x=x1=0.283

x=0.000

447

b

c a

c

Figure 5. Charge density isosurface change in (a) Al; (b) Cu during the shear deformation in ¯ direction. the {111}112

near future, the most interesting studies incorporating these analyses might address not only the mechanical behavior of materials under deformation and loading, but also the relation between mechanical deformation and loading, and physical and chemical reactions, such as stress corrosion. For this purpose, ab initio methods are the most powerful and reliable tools.

References [1] M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, and J.D. Joannopoulos. “Iterative minimization techniques for ab initio total-energy calculations – molecular dynamics and conjugate gradients,” Rev. Mod. Phys., 64, 1045–1097, 1992. [2] G. Kresse and J. Hafner, “Ab initio molecular dynamics for liquid metals,” Phys. Rev. B, 47, RC558–RC561, 1993. [3] G. Kresse and J. Furthmüller, “Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set,” Phys. Rev. B, 54, 11169–11186, 1996. [4] L. Kleinman and D.M. Bylander “Efficacious form for model pseudopotentials,” Phys. Rev. Lett., 48, 1425–1428, 1982.

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S. Ogata [5] X. Gonze, P. Kackell, and M. Scheffler, “Ghost states for separable, norm-conserving, ab initio pseudopotential,” Phys. Rev. B, 41, 12264–12267, 1990. [6] S.L. Dudarev, G.A. Botton, S.Y. Savrasov, C.J. Humphreys, and A.P. Sutton, “Electron-energy-loss spectra and the structural stability of nickel oxide: An LSDA+ U study,” Phys. Rev. B, 57, 1505–1509, 1998. [7] H.J. Monkhorst and J.D. Pack, “Special points for Brillouin zone integrations,” Phys. Rev. B, 13, 5188–5192, 1976. [8] D.J. Chadi, “Special points in the Brillouin zone integrations,” Phys. Rev. B, 16, 1746–1747, 1977. ˇ [9] M. Sob, M. Friák, D. Legut, J. Fiala, and V. Vitek, “The role of ab initio electronic structure calculations,” Mat. Sci. Eng. A, to be published, 2004. [10] S. Ogata, J. Li, and S. Yip, “Ideal pure shear strength of aluminum and copper,” Science, 298, 807–811, 2002. [11] J.P. Perdew and Y. Wang, “Atoms, molecules, solids, and surfaces: application of the generalized gradient approximation for exchange and correlation,” Phys. Rev. B, 46, 6671–6687, 1992. [12] D. Vanderbilt, “Soft self-consistent pseudopotentials in a generalized eigenvalue formalism,” Phys. Rev. B, 41, 7892–7895, 1990. [13] M. Methfessel and A. T. Paxton, “High-precision sampling for Brillouin zone in metals,” Phys. Rev. B, 40, 3616–3621, 1989.

2.1 INTRODUCTION: ATOMISTIC NATURE OF MATERIALS Efthimios Kaxiras1 and Sidney Yip2 1

Department of Nuclear Science and Engineering and Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2 Department of Physics, Harvard University, Cambridge, MA 02138, USA

Materials are made of atoms. The atomic hypothesis was put forward by the Greek philosopher Demokritos about 25 centuries ago, but was only proven by quantitative arguments in the 19th and 20th centuries, beginning with the work of John Dalton (1766–1844) and through the development of quantum mechanics, the theory that provided a complete and accurate description of the properties of atoms. The very large number of atoms encountered in a typical material (of order ∼1024 or more) precludes any meaningful description of its properties based on a complete account of the behavior of each and every atom that comprises it. Special cases, such as perfect crystals, are exceptions where symmetry reduces the number of independent atoms to very few; in such cases, the properties of the solid are indeed describable in terms of the behavior of the few independent atoms and this can be accomplished using quantum mechanical methods. However, this is only an idealized model of actual solids in which perfect order is broken either by thermal disorder or by the presence of defects that play a crucial role in determining the physical properties of the system. An example of a crystal defect is dislocations, which determine the mechanical behavior of solids (their tendency for brittle or ductile response to external loading); these defects have a core which can only be described properly by its atomic scale structure, but they also have long range strain and stress fields which are adequately described by continuum elasticity theory (see Chapters 3 and 7). This situation typifies the dilemma of describing the behavior of real materials: the majority of atoms, far from the defect regions, behave in a manner consistent with a macroscopic, continuum description, where the atomic hypothesis is not important, while a small minority of atoms, in the immediate neighborhood of the defects, do not follow this rule and need to be 451 S. Yip (ed.), Handbook of Materials Modeling, 451–458. c 2005 Springer. Printed in the Netherlands.

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described individually. Neither aspect, atomistic or macroscopic, can provide by itself a satisfactory description of the defect and its role in determining the material’s behavior. The example of dislocations is representative: any type of crystal defect (vacancies, interstitials, impurities, grain boundaries, surfaces, interfaces, etc.) requires, at some level, atomic scale representation in order to fully understand its effect on the properties of the material. Similarly, disorder induced by thermal motion and other external agents (pressure, irradiation) can lead to changes in the stucture of a solid, possibly driving it to new phases, which also requires a detailed atomistic description (see Chapters 2.29 and 6.11). Finally, the case of fluids or solids like polymers, in which there is no order at the atomic scale, is another example of where atomistic scale description is necessary to provide invaluable information for a comprehensive picture of the system’s behavior (see Chapters 8.1 and 9.1). These considerations provide the motivation for the description of materials properties based on atomistic simulations, by judiciously choosing the aspects that need to be explicitly modeled at the atomic scale. The term “atomistic simulations” has acquired a particular meaning: it refers to computational studies of materials properties based on explicit treatment of the atomic degrees of freedom within classical mechanics, either deterministically, that is, in accordance with the laws of classical dynamics (the so-called Molecular Dynamics or MD approach, see Chapter 2.8), or stochastically, that is, by appropriately sampling distributions from a chosen ensemble (the so called Monte Carlo or MC approach, see Chapter 2.10). The energy functional underlying the calculation of forces for the dynamics of atoms or the ensemble distribution, can be based either on a classical description or a quantum mechanical one. We will discuss briefly the issues that arise from the various approaches and then elaborate on what these approaches can provide in terms of a detailed understanding of the behavior of materials.

1.

The Input to Atomistic Simulation

The energy of a system as a function of atomic positions should ideally be treated within quantum mechanics, with the valence electrons providing the interactions between atoms that hold the solid together. The development of Density Functional Theory [1, 2] and of pseudopotential theory (for a comprehensive review see, e.g., Ref. [3]) has produced a computational methodology which is accurate and efficient, and has the required chemical versatility to describe a very wide range of materials properties, fully within the quantum mechanical framework [4]. However, this is an approach which puts exceptionally large demands on computational resources for systems larger than a few tens of atoms, a situation that arises frequently in the descriptions of

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realistic systems (the dislocation core is a case in point), and this limitation applies to a single atomic configuration. The description of systems comprising of thousands to millions of atoms, and including a large number of atomistic configurations (as a molecular dynamics or a Monte Carlo simulation would require) is beyond current and anticipated computational capabilities. Consequently, alternative approaches have been pursued in order to be able to model such systems, which, though large on the atomistic scale, are still many orders of magnitude smaller than typical meterials. The basic idea is to employ either a simplified quantum mechanical approach for the electrons, or a purely classical one in which the electronic degrees of freedom are completely eliminated and the interactions between atoms are modeled by an effective potential; in both cases, the computational resources required are greatly reduced, permitting the treatment of much larger systems and more extensive exploration of their configurational space (more time steps in a MD simulation or more samples in a MC simulation). The strategies for reducing the computational cost, whether quantum mechanical or classical in nature, are usually distinctly different when applied to systems with covalent versus those with metallic bonding, because of the difference in the nature of electronic bonds in these two situations. In the quantum case, covalent systems are typically modeled by a so-called tight-binding hamiltonian, which restricts the electronic wavefunctions to linear combinations of localized atomic orbitals; this approach is adequate to describe the nature of the covalent bonds (see Chapters 1.14 and 1.15), but can also be extended to capture metallic systems. The restricted variational freedom of electronic wavefunctions greatly reduces the computational cost involved in finding the proper solution. For simple metallic systems, an approach based on density functional theory but without requiring electronic orbitals has also been employed to approximate their properties, again with very substantial reduction in computational cost. These developments have made possible the quantum mechanical, atomistic scale simulation of systems consisting of up to a few thousand atoms (see Ref. [3] for examples). An altogether different methodology is to maintain a strictly classical description with interactions between the atoms provided by an effective potential which somehow encapsulates all the effects of valence electrons. The methodology used in this type of approach is again determined by the type of system to which it is applied. Specifically, for covalently bonded systems, the emphasis of the potential is to reproduce the energy cost of distorting the length of covalent bonds, the angles between them and the torsional angles, which are the basic features characterizing structures with predominantly covalent bonding; a characteristic example of such approaches is silicon, the prototypical covalently bonded solid, for which many attempts have been made to produce a reliable effective interactomic potential with various

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degrees of success [5–7]. In contrast to this, for metallic systems the emphasis of the potential is to describe realistically the environment of an atom embedded in the background of valence electrons of the host solid; the approaches here often employ an effective (but not necessarily realistic) representation of the valence electron density and are referred to as the embedded atom method [for a review see, Ref. [8], see Chapter 2.2]. In both types of approaches, great care is given to ensuring that the potential reproduces accurately the energetics of at least a set configurations, by fitting it to a database produced by the more elaborate and accurate quantum mechanical methods. Finally, there are also cases where a more generic type of approach can be employed, modeling for instance the interaction between atoms as a simple potential derived by heuristic arguments without fitting to any particular system. Examples of such potentials are the well known van der Waals and Morse potentials, which have the general behavior of an attractive tail, a well-defined minimum and a repulsive core, as a function of the distance between two atoms (see Chapters 2.2–2.6, and 9.2). While not specific to any given material or system, these potentials can provide great insight as far as generic behavior of solids is concerned, including the role of defects in fairly complex contexts (see Chapters 6.1 and 7.1).

2.

Unique Properties of Molecular Dynamics and Monte Carlo

There are certain aspects of atomistic simulation, particularly molecular dynamics and Monte Carlo, which make this approach quite unique. The basic underlying concept here is particle tracking. Without going into the distinction between the two methods of simulation, we make the following general observations. (i) A few hundred particles are often sufficient to simulate bulk properties. Bulk or macroscopic properties like the system pressure and temperature can be determined with a simulation cell containing less than a thousand atoms, even though the number of atoms in a typical macroscopic system is of order of Avogadro’s number, 6 × 1023 . (ii) Simulation allows a unified study of all physical properties. A single simulation can generate the basic data, particle trajectories or configurations, with which one can calculate all the materials properties of interest, structural, thermodynamic, vibrational, mechanical, transport, etc. (iii) Simulation provides a direct connection between the fundamental description of a material system, such as internal energy and atomic structure, and all the physical properties of interest. In essence, it is a “numerical theory of matter”.


455

(iv) In simulation one has complete control over the conditions under which the system study is carried out. This applies to the specification of interatomic interactions and the initial and boundary conditions. With this information and the simulation output one has achieved a precise characterization of the material being simulated. (v) Simulation can give properties that cannot be measured. This can be a very significant feature with regard to testing theory. In situations where the most clean-cut test involves systems or properties not accessible by laboratory experiments, simulation can play the role of experiment and provide this information. Conversely, in those cases where there are no theories to interpret an experiment, simulation can play the role of theory. (vi) Simulation makes possible the direct visualization of physical phenomena of interest. Visualization can play a very important role in modeling and simulation at all scales, for communication of results, gaining physical insights, and discovery. While its potential is recognized, its practical use remains underdeveloped. We recall here an oft quoted sentiment: “Certainly no subject is making more progress on so many fronts than biology, and if we were to name the most powerful assumption of all, which leads one on and on in an attempt to understand life, it is that all things are made of atoms, and that everything that living things do can be understood in terms of the jiggling and wiggling of atoms.” Richard Feynman, Lectures on Physics, vol. 1, p. 3–6 (1963)

3.

Limitations of Atomistic Simulation

To balance the usefulness of molecular dynamics and Monte Carlo, it is appropriate to acknowledge at the same time the inherent limitations of atomistic simulation. As mentioned earlier, the first-principles, quantum mechanical description of atomic bonding in solids is restricted to very few (by macroscopic standards) atoms and for exteremely short time scales: barely a few hundred atoms can be handled, for periods of few hundreds of femto seconds. Extending this fundamental description to larger systems and longer times of simulation requires the introduction of approximations in the quantum mechanical method (such as tight binding or orbital-free approaches), which significantly limit the accuracy of the quantum mechanical approach. With such restrictions on size and time-span of the simulation, the scope of applications to real materials properties is rather limited. The alternative is to use a purely classical description, based on empirical interatomic potentials to describe the interactions of atoms. This, however, introduces more severe approximations, which limit the

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ability of the approach to capture realistically how the bonds between atoms are formed and dissolved during a simulation. Such uncertainties put bounds on the scope of physical phenomena that can be successfuly addressed by simulations. The other limitation is a practical issue, that is, the finite capabilities of computers no matter how large they are. This translates into limits on the spatial size (usually identified with the number of atoms N in the model) and the temporal extent of simulations, which often fall short of desired values. It is quite safe to say that the upper bounds on system size and run time, whatever they are, will be pushed out further with time, because computer power is certain to increase in the foreseeable future. Probably more important in extending the effective size of simulations are novel algorithmic developments, which are likely to produce computational gains in the simulation size and duration much larger than any direct gains by raw increases in computer power. As an example of new approaches, we mention multiscale simulations of materials, which combine the different types of system description (quantum, classical and continuum) into a single method. Several approaches of this type have appeared in the last few years, and their development is at present a very active field which holds promise for bringing to fruition the full potential of atomistic simulations.

4.

A Brief Survey of the Chapter Contents

The diversity of atomistic simulations, regarding either methods or applications, makes any attempt at a complete coverage a practically impossible task. The contributions that have been brought together here should give the reader a substantial overview of the basic capabilities of the atomistic simulation approach, along with emphasis on certain unique features of modeling and simulation at this scale from the standpoint of multiscale modeling. Leading off the discussions are five articles describing the development of interatomic potentials for specific classes of materials – metals (Chapter 2.2), ionic (Chapter 2.3) and covalent (Chapter 2.4) solids, molecules (Chapter 2.5), and ferroelectrics (Chapter 2.6). From these the reader gains an appreciation of the physics and the database that go into the models, and how the resulting potentials are validated. Immediately following are articles on the simulation methods where the potentials are the necessary inputs, energy minimization (Chapter 2.7), molecular dynamics (Chapters 2.8, 2.9, 2.11), Monte Carlo (Chapter 2.10), and methods at the mesoscale which incorporate atomistic information (Chapters 2.12, 2.13). In the next set of articles emphasis is directed at applications, beginning with free-energy calculations (Chapters 2.14, 2.15) for which atomistic simulations are uniquely well suited, followed by studies of elastic constants (Chapter 2.16), transport coefficients (Chapters 2.17, 2.18),


457

mechanical behavior (Chapter 2.19), dislocations (Chapters 2.20, 2.21, 2.22), fracture in metals (Chapter 2.23), and semiconductors (Chapter 2.24). The next two articles deal with large scale simulations, on metallic and ceramic nanostructures (Chapter 2.25) and biological membranes (Chapter 2.26), followed by three articles on studies in radiation damage to which atomistic modeling and simulations have made significant contributions (Chapters 2.27, 2.28, 2.29). The next article, on thin-film deposition (Chapter 2.30), is an example of how simulation can address problems of technological relevance. The chapter concludes with an article on visualization at the atomistic level (Chapter 2.31), a topic which is destined to grow in recognized importance as well as opportunities for software innovation. The contents of this chapter clearly have a great deal of overlap with the rest of the Handbook. The connection between atomistic simulations using classical potentials and electronic structure calculations (Chapter 1.1) permeates throughout the present chapter, since the potentials used in MD/MC simulations rely on the first-principles quantum mechanical calculations for inspiration of functional form of the potentials, for the database used to determine parameter values, and for benchmark results in model validation. The connection to the mesoscale (Chapter 3.1) is clearly also very intimate since this is the next level of length/time scale. Since atomistic simulation methods and results are used liberally throughout the Handbook, one may be tempted to say that this chapter serves as perhaps the most central link to the different parts of the volume. If we may be allowed another quote from R.P. Feynman, the following is a different way of expressing the centrality of the chapter. “If, in some cataclysm, all of scientific knowledge were to be destroyed, and only sentence passed on to the next generatios of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, whatever you wish to call it) that all things are made of atoms – little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon squeezed into one another. In that one sentence, you will see, there is enormous amount of information about the world, if just a little imagination and thinking are applied.” Richard P. Feynman, Six Easy Pieces, (Addison-Wesley, Reading, 1963), p. 4.

References [1] P. Hohenberg and W. Kohn, “Inhomogeneous electron gas,” Phys. Rev., 136, B864– 871, 1964. [2] W. Kohn and L.J. Sham, “Self-consistent equations including exchange and correlation effects,” Phys. Rev. A, 140, 1133–1138, 1965. [3] R.M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge University Press, Cambridge, 2004.

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E. Kaxiras and S. Yip [4] R. Car and M. Parrinello, “Unified approach for molecular dynamics and densityfunctional theory,” Phys. Rev. Lett., 55, 2471–2474, 1985. [5] F.H. Stillinger and T.A. Weber, “Computer simulation of local order in condensed phases of silicon,” Phys. Rev. B, 31, 5262–5271, 1985. [6] J. Tersoff, “New empirical model for the structural properties of silicon,” Phys. Rev. Lett., 56, 632–635, 1986. [7] J. Justo, M.Z. Bazant, E. Kaxiras, V.V. Bulatov, and S. Yip, “Interatomic potential for silicon defects and disordered phases,” Phys. Rev. B, 58, 2539–2550, 1998. [8] A.F. Voter, Intermetallic Compounds, vol. 1, Wiley, New York, pp. 77, 1994.

2.2 INTERATOMIC POTENTIALS FOR METALS Y. Mishin George Mason University, Fairfax, VA, USA

Many processes in materials, such as plastic deformation, fracture, diffusion and phase transformations, involve large ensembles of atoms and/or require statistical averaging over many atomic events. Computer modeling of such processes is made possible by the use of semi-empirical interatomic potentials allowing fast calculations of the total energy and classical interatomic forces. Due to their computational efficiency, interatomic potentials give access to systems containing millions of atoms and enable molecular dynamics simulations for tens or even hundreds of nanoseconds. State-ofthe-art potentials capture the most essential features of interatomic bonding, reaching the golden compromise between computational speeds and accuracy of modeling. This article reviews interatomic potentials for metals and metallic alloys. The basic concepts used in this area are introduced, the methodology commonly applied to generate atomistic potentials is outlined, and capabilities as well as limitations of atomistic potentials are discussed. Expressions for basic physical properties within the embedded-atom formalism are provided in a form convenient for computer coding. Recent trends in this field and possible future developments are also discussed.

1.

Embedded-atom Potentials

Molecular dynamics, Monte Carlo, and other simulation methods require multiple evaluations of Newtonian forces Fi acting on individual atoms i or (in the case of Monte Carlo simulations) the total energy of the system, E tot . Atomistic potentials, also referred to as force fields, parameterize the configuration space of a system and represent its total energy as a relatively simple function of configuration point. The interatomic forces are then obtained as coordinate derivatives of E tot , Fi = −∂ E tot /∂ri , ri being the radius-vector of an 459 S. Yip (ed.), Handbook of Materials Modeling, 459–478. c 2005 Springer. Printed in the Netherlands.

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atom i. This calculation of E tot and Fi is a simple and fast numerical procedure that does not involve quantum-mechanical calculations, although the latter are often used when generating potentials as will be discussed later. Potential functions contain fitting parameters, which are adjusted to give desired properties of the material known from experiment and/or first-principles calculations. Once the fitting procedure is complete, the parameters are not subject to any further changes and the potential thus defined is used in all subsequent simulations of the material. The underlying assumption is that a potential providing accurate energies/forces at configuration points used in the fit will also give reasonable results for configurations between and beyond them. This property of potentials, often refereed to as “transferability,” is probably the most adequate measure of their quality. Early atomistic simulations employed pair potentials, usually of the Morse or Lennard-Jones type [1, 2]. Although such potentials have been and still are a useful model for fundamental studies of generic properties of materials, the agreement between simulation results and experiment can only be qualitative at best. While such potential can be physically justified for inert elements and perhaps some ionic solids, they do not capture the nature of atomic bonding even in simple metals, not to mention transition metals or covalent solids. Daw and Baskes [3] and Finnis and Sinclair [4] proposed a more advanced potential form that came to be known as the embedded atom method (EAM). In contrast to pair potentials, EAM incorporates, in an approximate manner, many-body interactions between atoms, which are responsible for a significant part of bonding in metals. The introduction of the many-body term has enabled a semi-quantitative, and in good cases even quantitative, description of metallic systems. In the EAM model, E tot is given by the expression E tot =

1 si s j (rij ) + Fsi (ρ¯i ). 2 i, j ( j =/ i) i

(1)

The first term is the sum of all pair interactions between atoms, si s j (rij ) being a pair-interaction potential between atoms i (of chemical sort si ) and j (of chemical sort s j ) at positions ri and r j = ri + rij , respectively. Function Fsi is the so-called embedding energy of atom i, which depends upon the host electron density ρ¯i at site i induced by all other atoms of the system. The host electron density is given by the sum ρ¯i =

ρs j (rij ),

(2)

j= /i

where ρs j (r) is the electron density function assigned to atom j . The second term in Eq. (1) represents the many-body effects. The functional form of Eq. (1) was originally derived as a generalization of the effective medium theory [5] and the second moment approximation to tight-binding theory [4, 6]. Later, however, it lost its close ties with the original physical meaning

Interatomic potentials for metals

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and came to be treated as a working semi-empirical expression with adjustable parameters. A complete EAM description of an n-component system requires n(n + 1)/2 pair interaction functions ss (r), n electron density functions ρs (r), ¯ (s = 1, . . . , n). An elemental metal is desand n embedding functions Fs (ρ) cribed by three functions (r), ρ(r) and F(ρ), ¯ 1 while a binary system A–B ¯ and requires seven function AA (r), AB (r), BB (r), ρA (r), ρB (r), FA (ρ), ¯ Notice that if potential functions for pure metals A and B are availFB (ρ). able, only the cross-interaction function AB (r) is needed for describing the respective binary system. Over the past two decades, EAM potentials have been constructed for many metals and a number of binary systems. Potentials for ternary systems are scares and their reliability is yet to be evaluated. The pair-interaction and electron-density functions are normally forced to turn to zero together with several higher derivatives at a cutoff radius Rc . Typically, Rc covers 3–5 coordination shells. EAM functions are usually defined by analytical expressions. Such expressions and their derivatives can be directly coded into a simulation program. However, a more common and computationally more efficient procedure is to tabulate each function at a large number of points (usually, a few thousand) and store it in the tabulated form for all subsequent simulations. In the beginning of each simulation run, the tables are read into the program, interpolated by a cubic spline, and the spline coefficients are used during the rest of the simulation for retrieving interpolated values of the functions and their derivatives for any desired value of the argument. It is important to understand that the partition of E tot into pair interactions and the embedding energy is not unique [7]. Namely, E tot defined by Eq. (1) is invariant under the transformations ¯ → Fs (ρ) ¯ + gs ρ, ¯ Fs (ρ) ss (r) → ss (r) − gs ρs (r) − gs ρs (r),

(3) (4)

where s, s = 1, . . . , n and gs are arbitrary constants. In addition, all functions ρs (r) can be scaled by the same arbitrary factor p with a simultaneous scaling of the argument of the embedding functions: ρs (r) → pρs (r), ¯ → Fs (ρ/ ¯ p). Fs (ρ)

(5) (6)

Thus, there is a large degree of ambiguity in defining EAM potential functions: the units of the electron density are arbitrary, the pair-interaction and electron-density functions can be mixed with each other, and the embedding energy can only be defined up to a linear function. It is important, however, that 1 For elemental metals, the chemical indices s are often omitted.

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the embedding function be non-linear, otherwise the second term in Eq. (1) can be absorbed by the first one, resulting in a simple pair potential. The non¯ reflects the bond-order character of atomic interactions by linearity of Fs (ρ) making the energy per nearest-neighbor bond decrease with increasing number ¯ must be positive of bonds. To capture this trend, the second derivative Fs (ρ) and thus Fs (ρ) ¯ a convex curve, at least around the equilibrium volume of the ¯ is proportional to crystal. Furthermore, in pure metals at equilibrium, F (ρ) the Cauchy pressure (c12 − c44 )/2, which is normally positive (cij are elastic constants). Notice that all pair potentials inevitably give c12 = c44 , a relation which is rarely followed by real materials. Given this arbitrariness of EAM functions, one should be careful when comparing EAM potentials developed by different research groups for the same material: functions looking very different may actually give close physical properties. As a common platform for comparison, potentials are often converted to the so-called effective pair format. To bring potential functions to this format, apply the transformations by Eqs. (3) and (4) with coefficients gs chosen as gs = −Fs (ρ), ¯ where the derivative is taken at the equilibrium lattice parameter of a reference crystal structure. For that structure, the transformed ¯ = 0 at equilibrium. In embedding functions will satisfy the condition Fs (ρ) other words, each embedding function Fs (ρ) ¯ will have a minimum at the host electron density arising at atoms of the respective sort s in the equilibrium reference structure. Together with the normalization condition ρ¯1 = 1 applied to sort s = 1 in that structure, the potential format is uniquely defined and different potentials can be conveniently compared with each other provided that their reference structures are identical. In elemental metals, the natural choice of the reference structure is the ground state, whereas for binary systems this choice is not unique and should always be specified by the author.

2.

Calculation of Properties with EAM Potentials

Below we provide EAM expressions for some basic physical properties of materials in a form convenient for computer coding. We are using a laboratory reference system with rectangular Cartesian coordinates, so that positions of indices of vectors and tensors are unimportant. We will reserve superscripts for Cartesian coordinates of atoms and subscripts for their labels (all atoms are assumed to be labeled) and chemical sorts (s-indices). The force acting on a particular atom i in a Cartesian direction α(α = 1, 2, 3) is given by the expression Fiα =

j= /i

f ij (rij )

rijα , rij

(7)


463

where f ij (rij ) = si s j (rij ) + Fsi (ρ¯i )ρs j (rij ) + Fs j (ρ¯ j )ρs i (rij ).

(8)

Notice that this force depends on the electron density on all neighboring atoms j , which in turn depends on positions of all neighbors of atom j . It follows that force coupling between atoms extends effectively over a distance of 2Rc and not just Rc as for pair potentials. EAM allows a direct calculation of the mechanical stress tensor for any atomic configuration: 1 αβ σ i , V i i

σ αβ =

(9)

where αβ

σ i i ≡

1 j= /i

2

si s j (rij ) + Fsi (ρ¯i )ρs j (rij )

β

rijα rij rij

.

(10)

Here, V = i i is the total volume of the system and i are atomic volumes assigned to individual atoms. A partition of V between atoms is somewhat arbitrary but adopting a reasonable approximation (for example, equipartiαβ tion) one can compute the local stress tensor σi on individual atoms. Analysis of stress distribution can be especially useful in atomistic simulations of dislocations, grain boundaries and other crystal defects. The condition of mechanical equilibrium of an isolated or periodic system can be expressed as σ αβ = 0 for all α and β: 1 i, j ( j = / i)

2

si s j (rij )

+

Fsi (ρ¯i )ρs j (rij )

β

rijα rij = 0. rij

(11)

In particular, equilibrium with respect to volume variations requires that the hydrostatic stress vanish, α σ αα = 0, which reduces Eq. (11) to 1 i, j ( j = / i)

2

si s j (rij ) + Fsi (ρ¯i )ρs j (rij ) rij = 0.

(12)

Analysis of stresses also allows us to formulate equilibrium conditions of a crystal with respect to tetragonal or any other homogeneous distortion. We now turn to elastic constants of an equilibrium prefect crystal. The elastic constant tensor C αβγ δ of a general crystal structure is given by C αβγ δ =

1 αβγ δ αβγ δ αβ γ δ Ui + Fsi (ρ¯i )Wi + Fsi (ρ¯i )Vi Vi , n b 0 i

(13)

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where 0 is the equilibrium atomic volume and

αβγ δ Ui

αβγ δ Wi

=

j= /i

αβ Vi

si s j (rij ) rijα rijβ rijγ rijδ 1 = si s j (rij ) − , 2 j =/ i rij (rij )2

ρsj (rij )

ρs (rij ) rijα rijβ rijγ rijδ − j , rij (rij )2

β rijα rij ρs j (rij ) rij j= /i

=

(14)

(15)

.

(16)

In Eq. (13), i is the summation over n b basis atoms defining the structure, while the summation j extends over all neighbors of atom i within its cutoff sphere. Expressions for contracted elastic constants cij can be readily developed from the above equations. It is important to remember that Eqs. (13)–(16) have been derived by applying to the crystal an infinitesimal homogeneous strain. These equations are, thus, not valid for structures (e.g., HCP or diamond cubic) where the lack of inversion symmetry gives rise to internal atomic relaxations under applied strains. EAM provides relatively simple expressions for force constants and the dynamical matrix [8]. For off-diagonal (i=/ j ) elements of the force-constant αβ matrix G ij we have αβ G ij

s s (rij ) rijα rijβ δαβ f ij (rij ) ≡ α β =− − si s j (rij ) − i j rij rij (rij )2 ∂ri ∂r j ∂ E tot

−

Fsi (ρ¯i )

ρsj (rij )

−

−Fs j (ρ¯ j ) ρsi (rij ) −

ρs j (rij ) rijα rijβ rij

+

k= / i, j

(rij )2 β

ρs i (rij ) rijα rij rij (rij )2

α β rij

+ Fsj (ρ¯ j )ρs i (rij )Q j

rij

− Fsi (ρ¯i )ρs j (rij )Q αi

Fsk (ρ¯k )ρs i (rik )ρs j (r j k )

β

rij rij

β

rikα r j k , rik r j k

(17)

where Q αi =

m= /i

ρs m (rim )

α rim rim

(18)


465 αβ

and f ij (rij ) is given by Eq. (8). For the diagonal elements G ii we have αβ G ii

∂ E tot

≡

β

∂riα ∂ri +

k= /i

+

k= /i

+

= δαβ

Fsi (ρ¯i )

f ik (rik ) k= /i

rik

ρsk (rik )

Fsk (ρ¯k )

ρsi (rik )

β Fsi (ρ¯i )Q αi Q i

+

k= /i

+

k= /i

−

ρs k (rik ) rik

si sk (rik )

(rik ) rikα rikβ − si sk rik (rik )2

β

rikα rik (rik )2

ρ (rik ) rikα rikβ − si rik (rik )2 Fsk (ρ¯k )

ρs i (rik )

2 r α r β ik ik

(rik )2

.

(19)

If the system is subject to periodic boundary conditions or if there are no αβ external fields, G ii can be simply found from the relation

αβ

αβ

G ij + G ii = 0,

(20)

j= /i

expressing the invariance of E tot with respect to arbitrary rigid translations of the system. Eqs. (17) and (19) reveal again that dynamic coupling between atoms in EAM extends over distances up to 2Rc . Notice that these equations are not limited to a perfect crystal and are valid for any equilibrium atomic configuration. αβ Knowing G ij , we can construct the dynamical matrix αβ Dij

αβ

G ij = , Mi M j

(21) αβ

Mi and M j being the atomic masses. A diagonalization of Dij gives us squares, ωn2 , of the normal vibrational frequencies ωn of our system. For a stable system all eigenvalues ωn2 are non-negative, which allows us to determine the normal frequencies. These, in turn, can be immediately plugged into the relevant statistical-mechanical expressions for the free energy and other thermodynamic functions associated with atomic vibrations. This procedure, with possible slight modifications, lies in the foundation of all harmonic and quasi-harmonic thermodynamics calculations with atomistic potentials [9, 10]. In particular, a minimization of the total free energy (vibrational free energy plus E tot ) with respect to volume provides a quasi-harmonic scheme of thermal expansion calculations [11]. Alternatively, for a perfect crystal it is straightforαβ ward to compute the Fourier transform, Dij (k), of the dynamical matrix for various k-vectors within the Brillouin zone (here i and j refer to basis atoms). αβ A diagonalization of Dij (k) permits a calculation of 3n b phonon dispersion relations ω(k).

466

Y. Mishin

If an EAM potential is used in the effective pair format and we need to αβ αβ ¯ =0 compute G ij or Dij for the equilibrium reference structure, then all Fs (ρ) and Eqs. (17) and (19) are somewhat simplified. But even without this simpliαβ fication, the computation of G ij directly from Eqs. (17) and (19) is a straightforward and relatively fast computational procedure. In fact, it is the diagonalization of the dynamical matrix rather than its construction that becomes the bottleneck of harmonic calculations for large systems. Finally, we will provide EAM expressions for the unrelaxed vacancy formation energy. The change in E tot accompanying the creation of a vacancy at a site i without relaxation equals E i = −

si s j (rij ) − Fsi (ρ¯i ) +

j= /i

Fs j (ρ¯j − ρi (rij )) − Fs j (ρ¯j ) ,

j= /i

(22) where ρ¯j is the host electron density at site j =/ i before the vacancy creation. The first two terms in Eq. (22) account for the energy of broken bonds and the loss of the embedding energy of atom i, whereas the third term represents the changes in embedding energies of neighboring atoms j due to the reduction in their host electron density upon removal of atom i. For an elemental metal whose crystal structure consists of symmetrically equivalent sites,2 the unrelaxed vacancy formation energy equals E v = E i + E 0 , where E0 =

1 (rij ) + F(ρ) ¯ 2 j =/ i

(23)

is the cohesive energy of the crystal (the choice of site i is unimportant). Thus, Ev = −

1 (rij ) + F(ρ¯ − ρ(rij )) − F(ρ) ¯ . 2 j =/ i j= /i

(24)

The relaxation typically decreases E v by 10–20%. For a pair potential, Eq. (24) leads to E v = −E 0 , a relation which overestimates experimental values of E v over a factor of two. For example, in copper E v = 1.27 eV while E 0 = −3.54 eV (both experimental numbers). The embedding energy terms in Eq. (24) make the agreement with experiment much closer. For an alloy or compound, Eq. (22) only gives the so-called “raw” formation energy of a vacancy [12]. This energy alone is not sufficient for calculating the equilibrium vacancy concentration but it serves as one of the ingredients required for such calculations. For an ordered intermetallic compound, “raw” energies of vacancies and antisite defects need to be computed for each sublattice. Expressions similar to Eq. (22) can be readily developed 2 Some structures, for example A15, contain nonequivalent sites.


467

for antisite defects. Another ingredient is the average cohesive energy of the compound, 



1 1 s s (rij ) + Fsi (ρ¯i ), E0 = n b i 2 j =/ i i j

(25)

where the summation i is over n b basis atoms and the summation j is over all neighbors of atom i. The set of all “raw” formation energies of point defects and E 0 provides input for statistical-mechanical models describing dynamic equilibrium among point defects and allowing a numerical calculation of their equilibrium concentrations [12, 13]. Although relaxations can reduce the “raw” energies significantly, fast unrelaxed calculations are very useful when generating potentials or making preliminary tests. EAM potentials serve as a workhorse in the overwhelming majority of atomistic simulations of metallic materials. They are widely used in simulations of grain boundaries and interfaces [14], dislocations [15], fracture [16], diffusion and other processes [17]. EAM potentials have a good record of delivering reasonable results for a wide variety of properties. For elemental metals, elastic constants and the vacancy formation energies are usually reproduced accurately. Surface energies tend to lie 10–20% below experiment, a problem that can hardly be solved within regular EAM. Surface relaxations and reconstructions usually agree with experiment at least qualitatively. Vacancy migration energies tend to underestimate experimental values unless specifically fit to them. Phonon dispersion curves, thermal expansion, melting temperatures, stacking fault energies, and structural energy differences may not come out accurate automatically but can be adjusted during the potential generation procedure (see below). For binary systems, experimental heats of phase formation and properties of individual ordered compounds can be fitted to with reasonable accuracy. For some binary systems, even basic features of phase diagrams can be reproduced without fitting to experimental thermodynamic data [18]. However, in systems with multiple intermediate phases, transferability across the entire phase diagram can be problematic [18].

3.

Generation and Testing of Atomistic Potentials

We will first discuss potential generation procedures for elemental metals. The EAM functions (r) and ρ(r) are usually described by analytical expressions containing five to seven fitting parameters each. Different authors use polynomials, exponents, Morse, Lennard-Jones or Gaussian functions, or their combinations. In the absence of strong physical leads, any reasonable function can be acceptable as long as it works. It is important, however, to keep the functions simple and smooth. Oscillations and wiggles can lead to

468

Y. Mishin

rapid changes or even discontinuities in higher derivatives and cause unphysical effect in phonon frequencies, thermal expansion and other properties. The risk increases when analytical forms are replaced by cubic splines (discontinuous third derivative), especially with a large number of nodes. Increasing the number of fitting parameters should be done with great caution. The observed improvement in accuracy of fit can be illusive as the potential may perform poorly for properties not included in the fit. Many sophisticated potentials contain hidden flaws that only reveal themselves under certain simulation conditions. As a rough rule of thumb, potentials whose (r) and ρ(r) together contain over 15 fitting parameters may lack reliability in applications. At the same time, using too few (say, < 10) parameters may not take full advantage of the capabilities of EAM. Since the speed of atomistic simulations does not depend on the complexity of potential functions or the number of fitting parameters,3 it makes sense to put efforts in optimizing them for the best accuracy and reliability. There are two ways of constructing the embedding function F(ρ). ¯ One way is to describe it by an analytical function (or cubic spline [19]) with adjustable parameters. Another way is to postulate an equation of state of the ground-state structure. Most authors use the universal binding curve [20], E(a) = E 0 (1 + αx) e−αx ,

(26)

where E(a) is the crystal energy per atom as a function of the lattice parameter a, x = (a/a0 − 1) (a0 being the equilibrium value of a),

α=

−

90 B , E0

and B is the bulk modulus. F(ρ) ¯ is then obtained by inverting Eq. (26). Namely, by varying the lattice parameter we compute ρ(a) ¯ and F(a) = E(a) − E p (a), where E(a) is given by Eq. (26) and E p (a) is the pair-interaction part of ¯ thus obtained parametrically define F(ρ). ¯ E tot . The functions F(a) and ρ(a) Notice that this procedure automatically guarantees an exact fit to E 0 , a0 and B. A slightly improved procedure is to add a higher-order term ∼βx 3 to the pre-exponential factor of Eq. (26) and use the additional parameter β to fit to an experimental pressure-volume relation under large compressions [21]. Even if we do not postulate Eq. (26) and treat F(ρ) ¯ as a function with parameters, E 0 , a0 , and B can still be matched exactly using Eq. (23) for E 0 , the lattice equilibrium condition 1 (rij )rij + F (ρ) ¯ ρ (rij )rij = 0 2 j =/ i j= /i 3 We assume that potential functions are used by the simulation program in a tabulated form.

(27)


469

(follows from Eq. (12)) and the expression for B, 90 B =

1 (rij )(rij )2 + F (ρ) ¯ ρ (rij )(rij )2 2 j =/ i j= /i

+ F (ρ) ¯

2

ρ (rij )rij

(28)

j= /i

(can be derived from Eqs. (13) and (27)). These three equations can be readily ¯ and F (ρ) ¯ at a = a0 . satisfied by adjusting the values of F(ρ), ¯ F (ρ) Fitting parameters of a potential are optimized by minimizing the weighted mean squared deviation of properties from their target values. The weights are used as a means of controlling the importance of some properties over others. Some properties are included with a very small weight that only prevents unreasonable values without pursuing an actual fit. While early EAM potentials were fit to experimental properties only, the current trend is to include into the fitting database both experimental and first-principles data [19, 21, 22]. In fact, some of the recent potentials are predominantly fit to firstprinciples data and only use a few experimental numbers, which essentially makes them a parameterization of first-principles calculations. The incorporation of first-principles data into the fitting database improves the reliability of potentials by sampling larger areas of configuration space, including atomic configurations away from those represented by experimental data. Experimental properties used for potential generation traditionally include E 0 , a0 , elastic constants cij , the vacancy formation energy, and often the stacking fault energy. Thermal expansion factors, phonon frequencies, surface energies, and the vacancy migration energy can also be included. Depending on the intended use of the potential, some of these properties are strongly enforced while others are only used for a sanity check (small weight). First-principles data usually come in the form of energy–volume relations for the ground-state structure and several hypothetical “excited” structures of the same metal. The role of these structures is to probe various local environments and atomic volumes of the metal. This sampling improves the transferability of potentials to atomic configurations occurring during subsequent atomistic simulations. Furthermore, first-principles energies along uniform deformation paths between different structures are often calculated, such as the tetragonal deformation path between the FCC and BCC structures (Bain path) or the trigonal deformation path FCC – simple cubic – BCC. Such deformations, however, are normally used for testing potentials rather than fitting. An alternative way of using first-principles data is to fit to interatomic forces drawn from snapshots of first-principles molecular dynamics simulations for solid as well as liquid phases of a metal (force matching method) [19]. The liquid-phase configurations can improve the accuracy of the potential in melting simulations.

470

Y. Mishin

To illustrate the accuracy achievable by modern EAM potentials, Table 1 summarizes selected properties of copper calculated with an EAM potential [23] in comparison with experiment. This particular potential was parameterized by simple analytical functions. A universal equation of state was not enforced and F(ρ) ¯ was described by a polynomial. The cutoff radius of the potential, Rc = 0.551 nm, covers four coordination shells but the contribution of the fourth shell is extremely small. Besides experimental properties indicated in Table 1, the fitting database included two experimental phonon frequencies at the zone-boundary point X , a high pressure–volume relation and, with a small weight, the dimer bond energy E d and thermal expansion factors at several temperatures. The first-principles data included energy–volume relations for several structures. Only the FCC, HCP, and BCC structures were used in the fit, while other structures were deferred for testing. The potential demonstrates excellent agreement with experiment for both fitted and predicted properties, except for the surface energies which are too low. Phonon dispersion relations and thermal expansion factors are also in accurate agreement with experiment (Fig. 1). The potential accurately reproduces firstprinciples energies of alternate structures not included in the fit, as well as energies along several deformation paths between them.

Table 1. Selected properties of Cu calculated with an embedded-atom potential [23] in comparison with experimental data (see [23] for experimental references). Notations: E vf and E vm – vacancy formation and migration energies, E if and E im – self-interstitial formation and migration energies, γSF – intrinsic stacking fault energy, γus – unstable stacking fault energy, γs – surface energy, γT – symmetrical twin boundary energy, Tm – melting temperature, Rd – dimer bond length, E d – dimer bond energy. All other notations are explained in the text. All defect energies were obtained by static relaxation at 0 K Property a0 (nm)a E 0 (eV)a c11 (GPa)a c12 (GPa)a c44 (GPa)a E vf (eV)a E vm (eV)a E if (eV) E im (eV)

Experiment

EAM

0.3615 −3.54 170.0 122.5 75.8 1.27 0.71 2.8–4.2 0.12

0.3615 −3.54 169.9 122.6 76.2 1.27 0.69 3.06 0.10

Property γSF (mJ/m2 )a γus (mJ/m2 ) γT (mJ/m2 ) γs (111) (mJ/m2 ) γs (110) (mJ/m2 ) γs (100) (mJ/m2 ) Tm (K) Rd (nm) E d (eV)d

a Used in the fit. b Average orientation. c Calculated by molecular dynamics (interface velocity method). d Used in the fit with a small weight.

Experiment

EAM

45 – 24 1790b 1790b 1790b 1357 0.22 −2.05

44.4 158 22.2 1239 1475 1345 1327 0.218 −1.93

Interatomic potentials for metals (a)

9

Γ

[q00]

471

X

K

Γ

[qq0]

[qqq]

L

EAM Experiment

8 7

T2

L

ψ(THz)

6 5

L

L

4 T

3

T1

2

T

1 0 0.00 0.25 0.50 0.75 1.00

0.75

q

(b)

0.50 q

0.25

0.00

0.25

0.50

q

EAM Monte Carlo Experiment

2.0

Linear expansion (%)

1.5

1.0

0.5

0.0

Tm

⫺0.5 0

200

400

600 800 1000 Temperature (K)

1200

1400

Figure 1. Comparison of embedded-atom calculations [23] with experimental data for Cu. (a) phonon dispersion curves, (b) linear thermal expansion relative to room temperature. The discrepancy in thermal expansion at low temperatures is due to quantum effects that are not captured by classical Monte Carlo simulations.

For a binary system A–B, the simplest potential generation scheme is to utilize existing potentials for two metals A and B and only construct a cross-interaction function AB (r).4 4 An alternative approach is to optimize all seven potential functions simultaneously, see for example, Ref. [24].

472

Y. Mishin

To win additional fitting parameters we take advantage of the fact that the transformations ¯ → FA (ρ) ¯ + gA ρ, ¯ FA (ρ) AA (r) → AA (r) − 2gA ρA (r), ¯ → FB (ρ) ¯ + gB ρ, ¯ FB (ρ) BB (r) → BB (r) − 2gB ρB (r), ρB (r) → pB ρB (r), ¯ → FB (ρ/ ¯ pB ) FB (ρ)

(29) (30) (31) (32) (33) (34)

leave the energies of elements A and B invariant while altering energies of binary alloys. Thus, pB , gA and gB can be treated as adjustable parameters. After the fit, the new potential functions can be converted to the binary effective pair format by applying the invariant transformations by Eqs. (3)–(6) with gA = −FA (ρ¯A ) and gB = −FB (ρ¯B ), ρ¯A , and ρ¯B being host electron densities in a reference compound. It should be remembered that the binary effective pair format thus obtained will produce elemental potential functions different from the initial ones. Thus, if the initial elemental potentials were in the effective pair format, it will generally be destroyed by the fitting process. Indeed, the reference state of an elemental potential is its ground state, while the reference state of the binary system is a particular binary compound. Physically, however, both elemental potentials will remain exactly the same. All these mathematical transformations should be carefully observed when comparing different potentials or reconstructing them from published parameters. Experimental properties used for optimizing a binary potential typically include E 0 , a0 , and cij of a chosen intermetallic compound. For structural intermetallics, energies of generalized planar faults involved in dislocation dissociations can also be used in the fit to improve the applicability of the potential to simulations of mechanical behavior [15]. Fracture simulations [16] may additionally require reasonable surface energies, which can be adjusted to some extent during the fitting procedure. On the other hand, for thermodynamic and diffusion simulations it is more important to reproduce the heat of the compound formation and point defect characteristics. As with pure metals, the current trend in constructing binary potentials is to incorporate first-principle data, usually in the form of energy–volume relations for experimentally observed and hypothetical compounds. The transferability of a potential can be significantly improved by including compounds with several different stoichiometries across the entire phase diagram [18, 21, 24]. Even if such compounds do not actually exist on the experimental diagram, they sample a broader area of configuration space and secure reasonable energies of various environments and chemical compositions that may occur locally during atomistic simulations, for example, in core regions of lattice


473

defects. Some of the recent binary potentials only use a few experimental numbers but otherwise heavily rely on first-principles input [18]. Besides structural energies, such input may include energies along deformation paths between compounds, energies of stable and unstable planar faults, point defect energies and other data. Some of this information can be deferred from the fitting database and used for testing the potential. The most critical test of transferability of a binary potential is its ability to reproduce the phase diagram at least qualitatively. Unfortunately, many existing potentials are nicely fit to specific properties of a particular compound but fail to describe other structures and compositions with any acceptable accuracy. Such potentials can easily produce incorrect structures of grain boundaries, interfaces or any other defects whose local chemical composition deviates significantly from the bulk composition. A challenge of future research is to establish a procedure for generating reliable EAM potentials for ternary systems. A carefully chosen model system A–B–C must be used as a testing ground. The first step would be to simply construct three binary potentials, A–B, B–C, and C–A, based on the same set of high-quality elemental potentials and capable of reproducing the relevant binary phase diagrams at least on a qualitative level. Such potentials should be based on extensive first-principles input and a smart procedure for a simultaneous optimization of the transformation parameters gs and ps relating to different binaries. The critical test of this potential set would be an evaluation of thermodynamic stability of ternary compounds existing on the experimental diagram. At the next step, calculated properties of such compounds can be improved by further adjustments of the binary potentials.

4.

Angular-dependent Potentials

EAM potentials work best for simple and noble metals but are less accurate for transition metals. The latter reflects an intrinsic limitation of EAM, which is essentially a central-force model that cannot capture the covalent component of bonding arising due to d-electrons in transition metals. Baskes et al. [25– 28] developed a non-central-force extension of EAM, which they called the modified embedded-atom method (MEAM). In MEAM, electron density is treated as a tensor quantity and the host electron density ρ¯i is expressed as a function of the respective tensor invariants. In the simplest approximation, ρ¯i is given by the expansion

(0) (ρ¯i )2 = ρ¯i

2

+ ρ¯i(1)

2

+ ρ¯i(2)

2

+ ρ¯i(3)

2

,

(35)

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Y. Mishin

where

ρ¯i(0)

2



=

j= /i

ρ¯i(1)

ρ¯i(2)

2

ρ¯i(3)

2

(36)

j= /i

sj

(37)

rij

 2  2 α β r r 1 ij ij   = ρ (2) (rij ) −  ρ (2) (rij ) , α,β

ρs(0) (rij ) , j

 2 α r ij  = ρ (1) (rij )  , α

2

2

=

α,β,γ

j= /i

 

j= /i

sj

ρs(3) (rij ) j

rij2

3

β γ

rijα rij rij rij3

j= /i

sj

(38)

2  .

(39)

The terms ρ¯i(k) (k = 0, 1, 2, 3) can be thought of as representing contributions of s, p, d, and f electronic orbitals, respectively. It should be emphasized, however, that the exact relation of these terms to electronic orbitals is not physically clear and Eqs. (35)–(39) can as well be viewed as ad hoc expressions whose only role is to introduce non-spherical components of bonding. The regular EAM is recovered by including only the electron density of “s-orbitals,” ρ¯i(0) , and neglecting all other terms. In comparison with regular EAM, MEAM introduces three new functions, ρs(1) (r), ρs(2) (r), and ρs(3) (r) for each species s, which are fit to experimental and first-principles data in much the same manner as in EAM. While EAM potentials are smoothly truncated at a sphere breaembracing several coordination shells, MEAM includes only one or two coordination shells but introduces a many-body “screening” procedure described in detail by Baskes [27, 29]. Computationally, MEAM is roughly a factor of five to six slower than EAM but can be more accurate for transition metals. It has even been successfully applied to covalent solids, including Si and Ge [27]. Advantages of MEAM over EAM are particularly strong for noncentrosymmetric structures and materials with a negative Cauchy pressure. The latter can be readily reproduced ¯ > 0. MEAM potentials have by angular-dependent terms while keeping F (ρ) been constructed for a number of metals [27, 29, 30] and intermetallic compounds [31, 32]. Pasianot et al. [33] proposed a slightly different way of incorporating angular interactions into EAM. In their so-called embedded-defect method (EDM), the total energy is written in the form E tot =

1 si s j (rij ) + Fsi (ρ¯i ) + G Yi , 2 i, j ( j =/ i) i i

(40)

Interatomic potentials for metals where ρ¯i =

475

ρs j (rij ),

(41)

j= /i

   2 β 2 rijα rij 1  ρs j (rij ) 2  −  ρs j (rij ) . Yi = α,β

3

rij

j= /i

(42)

j= /i

Expression (40) was originally derived from physical considerations different from those underlying MEAM. Mathematically, however, Eqs. (40)–(42) present a particular case of Eqs. (35)–(39) in which ρ¯i(1) and ρ¯i(3) are neglected, F(ρ¯i ) is approximated by a linear expansion in terms of the small 2 perturbation ρ¯i(2) , and the later is expressed through the undisturbed electron density function ρs (r): ρs(2) (r) ≡ρs (r). In comparison with EAM, EDM introduces only one additional parameter, G. Like EAM, EDM uses cutoff functions, thus avoiding the MEAM screening procedure. EDM potentials have been successfully constructed for several HCP [33] and BCC transition metals [33, 35–37]. While EDM is computationally faster than MEAM, it is less general and offers less fitting parameters for the angular part. However, the original EDM formulation can be readily generalized by including more angular-dependent terms: E tot =

1 si s j (rij ) + Fsi (ρ¯i ) 2 i, j ( j =/ i) i

+

2 ρ¯i(1)

+

2 ρ¯i(2)

+

2 ρ¯i(3)

,

(43)

i

where ρ¯i(k) are expressed through parameterized functions ρs(k) (r) by Eqs. (37)–(39). Overall, MEAM, EDM, and Eq. (43) are all equally legitimate empirical expressions introducing angular-dependent forces. The role of ρ¯i(k) ’s is to simply penalize E tot for deviations from local cubic symmetry. These terms do not affect the energy–volume relations for cubic crystals but are important for structures with broken local cubic symmetry. Thus, energies of many common crystal structures such as L12 , L10 , and L11 , depend of the “quadrupole” term ρ¯i(2) . This dependence opens new degrees of freedom for reproducing structural energies of intermetallic compounds. Since nonhydrostatic strains break cubic symmetry, ρ¯i(2) also affects elastic constants, which enables their more accurate fit and a reproduction of negative Cauchy pressures. In some structures, such as diamond and some binary compounds, elastic constants are also affected by the “dipole” term ρ¯i(1) . Areas of broken symmetry inevitably

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Y. Mishin

exist around lattice defects. Due to the additional penalty arising from angular terms, defect energies can be larger than in EAM. In particular, it becomes possible to reproduce higher surface energies and a more accurate vacancy migration energy. In sum, angular-dependent terms can improve the accuracy of fit of potentials in comparison with regular EAM. However, the effect of such terms on the transferability of potential needs to be studied in more detail.

5.

Outlook

Embedded-atom potentials provide a reasonable description of a broad spectrum of properties of metallic systems and enable fast atomistic simulations of a variety of processes ranging from thermodynamic functions and diffusion to plastic deformation and fracture. There are intrinsic limitations of EAM, which is still a semi-empirical model based on central-force interactions. Such limitations set boundaries to the accuracy achievable within this method. However, the accuracy and robustness of EAM potentials gradually improve, within those boundaries, by developing more efficient fitting and testing procedures, using larger data sets, and most importantly, increasing the weight of first-principles data. The latter trend may eventually transform the method to a parameterization, or mapping, of first-principles data. Much work needs to be done to improve transferability of binary EAM potentials. This, again, can be achieved by further optimizing the potential generation procedures and using more first-principle data. The most severe test of a binary potential is its ability to predict the correct phase stability across the entire phase diagram. It is not quite clear at this point how far EAM can be pushed in that direction, but this certainly deserves to be explored. Reliable ternary potentials remain a grand challenge of future research. Presently, the only way of generalizing EAM to include non-central interactions is to introduce energy penalties for local deviations from cubic symmetry. This can be achieved by calculating local dipole, quadrupole, and perhaps higher order tensors and making the energy a function of their invariants. Depending on the initial physical motivation behind such tensors and some technical details (such as cutoff functions versus screening), this idea has been implemented first in MEAM and later in EDM. It should be emphasized, however, that other equally legitimate forms of an angular-dependent potential can be readily constructed in the same spirit, Eq. (43) being just one example. Since there is no unique physical justification for those different forms, they all can simply be viewed as useful empirical expressions. Both MEAM and EDM potentials have been developed for a number of transition metals and have demonstrated an improved accuracy in reproducing their properties. MEAM has also been applied, with significant success, to


477

intermetallic compounds and even covalent solids. Future work may further develop this group of methods towards binary and eventually ternary systems.

References [1] D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, 2nd edn., Academic, San Diego, 2002. [2] D.P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, Cambridge, 2000. [3] M.S. Daw and M.I. Baskes, “Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals,” Phys. Rev. B, 29, 6443–6453, 1984. [4] M.W. Finnis and J.E. Sinclair, “A simple empirical N-body potential for transition metals,” Philos. Mag. A, 50, 45–55, 1984. [5] J.K. Nørskov, “Covalent effects in the effective-medium theory of chemical binding: Hydrogen heats of solution in the 3d metals,” Phys. Rev. B, 26, 2875–2885, 1982. [6] D.G. Pettifor, Bonding and Structure of Molecules and Solids, Clarendon Press, Oxford, 1995. [7] M.S. Daw, “Embedded-atom method: many-body description of metallic cohesion,” In: V. Vitek and D.J. Srolovitz (eds.), Atomistic Simulation of Materials: Beyond Pair Potentials, Plenum Press, New York, pp. 181–191, 1989. [8] M.S. Daw and R.L. Hatcher, “Application of the embedded atom method to phonons in transition metals,” Solid State Comm., 56, 697–699, 1985. [9] A. Van de Walle and G. Ceder, “The effect of lattice vibrations on substitutional alloy thermodynamics,” Rev. Mod. Phys., 74, 11–45, 2002. [10] J.M. Rickman and R. LeSar, “Free-energy calculations in materials research,” Annu. Rev. Mater. Res., 32, 195–217, 2002. [11] S.M. Foiles, “Evaluation of harmonic methods for calculating the free energy of defects in solids,” Phys. Rev. B, 49, 14930–14938, 1994. [12] Y. Mishin and C. Herzig, “Diffusion in the Ti-Al system,” Acta Mater., 48, 589–623, 2000. [13] M. Hagen and M.W. Finnis, “Point defects and chemical potentials in ordered alloys,” Philos. Mag. A, 77, 447–464, 1998. [14] D. Wolf, Handbook of Materials Modeling, vol. 1, Chapter 8, Interfaces, 2004. [15] W. Cai, “Modeling dislocations using a periodic cell,” Article 2.21, this volume. [16] D. Farkas and R. Selinger, “Atomistics of fracture,” Article 2.33, this volume. [17] A.F. Voter, “The embedded-atom method,” In: J.H. Westbrook and R.L. Fleischer (eds.), Intermetallic Compounds, vol. 1, John Wiley & Sons, New York, pp. 77–90, 1994. [18] Y. Mishin, “Atomistic modeling of the γ and γ phases of the Ni-Al system,” Acta Mater., 52, 1451–1467, 2004. [19] F. Ercolessi and J.B. Adams, “Interatomic potentials from first-principles calculations: the force-matching method,” Europhys. Lett., 26, 583–588, 1994. [20] J.H. Rose, J.R. Smith, F. Guinea, and J. Ferrante, “Universal features of the equation of state of metals,” Phys. Rev. B, 29, 2963–2969, 1984.

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Y. Mishin [21] R.R. Zope and Y. Mishin, “Interatomic potentials for atomistic simulations of the Ti-Al system,” Phys. Rev. B, 68, 024102, 2003. [22] Y. Mishin, D. Farkas, M.J. Mehl, and D.A. Papaconstantopoulos, “Interatomic potentials for monoatomic metals from experimental data and ab initio calculations,” Phys. Rev. B, 59, 3393–3407, 1999. [23] Y. Mishin, M.J. Mehl, D.A. Papaconstantopoulos, A.F. Voter, and J.D. Kress, “Structural stability and lattice defects in copper: ab initio, tight-binding and embeddedatom calculations,” Phys. Rev. B, 63, 224106, 2001. [24] Y. Mishin, M.J. Mehl, and D.A. Papaconstantopoulos, “Embedded-atom potential for B2-NiAl,” Phys. Rev. B, 65, 224114, 2002. [25] M.I. Baskes, “Application of the embedded-atom method to covalent materials: a semi-empirical potential for silicon,” Phys. Rev. Lett., 59, 2666–2669, 1987. [26] M.I. Baskes, J.S. Nelson, and A.F Wright, “Semiempirical modified embedded-atom potentials for silicon and germanium,” Phys. Rev. B, 40, 6085–6110, 1989. [27] M.I. Baskes, “Modified embedded-atom potentials for cubic metals and impurities,” Phys. Rev. B, 46, 2727–2742, 1992. [28] M.I. Baskes, J.E. Angelo, and C.L. Bisson, “Atomistic calculations of composite interfaces,” Modelling Simul. Mater. Sci. Eng., 2, 505–518, 1994. [29] M.I. Baskes, “Determination of modified embedded atom method parameters for nickel,” Mater. Chem. Phys., 50, 152–158, 1997. [30] M.I. Baskes and R.A. Johnson, “Modified embedded-atom potentials for HCP metals,” Modelling Simul. Mater. Sci. Eng., 2, 147–163, 1994. [31] M.I. Baskes, “Atomic potentials for the molybdenum–silicon system,” Mater. Sci. Eng. A, 261, 165–168, 1999. [32] D. Chen, M. Yan, and Y.F. Liu, “Modified embedded-atom potential for L10 -TiAl,” Scripta Mater., 40, 913–920, 1999. [33] R. Pasianot, D. Farkas, and E.J. Savino, “Empirical many-body interatomic potentials for bcc transition metals,” Phys. Rev. B, 43, 6952–6961, 1991. [34] J.R. Fernandez, A.M. Monti, and R.C. Pasianot, “Point defects diffusion in α-Ti,” J. Nucl. Mater., 229, 1–9, 1995. [35] G. Simonelli, R. Pasianot, and E.J. Savino, “Point-defect computer simulation including angular forces in bcc iron,” Phys. Rev. B, 50, 727–738, 1994. [36] G. Simonelli, R. Pasianot, and E.J. Savino, “Phonon-dispersion curves for transition metals within the embedded-atom and embedded-defect methods,” Phys. Rev. B, 55, 5570–5573, 1997. [37] G. Simonelli, R. Pasianot, and E.J. Savino, “Self-interstitial configuration in BCC metals. An analysis based on many-body potentials for Fe and Mo,” Phys. Status Solidi (b), 217, 747–758, 2000.

2.3 INTERATOMIC POTENTIAL MODELS FOR IONIC MATERIALS Julian D. Gale Nanochemistry Research Institute, Department of Applied Chemistry, Curtin University of Technology, Perth, 6845, Western Australia

Ionic materials are present in many key technological applications of the modern era, from solid state batteries and fuel cells, nuclear waste immobiliza tion, through to industrial heterogeneous catalysis, such as that found in automotive exhaust systems. With the boundless possibilities for their utilization, it is natural that there has been a long history of computer simulation of their structure and properties in order to understand the materials science of these systems at the atomic level. The classification of materials into different types is, of course, an arbitrary and subjective decision. However, when a binary compound is composed of two elements with very different electronegativities, as is the case for oxides and halides in particular, then it is convenient to regard it as being an ionic solid. The implication is that, as a result of charge transfer from one element to the other, the dominant binding force between particles is the Coulombic attraction between opposite charges. Such materials tend to be characterized by close-packed, dense structures that show no strong directionality in the bonding. Typically, most ionic materials possess a large band gap and are therefore insulating. As a consequence, the notion that the solid is composed of spherical ions whose interactions can be represented by simple distance-dependent functional forms is quite a reasonable one, since overtly quantum mechanical effects are lesser than in materials where covalent bonding occurs. Thus it is possible to develop force fields that are specific for ionic materials, and this approach can be surprisingly successful considering the simplicity of the interatomic potential model. When considering how to construct a force field for ionic materials, the starting point, as is the case for all types of system, is to assume that the total 479 S. Yip (ed.), Handbook of Materials Modeling, 479–497. c 2005 Springer. Printed in the Netherlands.

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energy, Utot, can be decomposed into interactions between different numbers of atoms: Utot =

1 1 1 Ui j + Ui j k + Ui j kl + · · · 2! i j 3! i j k 4! i j k l

Here, Ui j is the energy of interaction between a pair of atoms, i and j , or so-called two-body interaction energy; Ui j k is the extra interaction that arises (beyond the sum of the three two-body energy components for the pairs i − j, j − k, and i − k) when a triad of atoms is considered, and so forth for higher order terms. Note that the inverse factorial prefactor is required to avoid double counting of interactions between particles. In principle, the above decomposition is exact if carried out to terms of high enough order. However, in practice it is necessary to truncate the expansion at some point. For many ionic materials it is often sufficient to only include the two-body term, though the extensions beyond this will be discussed later. Imagining an ionic solid as being composed of cations and anions whose electron densities are frozen, which represents the simplest possible case, the physical interactions present can be intuitively understood. There will obviously be a Coulombic attraction between ions of opposite charge, with a corresponding repulsive force between those of like nature. Because ions are arranged such that the closest neighbours are of opposite sign, this gives rise to a strong net attractive energy that will tend to contract the solid in order to lower the energy. In order that an equilibrium structure is obtained there must be a counterbalancing repulsive force. This arises from the overlap of the electron densities of two ions, regardless of the sign of their charge, and has its origin in the Pauli repulsion between electrons. Hence, we can write the breakdown of the two-body energy in general terms as: repulsive

+ Ui j Ui j = UiCoulomb j

While real spherical ions will have a radial electron density distribution, it is convenient to treat the ions as point charges – i.e., as though all the electron density is situated at the nucleus. Within this approximation, the electrostatic interaction of two charged particles is just given by Coulomb’s law; = UiCoulomb j

qi q j 4π 0ri j

or, if written in atomic units, as will subsequently be done, we can drop the constant factor of 4π 0 : = UiCoulomb j

qi q j ri j

Interatomic potential models for ionic materials

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The error in the electrostatic energy arising from the point charge approximation is usually subsumed into the repulsive energy contribution, since this latter term is usually derived by a fitting procedure, rather than from direct theoretical considerations.

1.

Calculating the Electrostatic Energy

Not only is the electrostatic energy the dominant contribution to the total value, but it turns out that it is actually the most difficult to evaluate. While it is easy to write down that the electrostatic energy is the sum over all pairwise interactions, including all periodic images of the unit cell, the complication arises because the sum must be truncated for actual computation. Unfortunately, the summation is an example of a conditionally convergent series, i.e., the value of the sum depends on how the truncation is made. The reason for this can be understood by considering the interactions of a single ion with all other ions within a given radius, r. The convergence of the energy of r , is given by the number of interactions, Nr , multiplied by the interaction, Utot magnitude of the interaction, U r : r Utot =

Nr U r

r

As r increases, the number of interactions rises in proportion to the surface area of the cut-off sphere: Nr ∝ 4πr 2 . However, the interaction itself only decreases as the inverse power of r, as has been shown previously. Consequently, the magnitude of interaction potentially increases as the cut-off radius is extended. The fact that the magnitude converges in practice relies on the fact that there is cancelation between interactions with cations and anions. It turns out that the electrostatic energy of a system actually depends on the macroscopic state of a crystal due to the long-ranged effect of Coulomb fields. In other words, it is not purely a property of the bulk crystal, but also depends, in general, on the nature of the surfaces and of the crystal morphology [3]. To make it feasible to define an electrostatic energy that is useful for the simulation of ionic materials, it is conventional to impose two conditions on the Coulomb summation: 1. The sum of the charges within the system must be equal to zero: i

qi = 0

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2. The total dipole moment of the system in all directions must also be equal to zero: µ x = µ y = µz = 0 If these conditions are satisfied, the electrostatic energy will always converge to the same value as the cut-off radius is incremented. It is also possible to define the electrostatic energy when the dipole moments along the three Cartesian axes differ from zero. This Coulomb energy is related to the value obtained when the dipole moment is zero, U 0 , according to the following expression; U = U0 +

2π 2 µx + µ2y + µ2z 3V

where V is the volume of the unit cell. Considering the expression for the dipole moment in a given direction, α; µα =

qi riα

i

where riα is the position of the ith ion projected on to this axis, then there is a complication. Because there are multiple images of the same ion, due to the presence of periodic boundary conditions, the dipole contribution of any given ion is an ambiguous quantity. The only way to determine the true dipole moment is to perform the sum over all ions within the entire crystal, which includes those ions at the surface. This is the origin of the electrostatic energy being a macroscopic property of the system. While it has been stated that the electrostatic energy is convergent if the above conditions are obeyed, it is not obvious how to achieve this in practice for a general crystal structure. Various methods have been proposed, the most reknown of which is that of Evjen who constructed charge neutral shells of ions about each interacting particle. However, this is more difficult to automate for a computational implementation and is best for high symmetry structures. Apart from the need to converge to a defined electrostatic energy, there is also the issue of how rapidly the sum converges, since it is required that the calculation be fast for numerical evaluation. By far the dominant approach to evaluating the electrostatic energy is through the use of the summation method due to Ewald which aims to accelerate the convergence by partially transforming the expression into reciprocal space. While the details of the derivation are beyond the scope of this text, and can be found elsewhere [2, 9], the concepts behind the approach and the final result will be given below. In Ewald’s approach, a Gaussian charge distribution of equal magnitude, but opposite sign, is placed at the position of every ion in the crystal. Because the charges cancel, all but for the contribution from the differing


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shape of the distribution, the resulting electrostatic interaction between ions is now rapidly convergent when summed out in real space and converges to the energy U real . In order to recover the original electrostatic energy it is then necessary to compute two further terms. Firstly, the interaction of the Gaussian charge distributions with each other must be subtracted. Because of the smooth nature of the electrostatic potential arising from such a distribution, it is possible to efficiently evaluate this term, U recip , by expanding the charge density in planewaves with the periodicity of the reciprocal lattice. Again, the energy contribution is rapidly convergent with respect to the cut-off radius within reciprocal space. Finally, there is the self-energy, U self , that arises from the interaction of the Gaussian with itself. Mathematically, the Ewald sum is derived by a Laplace transform of the Coulomb energy and the final expressions are given below; U Coulomb = U real + U recip + U self 1 1 qi q j U real = er f c η 2 ri j 2 R i j ri j U recip =

1 4π exp −(G 2 /4η) q q exp G.r) (i i j 2 G i j V 2 G2

U self = −

i

1

qi2

η π

2

where R denotes a real space lattice vector, G represents a reciprocal lattice vector and η is a parameter that determines the width of the Gaussian charge distribution. Note that the summation over reciprocal lattice vectors excludes the case when G = 0. The key to rapid convergence of the Ewald sum is to choose the optimal value of η. If the value is small, then the Gaussians are narrow and so the real space expression converges quickly, while the reciprocal space sum requires a more extensive summation due to the higher degree of curvature of the charge density. Choosing a large value of η obviously leads to the inverse situation. One approach to choosing the convergence parameter is to derive an expression for the total number of terms to be evaluated in real and reciprocal space for a given accuracy and then to find the stationary point where this quantity is at a minimum. The choice of ηopt is then given by;

ηopt =

Nπ3 V

1 3

where N is the number of particles within the unit cell. If the target accuracy, A, is represented by the given fractional degree of convergence (e.g.,

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A = 0.001 would imply that the energy is converged to within 0.1%), then the cut-off radii in real and reciprocal space are given as follows:

max ropt

−ln A = η

12 1

2 G max opt = 2(−η ln A)

Before leaving the evaluation of the electrostatic energy, it is important to comment on other dimensionalities than three-dimensional (3-D) periodic boundary conditions. There is also an analogous approach involving a partial reciprocal space transformation in two dimensions, due to Parry, which can be employed for slab or surface calculations [6]. For the 1-D case of a polymer, the Coulomb sum is now absolutely convergent for a charge neutral system. However, it is still beneficial to use methods that accelerate the convergence, though there is less concensus as to the most efficient technique.

2.

Non-electrostatic Contributions to the Energy

While the electrostatic energy often accounts for the majority of the binding, the non-Coulombic contributions are equally critical since they determine the position and shape of the energy minimum. As previously mentioned, there must always be a short-ranged repulsive force between ions to counter the Coulomb attraction and therefore prevent the collapse of the solid. Most work has followed the pioneering work in the field, as embodied in the Born– Meyer and Born–Lande equations for the lattice energy, by utilizing either an exponential or inverse power-law repulsive term. This gives rise to two widely employed functional forms, namely the Buckingham potential; short−ranged

Ui j

= Ai j exp −

ri j ρi j

−

Ci j ri6j

and that due to Lennard–Jones: Bi j Ci j short−ranged = m − n Ui j ri j ri j For the Lennard–Jones potential, the exponents m and n are typically 9–12 and 6, respectively. This latter potential can also be recast in many different forms by rewriting in terms of the well-depth, ε, and either the distance at repulsive which the potential intercepts the Ui j = 0 axis, r0 , or the position of the minimum, req . Both the Buckingham and Lennard–Jones potentials have the same common features – a short-ranged repulsive term and a slightly longerranged attractive term. The latter contribution, often referred to as the C6 term,


485

arises as the leading term in the expansion of the dispersion energy between two non-overlapping charge densities. When choosing between the use of Buckingham and Lennard–Jones potentials, there are arguments for and against both. Physically, the exponential form of the Buckingham potential should be more realistic because electron densities of ions decay with this shape and so it would seem natural that the repulsion follows the magnitude of the interacting ion densities, at least for weak overlap. However, in the limit of ri j → 0 the repulsive Buckingham potential tends to Ai j , i.e., a constant value that is unphysically low for nuclear fusion! Worse still, if the coefficient Ci j is non-zero, then the potential, while initially repulsive, goes through a maximum and then tends to −∞ – a result that is physically absurd. In contrast, the Lennard-Jones potential behaves sensibly and tends to +∞ as long as m > n. While the false minimum of the Buckingham potential is not usually a problem for energy minimization studies, it can be an issue in molecular dynamics where there is a finite probability of the system gaining sufficient kinetic energy to overcome the repulsive barrier. There is a further solution to the problems with the Buckingham potential at small distances. The problems arise due to the simple power-law expression for the dispersion energy. However, this is also incorrect at short-range since the electron densities begin to overlap leading to a reduction of the dispersion contribution. This can be accounted for by explicitly damping the C6 term as the distance tends to zero, and the most widely used approach for doing this is to adopt the form proposed by Tang and Toennies:

UiC6 j

=− 1−

6

bri j k k=0

k!

exp −bri j

Ci j

ri6j

Occasionally other short-ranged, two-body potentials are choosen, such as the Morse or a harmonic potential. However, these are normally selected when acting between two atoms that are bonded. In this situation, the potential is usually Coulomb-subtracted too, in order that the parameters can be directly equated with the bond length and curvature. All the above short-ranged potentials are pairwise in form. However, there are instances where it is useful to include higher order contributions. For example, in the case of semi-ionic materials, such as silicates, where there is a need to reproduce a tetrahedral local coordination geometry, it is common to include three-body terms that act as a constraint on an angle: 2 1 Ui j k = k3 θi j k − θi0j k 2

There are also many variants on this, such as including higher powers of the deviation of the angle from the equilibrium value and the addition of an

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exponential dependence on the bond lengths so that the potential becomes smooth and continuous with respect to coordination number changes. For systems containing particularly polarizable ions, there is also the possibility of including the three-body contribution to the dispersion energy, as embodied in the Axilrod–Teller potential. As with all materials, it is necessary to select the most approriate force field functional form based on the physical interactions that are likely to dominate in an ionic material. While this will often consist of just the electrostatic term and a two-body short-ranged contribution for dense close-packed materials, it may be necessary to contemplate adding further terms as the degree of covalency and structural complexity increases.

3.

Ion Polarization

Up to this point we have considered ions to have a frozen spherical electron density that may be represented by a point charge. While this is a reasonable representation of many cations, it is not as accurate a description for anions which tend to be much more polarizable. This can be readily appreciated for the oxide ion, O2− in particular. In this case, the first electron affinity of oxygen is favourable, while the second electron affinity is endothermic due to the Coulomb repulsion between electrons. Consequently, the second electron is only bound by the electrostatic potential due to the surrounding cations, and therefore the distribution of this electron will be strongly perturbed by the local environment. It is therefore natural to include the polarizability of anions, and even some larger cations, in ionic potential models when reliable results are required. While polarization may occur to arbitrary order, here the focus will be on the dipole polarizability, α, which is typically the dominant contribution. In the presence of an electric field, E, the dipole moment, µ, generated is given by; µ = αE and the polarization energy, U dipolar, that results is: U dipolar = − 12 α E 2 The electric field at an ion is given by the first derivative of the electrostatic potential with respect to the three Cartesian directions, and therefore can be calculated from the Ewald summation for a bulk material. In principle, it is then straightforward to apply the above point ion polarizability correction to the total energy of a simulation. However, it introduces extra complexity since


487

the induced dipole moments will also generate an electric field at all other ions in the system. Hence, it is necessary to consider the charge–dipole and dipole–dipole interactions as well. The whole procedure involves iteratively solving for the dipole moments on the ions until self-consistency is achieved in a manner analogous to the self-consistent field procedure that occurs in quantum mechanical methods. There is one disadvantage to the use of point ion polarizabilities, as described above, which is that the value of α is a constant. Physically, the more polarized an ion becomes, the harder it should be to polarize it further, and so the induced dipole is prevented from reaching extreme values. If the polarizablity is a constant, a so-called polarization catastrophe can occur in which the total electrostatic energy becomes exothermic faster than the repulsive energy increases leading to the collapse of two ions onto each other. This is particularly problematic with the Buckingham potential since the energy at zero distance tends to −∞. An alternative description of dipolar ion polarization that addresses the above problem is the shell model introduced by Dick and Overhauser [4]. Their approach is to create a simple mechanical model for polarization by dividing each ion into two particles, known as the core and the shell. Here the core can be conceptually thought of as representing the nucleus and core electrons, while the shell represents the more polarizable valence electrons. Thus the core is often positively charged, while the shell is negatively charged, though when utilizing a shell model for a cation it is not uncommon for both core and shell to share the positive charge. Both particles are Coulombically screened from each other and only interact via a harmonic restoring force: 2 U core−shell = 12 kcsrcs

where rcs is the distance between the core and shell. There are two important consequences of the shell model approach. Firstly, because the shell enters the simulation as a point particle, the achievement of electrostatic self-consistency is transformed into a minimization of the shell coordinates. Consequently, this is achieved concurrently with the optimization of the real atomic positions (namely the core positions), though at the cost of doubling the number of variables. While this significantly increases the time required to invert the Hessian matrix, assuming Newton–Raphson optimization is being employed, the convergence rate is also enhanced through all the information on the coupling of coordinates with the polarization being utilized. Secondly, it is the usual convention for the short-ranged potentials to act on the shell of a particle, rather than on the core, which leads to the polarizability becoming environment dependent. If the force constant (second derivative) of the short-range potential acting on the shell is kSR and the shell charge is

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qshell , the polarizability of the ion is equal to: α=

2 qshell kcs + kSR

Special handling of the shell model is required in some simulations. In particular, for molecular dynamics the presence of a particle with no mass potentially complicates the solution of Newton’s equations of motion. However, there are two solutions to this that parallel the techniques found in electronic structure methods. One approach is to divide the atomic mass so that a small fraction is attributed to the shell instead of the core. If chosen to be small enough, the frequency spectra for the shells is higher than any mode of the real material, such that the shells are largely decoupled from the nuclear motions. The disadvantage of this is that a smaller timestep is required in order to achieve an accurate integration. Alternatively, the shells can be minimized at every timestep in order to follow the adiabatic surface. Although the same timestep can now be used as per core-only dynamics, the cost per move is greatly increased. Similarly in lattice dynamics, it is also necessary to consider the contribution from relaxation of the shell positions to the dynamical matrix, which will act to soften the energy surface. Both point ion polarizabilities and the shell model have benefits for interatomic potential simulations of ionic materials. Firstly, they act to stabilize lower symmetry structures and hence it would not be possible to reproduce the structural distortion of various materials without their inclusion. Secondly, they make it possible to determine many materials properties that intrinsically have a strong electronic component. For instance, both the low and high frequency dielectric constant tensors may be calculated, where the former is determined by both the electronic and nuclear contributions, while the latter is purely dependent on the contribution from the polarization model.

4.

Derivation of Ionic Potentials

So far, the typical functional form of the interaction energy in ionic materials has been described, without discussing how the parameter values are arrived at within the model. Many aspects are similar to general forcefield derivation as practiced for organic and inorganic systems, be they ionic or not. However, there are a few differences also that will be highlighted below. Given the dominance of the electrostatic contribution for ionic materials, the starting point for any force field is to determine the nature of the point charges to be employed. There are two broad approaches – either to employ the formal valence charge or to chose smaller partial charges. The main advantages of formal charges are that they remove a degree of freedom from the fitting process and also ensure wide compatability of force fields, in


489

that parameters from binary compounds can be combined to model ternary or more complex phases where the cations do not have the same formal valence charge. Furthermore, when studying defects in materials the vacancy, interstitial or impurity will be guaranteed to carry the correct total charge. On the other hand, for materials with a formal valence of greater than +2 it is argued that formal charges are unrealistic and so partial charges must be used. Indeed, Mulliken charges from ab initio calculations do suggest that such materials are not fully ionic. However, the Mulliken charge is only one of several charge partitioning schemes. Arguably more pertinent measures of ionicity are the Born effective charges that describe the response of the charge density to an electric field. For a solid, where it is not possible to determine the charges that best reproduce the external electrostatic potential, as would be the case for molecules, considering the dipolar response is the next best thing. It is often the case that formal charges, in combination with a shell model for polarization, yield very similar Born effective charges to periodic density functional calculations [6]. Consequently, for low symmetry structures at least, both formal and partial charges can be equally valid in a well derived model. Having determined the charge states of the ions, it is then necessary to derive the short-range and other parameters for the force field by fitting. Parameter derivation falls into one of two classes, either being based on the use of theoretical or experimental data. While truly ab initio parameter derivation is desirable, most theoretical procedures are subject to systematic errors and so empirical fitting to experimental information has tended to be prevalent. Fitting consists of specifying a training set of observable quantities, that may be derived theoretically or experimentally, and then varying the parameters in a least squares procedure in order to minimize the discrepancy between the calculated and observed values [5]. Typically, the training set would consist of one or more structures that represent local energy minima (i.e., stable states with zero force) and data that provide information as to the curvature of the energy surface about these minima, such as bulk moduli, elastic constants, dielectric constants, phonon frequencies, etc. Ideally, multiple structures and as much data as possible should be included in the procedure in order to maximize transferability and to constrain the parameters to physically sensible values. Because it is possible to weight the observables according to their reliability or importance there can never be a single unambiguous fit. In the above brief statement of what fitting is, it is given that the structural data is to be used as an observable. However, there are several distinct ways in which this can be done. If the force field is a perfect fit then the forces calculated at the observed experimental, or theoretically optimized, structure should be zero. Hence it is common to use the forces determined at this point as the observable for fitting, rather than the structure per se, since they are straight forward to calculate. In practice, the quality of the fit is usually imperfect and so there will be residual forces. Lowering the forces does not guarantee that the

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discrepancy in the optimized structural parameters will be minimized though, since this also depends on the curvature. Assuming that the system is within the harmonic region, the errors in the structure, x, will be related to residual force vector, f resid , according to x = H −1 f resid where H is the Hessian matrix containing the second derivatives. Thus one approach to directly fitting the structure is to use the above expression for the errors in the structure. Alternatively, the structure can be fully optimized for each evaluation of the fit quality, which is considerably more expensive, but guaranteed to be reliable regardless of whether the energy surface is quadratic or not. This latter method, referred to as relaxed fitting, also possesses the advantage that any curvature related properties can be evaluated for the structure of zero force, such that the harmonic expressions employed are truly valid. The case of a shell model fit deserves special mention here, since the issues do not usually arise during fits to other types of model. Because of the mapping of dipoles to a coordinate space representation there is the question of how to handle the shell positions during a fit. Given that the cores are equated with the nuclear position, and that it is difficult to ascribe atom-centered dipoles in a crystal, there is rarely any information on where the shells should be sited. In a relaxed fit the issue disappears, since the shells just optimize to the position of minimum force. For a conventional force-based fit then the shells must either still be relaxed explicitly at each evaluation of the sum of squares, or their coordinates can be included as variable parameters such that the relaxation occurs concurrently with the fitting process. Theoretical derivation of parameters can either closely resemble empirical fitting, by inputing calculated observables, or alternatively an energy hypersurface can be utilized. In this latter case many different structures, usually sampled from around the energy minima, are specified along with their corresponding energies. As a result, the curvature of the energy surface is fitted directly rather than by assuming harmonic behavior about the minimum. Again the issue of weighting is particularly important since it tends to be more crucial to ensure a good quality of fit close to the minimum at the expense of points that are further away. To date it has been more common to utilize quantum mechanical data for finite clusters in potential derivation, rather than directly fitting solid state ab initio information. However, this introduces uncertainties, since it is not clear how transferable the gas phase cluster data will be to bulk materials since they are dominated by surface effects. There are two further theoretical methods for parameter derivation that deserve a mention, namely electron gas methods and rule-based methods. The first is particularly significant since it was a popular approach in the early days of the computer simulation of ionic materials at the atomistic level. In the electron gas method, the energy of overlapping frozen ion electron densities


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is calculated according to density functional theory as a function of distance. These energies can then be used directly via splines or fitted to a functional form. Given that not all ions, such as O2− , are stable in vacu, the ion densities were usually determined in an appropriate potential well to mimic the lattice environment. The results obtained directly from this procedure where not always accurate, given the limitations of density functional theory, so often the distance dependence was shifted to improve the position of the minimum. The second alternative theoretical approach is to use rules that encapsulate how to determine interactions from atomic properties, such as the polarizability and atomic radius, in order to generate force fields of universal applicability. Of course, this compromises the accuracy of the results for any given system, but can be useful for systems were there is little known data to fit to.

5.

Applications of Ionic Potentials

Having defined the appropriate force field for a material, it is then possible to calculate many different properties in a very straight forward fashion. Simulations can be broadly divided into two categories – static and dynamic. In a static calculation, the structure of a material is optimized to the nearest local minimum, which may represent one desired polymorph of a system, as opposed to the global minimum, and then the properties are derived by consideration of the curvature about that position. For example, many of the mechanical, vibrational and electrical response properties are all functions of the second derivatives of the energy with respect to atomic coordinates and lattice strains. For pair potentials, the determination of these properties is not dramatically more expensive than the evaluation of the forces, with the exception of matrix inversions that may be required once the second derivative matrix has been calculated. This is in contrast to quantum mechanical methods where the determination of the wavefunction derivatives makes analytical property calculations almost as expensive as finite difference procedures. In a dynamical simulation, the probability distribution, composed of many different nuclear configurations, is sampled to provide averaged properties that depend on temperature. This usually involves performing either molecular dynamics (in which case the time correlation between data is known) or Monte Carlo (where configurations are selected randomly according to the Boltzmann distribution). Fundamentally static and dynamic methods differ because the former are founded within the harmonic approximation, while the latter allow for anharmonicity. For the purposes of this section, the focus will be placed on the static information that can be obtained from ionic potentials, but stoichastic simulations would also be equally as applicable. The first information to be yielded by an energy minimization is the equilibrium structure. Given that many potentials are

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fitted to such data, it is not surprising that the quality of structural reproduction, at least for simple binary materials, is usually high. Many force fields are derived with out explicit reference to temperature, so consequently the structure that is calculated may contain implicit temperature effects even though the optimization was performed nominally at zero Kelvin. As an example of the application of the formal charge, shell model potential a set of parameters has been derived for alumina. The observables used consisted of the structure of corundum and its elastic and dielectric constants. As a starting model, the parameters originally derived by Catlow et al. [1] were used and subjected to the relax fitting approach. Alumina is a material that has been much studied already, so the aim here is just to illustrate typical results yielded by a fit to such a material and some of the related issues. Values of the calculated properties for corundum, α-Al2 O3 are given in Table 1, along with the comparison against experiment, using the potentials derived, which are given in Table 2. Before considering the results, let us consider the parameters that resulted from the fit since they highlight a number of points. Firstly, by looking at the shell charges and spring constants it can be seen that the oxide ion is responsible for most of the polarizability of the system as would be expected. This is a natural result of the fitting process since the charge distribution between core and shell, as well as the spring constant, was allowed to vary. Secondly, in accord with this picture the attractive dispersion term for Al–O is set to zero, though even if allowed to vary it remains small. Finally, the oxygen–oxygen Table 1. Calculated versus experimental structure and properties for aluminium oxide in the corundum structure based on a shell model potential fitted to the same experimental data Observable

Experiment

Calculated

a (Å) c (Å) Al z (frac) O x (frac) C11 (GPa) C12 (GPa) C13 (GPa) C14 (GPa) C33 (GPa) C44 (GPa) C66 (GPa) 0 ε11 0 ε33 ∞ ε11 ∞ ε33

4.7602 12.9933 0.3522 0.3062 496.9 163.6 110.9 −23.5 498.0 147.4 166.7 9.34 11.54 3.1 3.1

4.9084 12.9778 0.3597 0.2987 567.1 224.6 158.1 −54.3 453.3 127.6 171.2 8.70 13.38 2.88 3.06


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Table 2. Interatomic potential parameters derived for alumina based on relax fitting to the experimental observables given in Table 1. The starting parameters were taken from Catlow et al. and a two-body cut-off distance of 16.0 Å was employed, while that for the core-shell interaction was 0.8 Å. All non-Coulombic interactions not explicitly given are implicitly zero. The shell charges for A1 and O were −0.0395 and −2.0816, respectively Species 1

Species 2

A (eV)

ρ (Å)

C (eV/Å6 )

kcs (eV/Å2 )

A1 shell O shell A1 core O core

O shell O shell A1 shell O shell

1012.17 22764.00 – –

0.32709 0.14900 – –

0.0 22.368 – –

– – 331.958 24.625

repulsive term is particularly short-ranged and only makes a minute contribution at the equilibrium structure. Consequently, the A and ρ values are rarely varied from the original starting values. The rhombohedral corundum structure is sufficiently complex that even though the potential was empirically fitted to this particular system it is still not possible to achieve a perfect fit. While for many dense high symmetry ionic compounds it is possible to obtain accuracy of better than 1% for structural parameters, the moment there are appreciable anisotropic effects it becomes more difficult. This is illustrated by corundum where it is impossible with the basic shell model to accurately describe the behavior in the ab plane and along the c axis simultaneously, leading to an error of 3% in the a and b cell parameters. Not only is this true for the structure, but it is even more valid for the curvature related properties. If the values of C11 and C33 are compared, which are indicative of the elastic behavior in the two distinct directions, the calculated values have to achieve a compromise by one value being higher than experiment, while the other is lower. In reality, alumina is elastically fairly isotropic, but a dipolar model cannot capture this. The above results for alumina also illustrate the fact that while it is usually possible to reproduce structural parameters to within a few percent, the errors associated with other properties can be considerably greater. As pointed out earlier, although a formal charge model for alumina was employed, the ions in fact behave as though the system is less than fully ionic due to the polarizability. The calculated Born effective charges show that aluminium has a reduced ionicity with a charge of +2.32 in the ab plane and a slightly higher value of +2.55 parallel to the c axis. These magnitudes are in good agreement with assessments of the degree of ionicity of corundum obtained from ab initio calculations. There are many more bulk properties that can be readily determined from interatomic potentials than those given above. For instance, phonon

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frequencies, dispersion curves and densities of states, acoustic velocities, thermal expansion coefficients, heat capacities, entropies and free energies can all be obtained from determining the dynamical matrix about an optimized structure [6]. Other important quantities can also be determined by creating defects in the system, such as vacancies, interstitials and grain boundaries, or by locating other stationary points, in particular transition states for ion diffusion. The possibilities are as boundless as the number of physical processes that can occur in a real material.

6.

Discussion

So far, the basic ionic potential approach to the modeling of solids has been portrayed. While this is very successful for many of the materials for which it was intended, and that composed the majority of the earlier studies, there are increasingly many situations where extensions and modifications are required in order to broaden the scope of the technique. These enhancements recognize the fact that many systems comprise atoms that are less than fully ionic and often non-spherical. One of the most limiting aspects of the ionic model is the use of fixed charges. It is often the case that potential parameters are derived for the bulk material alone where a compound is at its most ionic. However, the ideal force field should also be transferable to lower coordination environments, such as surfaces and even gas phase clusters. Fundamentally, the problem with any fixed charge model, be it formally or partially charged, is that it cannot reproduce the proper dissociation limit of the interaction. Ultimately, if sufficiently far removed from each other, an ionic structure should transform into separate neutral atoms. There is a more sophisticated way of determining partial charges within a force field that addresses the above issue, which is to calculate them as an explicit function of geometry. While this has only been sparsely utilized to date, due to the extra complexity, it has the potential to capture, through chargetransfer, many of the higher order polarizabilities beyond the dipole level, as well as yielding the proper dissociation behavior. The predominant approach to determining the charges has been via electronegativity equalization [8]. Here the self energy of an ion is expressed as a quadratic function of the charge in terms of the electronegativity, χ, and hardness, µ: Uiself (q) = Uiself (0) + χi q + 12 µi q 2 When coupled to the electrostatic energy of interaction between the ions, and solved subject to the condition of charge neutrality for the unit cell, this


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determines the charges on the ions. The main variation between schemes is the form selected for the Coulomb interaction between ions. While some workers have used the limiting point-charge interaction of 1r at all distances, it has been argued that damped interactions should be used that more realistically mimic the nature of two-centre integrals (i.e., tend to a constant value as r → 0). Variable charge schemes have shown some promise, and have clear advantages since they allow multiple oxidation states to be treated with a single set of parameters, at least in principle. This simplifies the study of materials where the same cation occurs in multiple oxidation states, since no prior assumption needs to be made as to the charge ordering scheme. However, there are still many challenges in this area since it appears that choosing the more formally correct screened Coulomb interaction leads to the electrostatics only contributing weakly to the interionic forces to an extent that is unrealistic. Looking beyond dipolar polarizability, which is a limitation of the most widely used form of ionic model, there are instances where higher order contributions are important. Here, we consider two examples that highlight the issues. Experimentally it is observed that many cubic rock salt structured materials exhibit a so-called Cauchy violation in that the elastic constants C12 and C44 are not equivalent. It has been demonstrated that two-body potential models are unable to reproduce this phenomenon, and inclusion of dipolar polarizability fails to improve the situation. The Cauchy violation actually requires a many-body coupling of the interactions through a higher order polarization. This can be handled through the inclusion of a breathing shell model. Here the shell is given a finite radius that is allowed to vary with a harmonic restoring force about an equilibrium value, with the repulsive short-ranged potential also acting on it. This non-central ion force generates a Cauchy violation, though always of one particular sign (C44 > C12 ), while the experimental values can be in either direction. A second example of the role of polarization, is in the stability of polymorphs of alumina. If the relative energies of alumina adopting different possible M2 O3 structures is examined using most standard interatomic potential models, including that given in the previous section, then it is found that the corundum structure (which is the experimental ground state under ambient conditions) is not the most stable, with the bixbyite form being preferred. Investigations have demonstrated that the inclusion of quadrupolar polarizability is essential here [7]. This can be readily achieved within the point ion approach, but is more difficult in the shell model case. While an elliptical breathing shell model can capture the effect, it highlights the fact that the extension of this mechanical approach to higher order terms becomes increasingly cumbersome. While most alkali and alkaline earth metals conform reasonably well to the ionic model, there are substantial problems with describing many of the remaining cations in the periodic table. In particular, transition metals ions

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are often non-spherical due to the partial occupancy of the d-orbitals. The classic example of this is when the anti-bonding eg∗ orbitals of an octahedral ion are half-filled for a particular spin, giving rise to a Jahn–Teller distortion, as is the case for Cu2+ . To describe this effect with a simple potential model is impossible, except by constructing a highly specific model with different short-ranged potentials for each metal–ligand interaction, regardless of the fact that they may be acting between the same species. So far, the only solution to the problem of ligand–field effects has been to resort to approaches that mimic the underlying quantum mechanics, but in an empirical fashion. Hence, most work has utilized the angular overlap model to describe a set of energy levels that are subsequently populated according to a Fermi–Dirac distribution, where the states are determined by diagonalizing a 5 × 5 matrix determined according to the local environment [11]. This approach has been successfully used to describe the manganate (Mn3+ , d4 ) cation, as well as other systems within a molecular mechanics framework. At the heart of the ionic potential method is the electrostatic energy, normally evaluated according to the Ewald sum when working within 3-D boundary conditions. However, this approach possesses the disadvantage that it scales 3 at best as N 2 , where N again represents the number of atoms within the simulation cell. In an era when very large scale simulations are being targeted, it is necessary to also reassess the underlying algorithms to ensure the optimal efficiency is attained. Consequently, the fundamental task of calculating the Coulomb energy is still an area of active research. Approaches currently being employed include the particle-mesh and cell multipole methods. The desirable characteristics of an algorithm are now that it should both scale linearly with system size and also be amenable to parallel computation. Both of these can be achieved as long as the method is local in real space, in some cases with complementary linear-scaling in reciprocal space, or if a hierarchical scheme is utlized within the cell multipole method to make the problem increasing coarse-grained the greater the distance of interaction is. Methods have been proposed that use a spherical cut-off in real space alone, which naturally satisfies both desirable criteria [10]. However, it becomes difficult to achieve the defined Ewald limiting value without a considerable prefactor.

7.

Outlook

The state of the art in force fields for ionic materials looks set for a gradual evolution that sees it take on board many concepts from other types of system, while retaining the aim of an accurate evaluation of the electrostatic energy at the core. For the very short-ranged interactions it is likely that bond order models, widely used in the semiconductor and hydrocarbon fields, and


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also closely related to the approach taken for metallic systems, will be blended with schemes that capture the variation of the charge and higher order multipole moments as a function of structure. The result will be force fields that are capable of simulating not only one category of material, but several distinct ones. Development of solid state quantum mechanical methods to increased levels of accuracy will increasingly provide the wealth of information required for parameterisation of more complex interatomic potentials for systems, especially where there is a paucity of experimental data. Ultimately, this will lead to a seamless transition to models capable of reliably describing interfaces between ionic and non-ionic systems – currently one of the most challenging problems in materials science.

References [1] C.R.A. Catlow, R. James, W.C. Mackrodt, and R.F. Stewart, “Defect energetics in α-Al2 O3 and rutile TiO2 ,” Phys. Rev. B, 25, 1006–1026, 1982. [2] C.R.A. Catlow and W.C. Mackrodt, “Theory of simulation methods for lattice and defect energy calculations in crystals,” Lecture Notes in Phys., 166, 3–20, 1982. [3] S.W. de Leeuw, J.W. Perram, and E.R. Smith, “Simulation of electrostatic systems in periodic boundary conditions. i. lattice sums and dielectric constants,” Proc. R. Soc. London, Ser. A, 373, 27–56, 1980. [4] B.G. Dick and A.W. Overhauser, “Theory of the dielectric constants of alkali halide crystals,” Phys. Rev., 112, 90–103, 1958. [5] J.D. Gale, “Empirical potential derivation for ionic materials,” Phil. Mag. B, 73, 3–19, 1996. [6] J.D. Gale and A.L. Rohl, “The general lattice utility program (GULP),” Mol. Simul., 29, 291–341, 2003. [7] P.A. Madden and M. Wilson, “ ‘Covalent’ effects in ‘ionic’ systems,” Chem. Soc. Rev., pp. 339–350, 1996. [8] W.J. Mortier, K. van Genechten, and J. Gasteiger, “Electronegativity equalization: applications and parameterization,” J. Am. Chem. Soc., 107, 829–835, 1985. [9] M.P. Tosi, “Cohesion of ionic solids in the Born model,” Solid State Phys., 16, 1–120, 1964. [10] D. Wolf, P. Keblinski, S.R. Philpot, and J. Eggebrecht, “Exact method for the simulation of Coulombic systems by spherically truncated, pairwise r −1 summation,” J. Chem. Phys., 110, 8254–8282, 1999. [11] S.M. Woodley, P.D. Battle, C.R.A. Catlow, and J.D. Gale, “Development of a new interatomic potential for the modeling of ligand field effects,” J. Phys. Chem. B, 105, 6824–6830, 2001.

2.4 MODELING COVALENT BOND WITH INTERATOMIC POTENTIALS Joa˜ o F. Justo Escola Politécnica, Universidade de São Paulo, São Paulo, Brazil

Atoms, the elementary carriers of chemical identity, interact strongly with each other to form solids. It is interesting that those interactions could be directly mapped to the electronic and structural properties of the resulting materials. This connection between microscopic and macroscopic worlds is appealing, and suggests that a theoretical atomistic model could help to model and build materials with predetermined properties. Atomistic simulations represent one of the tools that can bridge those two worlds, accessing to information on the microscopic mechanisms which, in many cases, could not be sampled out by experiments. One of the most important elements in an atomistic simulation is the model describing the interatomic interactions. In principle, such model should take into account all the particles (electrons and nuclei) of the system. Quantum mechanical (or ab initio) methods provide a precise description of those interactions, but they are computationally prohibitive. As a result, simulations would be restricted to systems involving only up to a thousand (or a few thousand) atoms, which is not enough to capture many important atomistic mechanisms. Some approximation, leading to less expensive models, should be implemented. A radical approach is to describe the interactions by classical potentials, in which the electronic effects are somehow integrated out, being taken into account only implicitly. The gain in computational efficiency comes with a price: a poorer description of the interactions. Ab initio methods will become increasingly important in materials science over the next decade. Even using the fastest computers, those methods will continue being computationally expensive. Therefore, there is a demand for less expensive models to explore a number of important phenomena, to provide a qualitative view, scan for trends or insights on atomistic events, which could be later refined using ab initio methods. Developing an interatomic potential involves a combination of intuitive thinking, which comes out from our 499 S. Yip (ed.), Handbook of Materials Modeling, 499–507. c 2005 Springer. Printed in the Netherlands.

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knowledge on the nature of the interatomic bonding, and theoretical input. However, there is no theory which would directly provide the functional form for an interatomic potential. As a result, depending on the bonding type, considerably distinct approaches have been devised to describe interatomic interactions [1, 2]. In any case, the functional form should have a physical motivation and enough flexibility, in terms of fitting parameters, to capture the essential aspects underlying the interatomic interactions. The next sections discuss the specific case of modeling the covalent bonding by interatomic potentials, and the elements which should be present to properly describe such interactions.

1.

Pair Potentials

The cohesive energy (E c ) is the relevant property which quantifies cohesion in a solid. It is given by E c (Rn , rm ), where Rn and rm represent the degrees of freedom of the n nuclei and m electrons, respectively. While E c could be computed by solving the quantum mechanical Schrödinger equation for the electrons of the system, one should inquire what kind of approximation could be performed to describe E c with less expensive methods. One strategy is to average the electronic effects out, but still keeping the electronic degrees of freedom explicitly. One of these approaches, called tight-binding method, provides a realistic description of bonding in solids. However, those models are still computationally too expensive, although simulations with a few thousand atoms could be performed. An extreme approach is to remove all the electronic degrees of freedom, and E c would be given by E c (Rn , rm ) ≈ E c (Rn ). In this last case, the electronic effects would be implicitly present in the functional form. Several interatomic potentials for covalent bonding have been developed over the years. Only for silicon, which is considered the prototypical covalent material, there are more than thirty models which have been extensively used and tested [3]. This and the following sections discuss the relevant elements of an interatomic potential to describe a typical covalent material. The discussion focuses on the two most important models which have been developed for silicon [4, 5]. Cohesive energy could be determined by the atomic arrangement, in terms of a many-body expansion [6] Ec =

n i

V1 (Ri ) +

n i, j

V2 (Ri , R j ) +

n

V3 (Ri , R j , Rk ) + · · · ,

(1)

i, j,k

in which the sums are over all the n atoms of the system. In principle, E c could be determined by an infinite many-body expansion, but the computational cost scales with n l , where l is the order in which the expansion is truncated. The one-body terms (V1 ) are generally neglected, but the two-body (V2 ) and

Modeling covalent bond with interatomic potentials

501

three-body (V3 ) terms carry most of the relevant effects underlying bonding. While the V2 and V3 have a simple physical interpretation, intuition for higher order terms is not so straightforward, and most models have avoided such terms. Could the expansion (1) be truncated in a two-body expansion and still capture the essential properties of covalent bonding? For a long period, pair potentials were used to investigate materials properties, and revealed a number of fundamental atomistic processes. Models including higher order expansions, later developed, provided results which were qualitatively consistent with those early investigations. This sets light on the discussion of pair potentials. Although they provide an unrealistic description of covalent bonding, they still capture some of the essential aspects of cohesion. A typical V2 function has a strong repulsive interaction at short interatomic separations, changing to an attractive interaction at intermediate separations which goes smoothly to zero at longer distances. The V2 interaction, between atoms i and j , can be written as combination of a repulsive (VR ) plus an attractive (V A ) interaction in terms of the interatomic distance, ri j = |Ri − R j |.

V2 / ε

1

0

⫺1 1

2

r/a Figure 1. The two-body interatomic potential. The figure presents V2 for two models: the Lennard–Jones (full line) and the Stillinger–Weber (dashed line) potentials. The functions are plotted normalized in terms of the minimum in energy and equilibrium separation (a).

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The Lennard–Jones potential, shown in Fig. 1, is an example of a pair potential used to model cohesion in a solid V2 (r) = VR (r) + V A (r) = 4ε

12

σ r

−

6

σ r

,

(2)

where ε and σ are free parameters which can be fitted to properties of the material. The equilibrium interatomic distance (a) is related to the crystalline lattice parameter, while the curvature of the potential near a is directly correlated to the macroscopic bulk modulus. The functional form in Eq. (2) is long ranged, and the computational cost scales with n 2 . On the other hand, this cost could scale linearly with n if a cut-off function f c (r) were used. This f c (r) function should not change substantially the interaction for the relevant region of bonding, near the minimum of V2 (r), and should vanish at a certain interatomic distance Rc , defined as the cut-off of the interaction. Therefore, the two-body interaction is described by an effective potential V2eff (r) = V2 (r) f c (r). The functional form of the Lennard–Jones potential can provide a realistic description of noble gases in condensed phases. Although pair potentials capture some essential aspects of bonding, there are still some important elements missing in order to properly describe covalent bonding. If interatomic interactions were described only by pair potentials, there would be a gain in cohesive energy if an atom increased its coordination (number of nearest neighbors). Since there is no energy penalty for increasing coordination, pair potentials will always lead to closed packed crystalline structures. However, atoms in covalent materials sit in much more open crystalline structures, such as hexagonal or the diamond cubic. Pair potentials alone cannot describe the covalent bonding, and many-body effects must be introduced in the description of cohesion.

2.

Beyond Pair Potentials

The many-body effects [6] could be introduced in E c by several procedures: inside the two-body expansion (pair functionals), by an explicit many-body expansion (cluster potentials), or a combination of both (cluster functionals). Models which have been successfully developed to describe covalent systems fit into one of these categories. The Stillinger–Weber [4] and the Tersoff [5] models can be classified as a cluster potential and as a cluster functional, respectively. In a description using only pair potentials, as given by Eq. (2), the cohesive energy of an individual bond inside a crystal is constant for any atomic coordination. However, this departs from a realistic description. Figure 2(a) shows the cohesive energy per bond as a function of atomic coordination for several crystalline structures of silicon. There is a weakening of the bond strength

Modeling covalent bond with interatomic potentials (a)

503

(b) 1.5

0

1

b(Z)

E c /bond

⫺1

⫺2

0.5 ⫺3

0

2

4

6

8

10

12

14

0

2

Z

4

6

8

10

12

14

Z

Figure 2. (a) Cohesive energy per bond (E c /bond) as a function of atomic coordination (Z ). Cohesive energies are taken from ab initio calculations (diamond), and the full and dashed lines represent fitting with a Z −1/2 and exp(−β Z 2 ), respectively. (b) Bond order term b(Z ) as a function atomic coordination taken from ab initio calculations (diamond), and fitted to Z −1/2 (full line) and exp(−β Z 2 ) (dashed line).

with increasing coordination, a behavior that is observed in any material. However, bond strength weakens very fast with coordination in molecular crystals and very slow in most metals. That is why molecular solids favor very low coordinations and metals favor high coordinations. Covalent solids fall between those two extremes. Cohesive energy can be written as a sum over all the individual bonds Vi j Ec =

1 1 Vi j = VR (ri j ) + bi j V A (ri j ) , 2 i, j 2 i, j

(3)

where the parameter bi j controls the strength of the attractive interaction in Vi j . The attractive interaction between two atoms, i.e., the interaction controlling cohesion, is a function of the local environment. This dependence could be translated into a physical quantity called local coordination (Z ). As the coordination increases, valence electrons should be shared with more neighbors, so the individual bond between an atom and its neighbors weakens. Using chemistry arguments, it can be shown that the bond order term (bi j ), can be given as a function of the local coordination (Z i ) in atom i by −1/2

bi j (Z i ) = η Z i

,

(4)

where η is a fitting parameter. Figure 2(b) shows the bond order term as a function of coordination for several crystalline structures. The Z −1/2 function is a good approximation for high coordinations, but fails for low coordinations. It has been recently shown [7] that an exponential behavior for bi j would be more adequate. The introduction of the bond order term in V2 considerably improves the description of cohesion in a covalent material. With this new

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term, the equilibrium distance and strength of a bond is also determined by the local coordination at each atomic center. Even using a bond order term, covalent bonding still requires a functional form with some angular dependence to stabilize the open crystalline structures. Angular functions could be introduced inside the bond order term b(Z ), as developed by Tersoff [5], which becomes b(Z , θ), where θ represents the angles between adjacent bonds around each atom of the system. Another procedure is to use an explicit three-body expansion [4]. In terms of energetics, there is a parallel between two-body and three-body potentials. In the former case, there is an energy penalty for interatomic distances differing from a certain equilibrium value. In the later case, there is a penalty for angles differing from a certain equilibrium value θ0 . The three-body potentials are generally positive, being null at an equilibrium angle. The interaction for the (i, j, k) set of atoms is described by V3 (ri j , rik , r j k ) = h(ri j )h(rik )g(θi j k ),

(5)

where the radial functions h(r) goes monotonically to zero with increasing the interatomic distance. Figure 3 shows the behavior of typical angular functions g(θ). The Stillinger–Weber model used a three-body expansion, and the V3 potential was developed as a penalty function with a minimum 2

i θ ijk

1.5

g(θ)

j

k

1

0.5

0

30

60

90

120

150

180

θ Figure 3. Angular function g(θ) from the Stillinger–Weber (full line) and Tersoff (dashed line) models.


505

at the tetrahedral angle (109.47◦ ). On the other hand, the Tersoff potential introduced an angular function inside the bond order term, and the minimum of the angular term was a fitting parameter.

3.

Models

Developing an interatomic potential involves several elements. The first one is the functional form, which should capture all the properties of covalent bonding. The functions should have enough flexibility, in terms of number of free parameters, to allow a description of a wide set of the materials properties. The second element is the fitting procedure used to find the set of free parameters that better describes a predetermined database. The database comprises a set of crystalline properties (such as cohesive energy, lattice parameter, elastic constants) and other specific properties (such as the formation energy of point defects) obtained from experiments or ab initio calculations. Additionally, the interatomic potential should be transferable, i.e., it should provide a realistic description of relevant configurations away from the database. Two interatomic potentials [4, 5] have prevailed over the others in studies of covalent materials. The Tersoff model is described by a two-body expansion, including a bond order term Ec =

1 Vi j , 2 i=/ j

(6)

Vij = f c (rij ) f R (rij ) + bij f A (ri j ) ,

(7)

where f R (ri j ) and f A (ri j ) are respectively, the repulsive and attractive terms given by f R (r) = A exp(−λ1r)

and

f A (r) = −B exp(−λ2r).

(8)

The f c (r) is a cut-off function which is one for the relevant region of bonding r < S, going smoothly to zero in the range S < r < R. The R and S, which control the range of interactions, are fitting parameters. The bij is the bond order term which is given by

bi j = 1 + β n ζinj ζi j =

1/2n

,

(9)

g(θi j k ) exp α 3 (ri j − rik )3 ,

(10)

c2 c2 − , d 2 d 2 + (h − cos θ)2

(11)

k= / i, j

g(θ) = 1 +

where θij k is the angle between i j and ik bonds.

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The Tersoff potential was fitted to several silicon polytypes, being extended to other covalent systems, including multi-component materials. The Brenner potential [8], a model which resembles the Tersoff potential, is widely used to study hydrocarbon systems. The Stillinger–Weber potential is the most used model for covalent materials. It was developed as a three-body expansion E=

V2 (ri j ) +

i, j

V3 (ri j , rik , r j k ).

(12)

i, j,k

The two-body term V2 (r) is given by

B V2 (r) = A ρ − 1 f c (r), r

(13)

where the cut-off function f c (r) is given by

f c (r) = exp µ/(r − R) ,

(14)

if r < R and null otherwise. The three-body potential V3 is given by: V3 (ri j , rik ) = h(ri j )h(rik )g(θi j k ), h(r) = exp γ /(r − R) , g(θ) = (cos θ + 1/3)2.

(15) (16) (17)

This model was fitted to properties of the diamond cubic structure and local order of liquid silicon. Other models have been developed to describe covalent materials. Those models have used different approaches, such as functional forms with up to 50 parameters and extensive database. Some of those models have been compared with each other, specially in the case of silicon [3]. Such comparisons revealed that no interatomic potential is suitable for all situations, such that there is still space for further developments. Recently, a new model for covalent materials was developed [7] and included the features of both the Tersoff and the Stillinger–Weber models. That model included explicitly bond order terms in the two-body and three-body interactions, which allowed a better description of covalent bonding as compared to previous models.

4.

Perspectives

Interatomic potentials will continue playing an important role in atomistic simulations. Although potentials have been successfully applied to investigate covalent materials, they still face several challenges. As new models are


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developed, theoretical input will increasingly prevail over empirical input. So far, the physical properties of bonding have been introduced by trial and error. Attempts to improve models have been in the direction of trying new functional forms, going to higher order expansions or increasing the number of fitting parameters. This will give place to more sophisticated approaches, in which the functional forms could be directly extracted from theory. Interatomic potentials also face the challenge to describe materials with mixed bonding character (metallic, covalent, and ionic altogether). The Tersoff potential, for example, has been extended to systems with some ionic character, but still with prevailing covalent character. That model would not work for materials with stronger ionic character, requiring at least the introduction of a long-ranged Coulomb interaction term. Finally, even if sophisticated interatomic potentials are developed, one should keep in mind that every model has its limited applicability and should always be used with caution.

References [1] A.F. Voter, “Interatomic potentials for atomistic simulations,” MRS Bulletin, 21(2), 17–19, (and additional references in the same issue, 1996). [2] R. Phillips, Crystals, Defects and Microstructures: Modeling Across Scales, Cambridge University Press, Cambridge, UK, 2001. [3] H. Balamane, T. Halicioglu, and W.A. Tiller, “Comparative study of silicon empirical interatomic potentials,” Phys. Rev. B, 46, 2250–2279, 1992. [4] F.H. Stillinger and T.A. Weber, “Computer simulation of local order in condensed phases of silicon,” Phys. Rev. B, 31, 5262–5271, 1985. [5] J. Tersoff, “New empirical-approach for the structure and energy of covalent systems,” Phys. Rev. B, 37, 6991–7000, 1988. [6] A.E. Carlsson, “Beyond pair potentials in elemental transition metals and semiconductors,” In: H. Ehrenreich and D. Turnbull (eds.), Solid State Physics, vol. 43, Academic Press, San Diego, pp. 1–91, 1990. [7] J.F. Justo, M.Z. Bazant, E. Kaxiras, V.V. Bulatov, and S. Yip, “Interatomic potential for silicon defects and disordered phases,” Phys. Rev. B, 58, 2539–2550, 1998. [8] D.W. Brenner, “Empirical potential for hydrocarbons for use in simulating the chemical vapor-deposition of diamond films,” Phys. Rev. B, 42, 9458–9471, 1990.

2.5 INTERATOMIC POTENTIALS: MOLECULES Alexander D. MacKerell, Jr. Department of Pharmaceutical Sciences, School of Pharmacy, University of Maryland, 20 Penn Street, Baltimore, MD, 21201, USA

Interatomic interactions between molecules dominate their behavior in condensed phases, including the aqueous phase in which biologically relevant processes occur [1]. Accordingly, it is essential to accurately treat interatomic interactions using theoretical approaches in order to apply such methods to study condensed phase phenomena. Typical condensed phase systems subjected to theoretical studies include thousands to hundreds of thousands of particles. Thus, to allow for calculations on such systems to be performed simple, computationally efficient functions, termed empirical or potential energy functions, are applied to calculate the energy as a function of structure. In this chapter an overview of potential energy functions used to study of condensed phase systems will be presented, with emphasis on biologically relevant systems. This overview will include information on the optimization of these models and address future developments in the field.

1.

Empirical Force Fields

Potential energy functions used for condensed phase simulation studies are comprised of simple functions to relate the structure, R, to the energy, V , of the system. An example of such a function is shown in Eqs. (1)–(3). The total potential V (R)total = V (R)internal + V (R)external V (R)internal =

K b (b − b0 )2 +

bonds

+

(1) K θ (θ − θ0 )2

angles

K χ (1 + cos (nχ − δ))

dihedrals


(2)

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A.D. MacKerell V (R)external





Rmin,ij εij  = rij nonbonded

12

−

Rmin,ij ri j

6 



qq + i j  ε D rij

atompairs

(3) energy, V (R)total, is separated into internal or intramolecular terms, V (R)internal and external, V (R)external terms. The latter are also referred to as intermolecular or nonbonded terms. While interatomic interactions are dominated by external terms, the internal terms also make a significant contribution to condensed phase properties, requiring their consideration in this chapter [2]. Furthermore, it is not just the potential energy function alone that is required for determination of the energy as a function of the structure, but the parameters in Eqs. (2) and (3) are also needed. The combination of the potential energy function along with the parameters is termed an empirical force field. Application of an empirical force field to a chemical system of interest, in combination with numerical approaches allowing for sampling of relevant conformations via, e.g., a molecular dynamics simulation (MD) [3] (see below), can be used to predict a variety of structural and thermodynamic properties via statistical mechanics [4]. Importantly, such approaches allow for comparisons with experimental thermodynamic data and the atomic details of interatomic interactions between molecules that dictate the thermodynamic properties can be obtained. Such atomic details are often difficult to access via experimental approaches, motivating the application of computational approaches. Equations (2) and (3) represent a compromise between simplicity and chemical accuracy. The structure or geometry of a molecule is simply represented by four terms, as shown in Fig. 1. The intramolecular geometry is based on bond lengths, b, valence angles, θ, and dihedral or torsion angles, χ, that describe the orientation of 1,4 atoms (i.e., atoms connected by 3 covalent bonds). Additional internal terms may be included in a potential energy function, as described elsewhere [5, 6]. The bond stretching and angle bending terms are treated harmonically; bond and angle parameters include b0 and θ0 , the equilibrium bond length and equilibrium angle, respectively, and K b and K θ are the force constants associated with the bond and angle terms, respectively. The use of harmonic terms for the bond and valence angles is typically sufficient for molecular distortions near ambient temperatures and in the absence of bond breaking or making events, due the bonds and angles staying close to their equilibrium values at room temperature. Dihedral or torsion angles represent the rotations that occur about a bond. These terms are oscillatory in nature (e.g., rotation about the central carbon– carbon bond in ethane changes the structure from a low energy staggered conformation, to a high energy eclipsed conformation, back to a low energy staggered conformation and so on), requiring the use of a sinusoidal function to accurately model them. The dihedral angle parameters (Eq. (2)) include the

Interatomic potentials: molecules

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Figure 1. Schematic diagram of the terms used to describe the structure of molecules in empirical force fields. Internal or intramolecular terms include bond lengths, b, valence angles, θ, and dihedral or torsion angles, χ. For the intermolecular interactions only the distance between atoms i and j, rij , is required.

force constant, Kχ , the periodicity or multiplicity, n, and the phase, δ. The magnitude of Kχ dictates the height of the barrier to rotation, such that Kχ associated with a double bond would be significantly larger that that for a single bond. The multiplicity, n, indicates the number of cycles per 360◦ rotation about the dihedral. In the case of an sp3–sp3 bond, as in ethane, n would equal three, while an sp2–sp2 C=C double bond would have n equal to two. The phase, δ, dictates the location of the maxima in the dihedral energy surface allowing for the location of the minima for a dihedral with n = 2 to be shifted from 0◦ to 90◦ and so on. Typically, δ is equal to 0 or 180, although recent extensions allow any value from 0 to 360 to be assigned to δ ◦ [7]. Each dihedral angle in a molecule may be treated with a sum of dihedral terms that have different multiplicities, as well as force constants and phases. The use of a summation of dihedral terms for a single torsion angle, a fourier series, greatly enhances the flexibility of the dihedral term allowing for more accurate reproduction of experimental and quantum mechanical (QM) energetic target data. Equation (3) describes the intermolecular, external or nonbond interaction terms which are dependent on the distance, rij , between two atoms i and j (Fig. 1). As stated above, these terms dominate the interactions between molecules and, accordingly, condensed phase properties. Intermolecular interactions are also important for the structure of biological macromolecules

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due to the large number of interactions that occur between different regions of biological polymers that dictate their 3D conformation (e.g., hydrogen bonds between Watson–Crick base pairs in DNA or between peptide bonds in α-helicies or β-sheets in proteins). Parameters associated with the external terms are the well depth, εij , between atoms i and j, the minimum interaction radius, Rmin,i j , and the partial atomic charge, qi . The dielectric constant, ε D , is generally treated as equal to one, the permittivity of vacuum, although exceptions do exist when implicit solvent models are used to treat the condensed phase environment [8]. The first term in Eq. (3) is used to treat the van der Waals (vdW) interactions. The particular form in Eq. (3) is referred to as the Lennard–Jones (LJ) 6-12 term. The 1/r 12 term represents the exchangerepulsion between atoms associated with overlap of the electron clouds of the individual atoms (i.e., the Pauli exclusion principle). The strong distance dependence of the repulsion is indicated by the 12th power of this term. Representing London’s dispersion interactions or instantaneous-dipole induceddipole interactions is the 1/r 6 term, which is negative indicating its favorable nature. In the LJ 6-12 equation there are two parameters; the well depth, εij , dictating the magnitude of the favorable London’s dispersion interactions between two atoms, i, j, and Rmin ,ij is the distance between atoms i and j at which the minimum LJ interaction energy occurs; the latter is related to the vdW radius of an atom. Typically, εij and Rmin ,ij are not determined for every possible interaction pair, i, j. Instead εi and Rmin,i parameters are determined for the individual atom types (e.g., sp2 carbon vs sp3 carbon) and then combining rules are used to create the ij cross terms. These combining rules are generally quite simple being either the arithmetic mean (i.e., Rmin,ij = (Rmin,i + √ Rmin, j )/2) or the geometric mean (i.e., εij = ( εi · ε j )), although other variations exist [9]. The use of combining rules greatly simplifies the determination of the εi and Rmin,i parameters. In special cases the force field can be supplemented by specific i, j LJ parameters, referred to as off-diagnol terms, to treat interactions between specific atom types that are poorly modeled by the use of combining rules. There are several commonly used alternate forms for treatment of the vdW interactions. The three primary alternatives to the LJ 6-12 term included in Eq. (3) are designed to “soften” the repulsive wall associated with Pauli exclusion, yielding better agreement with high-level QM data [9]. For example, the Buckingham potential [10] uses an exponential term to treat repulsion while a buffered 14-7 term is used in the MMFF force field [11–13]. A simple alternative is to replace the r 12 repulsion with an r 9 term. The final term contributing to the intermolecular interactions is the electrostatic or Coulombic term. This term involves the interaction between partial atomic charges, qi and q j , on atoms i and j divided by the distance, rij , between those atoms with the appropriate dielectric constant taken into account. Use of a charge representation for the individual atoms, or monopoles,


513

effectively includes all higher order electronic interactions, such as dipoles and quadrapoles. As will be discussed below, the majority of force fields treat the partial atomic charges as static in nature, due to computational considerations. These are referred to as non-polarizable or additive force fields. Finally, the use of a dielectric constant, ε, of one is appropriate when the condensed phase environment is treated explicitly (i.e., use of explicit water molecules to treat an aqueous condensed phase). Combined, the Lennard–Jones and Coulombic interactions have been shown to produce an accurate representation of the interaction between molecules, including both the distance and angle dependencies of hydrogen bonds [14]. This success has allowed for the omission of explicit terms to treat hydrogen bonding from the majority of empirical force fields. It is important to emphasize that the LJ and electrostatic parameters are highly correlated, such that LJ parameters determined for a set of partial atomic charges will not be applicable to another set of charges. In addition, the values of the internal parameters are dependent on the external parameters. For example, the barrier to rotation about the C–C bond in ethane includes electrostatic and vdW interactions between the hydrogens as well as contributions from the bond, angle and dihedral terms. Accordingly, if the LJ parameters or charges are changed, the internal parameters will have to be adjusted to reproduce the correct energy barrier. Finally, condensed phase properties obtained from empirical force field calculations contain contributions for the conformations of the molecules being studied as well as interatomic interactions between those molecules, emphasizing the importance of both internal and external portions of the force field for accurate condensed phase simulations.

2.

Parameter Optimization

Due to the simplicity of the potential energy function used in empirical force fields it is essential that the parameters in the function be optimized allowing for the force field to yield accurate results as judged by their quality in reproducing the experimental regimen. Parameter optimization is based on reproducing a set of target data. The target data may be obtained from QM calculations or experimental data. QM data is generally readily accessible for most molecules; however, limitations in QM level of theory, especially with respect to the treatment of dispersion interactions [15, 16], require the use of experimental data when available [6]. In the rest of this article, we will focus on intermolecular parameter optimization due to their dominant role in the interactions between molecules. Readers can obtain information on the optimization of internal parameters elsewhere [5, 11–13, 16, 17]. A large number of studies have focused on the determination of the electrostatic parameters; the partial atomic charges, qi . The most common charge

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determination methods are the supramolecular and QM electrostatic potential (ESP) approaches. Other variations include bond charge increments [19, 20] and electronegativity equilization methods [21]. An important consideration with the determination of partial atomic charges, related to the Coulombic treatment of electrostatics in Eq. (3), is the omission of explicit electronic polarizability or induction. Thus, it is necessary for static charges to reproduce the polarization that occurs in the condensed phase. To do this, the partial atomic charges of a molecule are “enhanced” leading to an overestimation of the dipole moment as compared to the gas phase value, yielding an implicitly polarized model. For example, many of the water models used in additive empirical force fields (e.g., TIP3P, TIP4P, SPC) have dipole moments in the vicinity of 2.2 debeye [22], vs. the gas phase value of 1.85 debeye for water. Such implicit polarizability allows for additive empirical force fields based on Eq. (3) to reproduce a wide variety of condensed phase properties [23]. However, such models are limited when treating molecules in environments of significantly different polar character. Determination of partial atomic charges via the supramolecular approach is used in the OPLS [24, 25] and CHARMM [26–29] force fields. In this approach, the charges are optimized to reproduce QM determined minimum interaction energies and geometries of a model compound with, typically, individual water molecules or for model compound dimers. Historically, the HF/6-31G* level of theory was used for the QM calculations. This level typically overestimates dipole moments [30], thereby approximating the influence of the condensed phase on the obtained charge distribution leading to the implicitly polarizable model. In addition, the supramolecular approach implicitly includes local polarization effects due to the charge induction caused by the two interacting molecules, facilitating determination of charge distributions appropriate for the condensed phase. With CHARMM it was found that an additional scaling of the QM interaction energies prior to charge fitting was necessary to obtain the correct implicit polarization for accurate condensed phase studies of polar neutral molecules [31]. Even though recent studies have shown that QM methods can accurately reproduce gas phase experimental interaction energies for a range of model compound dimers [32, 33], it is important to maintain the QM level of theory that was historically used for a particular force field when extending that force field to novel molecules. This assures that the balance of the nonbond interactions between different molecules in the system being studied is maintained. Finally, an advantage of charges obtained from the supramolecular approach is that they are generally developed for functional groups, such that they may be transferred between molecules allowing for charge assignment to novel molecules to readily be performed. ESP charge fitting methods are based on the adjustment of charges to reproduce a QM determined ESP mapped onto a grid surrounding the model


515

compound. Such methods are convenient and a number of charge fitting methods based on this approach have been developed [34–38]. However, there are limitations in ESP fitting methods. First, the ability to unambiguously fit charges to an ESP is not trivial [37] and charges on “buried” atoms (e.g., a carbon to which three or four nonhydrogen atoms are covalently bound) tend to be underdetermined, requiring the use of restraints during fitting [36]. The latter method is referred to as Restrained ESP (RESP). Third, since the charges are based on a gas phase QM wave function, they may not necessarily be consistent with the condensed phase, although recent developments are addressing this limitation [39]. Finally, considerations of multiple conformations of a molecule, for which different charge distributions typically exist, must be taken into account [30]. It should be noted that the last two problems must also be considered when using the supramolecular approach. As with the supramolecular approach, the QM level of theory was often the HF/6-31G*, as in the AMBER force fields [41], due to that level typically overestimating the dipole moment. More recently, higher level QM calculations have been applied in conjunction with the RESP approach [42], although their ability to reproduce condensed phase thermodynamic properties has not been tested. Clearly, both the supramolecular and ESP methods are useful for the determination of partial atomic charges. Which one is used, therefore, should be based on compatibility with that used for the remainder of the force field being applied. Accurate optimization of the LJ parameters is one of the most important aspects in the development of a force field for condensed phase simulations. Due to limitations in QM methods for the determination of dispersion interactions, optimization of LJ parameters is dominated by the reproduction of thermodynamics properties in condensed phase simulations, generally neat liquids [43, 44]. Typically, the LJ parameters for a model compound are optimized to reproduce experimentally measured values such as heats of vaporization, densities, isocompressibilities and heat capacities. Alternatively, heats or free energies of aqueous solvation, partial molar volumes or heats of sublimation and lattice geometries of crystals [45, 46] can be used as the target data. These methods have been applied extensively for development of the force fields associated with the programs AMBER, CHARMM, and OPLS. However, it should be noted that LJ parameters are typically underdetermined due to only a few experimental observations being available for the optimization of a significantly larger number of LJ parameters. This enhances the parameter correlation problem where LJ parameters for different atoms in a molecule (e.g., H and C in ethane) can compensate for each other such that it is difficult to accurately determine the “correct” LJ parameters of a molecule based on reproduction of condensed phase properties alone [5]. To overcome this approach a method has been developed that determines the relative value of the LJ parameters based on high level QM data [47] and the absolute values

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based on the reproduction of experimental data [16, 49]. This approach is tedious as it requires supramolecular interactions involving rare gases; however, once satisfactory LJ parameters are optimized for atoms in a class of functional groups they can often be directly transferred to other molecules with those functional groups without further optimization.

3.

Considerations for Condensed Phase Simulations

Proper application of an empirical force field is obviously essential for success of a condensed phase calculation. An important consideration is the inclusion of all nonbond interactions between all atom-atom pairs For the electrostatic interactions this can be achieved via Ewald methods [49], including the particle Mesh Ewald approach [50], for periodic systems while reaction field methods can be used to simulation finite (e.g., spherical) systems [51– 53]. For the LJ interactions, long-range corrections exist that treat the interactions beyond the atom-atom truncation distance (i.e., those beyond a distance were the atom–atom interactions are calculated explicitly) as homogenous in nature [54, 55]. Another important consideration is the use of integrators that generate proper ensembles in MD simulations, allowing for direct comparison with experimental data [3, 57–60]. In addition, a number of methods are available to increase the sampling of conformational space [60–62]. The available and proper use of these different methods greatly facilitates investigations of molecular interactions via condensed phase simulations.

4.

Available Empirical Force Fields

A variety of empirical force fields have been developed. Force fields that focus on biological molecules include AMBER [18, 42] CHARMM [26–29], GROMOS [63, 64], and OPLS [24, 25], All of these force fields have been parametrized to account for condensed phase properties, such that they all treat molecular interactions with a reasonably high level of accuracy [65, 66]. However, these force fields, to varying extents, do not treat the full range of pharmaceutically relevant compounds. Force fields designed for a broad range of compounds include MMFF [11–13, 67], CVFF [17, 68], the commercial CHARMm force field [69], CFF [70], COMPASS [71], the MM2/MM3/MM4 series [72–74], UFF [75], Drediing [76], the Tripos force field (Tripos, Inc.), among others. However, these force fields have been designed primarily to reproduce internal geometries, vibrations and conformational energies, often sacrificing the quality of the nonbond interactions [65]. Exceptions are MMFF and COMPASS where nonbond parameters have been investigated at a reasonable level of detail. With all force fields the user is advised to perform tests


517

on molecules for which experimental data is available to validate the quality of the model.

5.

Electronic Polarizability

Future improvements in the treatment of interatomic interactions between molecules will be based on the extension of the treatment of electrostatics to include explicit treatment of electronic polarizability [77, 78]. There are several methods by which electronic polarizability may be included in a potential energy function. These include fluctuating charge models [79–85], induced dipole models [85–89], or a combination of those methods [90, 91]. The classic Drude oscillator is an alternative method [92, 93] in which a “Drude” particle is attached to the nucleus of each atom and, by applying the appropriate charges to the atoms and “Drude” particles, the polarization response can be modeled. This method is also referred to as the shell model and has only been used in a few studies thus far [94–96]. In all of these approaches, the polarizability is solved analytically, iteratively or, in the case of MD simulations via extended Lagrangian methods [3, 77]. In extended Lagrangian methods the polarizability is treated as a dynamic variable in MD simulations. Extended Lagrangian methods are important for the inclusion of polarizability in empirical force fields as they offer the necessary computational efficiency to perform simulations on large systems. To date, work on water has dominated the application of polarizable force fields to molecular interactions. Polarizable water models have been shown to accurately treat both the gas and condensed phase properties [78, 86–89, 95, 97–99]. The ability to treat both the gas and condensed phases accurately marks a significant improvement over force fields where polarizability is not included explicitly. Other examples, where the inclusion of electronic polarization has been shown to increase the accuracy of the treatment of molecular interactions includes the solvation of ions [79, 85, 100, 101], ion-pair interactions in micellar systems [102], condensed phase properties of a variety of small molecules [78, 83, 103–107], cation–π interactions [103, 104], and in interfacial systems [108]. With respect to biological macromolecules, only a few successful applications have been made thus far [109–111]. Thus, explicit treatment of electronic polarizability in empirical force fields, although computationally more expensive then nonpolarizable models, is anticipated to make a significant contribution to the understanding molecular interactions at an atomic level of detail. An interesting observation with electronic polarizability is the apparent inability to apply gas phase polarizabilities to condensed phase systems, as evidenced in studies on water [95]. This phenomenom appears to be associated with the Pauli exclusion principle such that the deformability of the electron

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cloud due to induction by the environment is hindered by the presence of adjacent molecules in the condensed phase [112]. This would lead to a decreased effective polarizability in the condensed phase. Such a phenomena has more recently been observed in QM studies of water clusters [113]. Further studies are required to better understand this phenomenon and properly treat it in empirical force fields.

6.

Summary

Interatomic interactions involving molecules dominate the properties of condensed phase systems. Due to the number of particles in such systems, it is typically necessary to apply computationally efficient empirical force fields to study them via theoretical methods. The success of empirical force field is based, in large part, on their accuracy in reproducing a variety of experimental observations; the accuracy being dictated by the quality of the optimization of the parameters that comprise the empirical force field. Proper optimization requires careful selection of target data as well as use of the appropriate optimization process. In cases where empirical force field parameters are being developed as an extension of an available force field, the optimization strategy must be selected to insure consistency with the previous parameterized molecules. These considerations will maximize the potential that the atomistic details obtained from condensed phase simulations will be representative of the experimental regimen. Finally, when analyzing results from condensed phase simulations, possible biases due to the parameters themselves must be considered when interpreting the data.

Acknowledgments Financial support from the NIH (GM51501) and the University of Maryland, School of Pharmacy, Computer-Aided Drug Design Center is acknowledged.

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2.6 INTERATOMIC POTENTIALS: FERROELECTRICS Marcelo Sepliarsky1, Marcelo G. Stachiotti1 , and Simon R. Phillpot2 1

Instituto de Física Rosario, Facultad de Ciencias Exactas, Ingeniería y Agrimensura, Universidad Nacional de Rosario, 27 de Febreo 210 Bis, (2000) Rosario, Argentina 2 Department of Materials Science and Engineering, University of Florida, Gainesville, FL 32611, USA

Ferroelectric perovskites are important in many areas of modern technology including memories, sensors and electronic applications, and are of fundamental scientific interest. The fascinating feature of perovskites is that they exhibit a wide variety of structural phase transitions. Generically these compounds have a chemical formula ABO3 , where A is a monovalent or divalent cation and B, a transition metal cation; perovskites in which both A and B are trivalent, such as LaAlO3 also exist, though we will not discuss them here. Although their high-temperature structure is very simple (Fig. 1), it displays a wide variety of structural instabilities, which may involve rotation and distortions of the oxygen octahedral as well as displacement of the ions from their crystallographically defined sites. The types of crystal symmetries manifested in these materials and the types of phase transitions behavior depend on the individual compound. Among the perovskites one finds ferroelectric crystals such as BaTiO3 , KNbO3 (displaying three solid-state phase transitions), and PbTiO3 (displaying only one transition), antiferroelectrics such as PbZrO3 , and materials such as SrTiO3 that exhibit other nonpolar instabilities involving the rotation of the oxygen octahedra [1]. In recent years, new applications have opened up for these materials as the systems exploited have become both chemically more complex, e.g., solid solutions and superlattices, and microstructurally more complex, e.g., thin films and nanocapacitors. While the overall properties of such systems can be relatively easily investigated experimentally, it is difficult to obtain microscopic information. There is thus a significant need for a simulation method which can provide atomic-level information on ferroelectric behavior, and yet is computationally efficient enough to allow materials problems to be addressed. Computer 527 S. Yip (ed.), Handbook of Materials Modeling, 527–545. c 2005 Springer. Printed in the Netherlands.

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O

B

Figure 1. Cubic perovskite-type structure, ABO3 .

simulations based on interatomic potentials can provide such microscopic insights. However, the validity of any simulation potential study depends on the quality of the interatomic potential used, to a considerable extent. Obtaining accurate interatomic potentials which are able to describe ferroelectricity in ABO3 perovskites constitutes a challenging problem, mainly due to the small energy differences (sometimes less than 10 meV/cell) involved in the lattice instabilities associated with the various phases. The theoretical investigation of ferroelectric materials can be addressed at different lenght scale and level of complexity, ranging from phe-nomenological theories (based on the continuous medium approximation) to first-principles methods. The traditional approach is based on Ginzburg–Landau–Devonshire (GLD) theory [2]. This mesoscale approach treats a ferroelectric as a continuum solid denned by components of polarization and by elastic strains or stresses. This approach has proved very successful in providing significant insights into the ferroelectric properties of perovskites. However, it cannot provide detailed microscopic information. Over the last decade, considerable progress has been made in first-principles calculations of ferroelectricity in perovskites [3, 4]. These calculations have contributed greatly to the understanding of the origins of structural phase transitions in perovskites and to the nature of the ferroelectric instability. These methods are based upon a full solution for the quantum mechanical ground state of the electron system in the framework of Density Functional Theory (DFT). While able to provide detailed information on the structural, electronic and lattice dynamical properties of single crystals, they also have limitations. In particular, due to the heavy computational load, only systems of up to approximately a hundred ions can be simulated. Moreover, at the moment such calculations cannot provide anything but static, zero temperature, properties. An effective Hamiltonian method has been used for the simulation of finite-temperature properties of

Interatomic potentials: ferroelectrics

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perovskites [3]. Here, a model Hamiltonian is written as a function of a reduced number of degrees of freedom (a local mode amplitude vector and a local strain tensor). The parameters of the Hamiltonian are determined in order to reproduce the spectrum of low-energy excitations of a given material as obtained from first-principles calculations. This approach has been applied with considerable success to several ferroelectric materials (pure compounds and solid solutions), producing results in very good qualitative agreement with experiments. However, some quantitative predictions are not so satisfactory; in particular, the calculated transition temperatures can differ from the experimental values by hundreds of degrees. Moreover, the lack of an atomistic description of the material makes the effective Hamiltonian approach inappropriate for the investigation of many interesting properties of perovskites, such as surface and interface effects. Atomistic modeling using interatomic potentials has a long and illustrious history in the description of ionic materials. The fundamental idea is to describe a material at the atomic level, with the interatomic interactions defined by classical potentials, thereby providing spatially much more detailed information than the GLD approach, yet without the heavy computational load associated with the first-principles methods. In the context of ionic materials, the interactions between the point ions are generally described via the Coulombic interactions between the atoms which provides cohesion. However, a neutral solid interacting purely by Coulombic interactions is unstable to a catastrophic collapse in which all the ions become arbitrarily close. Thus, to mimic the physical short-ranged repulsion that prevents such a collapse, an empirical largely repulsive interaction is added. One standard choice for this function is the Buckingham potential, which consists of a purely repulsive, exponential decaying Born–Mayer term between shells and a van der Waals attractive term to account for covalency effects: V (r) = ae(−r/ρ) − (c/r 6 ). This is the so-called rigid ion model. In the shell model, an important improvement over the rigid-ion model, atomic polarizability is accounted for by defining a core and a shell for each ion (representing the ion core with the closed shells of electrons, and the valence electrons, respectively), which interact with each other through a harmonic spring (characterizing the ionic polarizability), and interact with the cores and shells of other ions via repulsive and Coulombic interactions. In some parameterizations, the ions (core plus shell) are assigned their formal charges. However, in ionic materials with a significant amount of covalency, such as perovskites, the incomplete transfer of electrons between the cations and anions can be accounted for by assigning partial charges (smaller than the formal charges) to the ions as well as the van der Waals term, which is non-zero only for the O–O interactions. For more details see the article “Interatomic potential models for ionic materials” by Julian Gale presented in this handbook.

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The success of the atomistic approach is evident from the large number of investigations on complex oxides crystals. Regarding ferroelectric perovskites, we note the early work of Lewis and Catlow, who derived empirical shellmodel potential parameters for the study of defect energies in cubic BaTiO3 [5, 18]. This model was subsequently used for more refined ab initio embeddedcluster calculations of impurities, as well as for the simulation of surface properties. For lattice dynamical properties, the most successful approach has been carried out in the framework of the nonlinear oxygen polarizability model [6]. In this shell model an anisotropic core–shell interaction is considered at the O2− ions, with a fourth-order core–shell interaction along the B–O bond. The potential parameters were obtained by fitting experimental phonon dispersion curves of the cubic phase. The main achievement of this model was the description of the soft mode temperature dependence (TO-phonon softening which is related with the ferroelectric transition). However, neither of these models, was able to simulate the ferroelectric phase behavior of the perovskites. Besides the traditional empirical approach, in which potentials are obtained by suitable fitting procedures to macroscopic physical properties, there is increasing interest in deriving pair potentials from first-principles calculations. In 1994, Donnerberg and Exner developed a shell model for KNbO3 , deriving the Nb–O short-range pair potential from Hartree–Fock calculations performed on a cluster of ions [7]. They showed that this ab initio pair potential was in good agreement with a corresponding empirical potential obtained from fitting procedures to macroscopic properties. Their model, however, was not able to simulate the structural phase transition sequence of KNbO3 either. They argued that the consideration of additional many-body potential contributions would enable them to model structural phase transitions. However, as we will see, it is in fact possible to simulate ferroelectric phase transitions just by using classical pairwise interatomic potentials fitted to first-principles calculations. Ab initio methods provide underlying potential surfaces and phonon dispersion curves at T = OK, thereby exposing the presence of structural instabilities in the full Bril-louin zone, and this information is indeed very useful for parameterizing classical potentials which can then be used in molecular dynamics simulations. In this way, finite-temperature simulations of ABO3 perovskites and the properties of chemically and microstructurally more complex systems can be addressed at the atomic level.

1.

Modeling Ferroelectric Perovskites

Among the perovskites BaTiO3 which can be considered as a prototypical ferroelectric is one of the most exhaustively studied [8]. At high temperatures, it has the classic perovskite structure. This is cubic centrosymmetric, with the


531

Ba at the corners, Ti at the center, and oxygen at the face centers (see Fig. 1). However, as the temperature is lowered, it goes through a succession of ferroelectric phases with spontaneous polarizations along the [001], [011], and [111] directions of the cubic cell. These polarizations arise from net displacements of the cations with respect to the oxygen octahedra along the above directions. Each ferroelectric phase involves also a small homogeneous deformation which can be thought of as an elongation of the cubic unit cell along the corresponding polarization direction. Thus the system becomes tetragonal at 393 K, orthorhombic at 278 K, and rhombohedral at 183 K. An anisotropic shell model with pairwise repulsive Buckingham potentials was developed for the simulation of ferroelectricity in BaTiO3 [9]. This model is a classical shell model where an anisotropic core–shell interaction is considered at the O2− ions, with a fourth-order core–shell interaction along the O–Ti bond. The Ba and Ti ions are considered to be isotropically polarizable. The set of seventeen shell model parameters were obtained by fitting phonon frequencies, lattice constant of the cubic phase, and underlying potential surfaces for various configurations of atomic displacements. In order to better quantify the ferroelectric instabilities of the cubic phase, a first-principles frozen-phonon calculation of the infrared active modes was performed. Once the eigenvectors at had been determined, the total energy as a function of the displacement pattern of the unstable mode was evaluated for different directions in the cubic phase, including also the effects of the strain. The first-principles total energy calculations were performed within DFT, using the highly precise full-potential Linear Augmented Plane Wave (LAPW) method. The energy surfaces of the model for different ferroelectric distortions is shown in Fig. 2, where they are compared with the first-principles results. A satisfactory overall agreement is achieved. The model yields clear ferroelectric instabilities with similar energies and minima locations as the LAPW calculations. Energy lowerings of ≈1.2, 1.65, and 1.9 mRy/cell are obtained for the (001), (011), and (111) ferroelectric mode displacements, respectively, which is consistent with the experimentally observed phase transitions sequence. Concerning the energetics for the (001) displacements, it can be also seen in the left panel that the effect of the tetragonal strain is to stabilize these displacements with a deeper minimum and with a higher energy barrier at the centrosymmetric positions. Phonon dispersion relations provide a global view of the harmonic energy surface around the cubic perovskite structure. In particular the unstable modes, which have imaginary frequencies, determine the nature of the phase transitions. A first-principles linear response calculation of the phonon dispersion curves of cubic BaTiO3 revealed the presence of structural instabilities with pronounced two-dimensional character in the Brillouin zone, corresponding to chains of displaced Ti ions oriented along the [001] directions [10]. The shell model reproduces these instabilities is illustrated in the calculated phonon

532

M. Sepliarsky et al. 1

E (mRy/cell)

[001]

[111]

[011]

0

1

2 0.00

c/ a =

0.05

1.01

0.00

0.05

0.00

0.05

Ti relative to Ba displacement (Å) Figure 2. Total energy as a function of the unstable mode displacements along the [001] (left panel), [011] (center panel), and [111] (right panel) directions. For the sake of simplicity, the mode displacement is represented through the Ti displacement relative to Ba; the oxygen ions are also displaced in a manner determined by the Ti ion displacement. Energies for [001] displacements in a tetragonal strained structure are also included in the left panel. First-principles calculations are denoted by squares (circles) for the unstrained (strained) structures. Full lines correspond to the shell model result.

dispersion curves in Fig. 3. Excellent agreement with the ab initio linear response calculation is achieved, particularly for the unstable phonon modes. Two transverse optic modes are unstable at the point, and they remain unstable along the –X direction with very little dispersion. One of them stabilizes along the –M and X–M directions; and both become stable along the –R and R–M lines. The Born effective charge tensor is conventionally defined as the proportionality coefficients between the components of the dipole moment per unit cell and the components of the κ sublattice displacement which give rise to the dipole moment ∗ = Z κ,αβ

∂ Pβ . ∂δκ,α

(1)

For the cubic structure of ABO3 perovskites, this tensor is fully characterized by four independent numbers. Experimental data had suggested that the amplitude of the Born effective charges should deviate substantially from the nominal static charges, with two essential features: the oxygen charge tensor is highly anisotropic (with two inequivalent directions either parallel or perpendicular to the B–O bond), and the Ti and O|| effective charges are anomalously large. This was confirmed by more recent first-principles calculations [3] demonstrating the crucial role played by the B(d)–O(2p) hybridization as a dominant mechanism for such anomalous contributions.


533

800

Frequency (cm-1 )

600

400

200

0

200 Γ

X

M

Γ

R

M

Figure 3. Phonon dispersion curves of cubic BaTiO3 calculated with the shell model. Imaginary phonon frequencies are represented as negative values.

Although the shell model does not explicitly include charge transfer between atoms, it takes into account the contribution of the electronic polarizability effects through the shell model. It is thus possible to evaluate the Born effective charge tensor by calculating the total dipole moment per unit cell created by the displacement of a given sublattice of atoms as a sum of two contributions Pα = Z κ δκ,α +

Yκ wκ,α .

(2)

κ

The first term is the sublattice displacement contribution while the second term is the electronic polarizability contribution. The calculated Born effective charges for cubic BaTiO3 are listed in Table 1 together with results obtained from different theoretical approaches. The two essential features of the Born effective charge tensor of BaTiO3 are satisfactorily simulated. To this point, we have shown that this anisotropic shell model for BaTiO3 reproduces the lattice instabilities and several zero-temperature properties which are relevant for this material. To investigate if the model can describe the temperature driven structural transitions of BaTiO3 constant-pressure molecular dynamics (MD) simulations were performed. Although an excellent overall agreement was obtained for the structural parameters, showing that the model reproduces the delicate structural changes involved along the transitions, the theoretically determined transition temperatures were much lower

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M. Sepliarsky et al. Table 1. Born effective charges of BaTiO3 in the cubic structure

Nominal Experiment First principles Shell model (nominal) Shell model (effective)

Z ∗Ba

Z T∗ i

Z ∗O

+2 +2.9 +2.75 +1.86 +1.93

+4 +6.7 +7.16 +3.18 +6.45

−2 −2.4 −2.11 −1.68 −2.3

⊥

Z ∗O

||

−2 −4.8 −5.69 −1.68 −3.79

than in experiment [9]. Interestingly, the effective Hamiltonian approach presents the same problem. Since ferroelectricity is very sensitive to volume, the neglect of thermal expansivity in the effective Hamiltonian approach was thought to be responsible for the shifts in the predicted transition temperatures. The MD simulations, however properly simulate the thermal expansion and, nevertheless, result in a similar anomaly in the transition temperatures. This indicates the presence of inherent errors in the first-principles LDA approach which tend to underestimate the ferroelectric instabilities. A recent study demonstrated that, in the effective Hamiltonian approach, there are at least two significant sources of errors: the improper treatment of the thermal expansion and the LDA error. Both types of errors may be of same magnitude [11]. While the anisotropic shell model for BaTiO3 does have the desired effect of describing the ferroelectric phase transition in perovskites it can only be used in crystallographic well-defined environment of O ions. Unfortunately, it is not always possible to unambiguously characterize the crystallographic environment of any given ion, for example, in the simulation of a grain boundary or other interface. For such systems isotropic models are required. Isotropic shell models have recently been developed, which describe the phase behavior of both KNbO3 [12] and BaTiO3 [13]. The isotropic shell model differs from the anisotropic one only in that the anisotropic fourthorder core–shell interaction on the O ions is replaced by an isotropic fourthorder core–shell interaction on both the transition metal and the O ions, which together stabilize the ferroelectric phases. Since the LDA-fitted shell model gives theoretically determined transition temperatures much lower than in experiment, the parameters of the potential were improved in an ad hoc manner to give better agreement. In this way, the model for KNbO3 displays the experimentally observed sequence of phases on heating: rhombohedral, orthorhombic, tetragonal and finally cubic with transition temperatures of 225 K, 475 K and 675 K, which are very close to the experimental values of 210 K, 488 K and 701 K, respectively. As shown in Fig. 4, for BaTiO3 , in comparison with the anisotropic model, the isotropic shell model gives transition temperature values (140 K, 190 K and 360 K) in better agreement with the experimental values (183 K, 278 K and 393 K).

Interatomic potentials: ferroelectrics BaTiO3

4.08

Lattice parameters (Å)

535

4.04

4

0

100

200

300

400

100

200

300

400

30

2

Polarization (µC/cm )

BaTiO3 20

10

0 0

Temperature (K) Figure 4. Phase diagram of BaTiO3 as determined by MD simulations for the isotropic shell model. Top panel: cell parameters as a function of temperature. Bottom panel: the three components of the average polarization (each one represented with a different symbol).

2.

Solid Solutions

The current keen interest in solid solutions of perovskites is driven by the idea of tuning the composition to create structures with properties unachievable in single component materials. Prototypical solid solutions are Bax Sr1−x TiO3 (BST), a solid solution of BaTiO3 and SrTiO3 , and KTax Nb1−x O3 , a solid solution of KTaO3 and KNbO3 . Both solutions exist for the whole concentration range and are mixtures of a ferroelectric with an incipient ferroelectric. We present briefly the main features of isotropic shell-model potentials developed to describe the structural behavior of BST.

536


In order to simulate BST solid solutions, it was also necessary to develop an isotropic model for SrTiO3 . From a computational point of view, the SrTiOs model must be compatible with the BaTiO3 model in that the only difference between the two can be in the different Ba–O and Sr–O interactions and the different polarizability parameters for Ba and Sr. The challenge is thus, by only changing these interactions, to reproduce the following main features of ST: (i) a smaller equilibrium volume, (ii) incipient ferroelectricity, and (iii) a tetragonal antiferrodistortive ground state. It is indeed possible to reproduce these three critical features. The equilibrium lattice constant of the resulting model in the cubic phase is a = 3.90 Å which reproduces the extrapolation to T = 0 K of the experimental lattice constant. Regarding the other two conditions, the low-frequency phonon dispersion curves of the cubic structure are shown in Fig. 5. The model reproduces the rather subtle antiferrodistortive instabilities, driven by the unstable modes at the R and M points. It also presents a subtle ferroelectric instability (unstable mode at the zone center). These detailed features of the dispersion of the unstable modes along different direction in the Brillouin zone are in good agreement with ab initio linear response calculations. Random solid solutions of BST of various compositions in the range x = 0 (pure SrTiO3 ) to x = 1 (pure BaTiO3 ) have been simulated. In the simulation supercell the A-sites of the ATiO3 perovskite are randomly occupied by Ba and Sr ions. The results of the molecular dynamics simulations on the phase behavior of BST are summarized in Fig. 6 (filled symbols connected by solid lines) as the concentration dependence of the transition temperatures.

Figure 5. Low-frequency phonon dispersion curves for cubic SrTiO3 . The negative values correspond to imaginary frequencies, characteristic of the ferroelectric instability at the point and the additional antiferrodistortive instabilities at the R and M points.


537

400 Ba xSr1- x TiO 3 Cubic

l

na

300

o ag

tr

T (K)

Te 200

Orthorhombic 100 Rhombohedral 0

0

0.2

0.6

0.4

0.8

1

x Figure 6. Concentration dependence of transition temperatures (solid symbols and dark lines) shows good agreement with experimental values (open symbols and dotted lines).

With increasing concentration of Sr (i.e., decreasing x), the Curie temperature decreases essentially linearly with x. The simulations showed that all four phases remain stable down to x ≈ 0.2 at which the three transition temperatures essentially coincide. Below x ≈ 0.2 only the cubic and rhombohedral phases appear in the phase diagram. These results are similar to the experimental data (open symbols and dotted lines), giving particularly good agreement for the concentration at which the tetragonal and orthorhombic phases disappear from the phase diagram. The above analyses demonstrate that the atomistic approach can reproduce the basic features of the phase behavior of perovskite solid solutions, on a semiquantitative basis. There are two fundamental structural effects associated with the solid solution: a concentration dependence of the average volume and large variations in the local strain arising from strong variations in the local composition [12, 13]. SrTiO3 is denser than BaTiO3 . Thus in the solid solution, the SrTiO3 cells tend to be under a tensile strain (which tends to encourage a ferroelectric distortion) while the BaTiO3 cells tend to be under a compressive strain (which tends to suppress the ferroelectric distortion). Indeed, the large tensile strain on the SrTiO3 cells has the effect of inducing a polarization. Remarkably, at a given concentration (fixed volume) the polarization of the SrTiO3

538


cells is actually larger than that of the BaTiO3 cells. There is also an additional effect associated with the local environment of each unit cell. In particular, the simulations show that the maximum and minimum values of polarization for the SrTiO3 cells correspond to the polarizations of SrTiO3 cells (of the same average volume as that of the solid solution) embedded completely in a matrix of SrTiO3 and BaTiO3 cells, respectively. Likewise, for the BaTiO3 cells the maximum and minimum polarizations correspond to SrTiO3 and BaTiO3 embeddings, respectively.

3.

Heterostructures

Superlattices containing ferroelectric offer another approach to achieving dielectric, and optical properties unachievable in the bulk. Among the heterostructures grown have been ferroelectric/paraelectric superlattices including BaTiO3 /SrTiO3 and KNbO3 / KTaO3 and ferroelectric/ferroelectric superlattices PbTiO3 /BaTiO3 . In comparison with the well-documented tunability of the properties of solid solutions, the tunability of the properties of multilayer heterostructures has been less well demonstrated. While there is experimental evidence for a strong dependence of the properties of such superlattices on modulation length, (the thickness of a KNbO3 / KTaO3 bilayer), the underlying physics controlling their properties is only poorly understood. Atomic-level simulations are ideal for the study of multilayers because the simulations can be carried out on the same length scale as the experimental systems. Moreover, the crystallography of the multilayer can be defined and the position of every ion determined, thereby providing atomic-level information on the ferroelectric and dielectric properties. Furthermore, once the nature of the interactions between ions and the crystallographic structure of the interface are defined, the atomic-level simulations will determine the local atomic structure and polarization at the interfaces. To that purpose, the structure and properties of coherent KNbO3 /KTaO3 superlattices were simulated using isotropic shell-model potentials for KNbO3 and KTaO3. Since the simulations were intended to model a superlattice on a KT substrate, as had been experimentally investigated, the in-plane lattice parameter was fixed to that of KT at zero temperature; however since the heterostructure is not under any constraint in the modulation direction, the length of the simulation cell in the z direction was allowed to expand or contract to reach zero stress. Figure 7 shows the variation in the polarization in the modulation direction Pz (solid circles) and in the x–y plane, Px = Py (open circles) averaged over unit-cell-thick slices through the = 36 superlattice. In analyzing these polarization profiles, we first address the strain effects produced by the KT substrate, which result in a compressive strain of 0.7% on the KN layers.


539

40 Pz

2

Polarization (µC/cm )

30 20

P x =P y

10 0 10 20 30

0

9

18

27

36

45

54

63

72

Z Figure 7. Components of polarization, Px (open circles) and Pz (solid circles), in unit-cellthick slices through the = 36 KN/KT superlattice on a KT substrate.

To compensate for this in-plane compression, the KN layers expand in the z direction thereby breaking the strict rhombohedral symmetry of the polarization of KN; however, these strains are not sufficient to force the KN to become tetragonally polarized. Similarly, the absence of any in-plane polarization for the KT layer is consistent with the absence of any strain arising from the KT substrate. The finite value of Pz in the interior of the KT layer, however, is different from the expected value of Pz =0 for this unstrained layer and arises from the very strong coupling of the electric field produces by the electric dipoles in the KNbO3 layers with the very large dielectric response of the KTaO3 [14, 15]. The switching behavior of ferroelectric heterostructures is of considerable interest. It was found that for = 6, the polarization in the KTaO3 layers is almost as large as in the KNbO3 layers; moreover, the coercive fields for the KNbO3 and KTaO3 layers are identical. This single value for the coercive fields and the weak spatial variation in the polarization indicates that the entire superlattice is essentially acting as a single structure, with properties different from either of its components. For = 36, the KNbO3 layer has a square hysteresis loop characteristic of a good ferroelectric; the polarization and coercive field are larger than for = 6, consistent with more bulk-like

540


behavior of a thicker KNbO3 layer. The KTO layer also displays hysteretic behavior. However, by contrast with the = 6 superlattice, the coercive field for the KTaO3 layers is much smaller than for the KNO layer, indicating that the KNbO3 and KTaO3 layers are much more weakly coupled than in the = 6 superlattice. The hysteresis loop for the KTO layers resembles the response of a poor ferroelectric; however, it was shown that it is actually the response of a paraelectric material under the combination of the applied electric field and the internal field produced by the polarized KNbO3 layers. The hysteretic behavior is, therefore, not an intrinsic property of the KTaO3 layer but arises from the switching of the KNbO3 layers under the large external electric field which, in turn, switches the sign of the internal field on the KTaO3 layers.

4.

Nanostructures

The causes of size effects in ferroelectrics are numerous, and it is difficult to separate true size effects from other factors that change with film thickness or capacitor size, such as microstructure, defect chemistry, and electrode interactions. For this reason, atomic-level investigations play a crucial role in determining their intrinsic behavior. The anisotropic shell model for BaTiO3 was used to determine the critical thickness for ferroelectricity in a free-standing BaTiO3 stress-free film (it was also shown that the model developed for the bulk material can also describe static surface properties [16] such as structural relaxations and surface energies, which are in quite good agreement with firstprinciples calculations). For this investigation a [001] TiO2 -terminated slab was chosen. The equilibrated zero-temperature structure of the films was determined by a zero-temperature quench. The size and shape of the simulation cell was allowed to vary to reach zero stress. Shown in the top panel of Fig. 8 is the cell-by-cell polarization profile pz (z) at T = 0 K of a randomly chosen chain perpendicular to the film surface for various film thicknesses. It is clear from this figure that the film of 2.8 nm width does not display ferroelectricity. As a consequence of surface atomic relaxations, the two unit cells nearest to the surface develop a small polarization at both sides of the slab, which are pointing inwards towards the bulk, so the net chain polarization vanishes. For the cases of 3.6 nm and 4.4 nm film thickness, however, the chains develop a net out-of-plane polarization. Although these individual chains display a perpendicular nonvanishing polarization, the net out-of-plane polarization of the film is zero due to the development of stripe-like domains, as is shown in the bottom panel of Fig. 8. It was demonstrated that the strain effect produced by the presence of a substrate can lead to the stabilization of a polydo-main ferroelectric state in films as thin as 2.0 nm [16].


541 d=2.8 nm d=3.6 nm d=4.4 nm

12

2

pz ( µ C/ cm )

9 6 3 0

3 6

0.0

0.8

1.6

2.4

3.2

4.0

z(nm) 2

Pz ( µ C/ cm ) 6 -- 8 4 -- 6 2 -- 4 0 -- 2 2 -- 0 4 -- 2 6 -- 4 8 -- 6

Figure 8. Top panel: Cell-by-cell out-of-plane polarization profile of a ramdomly chosen chain perpendicular to the film surface for different slab thickness. Bottom panel: top view of the out-of-plane polarization pattern for the case d = 4.4 nm showing stripe-like domains. A similar picture is obtained for d = 3.6 nm.

To investigate to what extent a decrease in lateral size will affect the ferroelectric properties of the film, the equilibrium atomic positions and local polarizations at T = 0 K for a stress-free cubic cell of 3.6 nm size were computed. The nanocell is constructed in such a way that the top and bottom faces (perpendicular to the z axis) are [001] TiO2 -planes and its lateral faces (parallel to the z axis) are [100] BaO-planes.

542


Shown in the top panel of Fig. 9 are the cell-by-cell polarization profiles pz (z) for three different chains along the z direction: one chain at an edge of the cell, one at the center of a face, and the last one inside the nanocell. It is clear from this figure that the total chain polarization at the edges and at the lateral faces is zero. The large local polarizations pointing in opposite directions, at both sides of the cell, are just a consequence of strong atomic relaxations at the nanocell surface. On the other hand, the chain inside the nanocell displays

edge face inside film

2

pz ( µ C/ cm )

40 20 0

20 40 0.0

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

z(nm) 2

Pz ( µ C/ cm ) 3 -- 5 1 -- 3 1 -- 1 3 -- 1 5 -- 3

Figure 9. Top panel: cell-by-cell polarization profiles ( pz (z)) of three chosen chains in the nanocell. The profile for the 3.6 nm slab is showed for comparison. Bottom panel: top view of the polarization pattern for the nanocell.


543

a net, nonvanishing, polarization of ≈ 5 µC/cm2 . For comparison we have also plotted in Fig. 9, the pz (z) profile of the stress-free film of 3.6 nm width. We can clearly see that the two profiles are very similar. This is an indication that the decrease in lateral size does not affect the original ferroelectric properties of the thin film. As in the film case, the net polarization of the nanocell is zero due to the development of domains with opposite polarizations, as is shown in the bottom panel of Fig. 9. It was further demonstrated that a nanocell with different lateral faces, TiO2 planes instead of Ba–O planes, present a different domain structure and polarization due to a strong surface effect [17].

5.

Outlook

First-principles calculations of ferroelectric materials can answer some important questions directly, but this approach by itself cannot address the most challenging materials-related and microstructure-related problems. Fortunately, first-principles methods can provide benchmarks for the validation of other conceptually less sophisticated approaches that, because of their low computational loads, can address such issues. The atomistic approach presented here demonstrates that enough of the electronic effects associated with ferroelectricity can be mimicked at the atomic level to allow the fundamentals of ferroelectric behavior to be reproduced. Moreover, the interatomic potential approach, firmly grounded by having its parameters computed on firstprinciples calculations, will be a very useful tool for the theoretical design of new materials for specific target applications. One important challenge in this field is the simulation of technologically important solid solutions which are more complex than the ones discussed here; for example, PbZrx Ti1−x O3 (PZT) and PbMg1/3 Nb2/3 O3 -PbTiO3 (PMNPT), which is a single crystal piezoelectric with giant electromechanical coupling. The difficult point here is the development of interatomic potentials suitable for such investigations. The simultaneous fitting of transferable potentials for the different pure materials is a way to develop interatomic potentials for the solid solutions. This could be done by using an extensive first-principles database to adjust the potential parameters. Although the methodology presented here is computationally efficient enough to allow materials problems to be addressed, clearly there are a lot of work to do in order to get a closer coupling with experiment. Real ferroelectric materials are frequently ceramics, and a critical role is often played by grain boundaries, impurities, surfaces, dislocations, domains walls, etc. Among the critical issues that atomic-level simulation should be able to address include the microscopic processes associated with ferroelectric switching by domainwall motion and the coupling of ferroelectricity and microstructure in such ceramics. There are exciting challenges in the simulation of ferroelectric

544


device structures. However, since such structures can involve ferroelectrics, electrodes (metallic or conducting oxide) and semiconductors, the development of atomic-level methods to simulate such chemically diverse materials will have to be developed; this is an exciting challenge for the future.

Acknowledgments We would like to thank S. Tinte, D. Wolf, and R.L. Migoni, who collaborated in the work described in this review.

References [1] M.E. Lines and A.M. Glass, Principles and Applications of Ferroelectric and Related Materials, Clarendon Press, Oxford, 1977. [2] A.F. Devonshire, “Theory of ferroelectrics,” Phil. Mag., (Suppl.) 3, 85, 1954. [3] D. Vanderbilt, “First-principles based modelling of ferroelectrics,” Current Opinion in Sol. Stat. Mater. Sci., 2, 701–705, 1997. [4] R. Cohen, “Theory of ferroelectrics: a vision for the next decade and beyond,” J. Phys. Chem. Sol., 61, 139–146, 2000. [5] G.V. Lewis and C.R.A. Catlow, “Potential model for ionic oxides,” J. Phys. C, 18, 1149–1161, 1985. [6] R. Migoni, H. Bilz, and D. Bäuerle, “Origin of Raman scattering and ferroelectricity in oxide perovskites,” Phys. Rev. Lett., 37, 1155–1158, 1976. [7] H. Donnerberg and M. Exner, “Derivation and application of ab initio Nb5+ –O2− short-range effective pair potentials in shell-model simulations of KNbO3 and KTaO3 ,” Phys. Rev. B, 49, 3746–3754, 1994. [8] F. Jona and G. Shirane, Ferroelectric Crystals, Dover Publications, New York, 1993. [9] S. Tinte, M.G. Stachiotti, M. Sepliarsky, R.L. Migoni, and C.O. Rodriguez, “Atomistic modelling of BaTiO3 based on first-principles calculations,” J.Phys.: Condens. Matter, 11, 9679–9690, 1999. [10] P.H. Ghosez, E. Cockayne, U.V. Waghmare, and K.M. Rabe, “Lattice dynamics of BaTiO3 , PbTiO3 and PbZrO3 : a comparative first-principle study,” Phys. Rev. B, 60, 836–843, 1999. [11] S. Tinte, J. Iniguez, K. Rabe, and D. Vanderbilt, “Quantitative analysis of the firstprinciples effective Hamiltonian approach to ferroelectric perovskites,” Phys. Rev. B, 67, 064106, 2003. [12] M. Sepliarsky, S.R. Phillpot, D. Wolf, M.G. Stachiotti, and R.L. Migoni, “Atomiclevel simulation of ferroelectricity in perovskite solid solutions,” Appl. Phys. Lett., 76, 3986–3988, 2000. [13] S. Tinte, M.G. Stachiotti, S.R. Phillpot, M. Sepliarsky, D. Wolf, and R.L. Migoni, “Ferroelectric properties of Bax Sr1−x TiO3 solid solutions by molecular dynamics simulation,” J. Phys.: Condens. Matt., 16, 3495–3506, 2004. [14] M. Sepliarsky, S. Phillpot, D. Wolf, M.G. Stachiotti, and R.L. Migoni, “Long-ranged ferroelectric interactions in perovskite superlattices,” Phys. Rev. B, 64, 060101 (R), 2001.


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[15] M. Sepliarsky, S. Phillpot, D. Wolf, M.G. Statchiotti, and R.L. Migoni, “Ferroelectric properties of KNbO3 /KTaO3 superlattices by atomic-level simulation,” J. Appl. Phys., 90, 4509–4519, 2001. [16] S. Tinte and M.G. Stachiotti, “Surface effects and ferroelectric phase transitions in BaTiO3 ultrathin films,” Phys. Rev. B, 64, 235403, 2001. [17] M.G. Stachiotti, “Ferroelectricity in BaTiO3 nanoscopic structures,” Appl. Phys. Lett., 84, 251–253, 2004. [18] G.V. Lewis and C.R.A. Catlow, “Defect studies of doped and undoped Barium Titanate using computer simulation techniques,” J. Phys. Chem. Sol., 47, 89–97, 1986.

2.7 ENERGY MINIMIZATION TECHNIQUES IN MATERIALS MODELING C.R.A. Catlow1,2 1

Davy Faraday Laboratory, The Royal Institution, 21 Albemarle Street, London W1S 4BS, UK 2 Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK

1.

Introduction

Energy minimization is one of the simplest but most widely applied of modeling procedures; indeed, its applications have ranged from biomolecular systems to superconducting oxides. Moreover, minimization is often the first stage in any modeling procedure. In this section, we review the basic concepts and techniques, before providing a number of topical examples. We aim to show both the wide scope of the method as well as its extensive limitations.

2.

Basics and Definitions

The conceptual basis of energy minimization (EM) is simple: an energy function E(r1 , . . . , r N ) is minimized with respect to the nuclear coordinates ri (or combinations of these) of a system of N atoms, which may be a molecule or cluster, or a system with 1, 2 or 3D periodicity; in the latter case, the minimization may be applied to the lattice parameter(s), in addition to the coordinates of the atom within the repeat unit. E may be calculated using a quantum mechanical method, although the term energy minimization is often associated with interatomic potential methods or some simpler procedures. The term “molecular mechanics” is essentially synonymous but refers to applications to molecular systems. The term “static lattice” methods is also widely used and normally implies a minimization procedure followed by the calculation of properties of the minimized configuration. EM methods may be extended to “free energy minimization” if the entropy contribution can be calculated 547 S. Yip (ed.), Handbook of Materials Modeling, 547–564. c 2005 Springer. Printed in the Netherlands.

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by configurational or by molecular or lattice dynamical procedures. But by definition, EM excludes any explicit treatment of thermal motions. EM methods normally involve the specification of a “starting point” or initial configuration and the subsequent application of a numerical algorithm to locate the nearest local minimum, from which there arises possibly the most fundamental limitation of the approach, i.e., the “local minimum” problem: minimization can never be guaranteed to find the global minimum of an energy (or any other) function. And straightforward implementations of the method are essentially refinements of approximately known structures. Indeed, for many complex systems, e.g., protein structures, unless the starting configuration is very close to the global minimum, a local minimum will invariably be generated by minimization. Procedures for attempting to identify global minima will be discussed later in the section. Although minimization by definition excludes dynamical effects, it is possible to apply the technique to rate processes (e.g., diffusion and reaction) using methods based on an Absolute Rate Theory, in which rates (ν) are calculated according to the expression: ν = ν0 exp(−G ACT /kT ),

(1)

where the pre-exponential factor, ν0 may be loosely related to a vibrational frequency and G ACT refers to the free energy of activation of the process, i.e., the difference between the free energy of the transition states for the process and the ground state of the system. If the transition states can be located via some search procedure (or can be postulated from symmetry or other considerations), then the activation energy and (much less commonly) activation free energy may be calculated. Such procedures have been widely used in modeling atomic transport in solids. In Section 2.1, we first consider the type of energy function employed; the methods used to identify minima are then discussed followed by a more detailed survey of methodologies. Recent applications are reviewed in the final sub-section. In all cases, the emphasis is on applications to materials, but many of the considerations apply generally to atomistic modeling.

2.1.

Energy Functions

As noted earlier, minimization may be applied to any energy function that may be calculated as a function of nuclear coordinates. In atomistic simulation studies, three types of energy function may be identified: (i) Quantum mechanically evaluated energies, where essentially we use the energy calculated by solving the Schrödinger equation at some level of approximation. Extensive discussions of such methods are, of course, available elsewhere in this volume.

Energy minimization techniques in materials modeling

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(ii) Interatomic potential based energy function. Here we use interatomic potentials to calculate the total energy of the system with respect to component atoms (i.e., the cohesive energy) or ions (the lattice energy), i.e., E=

N N N N N 1 1 Vi2j (ri j ) + V 3 (ri r j rk ) . . . . 2 i j =/ i 3 i j =/ i k/= j =/ i i j k

(2)

where the Vi j are the pair potential components, Vi j k the three-body term and of course the series continues in principle to higher order terms. The sum is over all N atoms in the system, but would normally be terminated beyond a “cut-off” distance (although note the case of the electrostatic term discussed later). In a high proportion of calculations (especially on non-metallic systems) only the two-body term is included, which allows the energy, E, for periodic systems to be written as: E=

Nc Ncut 1 Vi j (ri j ), 2 i=1 j =/ i

(3)

where the first summation refers to all atoms in the unit cell where interactions with all other atoms are summed up to the specified cut-off. It is common to separate off the electrostatic contributions Vi j , i.e., Vi j (ri j ) =

qi q j + ViSR j (ri j ), ri j

(4)

where qi and q j are atomic or ion charges and V SR is the remaining, “shortrange” component of the potential. This allows us to write: E = Ec +

Nc Ncut

Vij (ri j ),

(5)

i=1 j = /i

where E c is the coulomb term, obtained by summing the r −1 terms, which should not be truncated in any accurate calculation. The short-range terms can, however, usually be safely truncated at a distance of 10–20 Å. The summation of the electrostatic term must be carefully undertaken, as it may be conditionally convergent if handled in real space. The most widely used procedure rests on the work of Ewald (see, e.g., [1]) which obtains rapid convergence by a partial transformation into reciprocal space. The procedure has been very extensively used and for applications to materials we refer to the articles in Ref. [2].

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2.2.

C.R.A. Catlow

Other Functions

In some cases, a simple “cost function” may be used based on geometrical criteria rather than energies. For example, the distance least squares (DLS) approach [3] is based on minimization of a cost function obtained by summing the squares of the distances between calculated and “standard” bond lengths for a structure. More complex cost functions include deviation from calculated and specified coordination numbers. We have also noted earlier that if entropy terms can be estimated, energy can be extended to free energy minimization. Such extensions will be discussed in detail for the case of periodic lattices.

2.3.

Identification of Minima

We recall that standard minimization methods aim to identify the energy minimum starting form a specified initial configuration, using algorithms which will be discussed later. And as argued earlier, it is impossible ever to guarantee that a global minimum has been achieved. However, a number of procedures are available to mitigate the effects of the local minimum problem, with the two main classes being: (i) Simulated Annealing (SA), where the approach is to use molecular dynamics (MD) or Monte Carlo (MC) systems initially at high temperature, thereby allowing the system to explore the potential energy surface and escape from local into the global minimum region. The normal procedure is to “cool” the system during the course of the simulation, which usually concludes with a standard minimization. SA has been used successfully and predictively in a number of cases in crystal structure modeling. If used carefully and appropriately, the method offers a good probability of identifying the global minimum; but there always remains a distinct possibility that the simulation will fail to locate regions of configurational space close to the global minimum, especially if there are substantial energy barriers between this and other regions. (ii) Genetic Algorithm methods (GA), which GA have been widely used in optimization studies, and where the approach is fundamentally different from SA. Instead of one starting point, there are many, which may simply be different random arrangements of atoms (with some overall constraint such as unit-cell dimensions). A cost function is specified, and is evaluated for each configuration. the population of configurations then evolves through successive generations. The “breeding” process involves exchange of features between different members of the population and is driven so as to generate a population with a low cost function.


551

At the end of the procedure, selected members of the population are subjected to energy minimization, giving a range of minimum structures from which the lowest energy one may be selected. GA methods again offer no guarantee that the global minimum has been located. Their particular merit is that they use a variety of initial configurations, rather than one as in SA. However, both approaches unquestionably have their value. A good account of the application of the GA method to periodic solids is given in Ref. [4].

3.

Methodologies

Minimization methods may be applied to periodic lattices, to defects within lattices, to surfaces and to clusters. The methodological aspects are similar in all these different areas. In this section, we pay the greatest attention to perfect lattice minimization. The field of defect calculations is reviewed in Chapter 6.4.

3.1.

Perfect Lattice Calculations

The first objective here is to calculate the lattice energy, in which the summation in Eq. (1) is taken over all atoms/ions in the unit cell interacting with all other species. The calculation is tractable via the use of the Ewald summation for the Coulombic terms and the cut-off for the short-range interactions. We note that the great majority of lattice energy calculations only include the two-body contribution to the short-range energy. One important matter of definition is that the lattice energy gives the energy of the crystal with respect to component ions at infinity. If it is desired to express the energy with respect to atoms at infinity (for which the more appropriate term is then the cohesive energy) then the appropriate ionization energies and electron affinities will be added. Lattice energy calculations are now routine, and may be carried out for very large unit cells containing several hundred atoms. The codes METAPOCS, THBREL and GULP undertake lattice energy calculations including both twoand three-body terms, using both bond-bending and triple-dipole formalisms. Lattice energy calculations provide valuable insight into the structures and stabilities of ionic and semi-ionic solids. The technique is most powerful when combined with energy minimization procedures, which generate the structure of minimum energy. These are discussed later after the calculation of entropies have been described. The results in Table 1 give a good illustration of the value of lattice energy studies. They are the energy minimum lattice energies calculated for a number of purely siliceous microporous zeolitic structures which

552

C.R.A. Catlow Table 1. Relative energies (per mol) of microporous siliceous structures with respect to quartz (after Ref. [5]) Structure

Energy (kJ/mol)

Silicalite Mordenite Faujasite

11.2 20.52 21.4

are compared with the lattice energy of α-SiO2 . The latter has the lowest value as would indeed be expected since the more porous structures are known to be metastable with respect to the dense α-SiO2 polymorph. Of greater interest is the observation that of the porous structures, silicalite has the greatest stability. This accords with the fact that this polymorph can only be prepared as a highly siliceous compound unlike the case with the other zeolitic structures which are normally synthesized with high aluminium contents. The calculations which are discussed in greater detail by Ooms et al. [5], suggest that this behavior has its origin at least in part in the thermodynamic stability of the compounds. We note that more recently very similar results were obtained by Henson et al. [6] who also showed that the calculated values were in excellent agreement with experiment. In addition to calculating energies, it is also possible to calculate routinely a range of crystal properties, including the lattice stability, the elastic and dielectric and piezoelectric constants, and the phonon dispersion curves. The techniques used which are quite standard require knowledge of both first and second derivatives of the energy with respect to the atomic coordinates. Indeed it is useful to describe two quantities: first the vector, g, whose components gα i are defined as: gα i =

∂E ∂xα i

(6)

i.e., the first derivative of the lattice energy with respect to a given Cartesian coordinate (α) of the ith atom. The second derivative matrix W has components αβ Wij ; defined by: 



∂ 2E  αβ Wij =  β ∂xα i ∂xj

(7)

The expressions used in calculating the properties referred to above from these derivatives are discussed in greater detail in Refs. [2] and [7]. For more detailed discussions of the calculation of phonon dispersion curves from the second derivative or “dynamical” matrix W , the reader should consult [8] and


553

Parker and Price [9]. Finally, we note that by the term “lattic stability” we refer to the equilibrium conditions both for the atoms within the unit cell, and for the unit cell as a whole. The former are available from the gradient vector g, while the latter are described in terms of the six components ε1 . . . ε1 which define the strain matrix ε, where  

ε= 

ε1

1 ε 2 4

1 ε 2 4 1 ε 2 5

ε2

1 ε 2 5 1 ε 2 6

1 ε 2 6

ε3

   

(8)

So when the unit cell as a whole is strained, we describe the modification of an arbitrary vector r in the unstrained matrix to a vector r in the strained matrix, using the equation: r = (1 + ε) r

(9)

where 1 is the unit matrix. The six derivatives of energy with respect to strain, [∂ E/∂εi ], therefore measure the forces acting on the unit-cell. The equilibrium condition for the crystal therefore requires that g = 0 and [∂ E/∂εi ] = 0 for all i.

3.2.

Entropy Calculations

The entropy in a solid arises first from configuration terms which for a perfect solid are zero; while for a solid showing orientational or translational disorder configurational expressions based on the Boltzmann expression S = k ln(W ) may be used. In this section we shall pay more attention to the second term, which is due to the population of the vibrational degrees of freedom of the solid. Thus the entropy of a solid may be written as:

Q

Svib = k

dQ

hνi

−1 hνi −hνi exp − 1 − ln 1 − exp kT kT kT

i

0

(10)

where the sum is over all phonon frequencies and the integral is over the Brillouin zone. In practice the integral is normally evaluated by sampling over the zone for which a variety of techniques are available. Vibrational terms also give a contribution to the lattice energy of the crystal:

Q

E vib = kT

dQ 0

hνi i

−1

hνi hνi exp −1 + 2kT kT kT

(11)

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C.R.A. Catlow

which results in the following expression for the crystal free energy with respect to ions at rest of infinity: F = E + kT

Q

dQ 0

hνi i

2kT

+ ln 1 − exp

hνi kT

(12)

where E is the lattice energy (omitting vibrational terms).

3.3.

Energy Minimization

Having evaluated energies and free energies of a crystal structure we are now able to implement these in an energy (or free energy) minimization procedure. Let us consider first the simple case of minimization to constant volume (i.e., within fixed cell dimensions). We write the energy of the crystal as a Taylor expansion in the displacements of the atoms, δ, from that current configuration giving: 1 E(δ) = E 0 + gδ + δW δ + . . . . 2

(13)

If we terminate this function at the second order term and minimize E with respect to δ, we obtain for the energy minimum: 0 = g + Wδ

i.e., δ = −gW −1

(14)

Displacement of the coordinates by δ as given in Eq. (14) will generate the energy minimum configuration. Of course, in practice, it will not be valid to truncate the summation at the quadratic term, except when very close to the minimum. However, Eq. (14) provides the basis of an effective iterative procedure for attaining the minimum. Indeed this “Newton Raphson” method is widely used in both perfect and defect lattice energy minimization, as it is generally rapidly convergent. Its main disadvantage is that it requires the calculation, inversion and storage of the second derivative matrix, W . Recalculation and inversion each iteration may be avoided by use of updating procedures (see e.g., [10]). The storage problem may become serious with very large structures owing to the high cpu memory requirements. Recourse may be made to gradient methods, e.g., the well known conjugate gradients technique, which make use only of first derivatives. Such methods are, however, more slowly converging. The increasing availability of very large cpu memories is, however, reducing the difficulties associated with the storage of the W matrix. For evaluation of the energy minimum with respect to constant pressure (i.e., with variable cell dimensions), first we note that we can define the six


555

components of the mechanical pressure acting on the solid, corresponding to the six strain components, defined in Eq. (8), i.e., P εi =

1 V

dUi dεi

(15)

where V is the unit cell volume. The strains can then be evaluated, using Hooke’s law, ε = PC −1

(16)

where C is the (6 × 6) elastic constant tensor, which may be calculated from W . Substitution of these calculated strain components into Eq. (16) then yields the new cell dimensions and atomic coordinates. Again, the procedure is iterative, as it is only strictly valid in the region of applicability of the harmonic approximation. With a sensible starting point, however, only a small number of iterations (typically 2–5) is required. The treatment above assumes that the pressure and corresponding strains are entirely mechanical in origin. However, at finite temperatures there will be a “kinetic pressure” arising from the changes in the vibrational free energy with volume. These may be written as: εi Pvib

1 = V

dFvib dεi

(17)

where Fvib is the vibrational free energy. These kinetic pressures are most simply evaluated by applying small arbitrary strains to the structure and calculating the corresponding changes in Fvib . If Pvib is added to the mechanical pressure P in Eq. (15), it enables us to carry out free energy minimization. (see e.g., [11]). A general computer code, PAPAPOCS, is available for such calculations and the same functionality is available in the GULP code [12]. A detailed discussion is given by Parter and Price [9] and Watson et al. [11] who also describe how the techniques may be used to calculate lattice expansivity, either directly or by calculating the cell dimension as a function of temperature or by calculation of the thermal Grüneisen parameter.

3.4.

Surface Simulations

The procedures here are closely related to those employed in perfect lattice calculations but adapted to 2D periodicity. The most widely used procedure is that pioneered by Tasker et al. [13], in which a slab is taken and divided into

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two regions. Full minimization is undertaken on the upper region which represents the relaxed surface structure and which is embedded in a rigid representation of the underlying lattice. The Ewald summation must be adapted for 2D periodicity using the formalism developed by Parry [14]. Surface simulations have been widely and successfully applied especially to the surfaces of ionic materials, and a number of standard codes are available, e.g., METADISE and MARVIN. The methods may also be readily adapted to study interfaces and other 2D periodic systems such as grain boundaries as will be discussed later in this chapter.

3.5.

Defect and Cluster Calculations

Defects simulations, as discussed in detail in Chapter 6.4, proceed by relaxation of an atomistically represented region of lattice which is embedded in a more approximate representation of the more distant regions of the lattice whose dilectric and/or elastic response to the defect is calculated. An increasingly widely used extension of the procedure is to describe the immediate environment of the defect, (the defect itself and a small number of surrounding coordination shells) quantum mechanically. The detailed discussion of such “embedded cluster” methods is beyond the scope of the present chapter; a recent review is available in Ref. [15]. Minimization of the energy of clusters is, of course, conceptually straightforward. Minimization algorithms are applied to the cluster energy (or free energy) obtained by direct summation. Considerable attention has been paid in this field to the use of global optimization techniques owing to the prevalence of multiple minima. A recent review of cluster simulations is available from Ref. [16].

4.

Discussion and Applications

Minimization methods have been extensively applied to metals, ceramics, silicates, semiconductors and molecular materials. In this section we will provide topical examples which will illustrate the current capabilities of the techniques.

4.1.

Predictions of the Structures of Microporous Materials

Microporous materials have been widely investigated over the last 50 years owing to their extensive range of applications in catalysis, gas separation and


557

ion exchange. Zeolites, (originally observed as minerals, but now extensively available as synthetic materials) are all silica or aluminosilicate materials, based on fully corner shared networks of SiO4 and AlO4 tetrahedra, but with structures that contain channels pores and voids of molecular dimensions; pore sizes are typically in the range 5–15 Å. The aluminosilicate materials contain exchangeable cations, while the microporous structures give rise to the applications in molecular sieving and sorption. Exchange of protons into the materials creates acid sites which promote catalytic reactions including cracking, isomerization and hydrocarbon synthesis; while metal ions in both framework and extraframework locations can act as active sites for partial oxidation reactions. Modeling techniques have been applied extensively and successfully to the study of microporous materials (see, e.g., the books edited by Catlow [17] and Catlow et al. [18]). And there have been a number of successful applications of minimization techniques to the accurate and indeed to the predictive modeling of microporous structures. Here we highlight a recent significant development, namely the prediction of new hypothetical structures. There have been many attempts to predict new microporous structures, most of which have rested on the fact that the very definition of these materials is based on geometry, rather than on precise chemical composition, occurence or function. In order to be considered as a zeolite, or zeolitetype material (zeotype), a mineral or synthetic material must possess a 3D four-connected inorganic framework, i.e., a framework consisting of tetrahedra which are all corner-sharing. There is an additional criterion that the framework should enclose pores or cavities which are able to accommodate sorbed molecules or exchangeable cations, which leads to the exclusion of denser phases. Topologically, the zeolite frameworks may thus be thought of as fourconnected nets, where each vertex is connected to its four closest neighbours. So far 139 zeolite framework types are known , either from the structures of natural minerals or from synthetically produced inorganic materials. In enumerating microporous structures, a number of fruitful approaches have been developed. Some have involved the decomposition of existing structures into their various structural subunits, and then recombining these in such ways as to generate novel frameworks . Methods which involve combinatorial, or systematic, searches of phase space have also been successfully deployed. Recently, an approach based on mathematical tiling theory has also been reported [19]. It was established that there are exactly 9, 117 and 926 topological types of fourconnected uninodal (i.e., containing one topologically distinct type of vertex), binodal and trinodal networks, respectively, derived from simple tilings (tilings with vertex figures which are tetrahedra), and at least 145 additional uninodal networks derived from quasi-simple tilings (the vertex figures of which are derived from tetrahedra, but contain double edges). In principle, the tiling

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C.R.A. Catlow

approach offers a complete solution to the problem of framework enumeration, although the number of possible nets is infinite. Potentially therefore we may be able to generate an unlimited number of possible zeolitic frameworks. Of these, only a portion is likely to be of interest as having desirable properties, with an even smaller fraction being amenable to synthesis in any given composition. It is this last problem, the feasibility of hypothetical frameworks, which is the key question in any analysis of such structures. The answer is not a simple one, since the factors which govern the synthesis of such materials are not fully understood. As discussed earlier, zeolites are metastable materials. Aside from this thermodynamic constraint, the precise identity of the phase or phases formed during hydrothermal synthesis is said to be under “kinetic control,” although there is increasing sophistication in targeting certain types of framework using various templating methods, fluoride media and other synthesis parameters . Additionally, certain structural motifs are more likely to formed within certain compositions, e.g., double four-rings in germanates, three-rings in beryllium-containing compounds and so on. A full characterization of any hypothetical zeolite must therefore include an analysis of framework topology and of the types of building unit present, as well as some estimate of the thermodynamic stability of the framework. Using an appropriate potential model, lattice energy minimization can, as shown above, provide a very good measure of this stability and well as optimizing structures to a high degree of accuracy. In the method adopted by Foster et al. [20], networks derived from tiling theory were first transformed into “virtual zeolites” of composition SiO2 by placing silicon atoms at the vertices of the nets, and bridging oxygens at the midpoints of connecting edges. The structures were then refined using the geometry-based DLS procedure, referred to above, before final optimization by lattice energy minimization. Among the 150 or so uninodal structures examined, all 18 known uninodal zeolite frameworks were found. Moreover, most of the unknown frameworks had been described by previous authors; in fact there a considerable degree of overlap between sets of uninodal structures generated by different methods. Most of the binodal and trinodal structures, however, are completely new. Using calculated lattice energy as an initial measure of feasibility, a number of the more interesting structures are shown in Fig. (1). The challenge is now to synthesize these structures.

4.2.

Grain Boundary Structures in Mantle Minerals

Grain boundaries are known to be a major factor controling mechanical and rheological properties of materials. Detailed knowledge of their structures is, however, limited. Simulation methods have made a major contribution over


559

detl_14

detl_19

detl_11

delt_71

delt_35

Figure 1. Illustrations of feasible uninodal zolite structures generated by tiling theory and modeled using lattice energy minimization.

the past 20 years in developing models for grain boundaries as in the work of Keblinski et al. [21] on metal systems and Duffy, Harding and Stoneham [22] on ionic systems. Recent work has explored grain boundary properties in the Mantle mineral forsterite Mg2 SiO4 , a member of the olivine group of minerals, which comprise a major proportion of the upper part of the Earth’s Mantle. Knowledge of the grain boundary structure of this material is vital for developing an improved

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C.R.A. Catlow

understanding of the rheology of the Mantle. Modeling boundaries in this material, however, presents substantial challenges owing to the complexity of the crystal structure. The recent work of de Leeuw et al. [23] investigated this problem using static lattice simulation techniques. They modeled the forsterite grain boundaries using empirical potential models for SiO2 and MgO. Atomistic simulation techniques are appropriate for these calculations because they are capable of modeling systems consisting of large numbers of ions which is necessary when modeling grain boundaries, as shown in many studies. Energy minimization techniques were used to investigate the structure and stability of the grain boundaries and the interactions between the lattice ions at the boundaries and adsorbed species, such as protons and dissociated water molecules, to identify the strength of interaction with specific boundary features. They employed the energy minimization code METADISE, which is designed to model dislocations, interfaces and surfaces . A grain boundary is created by fitting two surface blocks together in different orientations. In the present case, two series of tilt grain boundaries (M1 and M2, defined by the type of cation site at the surface) were created from appropriate models of stepped forsterite (010) surfaces at increasing boundary angles. Both boundary and adhesion energies were calculated, which describe the stability of the boundary with respect to the bulk material and free surfaces, respectively. Results are reported in Table 2 and Fig. 2. The atomistic models generated are shown in Fig. 3. The larger grain boundaries do not form a continuously disordered interface but rather a series of open channels in the interfacial region with practically bulk termination of the two mirror planes (Fig. 3). We would expect that physical processes such as melting and diffusion of ions and molecules, e.g., oxygen or water, will be enhanced especially at the larger-terraced boundaries due to the low density of these regions compared to the bulk crystal. The minima in the adhesion energies at φ = ∼ 200 (M1) or ∼ 300 (M2) (Fig. 2) Table 2. Calculated boundary energies of (010) tilt grain boundaries in forsterite Boundary

Boundary angle (◦ )

Boundary energy (Jm−2 )

M2

65 47 36 28 23 60 41 30 23 19

1.32 2.72 3.57 3.50 3.09 2.12 3.13 3.19 2.94 2.88

M1


561

adhesion energy (J/m2)

5

4

3

2

1

0 0

20

40

60

80

angle (degrees) M2

M1

Figure 2. Adhesion energies as a function of grain boundary tilt angle.

indicate the boundaries which are most easily cleaved and are due to the relative stabilitities of the grain boundaries and corresponding free surfaces. Overall, the results show the ability of simulation methods to generate realistic models for these complex interfaces.

4.3.

Nanocluster Structures in ZnS

Our final example is an intriguing case study in cluster chemistry. As part of an extensive study aimed at identifying the structures of the critical growth nuclei in the growth of ZnS crystals Spano et al. [24, 25] have identified a whole series of stable open cluster structures for (ZnS)n clusters with n ranging from 1 to 80. They have employed simulated annealing and minimization techniques using interatomic potentials but with critical structures also being modeled by Density Functional Theory electronic structure methods, (the results of which validate the interatomic potential based simulations.) The cluster structures have quite different topologies from bulk ZnS. A particularly interesting example is shown in Fig. 4. It is an onion like cluster with an inner core and outer shell. Work is in progress aimed at detecting these structures experimentally.

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Figure 3. Relaxed structures of tilt grain boundaries with (010) mirror terraces, top (100) step wall showing two round channels per terrace, bottom (001) step wall with one triangular channel per terrace.

5.

Conclusions

This chapter has surveyed the essential methodological aspects of minimization techniques and has illustrated the scope of the field by a number of recent examples. Despite their simplicity, minimization methods will remain powerful tools in materials simulation.


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Figure 4. Predicted onion-like structure for (ZnS)60 .

Acknowledgments I am grateful to many colleagues for their contributions to the work discussed in this chapter, but special thanks go to Robert Bell, Martin Foster, Nora de Leeuw, Stephen Parker and Said Hamad, whose recent work was highlighted in the applications section.

References [1] M.P. Tosi, Solid State Phys., 16, 1, 1964. [2] C.R.A. Catlow (ed.), Computer Modelling in Inorganic Crystallograpy, Academic Press, London, 1997. [3] W.M. Meier and H. Villiger, Z. Kristallogr, 128, 352, 1969. [4] S.M. Woodley, In: R.L. Johston (ed.), Structure and Bonding, vol. 110, Springer, Heidelberg, 2004. [5] G. Ooms, R.A. van Santen, C.J.J. den Ouden, R.A. Jackson, and C.R.A. Catlow, J. Phys. C: Condensed Matter., 92, 4462, 1988. [6] N.J. Henson, A.K. Cheetham, and J.D. Gale, Chem. Mater., 6, 1647, 1994. [7] C.R.A. Catlow and W.C. Mackrodt (eds.), “Computer simulation of solids,” Lecture Notes in Physics, vol. 166, Springer, Berlin, 1982. [8] W. Cochran, Crit. Rev. Solid Sci., 2, 1, 1971. [9] S.C. Parker and G.D. Price, In: C.R.A. Catlow (ed.), Advanced Solid State Chemistry, vol. 1, JAI Press, 1990.

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[10] M.J. Norgett and R. Fletcher, J. Phys. C: Condensed Matter, 3, L190, 1970. [11] Watson et al., In: C.R.A. Catlow (ed.), Computer Modelling in Inorganic Crystallography, Academic Press, London, p. 55, 1997. [12] J.D. Gale, J. Chem Soc. Faraday Trans., 93, 629, 1997. [13] P.W. Tasker, J. Phys. C: Condensed Matter., 12, 4977, 1979. [14] D.E. Parry, Surf. Sci., 49, 433, 1975. [15] P. Sherwood et al., J. Mol. Struct. – Theochem, 632, 1, 2003. [16] R.L. Johnston, Dalton Trans., 22, 4193, 2003. [17] C.R.A. Catlow (ed.), Modelling of Structure and Reactivity in Zeolites, Academic Press, London, 1992. [18] C.R.A. Catlow, B. Smit, and R.A. van Santen (eds.), Modelling Microporous Materials, Elsevier, Amsterdam, 2004. [19] O. Delgado Friedrichs, A.W.M. Dress, D.H. Huson, J. Klinowski, and A.L. Mackay, Nature, 400, 644, 1999. [20] M.D. Foster, A. Simpler, R.G. Bell, O. Delgado Friedrichs, F.A. Almeida Paz, and J. Klinowski, Nature Mat., 3, 234, 2004. [21] P. Keblinski, D. Wolf, S.R. Phillpot, and H. Gleiter, Philos. Mag. A., 79, 2735, 1999. [22] D.M. Duffy, J.H. Harding, and A.M. Stoneham, Philos. Mag. A, 67, 865, 1993. [23] N.H. De Leeuw, S.C. Parker, C.R.A. Catlow, and G.D. Price, Am. Mineral, 85, 1143, 2000. [24] E. Spano, S. Hamad, and C.R.A. Catlow, J. Phys. Chem. B, 107, 10337, 2003. [25] E. Spano, S. Hamad, and C.R.A. Catlow, Chem. Commun., 864, 2004.

2.8 BASIC MOLECULAR DYNAMICS Ju Li Department of Materials Science and Engineering, Ohio State University, Columbus, OH, USA

A working definition of molecular dynamics (MD) simulation is technique by which one generates the atomic trajectories of a system of N particles by numerical integration of Newton’s equation of motion, for a specific interatomic potential, with certain initial condition (IC) and boundary condition (BC). Consider, for example, a system with N atoms in a volume . We can define its internal energy: E ≡ K + U , where K is the kinetic energy, K ≡

N 1 i=1

2

m i |˙xi (t)|2 ,

(1)

and U is the potential energy, U = U (x3N (t)).

(2)

x3N (t) denotes the collective of 3 D coordinates x1 (t), x2 (t), . . . , x N (t). Note that E should be a conserved quantity, i.e., a constant of time, if the system is truly isolated. One can often treat a MD simulation like an experiment (Fig. 1). Below is a common flowchart of an ordinary MD run: [system setup] sample selection (pot., N , IC, BC)

→

[equilibration] sample preparation (achieve T, P)

→

[simulation run] property average (run L steps)

→

[output] data analysis (property calc.)

in which we fine-tune the system until it reaches the desired condition (here, temperature T and pressure P), and then perform property averages, for instance calculating the radial distribution function g(r) [1] or thermal conductivity [2]. One may also perform a non-equilibrium MD calculation, during which the system is subjected to perturbational or large external driving forces, 565 S. Yip (ed.), Handbook of Materials Modeling, 565–588. c 2005 Springer. Printed in the Netherlands.

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N particles

xi(t) z

y x

Figure 1. Illustration of the MD simulation system.

and we analyze its non-equilibrium response, such as in many mechanical deformation simulations. There are five key ingredients to a MD simulation, which are boundary condition, initial condition, force calculation, integrator/ensemble, and property calculation. A brief overview of them is given below, followed by more specific discussions. Boundary condition. There are two major types of boundary conditions: isolated boundary condition (IBC) and periodic boundary condition (PBC). IBC is ideally suited for studying clusters and molecules, while PBC is suited for studying bulk liquids and solids. There could also be mixed boundary conditions such as slab or wire configurations for which the system is assumed to be periodic in some directions but not in the others. In IBC, the N -particle system is surrounded by vacuum; these particles interact among themselves, but are presumed to be so far away from everything else in the universe that no interactions with outside occur except perhaps responding to some well-defined “external forcing.” In PBC, one explicitly keeps track of the motion of N particles in the so-called supercell, but the supercell is surrounded by infinitely replicated, periodic images of itself. Therefore a particle may interact not only with particles in the same supercell but also with particles in adjacent image supercells (Fig. 2). While several polyhedra shapes (such as hexagonal prism and rhombic dodecahedron from Wigner–Seitz construction) can be used as the space-filling unit and thus can serve as PBC supercell, the simplest and most often used supecell shape is a parallelepiped, specified by its three edge vectors h1 , h2 and h3 . It should be noted that IBC can most often be well mimicked by a large enough PBC supercell so the images do not interact. Initial condition. Since Newton’s equations of motion are second-order ordinary differential equations (ODE), IC basically means x3N (t = 0) and

Basic molecular dynamics

567

rc h2

h1

Figure 2. Illustration of periodic boundary condition (PBC). We explicitly keep track of trajectories of only the atoms in the center cell called the supercell (defined by edge vectors h1 , h2 and h3 ), which is infinitely replicated in all three directions (image supercells). An atom in the supercell may interact with other atoms in the supercell as well as atoms in the surrounding image supercells. rc is a cut-off distance of the interatomic potential beyond which interaction may be safely ignored.

x˙ 3N (t = 0), the initial particle positions and velocities. Generating the IC for crystalline solids is usually quite easy, but IC for liquids needs some work, and even more so for amorphous solids. A common strategy creating a proper liquid configuration is to melt a crystalline solid. And if one wants to obtain an amorphous configuration, a strategy is to quench the liquid during the MD run. Let us focus on IC for crystalline solids. For instance, x3N (t = 0) can be a fcc perfect crystal (assuming PBC), or an interface between two crystalline phases. For most MD simulations, one needs to write a structure generator. Before feeding the initial configuration thus created into a MD run, it is a good idea to visualize it first, checking bond lengths and coordination numbers, etc. [3]. A frequent cause of MD simulation breakdown is pathological initial condition, as the atoms are too close to each other initially, leading to huge forces. According to the equipartition theorem [4], each independent degree of freedom should possess kB T /2 kinetic energy. So, one should draw each

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component of the 3N -dimensional x˙ 3N (t =0) vector from a Gaussian–Maxwell normal distribution N (0, kB T /m i ). After that, it is a good idea to eliminate the center of mass velocity, and for clusters, the net angular momentum as well. Force calculation. Before moving into details of force calculation, it should be mentioned that two approximations underly the use of the classical equation of motion mi

∂U d2 xi (t) = fi ≡ − , 2 dt ∂xi

i = 1, . . . , N.

(3)

to describe the atoms. The first is the Born–Oppenheimer approximation [5] which assumes the electronic state couples adiabatically to nuclei motion. The second is that the nucleus motion is far removed from the Heisenberg uncertainty lower bound: Et h¯ /2. If we plug in E = kB T /2, the kinetic energy, and t = 1/ω, where ω is a characteristic vibrational frequency, we obtain kB T /h¯ ω 1. In solids, this means the temperature should be significantly greater than the Debye temperature, which is actually quite a stringent requirement. Indeed, large deviations from experimental heat capacities are seen in classical MD simulations of crystalline solids [2]. A variety of schemes exist to correct this error [1], for instance the Wigner–Kirkwood expansion [6] and path integral molecular dynamics [7]. The evaluation of the right-hand side of Eq. (3) is the key step that usually consumes most of the computational time in a MD simulation, so its efficiency is crucial. For long-range Coulomb interactions, special algorithms exist to break them up into two contributions: a short-ranged interaction, plus a smooth, field-like interaction, both of which can be computed efficiently in separate ways [8]. In this contribution we focus on issues concerning shortrange interactions only. There is a section about the Lennard–Jones potential and its trunction schemes, followed by a section about how to construct and maintain an atom–atom neighborlist with O(N ) computational effort per step. Finally, see Chap. 2.4 and 2.5 for the development of interatomic potential U (x3N ) functions for metallic and covalent materials, respectively. Integrator/ensemble. Equation (3) is a set of second-order ODEs, which can be strongly nonlinear. By converting them to first-order ODEs in the 6N dimensional space of {x N , x˙ N }, general numerical algorithms for solving ODEs such as the Runge–Kutta method [9] can be applied. However, these general methods are rarely used in practice, because the existence of a Hamiltonian allows for more accurate integration algorithms, prominent among which are the family of predictor-corrector integrators [10] and the family of symplectic integrators [8, 11]. A section in this contribution gives a brief overview of integrators. Ensembles such as the micro-canonical, canonical, and grand-canonical are concepts in statistical physics that refer to the distribution of initial conditions. A system, once drawn from a certain ensemble, is supposed to follow strictly


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the Hamiltonian equation of motion Eq. (3), with E conserved. However, ensemble and integrator are often grouped together because there exists a class of methods that generates the desired ensemble distribution via time integration [12, 13]. Equation (3) is modified in these methods to create a special dynamics whose trajectory over time forms a cloud in phase space that has the desired distribution density. Thus, the time-average of a single-point operator on one such trajectory approaches the thermodynamic average. However, one should be careful in using it to calculate two-point correlation function averages. See Chap. 2.4 for detailed description of these methods. Property calculation. A great strength of MD simulation is that it is “omnipotent” at the level of classical atoms. All properties that are well-posed in classical mechanics and statistical mechanics can in principle be computed. The remaining issue is computational efficiency. The properties can be roughly grouped into four categories: 1. Structural characterizations. Examples include radial distribution function, dynamic structure factor, etc. 2. Equations of state. Examples include free-energy functions, phase diagrams, static response functions like thermal expansion coefficient, etc. 3. Transport. Examples include viscosity, thermal conductivity (electronic contribution excluded), correlation functions, diffusivity, etc. 4. Non-equilibrium response. Examples include plastic deformation, pattern formation, etc.

1.

The Lennard–Jones Potential

The solid and liquid states of rare gas elements Ne, Ar, Kr, Xe are better understood than other elements because their closed-shell electron configurations do not allow them to participate in covalent or metallic bonding with neighbors, which are strong and complex, but only to interact via weak van der Waals bonds, which are perturbational in nature in these elements and therefore mostly additive, leading to the pair-potential model: U (x3N ) =

N

V (|x j i |),

x j i ≡ x j − xi ,

(4)

j >i

where we assert that the total potential energy can be decomposed into the direct sum of individual “pair-interactions.” If there is to be rotational invariance in U (x3N ), V can only depend on r j i ≡ |x j i |. In particular, the Lennard–Jones potential V (r) = 4

12

σ r

−

6

σ r

,

(5)

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is a widely used form for V (r), that depends on just two parameters: a basic energy-scale parameter , and a basic length-scale parameter σ . The potential is plotted in Fig. 3. There are a few noteworthy facts about the Lennard–Jones potential: • V (r = σ ) = 0, at which point the potential is still repulsive, meaning V (r = σ ) > 0 and two atoms would repel each other if separated at this distance. • The potential minimum occurs at rmin = 21/6 σ , and Vmin = −. When r > rmin the potential switches from being repulsive to being attractive. • As r → ∞, V (r) is attractive and decays as r −6 , which is the correct scaling law for dispersion (London) forces between closed-shell atoms. To get a feel for how fast V (r) decays, note that V (r =2.5σ )=−0.0163, V (r = 3σ ) = −0.00548, and V (r = 3.5σ ) = −0.00217. • As r → 0, V (r) is repulsive as r −12 . In fact, r −12 blows up so quickly that an atom seldom is able to penetrate r < 0.9σ , so the Lennard– Jones potential can be considered as having a “hard core.” There is no conceptual basis for the r −12 form, and it may be unsuitable as a model for certain materials, so it is sometimes replaced by a “soft core” of the form exp(−kr), which combined with the r −6 attractive part is called the Buckingham exponential-6 potential. If the attractive part is also of an exponential form exp(−kr/2), then it is called a Morse potential.

2

VLJ(r)/ε

1.5 1 0.5 0 ⫺0.5 ⫺1

1

1.5

2 r/σ

Figure 3. The Lennard–Jones potential.

2.5


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For definiteness, σ = 3.405 Å and = 119.8 kB = 0.01032 eV for Ar. The mass can be taken to be the isotopic average, 39.948 a.m.u.

1.1.

Reduced Units

Unit systems are invented to make physical laws look simple and numerical calculations easy. Take Newton’s law: f =ma. In the SI unit system, this means that if an object of mass x (kg) is undergoing an acceleration of y (m/s2 ), the force on the object must be x y (N). However, there is nothing intrinsically special about the SI unit system. One (kg) is simply the mass of a platinum–iridium prototype in a vacuum chamber in Paris. If one wishes, one can define his or her own mass unit – ˜ which say is 1/7 of the mass of the Paris prototype: 1 (kg) = 7 (kg). ˜ (kg), ˜ If (kg) is one’s choice of the mass unit, how about the unit system? One really has to make a decision here, which is either keeping all the other units ˜ transition, or, changing some unchanged and only making the (kg) → (kg) ˜ other units along with the (kg) → (kg) transition. Imagine making the first choice, that is, keeping all the other units of the SI system unchanged, including the force unit (N), and only changes the mass unit ˜ That is all right, except in the new unit system the Newton’s from (kg) to (kg). ˜ law must be re-expressed as F = ma/7, because if an object of mass 7x (kg) 2 is undergoing an acceleration of y (m/s ), the force on the object is x y (N). There is nothing inherently wrong with the F = ma/7 expression, which is just a recipe for computation – a correct one for the newly chosen unit system. Fundamentally, F = ma/7 and F = ma describe the same physical law. But it is true that F = ma/7 is less elegant than F = ma. No one likes to memorize extra constants if they can be reduced to unity by a sensible choice of units. The SI unit system is sensible, because (N) is picked to work with other SI units to satisfy F = ma. ˜ as the mass unit? How may we have a sensible unit system but with (kg) ˜ ˜ ˜ Simple, just define (N) = (N)/7 as the new force unit. The (m)–(s)–(kg)–( N)– unit system is sensible because the simplest form of F = ma is preserved. Thus we see that when a certain unit in a sensible unit system is altered, other units must also be altered correspondingly in order to constitute a new sensible unit system, which keeps the algebraic forms of all fundamental physical laws unaltered. (A notable exception is the conversion between SI and Gaussian unit systems in electrodynamics, during which a non-trivial factor of 4π comes up.) In science people have formed deep-rooted conventions about the simplest algebraic forms of physical laws, such as F = ma, K = mv 2 /2, E = K + U , P = ρ RT , etc. Although nothing forbids one from modifying the constant coefficients in front of each expression, one is better off not to. Fortunately, as long as one uses a sensible unit system, these algebraic expressions stays invariant.

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Now, imagine we derive a certain composite law from a set of simple laws. On one side, we start with and consistently use a sensible unit system A. On the other side, we start with and consistently use another sensible unit system B. Since the two sides use exactly the same algebraic forms, the resultant algebraic expression must also be the same, even though for a given physical instance, a variable takes on two different numerical values on the two sides as different unit systems are adopted. This means that the final algebraic expression describing the physical phenomena must satisfy certain concerted scaling invariance with respect to its dependent variables, corresponding to any feasible transformation between sensible unit systems. This strongly limits the form of possible algebraic expressions describing physical phenomena, which is the basis of dimensional analysis. As mentioned, once certain units are altered, other units must be altered correspondingly to make the algebraic expressions of physical laws look invariant. For example, for a single element Lennard–Jones system, one can ˜ = (J), new length unit (m) ˜ = σ (m), and new mass define new energy unit (J) ˜ unit (kg) = m a (kg) which is the atomic mass, where , σ and m a are pure ˜ unit system, the potential energy function is, ˜ m)–( ˜ kg) numbers. In the (J)–( V (r) = 4(r −12 − r −6 ),

(6)

and the mass of an atom is m = 1. Besides that, all physical laws must remain invariant. For example, K = mv 2 /2 in the SI system, and it still should hold ˜ unit system. This can only be achieved if the derived time ˜ kg) in the (J˜)–(m)–( unit (also called reduced time unit), (˜s) = τ (s), satisfies,

m aσ 2 . (7) ˜ v = 1 (m)/(˜ ˜ s), and K = 1/2 (J˜) is a solution To see this, note that m = 1 (kg), 2 ˜ ˜ ˜ kg) unit system, but must also be a solution to to K = mv /2 in the (J)–(m)–( K = mv 2 /2 in the SI system. For Ar, τ turns out to be 2.156 × 10−12 , thus the reduced time unit [˜s] = 2.156 [ps]. This is roughly the timescale of one atomic oscillation period in Ar. = m a σ 2 /τ 2 ,

1.2.

or τ =

Force Calculation

For pair potential of the form (4), there is, fi = −

∂ V (ri j ) j =/i

=

j =/i

∂xi

=

j =/i

1 ∂ V (r) − r ∂r r=ri j

∂ V (r) − ∂r r=ri j

xˆ i j

xi j ,

(8)


573

where xˆ i j is the unit vector, xˆ i j ≡

xi j , ri j

xi j ≡ xi − x j .

(9)

One can define force on i due to atom j ,

fi j ≡

1 ∂ V (r) − r ∂r r=ri j

xi j ,

(10)

and so there is, fi =

fi j .

(11)

j =/i

It is easy to see that, fi j = −f j i .

(12)

MD programs tend to take advantage of symmetries like the above to save computations.

1.3.

Truncation Schemes

Consider the single-element Lennard–Jones potential in (5). Practically we can only carry out the potential summation up to a certain cutoff radius. There are many ways to truncate, the simplest of which is to modify the interaction as

V0 (r) =

V (r) − V (rc ), r < rc . 0, r ≥ rc

(13)

However, V0 (r) is discontinuous in the first derivative at r = rc , which causes large error in time integration (especially with high-order algorithms and large time steps) if an atom crosses rc , and is detrimental to calculating correlation functions over long time. Another commonly used scheme

V1 (r) =

V (r) − V (rc ) − V (rc )(r − rc ), r < rc 0, r ≥ rc

(14)

makes the force continuous at r = rc , but also makes the potential well too shallow (see Fig. 4). It is also slightly more expensive because we have to compute the square root of |xij |2 in order to get r. An alternative is to define V˜ (r) =

V (r) exp(rs /(r − rc )), r < rc 0, r ≥ rc

574

J. Li LJ6-12 potential and its truncated forms

E [ε]

0

⫺0.5

V(r) V0(r) V1(r) W(r)

⫺1 1

1.5

2

2.5

r [σ] Figure 4. Lennard–Jones potential and its modified forms with cutoff rc = 2.37343 σ . Black lines indicate positions of neighbors in a single-element fcc crystal at 0 K.

which has all derivatives continuous at r = rc . However, this truncation scheme requires another tunable parameter rs . The following truncation scheme,  6 18 12 12  σ σ σ σ   4ε − + 2 −    r r rc rc  6 12 6 W (r) = r σ σ  × −3 +2 ,    σ rc rc   

0,

r < rc

(15)

r ≥ rc

is recommended. W (r), V (r), V0 (r) and V1 (r) are plotted in Fig. 4 for comparison. rc is chosen to be 2.37343σ , which falls exactly at the 2/3 interval between the fourth and fifth neighbors at equilibrated fcc lattice of 0 K. There is clearly a tradeoff in picking rc . If rc is large, the effect of the artificial truncation is small. On the other hand, maintaining and summing over a large neighbor list (size ∝ rc3 ) costs more. For a properly written O(N ) MD code, the cost versus neighbor number relation is almost linear. Let us see what is the minimal rc for a fcc solid. Figure 5 shows the neighboring atom shells and their multiplicity. Also drawn are the three glide planes.


575 fcc neighboring shells 68; 86

748; 134

4 12; 54

324; 42

112; 12

origin

524; 78

26; 18

Figure 5. FCC neighboring shells. For example, label “68; 86 ” means there are eight sixth nearest neighbors of the type shown in figure, which adds up to 86 neighbors in all if included. The ABC stacking planes are also shown in the figure.

With (15), once the number of interacting neighbor shells are determined, we can evaluate the equilibrium volume and bulk modulus of the crystal in closed form. The total potential energy of each atom is r j i 1/2) is that we expect thermal expansion at finite temperature. If one is after converged Lennard–Jones potential results, then rc = 4σ is recommended. However, it is about five times more expensive per atom than the minimum-cutoff calculation with rc = 2.37343σ .

2.

Integrators

An integrator serves the purpose of advancing the trajectory over small time increments t: x3N (t0 ) → x3N (t0 + t) → x3N (t0 + 2t) → · · · → x3N (t0 + Lt) where L is usually ∼104 − 107 . Here we give a brief overview of some popular algorithms: central difference (Verlet, leap-frog, velocity Verlet), Beeman’s algorithm [14], predictor-corrector [10], and symplectic integrators [8, 11].

2.1.

Verlet Algorithm

Assuming x3N (t) trajectory is smooth, perform Taylor expansion xi (t0 + t) + xi (t0 − t) = 2xi (t0 ) + x¨ i (t0 )(t)2 + O((t)4 ).

(19)

Since x¨ i (t0 ) = fi (t0 )/m i can be evaluated given the atomic positions x3N (t0 ) at t = t0 , x3N (t0 + t) in turn may be approximated by,

xi (t0 + t) = −xi (t0 − t) + 2xi (t0 ) +

fi (t0 ) (t)2 + O((t)4 ). mi (20)


577

By throwing out the O((t)4 ) term, we obtain a recursion formula to compute x3N (t0 + t), x3N (t0 + 2t), . . . successively, which is the Verlet [15] algorithm. The velocities do not participate in the recursion but are needed for property calculations. They can be approximated by vi (t0 ) ≡ x˙ i (t0 ) =

1 [xi (t0 + t) − xi (t0 − t)] + O((t)2 ). 2t

(21)

To what degree does the outcome of the above recursion mimic the real trajectory x3N (t)? Notice that in (20), assuming xi (t0 ) and xi (t0 − t) are exact, and assuming we have a perfect computer with no machine error storing the relevant numbers or carrying out floating-point operations, the computed xi (t0 + t) would still be off from the real xi (t0 + t) by O((t)4 ), which is defined as the local truncation error (LTE). LTE is an intrinsic error of the algorithm. Clearly, as t → 0, LTE → 0, but that does not guarantee the algorithm works, because what we want is x3N (t0 +t ) for a given t , not xi (t0 +t). To obtain x3N (t0 + t ), we must integrate L = t /t steps, and the difference between the computed x3N (t0 + t ) and the real x3N (t0 + t ) is called the global error. An algorithm can be useful only if when t → 0, the global error → 0. Usually (but with exceptions), if LTE in position is ∼ (t)k+1 , the global error in position should be ∼ (t)k , in which case we call the algorithm a k-th order method. The Verlet algorithm is third order in position and potential energy, but only second order in velocity and kinetic energy. This is only half the story because the order of an algorithm only characterizes its performance when t → 0. To save computational cost, most often one must adopt a quite large t. Higher-order algorithms do not necessarily perform better than lower-order algorithms at practical t’s. In fact, they could be much worse by diverging spuriously (causing overflow and NaN), while a more robust method would just give a finite but manageable error for the same t. This is the concept of the stability of a numerical algorithm. In linear ODEs, the global error e of a certain normal mode k can always be written as e(ωk t, T /t) by dimensional analysis, where ωk is the mode’s frequency. One then can define the stability domain of an algorithm in the ωt complex plane as the border where e(ωk t, T /t) starts to grow exponentially as a function of T /t. To rephrase, a higher-order algorithm may have a much smaller stability domain than the lower-order algorithm even though its e decays faster near the origin. Since e is usually larger for larger |ωk t|, the overall quality of an integration should be characterized by e(ωmax t, T /t) where ωmax is the maximum intrinsic frequency of the molecular system that we explicitly integrate. The main reason behind developing constraint MD [1, 8] for some molecules is so that we do not have to integrate its stiff intramolecular vibrational modes, allowing one to take a larger t, so one can follow longer the “softer modes” that we are more interested in. This is also

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the rationale behind developing multiple time step integrators like r-RESPA [11]. In addition to LTE, there is round-off error due to the computer’s finite precision. The effect of round-off error can be better understood in the stability domain: (1) In most applications, the round-off error LTE, but it behaves like white noise which has a very wide frequency spectrum, and so for the algorithm to be stable at all, its stability domain must include the entire real ωt axis. However, as long as we ensure non-positive gain for all real ωt modes, the overall error should still be characterized by e(ωk t, T /t), since the white noise has negligible amplitude. (2) Some applications, especially those involving high-order algorithms, do push the machine precision limit. In those cases, equating LTE ∼ where is the machine’s relative accuracy, provides a practical lower bound to t, since by reducing t one can no longer reduce (and indeed would increase) the global error. For single-precision arithmetics (4 bytes to store one real number), ∼ 10−8 ; for double-precision arithmetics (8 bytes to store one real number), ≈ 2.2 × 10−16 ; for quadrupleprecision arithmetics (16 bytes to store one real number), ∼ 10−32 .

2.2.

Leap-frog Algorithm

Here we start out with v3N (t0 − t/2) and x3N (t0 ), then,

vi t0 +

t 2

t 2

= vi t0 −

+

fi (t0 ) t + O((t)3 ), mi

(22)

followed by,

xi (t0 + t) = xi (t0 ) + vi

t t0 + 2

t + O((t)3 ),

(23)

and we have advanced by one step. This is a second-order method. The velocity at time t0 can be approximated by,

vi (t0 ) =

2.3.

1 t vi t0 − 2 2

+ vi t0 +

t 2

+ O((t)2 ).

(24)

Velocity Verlet Algorithm

We start out with x3N (t0 ) and v3N (t0 ), then, xi (t0 + t) = xi (t0 ) + vi (t0 )t +

1 2

fi (t0 ) (t)2 + O((t)3 ), mi

(25)


579

evaluate f3N (t0 + t), and then,

1 fi (t0 ) fi (t0 + t) + t + O((t)3 ), vi (t0 + t) = vi (t0 ) + 2 mi mi

(26)

and we have advanced by one step. This is a second-order method. Since we can have x3N (t0 ) and v3N (t0 ) simultaneously, it is very popular.

2.4.

Beeman’s Algorithm

It is similar to the velocity Verlet algorithm. We start out with x3N (t0 ), f3N (t0 − t), f3N (t0 ) and v3N (t0 ), then,

4fi (t0 ) − fi (t0 − t) (t)2 xi (t0 + t) = xi (t0 ) + vi (t0 )t + mi 6 4 + O((t) ),

(27)

evaluate f3N (t0 + t), and then,

2fi (t0 + t) + 5fi (t0 ) − fi (t0 − t) t vi (t0 + t) = vi (t0 ) + , (28) mi 6 and we have advanced by one step. This is a third-order method.

2.5.

Predictor-corrector Algorithm

Let us take the often used 6-value predictor-corrector algorithm [10] as an example. We start out with 6 × 3N storage: x3N(0) (t0 ), x3N(1) (t0 ), x3N(2) (t0 ), . . . , x3N(5) (t0 ), where x3N(k) (t) is defined by,

x(k) i (t)

≡

dk x(ti ) dt k

(t)k k!

.

(29)

The iteration consists of prediction, evaluation, and correction steps:

2.5.1. Prediction step (0) (1) (2) (3) (4) (5) x(0) i = xi + xi + xi + xi + xi + xi , (1) (2) (3) (4) (5) x(1) i = xi + 2xi + 3xi + 4xi + 5xi , (2) (3) (4) (5) x(2) i = xi + 3xi + 6xi + 10xi , (3) (4) (5) x(3) i = xi + 4xi + 10xi , (4) (5) x(4) i = xi + 5xi .

(30)

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J. Li

The general formula for the above is x(k) i =

M−1 k =k

) k ! x(k i , (k − k)!k!

k = 0, . . . , M − 2,

(31)

with M = 6 here. The evaluation must proceed from 0 to M − 2 sequentially.

2.5.2. Evaluation step Evaluate force f3N using the newly obtained x3N(0) .

2.5.3. Correction step Define the error e3N as, ei ≡

x(2) i

−

fi mi

(t)2 . 2!

(32)

Then apply corrections, (k) x(k) i = xi − C Mk ei ,

k = 0, . . . , M − 1,

(33)

where C Mk are constants listed in Table 2. It is clear that the LTE for x3N is O((t) M ) after the prediction step. But one can show that the LTE is enhanced to O((t) M+1 ) after the correction step if f3N depends on x3N only (i.e., is conservative). And so the global error would be O((t) M ).

2.6.

Symplectic Integrators

In the absence of round-off error, certain numerical integrators rigorously maintain the phase space volume conservation property (Liouville’s theorem) of Hamiltonian dynamics, which are then called symplectic. This severely limits the possibilities of mapping from initial to final states, and for this reason symplectic integrators tend to have much better total energy conservation in Table 2. Gear predictor-corrector coefficients C Mk M M M M M

k=0

k=1

=4 1/6 5/6 =5 19/120 3/4 =6 3/20 251/360 = 7 863/6048 665/1008 = 8 1925/14112 19087/30240

k =2 1 1 1 1 1

k=3

k=4

k=5

k=6

k=7

1/3 1/2 1/12 11/18 1/6 1/60 25/36 35/144 1/24 1/360 137/180 5/16 17/240 1/120 1/2520


581 Integration of 1000 periods of Kepler orbitals with eccentricity 0.5

Integration of 100 periods of Kepler orbitals with eccentricity 0.5 0

10

0

10

⫺1

10 ⫺1

10

⫺2

II final (p,q) error II2

II final (p,q) error II2

10 ⫺2

10

⫺3

10

⫺4

10

⫺5

10

⫺6

Ruth83 Schlier98_6a Tsitouras99 Calvo93 Schlier00_6b Schlier00_8c 4th Runge-Kutta 4th Gear 5th Gear 6th Gear 7th Gear 8th Gear

⫺4

10

⫺5

10

⫺6

10

⫺7

10

⫺8

Ruth83 Schlier98_6a Tsitouras99 Calvo93 Schlier00_6b Schlier00_8c 4th Runge-Kutta 4th Gear 5th Gear 6th Gear 7th Gear 8th Gear

10

10

100

⫺3

10

150

200

300

400

500

number of force evaluations per period

600

700

800 900 1000

150

200

300

400

500

600

700 800 900 1000

1200 1400 16001800 2000

number of force evaluations per period

Figure 6. (a) Phase error after integrating 100 periods of Kepler orbitals. (b) Phase error after integrating 1000 periods of Kepler orbitals.

the long run. The velocity Verlet algorithm is in fact symplectic, followed by higher-order extensions [16, 17]. As with the predictor-corrector method which can be derived up to order 14 following the original construction scheme [10], suitable for double-precision arithmetics, symplectic integrators also tend to perform better at higher orders even on a per cost basis. We have benchmarked the two families of integrators (Fig. 6) by numerically solving the two-body Kepler’s problem (eccentricity 0.5) which is nonlinear and periodic, and comparing with the exact analytical solution. The two families have different global error versus time characteristics: non-symplectic integrators all have linear energy error (E ∝ t) and quadratic phase error (|| ∝ t 2 ), while symplectic integrators have constant (fluctuating) energy error (E ∝ t 0 ) and linear phase error (|| ∝ t), with respect to time. Therefore the asymptotic long-term performance of a symplectic integrator is always superior to that of a non-symplectic integrator. But, it is found that for a reasonable integration duration, say 100 Kepler periods, high-order predictorcorrector integrators can have a better performance than the best of the symplectic integrators at large integration timesteps (small number of force evaluations per period). This is important, because it means that in a real system if one does not care about the autocorrelation of a mode beyond 100 oscillation periods, then high-order predictor-corrector algorithms can achieve the desired accuracy at a lower computational cost.

3.

Order- N MD Simulation With Short-ranged Potential

We outline here a linked-bin algorithm that allows one to perform MD simulation in a PBC supercell with O(N ) computational effort per time step, where N is the number of atoms in the supercell (Fig. 7). Such approach

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J. Li

(a)

each timestep: N

2

(b)

(c) 1

2

3

rc

2D usage ratio: 35% ? ?

3D usage ratio: 16% (!)

Figure 7. There are N atoms in the supercell. (a) The circle around a particular atom with radius rc indicates the range of its interaction with other atoms. (b) The supercell is divided into a number of bins, which have dimensions such that an atom can only possibly interact with atoms in adjacent 27 bins in 3D (nine in 2D). (c) This shows that an atom–atom list is still necessary because on average there are only 16% of the atoms in 3D in adjacent bins that interact with the particular atom.

is found to outperform the brute-force Verlet neighbor-list update algorithm, which is O(N 2 ), when N exceeds a few thousand atoms. The algorithm to be introduced here allows for arbitrary supercell deformation during a simulation, and is implemented in large-scale MD and conjugate gradient relaxation programs as well as a visualization program [3]. Denote the three edges of a supercell in Cartesian frame by row vectors h1 , h2 , h3 , which stack together to form a 3 × 3 matrix H. The inverse of the H matrix B ≡ H−1 satisfies I = HB = BH.

(34)

If we define row vectors b1 ≡ (B11, B21, B31),

b2 ≡ (B12, B22, B32 ),

b3 ≡ (B13, B23 , B33), (35)

then (34) is equivalent to hi · b j ≡ hi bTj = δi j .

(36)

Since b1 is perpendicular to both h2 and h3 , it must be collinear with the normal direction n of the plane spanned by h2 and h3 : b1 ≡ |b1 |n. And so by (36), 1 = h1 · b1 = h1 · (|b1 |n) = |b1 |(h1 · n).

(37)


583

But |h1 · n| is nothing other than the thickness of the supercell along the h1 edge. Therefore, the thicknesses (distances between two parallel surfaces) of the supercell are, d1 =

1 1 1 , d2 = , d3 = . |b1 | |b2 | |b3 |

(38)

The position of atom i is specified by a row vector, si = (si1 , si2 , si3 ), with siµ satisfying 0 ≤ siµ < 1, µ = 1, . . . , 3,

(39)

and the Cartesian coordinate of this atom, xi , also a row vector, is xi = si1 h1 + si2 h2 + si3 h3 = si H,

(40)

where siµ has the geometrical interpretation of the fraction of the µth edge in order to build xi . We will simulate particle systems that interact via shortranged potentials of cutoff radius rc (see previous section for potential truncation schemes). In the case of multi-component system, rc is generalized to a matrix rcαβ , where α ≡ c(i), β ≡ c( j ) are the chemical types of atom i and j , respectively. We then define xji . (41) x j i ≡ x j − xi , r j i ≡ |x j i |, xˆ j i ≡ r ji The design of the program should allow for arbitrary changes in H that include strain and rotational components (see Section 2.5). One should use the Lagrangian strain η, a true rank-2 tensor under coordinate frame transformation, to measure the deformation of a supercell. To define η, one needs a reference H0 of a previous time, with x0 = sH0 and dx0 = (ds)H0 , and imagine that with s fixed, dx0 is transformed to dx = (ds)H, under H0 → H ≡ H0 J. The Lagrangian strain (see Chap 2.4) is defined by the change in the differential line length, dl 2 = dx dxT ≡ dx0 (I + 2η)dxT0 ,

(42)

where by plugging in dx = (ds)H = (dx0 )H−1 0 H = (dx0 )J, η is seen to be

η=

1 2

T −T H−1 0 HH H0 − I =

1 2

JJT − I .

(43)

Because η is a symmetric matrix, it always has three mutually orthogonal eigen-directions x1 η = x1 η1 , x2 η = x2 η√ 2 , x3 η = x3 η√ 3 . Along those √ directions, the line lengths are changed by factors 1 + 2η1 , 1 + 2η2 , 1 + 2η3 , which achieve extrema among all line directions. Thus, as long as η1 , η2 and η3 oscillate between [−ηbound , ηbound] for some √ chosen ηbound, any line segment at H0 can√be lengthened by no more than 1 + 2ηbound and shortened by no less than 1 − 2ηbound . That is, if we define length measure √ (44) L(s, H) ≡ sHHT sT ,

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J. Li

then so long as η1 , η2 , η3 oscillate between [ηmin , ηmax ], there is

1 + 2ηmin L(s, H0 ) ≤ L(s, H) ≤

1 + 2ηmax L(s, H0 ).

(45)

One can use the above result to define a strain session, which begins with H0 = H and during which no line segment is allowed to shrink by less than a threshold f c ≤ 1, compared to its length at H0 . This is equivalent to requiring that, f ≡

1 + 2 (min(η1 , η2 , η3 )) ≤ f c .

(46)

Whenever the above condition is violated, the session terminates and a new session starts with the present H as the new H0 , and triggers a repartitioning of the supercell into equal-size bins, which is called a strain-induced bin repartitioning. The purpose of bin partition is the following: it can be a very demanding task to determine if atoms i, j are within rc or not, for all possible i j combinations. Formally, this requires checking r j i ≡ L(s j i , H) ≤ rc .

(47)

Because si , s j and H are all moving – they differ from step to step, it appears that we have to do this at each step. This O(N 2 ) complexity would indeed be the case but for the observation that, in most MD, MC and static minimization procedures, si ’s of most atoms and H often change only slightly from the previous step. Therefore, once we ensured that (47) hold at some previous step, we can devise a sufficient condition to test if (47) still must hold now, at a much smaller cost. Only when this sufficient condition breaks down do we resort to a more complicated search and check in the fashion of (47). As a side note, it is often more efficient to count interaction pairs if the potential function allows for easy use of such half-lists, such as pair- or EAM potentials, which achieves 1/2 saving in memory. In these scenarios we pick a unique “host” atom among i and j to store the information about the i j -pair, that is, a particle’s list only keeps possible pairs that are under its own care. For load-balancing it is best if the responsibilities are distributed evenly among particles. We use a pseudo-random choice of: if i + j is odd and i > j , or if i + j is even and i < j , then i is the host; otherwise it is j . As i > j is “uncorrelated” with whether i + j is even or odd, significant load imbalance is unlikely to occur even if the indices correlate strongly with the atoms’ positions. The step-to-step small change is exploited as follows: one associates each si with a semi-mobile reduced coordinate sai called atom i’s anchor (Fig. 8). At each step, one checks if L(si − sai , H), that is, the current distance between 0 or not. If it is not, then sai i and its anchor, is greater than a certain rdrift ≥ rdrift a does not change; if it is, then one redefines si ≡ si at this step, which is called


585

atom trajectory

d L

anchor trajectory

d

Usually,

d = 0.05rc

Figure 8. This illustrates the concepts of an anchor, which is the relative immbobile part of an atom’s trajectory. Using an anchor–anchor list, we can derive a “flash” condition that locally updates an atom’s neighbor-list when the atom drifts sufficiently far away from its anchor.

atom i’s flash incident. At atom i’s flash, it is required to update records of all atoms (part of the records may be stored in j ’s list, if 1/2-saving is used and j happens to be the host of the i j pair) whose anchors satisfy L(saj − sai , H0 ) ≤ rlist ≡

0 rc + 2rdrift . fc

(48)

Note that the distance is between anchors instead of atoms (sai = si , though), and the length is measured by H0 , not the current H. (48) nominally takes O(N ) work per flash, but we may reduce it to O(1) work per flash by partitioning the supercell into m 1 × m 2 × m 3 bins at the start of the session, whose thicknesses by H0 (see (38)) are required to be greater than or equal to rlist : d1 (H0 ) d2 (H0 ) d3 (H0 ) , , ≥ rlist . m1 m2 m3

(49)

The bins deform with H and remains commensurate with it, that is, its s-width 1/m 1 , 1/m 2 , 1/m 3 remains fixed during a strain session. Each bin keeps an updated list of all anchors inside. When atom i flashes, it also updates the bin-anchor list if necessary. Then, if at the time of i’s flash two anchors are separated by more than one bin, there would be L(saj − sai , H0 ) >

d1 (H0 ) d2 (H0 ) d3 (H0 ) , , ≥ rlist, m1 m2 m3

(50)

and they cannot possibly satisfy (48). Therefore we only need to test (48) for anchors within adjacent 27 bins. To synchronize, all atoms flash at the start of a strain session. From then on, atoms flash individually whenever L(si −sai , H) > rdrift . If two anchors flash at the same step in a loop, the first flash may get it wrong – that is, missing the second anchor, but the second flash will correct the mistake. The important thing here is not to lose an interaction. We see that to maintain anchor lists that captures all solutions to (48) among the latest anchors, it takes only O(N ) work per step, and the pre-factor of which is also 0 . small because flash events happen quite infrequently for a tolerably large rdrift

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The central claim of the scheme is that if j is not in i’s anchor records (suppose i’s last flash is more recent than j ’s), which was created some time ago in the strain session, then r j i > rc . The reason is that the current separation 0 between the anchor i and anchor j , L(saj − sai , H), is greater than rc + 2rdrift , since by (45), (46) and (48), L(saj − sai , H) ≥ f · L(saj − sai , H0 ) > f · rlist ≥ f c · rlist = f c ·

0 rc + 2rdrift . fc (51)

So we see that r j i > rc maintains if neither i or j currently drifts more than f · rlist − rc 0 ≥ rdrift , (52) 2 from respective anchors. Put it another way, when we design rlist in (48), we take into consideration both atom drifts and H shrinkage which both may bring i j closer than rc , but since the current H shrinkage has not yet reached the designed critical value, we can convert it to more leeway for the atom drifts. For multi-component systems, we define rdrift ≡

αβ

rlist ≡

0 rcαβ + 2rdrift , fc

(53)

0 0 where both f c and rdrift are species-independent constants, and rdrift can be thought of as putting a lower bound on rdrift , so flash events cannot occur too frequently. At each bin repartitioning, we would require

d1 (H0 ) d2 (H0 ) d3 (H0 ) αβ , , ≥ max rlist . α,β m1 m2 m3

(54)

And during the strain session, f ≥ f c , we have

α rdrift

≡ min min β

αβ

f · rlist − rcαβ , min β 2

βα

f · rlist − rcβα 2

,

(55)

a time- and species-dependent atom drift bound that controls whether an atom of species α needs to flash.

4.

Molecular Dynamics Codes

At present there are several high-quality molecular dynamics programs in the public domain, such as LAMMPS [18], DL POLY [19, 20], Moldy [21], and some codes with biomolecular focus, such as NAMD [22, 23] and Gromacs [24, 25]. CHARMM [26] and AMBER [27] are not free but are standard and extremely powerful codes in biology.


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References [1] M. Allen and D. Tildesley, Computer Simulation of Liquids, Clarendon Press, New York, 1987. [2] J. Li, L. Porter, and S. Yip, “Atomistic modeling of finite-temperature properties of crystalline beta-SiC - II. Thermal conductivity and effects of point defects,” J. Nucl. Mater., 255, 139–152, 1998. [3] J. Li, “AtomEye: an efficient atomistic configuration viewer,” Model. Simul. Mater. Sci. Eng., 11, 173–177, 2003. [4] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, New York, 1987. [5] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, 2nd edn., Clarendon Press, Oxford, 1954. [6] R. Parr and W. Yang, Density-functional Theory of Atoms and Molecules, Clarendon Press, Oxford, 1989. [7] S.D. Ivanov, A.P. Lyubartsev, and A. Laaksonen, “Bead-Fourier path integral molecular dynamics,” Phys. Rev. E, 67, art. no.–066710, 2003. [8] T. Schlick, Molecular Modeling and Simulation, Springer, Berlin, 2002. [9] W. Press, B. Flannery, S. Teukolsky, and W. Vetterling, Numerical Recipes in C: the Art of Scientific Computing, 2nd edn., Cambridge University Press, Cambridge, 1992. [10] C. Gear, Numerical Initial Value Problems in Ordinary Differential Equation, Prentice-Hall, Englewood Cliffs, NJ, 1971. [11] M.E. Tuckerman and G.J. Martyna, “Understanding modern molecular dynamics: techniques and applications,” J. Phys. Chem. B, 104, 159–178, 2000. [12] S. Nose, “A unified formulation of the constant temperature molecular dynamics methods,” J. Chem. Phys., 81, 511–519, 1984. [13] W.G. Hoover, “Canonical dynamics – equilibrium phase-space distributions,” Phys. Rev. A, 31, 1695–1697, 1985. [14] D. Beeman, “Some multistep methods for use in molecular-dynamics calculations,” J. Comput. Phys., 20, 130–139, 1976. [15] L. Verlet, “Computer “experiments” on classical fluids. I. Thermodynamical properties of Lennard–Jones molecules,” Phys. Rev., 159, 98–103, 1967. [16] H. Yoshida, “Construction of higher-order symplectic integrators,” Phys. Lett. A, 150, 262–268, 1990. [17] J. Sanz-Serna and M. Calvo, Numerical Hamiltonian Problems, Chapman & Hall, London, 1994. [18] S. Plimpton, “Fast parallel algorithms for short-range molecular-dynamics,” J. Comput. Phys., 117, 1–19, 1995. [19] W. Smith and T.R. Forester, “DL POLY 2.0: a general-purpose parallel molecular dynamics simulation package,” J. Mol. Graph., 14, 136–141, 1996. [20] W. Smith, C.W. Yong, and P.M. Rodger, “DL POLY: application to molecular simulation,” Mol. Simul., 28, 385–471, 2002. [21] K. Refson, “Moldy: a portable molecular dynamics simulation program for serial and parallel computers,” Comput. Phys. Commun., 126, 310–329, 2000. [22] M.T. Nelson, W. Humphrey, A. Gursoy, A. Dalke, L.V. Kale, R.D. Skeel, and K. Schulten, “NAMD: a parallel, object oriented molecular dynamics program,” Int. J. Supercomput. Appl. High Perform. Comput., 10, 251–268, 1996. [23] L. Kale, R. Skeel, M. Bhandarkar, R. Brunner, A. Gursoy, N. Krawetz, J. Phillips, A. Shinozaki, K. Varadarajan, and K. Schulten, “NAMD2: Greater scalability for parallel molecular dynamics,” J. Comput. Phys., 151, 283–312, 1999.

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J. Li [24] H.J.C. Berendsen, D. Vanderspoel, and R. Vandrunen, “Gromacs – a messagepassing parallel molecular-dynamics implementation,” Comput. Phys. Commun., 91, 43–56, 1995. [25] E. Lindahl, B. Hess, and D. van der Spoel, “GROMACS 3.0: a package for molecular simulation and trajectory analysis,” J. Mol. Model., 7, 306–317, 2001. [26] B.R. Brooks, R.E. Bruccoleri, B.D. Olafson, D.J. States, S. Swaminathan, and M. Karplus, “Charmm – a program for macromolecular energy, minimization, and dynamics calculations,” J. Comput. Chem., 4, 187–217, 1983. [27] D.A. Pearlman, D.A. Case, J.W. Caldwell, W.S. Ross, T.E. Cheatham, S. Debolt, D. Ferguson, G. Seibel, and P. Kollman, “Amber, a package of computer-programs for applying molecular mechanics, normal-mode analysis, molecular-dynamics and freeenergy calculations to simulate the structural and energetic properties of molecules,” Comput. Phys. Commun., 91, 1–41, 1995.

2.9 GENERATING EQUILIBRIUM ENSEMBLES VIA MOLECULAR DYNAMICS Mark E. Tuckerman Department of Chemistry, Courant Institute of Mathematical Science, New York University, New York, NY 10003

Over the last several decades, molecular dynamics (MD) has become one of the most important and commonly used approaches for studying condensed phase systems. MD calculations generally serve two often complementary purposes. First, an MD simulation can be used to study the dynamics of a system starting from particular initial conditions. Second, MD can be employed as a means of generating a collection of classical microscopic configurations in a particular equilibrium ensemble. The latter of these uses shows that MD is intimately connected with statistical mechanics and can serve as a computational tool for solving statistical mechanical problems. Indeed, even when MD is used to study a system’s dynamics, one never uses just a single trajectory (generated from a single initial condition). Dynamical properties in the linear response regime, computed according to the rules of statistical mechanics from time correlation functions, require an ensemble of trajectories starting from an equilibrium distribution of initial conditions. These points underscore the importance of having efficient and rigorous techniques capable of generating equilibrium distributions. Indeed while the problem of producing classical trajectories from a distribution of initial conditions is relatively straightforward – one simply integrates Hamilton’s equations of motion – the problem of generating the equilibrium distribution for a complex system is an immense challenge for which advanced sampling techniques are often required. Whether or not one is employing MD on its own or combining it with one of a variety of advanced sampling methods, the underlying MD scheme must be tailored to generate the desired distribution. Once such a scheme is in place, it can be employed as is or adapted for advanced sampling techniques such as umbrella sampling [1], the bluemoon ensemble approach [2, 3], or variable transformations [4]. In this contribution, our focus will be on the underlying MD schemes, themselves, and the problem of generating numerical integrators 589 S. Yip (ed.), Handbook of Materials Modeling, 589–611. c 2005 Springer. Printed in the Netherlands.

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for these schemes. The latter is still an open area of research in which a number of important theoretical questions remain unanswered. Thus, we will discuss the current state of knowledge and allude to the outstanding issues as they arise. At this point, it is worth mentioning that equilibrium ensemble distributions are not the sole domain of MD. Monte Carlo (MC) methods and hybrid MD/MC approaches can also be employed. Moreover, advanced sampling techniques designed to work with MC, such as configurational bias MC [5], and with hybrid methods, such as hybrid MC [6], exist as well. To some extent, the choice between MC, MD and hybrid MD/MC approaches is a matter of taste. Each has particular advantages and disadvantages and both allow for creative innovations within their respective frameworks. A particular advantage of the MD and hybrid MD/MC approaches lies in the fact that they lend themselves well to scalable parallelization, allowing large systems and long time scales to be accessed. Indeed, efficient parallel algorithms for MD have been proposed [7] and a wide variety of parallel MD codes are available to the community via the Web, such as the NAMD (www.ks.uiuc.edu/Research/namd) and PINY MD (homepages.nyu.edu/˜mt33/PINY MD/PINY.html) codes, to name just a few. In thermodynamics, one divides the thermodynamic universe into the system and its surroundings. How the system interacts with its surroundings determines the particular ensemble distribution the system will obey. The interaction between the system and its surroundings causes certain thermodynamic variables to fluctuate and others to remain fixed. For example, if the system can exchange thermal energy with its surroundings, its internal energy will fluctuate, however, its temperature will, when equilibrium is reached, be fixed at the temperature of the surroundings. Thermodynamic variables of the system that are fixed due its interaction with the surroundings can be viewed as “control variables,” since they can be adjusted via the surroundings (e.g., changing the temperature of the surroundings will change the temperature of the system if the two can exchange thermal energy). These control variables, therefore, characterize the ensemble. Let us begin our discussion with the simplest possible case, that of a system that has no interaction with its surroundings. Let the system contain N particles in a container of volume V . Let the positions of the N particles at time t be designated r1 (t), . . . , r N (t) and velocities v1 (t), . . . , v N (t), and let the particles have masses m 1 , . . . , m N . In general, the time evolution of any classical system is given by Newton’s equations of motion m i r¨ i = Fi

(1)

where Fi is the total force on the ith particle, and the overdot notation signifies time differentiation, i.e., r˙ i = dri /dt = vi . Thus, r¨ i is the acceleration of the ith particle. Since Newton’s equations constitute a set of 3N coupled second order differential equations, if an initial condition on the positions and

Generating equilibrium ensembles via molecular dynamics

591

velocities r1 (0), . . . , r N (0), v1 (0), . . . , v N (0) is specified, the solution to Newton’s equations will be a unique function of time. For a system isolated from its surroundings, the force on each particle will only be due to its interaction with all of the other particles in the system. Thus, the forces F1 , . . . , F N will be functions only of the particle positions, i.e., Fi = Fi (r1 , . . . , r N ), and, in addition, they will be conservative, meaning that they can be expressed as the gradient of a scalar potential energy function U (r1 , . . . , r N ): ∂ (2) Fi (r1 , . . . , r N ) = − U (r1 , . . . , r N ) ∂ri If a conservative force is taken to act over a closed path that brings a particle back to its point of origin, no net work is done. When only conservative forces act within a system, the total energy E=

N 1 m i v2i + U (r1 , . . . , r N ) 2 i=1

(3)

is conserved by the motion. Given the law of conservation of energy, the equations of motion for an isolated system can be cast in a way that is particularly useful for establishing the connection to equilibrium ensembles, namely, in terms of the classical Hamiltonian. The Hamiltonian is nothing more than the total energy E expressed as a function of the positions and momenta, pi = m i vi . Thus, the Hamiltonian H is a function of these variables, i.e., H = H (p1 , . . . , p N , r1 , . . . , r N ). Introducing the shorthand notation r ≡ r1 , . . . , r N , p ≡ p1 , . . . , p N , and substituting vi = pi /m i into Eq. (3), the Hamiltonian becomes H (p, r) =

N p2i i=1

2m i

+ U (r, . . . , r N )

(4)

The equations of motion for the positions and momenta are then given by Hamilton’s equations ∂ H pi ∂H ∂U = =− (5) p˙ i = − r˙ i = ∂pi m i ∂ri ∂ri It is straightforward to show, by substituting the time derivative of the equation for r˙ i into the equation for p˙ i , that Hamilton’s equations are mathematically equivalent to Newton’s equations (1). It is also straightforward to show that H (p, r) is conserved by simply computing dH/dt via the chain rule:

N ∂H ∂H dH = · r˙ i + · p˙ i dt ∂ri ∂pi i=1

=

N ∂H i=1

=0

∂H ∂H ∂H · − · ∂ri ∂pi ∂pi ∂ri

(6)

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M.E. Tuckerman

(It is important to note that the form of Hamilton’s equations is valid in any set of generalized coordinates q1 , . . . , q3N , p1 , . . . , p3N , i.e., q˙k = ∂ H/∂ pk , p˙ k = −∂ H/∂qk .) Just as for Newton’s equations, given an initial condition, (p(0), r(0)), Hamilton’s equations will generate a unique solution (r(t), p(t)) that conserves the total Hamiltonian, i.e., that satisfies H (p(t), r(t))=constant. This condition tells us that the positions and momenta are not all independent variables. In order to understand what this means, let us introduce an abstract 6N -dimensional space, known as phase space, in which 3N of the mutually orthogonal axes are labeled by the 3N position variables and the other 3N axes are labeled by the 3N momentum variables. Since a classical system is completely specified by specifying all of the positions and momenta, a classical microscopic state, or classical microstate, is represented by a single point in the phase space. The condition H (p, r) = constant defines a (6N − 1)dimensional hypersurface in the phase space known as the constant energy hypersurface. It, therefore, becomes clear that any solution to Hamilton’s equations will, for all time, remain on a constant energy hypersurface determined by the initial conditions. If the dynamics is such that the trajectory is able to visit every point of the constant energy hypersurface given an infinite amount of time, then the trajectory is said to be ergodic. There is no general way to prove that a given trajectory is ergodic, and, indeed, in many cases, an arbitrary solution of Hamilton’s equations will not be ergodic. However, if a trajectory is ergodic, then it will generate a sampling of classical microscopic states corresponding to constant total energy, E. Moreover, since the system is in isolation, the particle number N and volume, V are trivially conserved. The collection of classical microscopic states corresponding to constant N , V , and E comprise the statistical mechanical ensemble known as the microcanonical ensemble. In the microcanonical ensemble, the classical microstates must be distributed according to f (p, r) ∝ δ(H (p, r) − E), which satisfies the equilibrium Liouville equation { f, H } = 0, where {. . . , . . .} is the classical Poisson bracket. Thus, an ergodic trajectory generates, not only the dynamics of the system, but also the complete microcanonical ensemble. This tells us that any physical observable expressible as an average A over the ensemble A =

MN (N, V, E)

dp

dr A(p, r)δ (H (p, r) − E)

(7)

D(V )

of a classical phase space function A(p, r) where M N = E 0 /(N !h 3N ), E 0 is a reference energy, h is Planck’s constant, D(V ) is the spatial domain defined by the containing volume, and (N, V, E) is the microcanonical partition function (N, V, E) = M N

dp D(V )

dr δ (H (p, r) − E)

(8)


593

can be computed from a time average over an ergodic trajectory 1 A = A¯ ≡ lim T →∞ T

T

dt A(p(t), r(t))

(9)

0

In Eq. (8), the phase space volume element dp dr = dp1 · · · dp N dr1 · · · dr N is a 6N -dimensional volume element. The Dirac delta-function δ(H (p, r) − E) restricts the integration over the phase space to only those points that lie on the constant energy hypersurface. Clearly, then, the microcanonical partition function corresponds to the total number of microscopic states contained in the microcanonical ensemble. It is, therefore, related to the entropy of the system S(N, V, E) via Boltzmann’s relation S(N, V, E) = k ln (N, V, E)

(10)

where k is Boltzmann’s constant. From this, it is clear that the partition function leads to other thermodynamic quantities via differentiation. The temperature, pressure and chemical potential, for example, are given by

∂S k∂ ln 1 = = T ∂ E N,V ∂E N,V P ∂S k∂ ln = = T ∂ V N,E ∂V N,E µ ∂S k∂ ln =− = T ∂ N V ,E ∂ N V ,E

(11)

The complexity of the forces in Hamilton’s equations is such that an analytical solution is not possible, and one must resort to numerical techniques. In constructing numerical integration schemes, it is important to preserve two properties characterized by Hamiltonian systems. The first is known as Liouville’s Theorem. For simplicity, let us denote the phase space trajectory (p(t), r(t)) simply by xt , known as the phase space vector. Since the solution, xt to Hamilton’s equations is a unique function of the initial condition x0 , we can express xt as a function of x0 , i.e., xt = xt (x0 ). This designation shows that Hamilton’s equations generate a transformation of the complete set of phase space variables from x0 −→ xt . If we consider a small volume element dxt in phase space, this volume element will transform according to dxt = J (xt ; x0 )dx0

(12)

where J (xt ; x0 ) is the Jacobian |∂ xt /∂ x0 | of the transformation. Liouville’s theorem states that J (xt ; x0 ) = 1 or equivalently that dxt = dx0

(13)

In other words, the phase space volume element is conserved. Liouville’s theorem is a consequence of the fact that Hamiltonian systems have a vanishing

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M.E. Tuckerman

phase space compressibility, κ(x) defined in an analogous manner to the usual hydrodynamic compressibility κ(x) = ∇ · x˙ = =

N ∂

i=1

∂ · p˙ i + · r˙ i ∂pi ∂ri

i=1

∂ ∂H ∂ ∂H − · + · ∂pi ∂ri ∂ri ∂pi

N

=0

(14)

The second property is the time reversibility of Hamilton’s equations. This property implies that if an initial condition x0 is allowed to evolve up to time t, at which point all of the momenta are reversed, the system will, in another time interval of length t, return to the point x0 . Any numerical integration scheme applied to Hamilton’s equations should respect these two properties, as they both ensure that all points of the constant energy hypersurface are given equal statistical weighting, as required by the equilibrium statistical mechanics. A class of integrators that satisfies these conditions are the so called symplectic integrators. In devising a numerical integrator for Hamilton’s equations, it is certainly possible to use a Taylor series approach and expand the solution xt for a short time t = t about t = 0. While this method is adequate for Hamiltonian systems described by Eq. (4), it generally fails for more complicated Hamiltonian forms as well as for non-Hamiltonian systems of the type we will be considering shortly for generating other ensembles. For this reason, we will introduce a more powerful and elegant approach based on operator calculus. This approach begins by recognizing that Hamilton’s equations can be case in a compact form as r˙ i = iLri

p˙ i = iLpi

where a linear operator iL has been introduced (i = iL = =

N ∂H i=1

∂ ∂H ∂ · − · ∂pi ∂ri ∂ri ∂pi

i=1

∂ ∂ · + Fi · m i ∂ri ∂pi

N pi

√

(15) −1) given by

(16)

This operator is known as the Liouville operator. Note that the operator L, itself, is Hermitian. Thus, the equations of motion can be cast in terms of the phase space vector as x˙ = iL x, which has the formal solution x t = eiLt x0

(17)


595

The unitary operator exp(iLt) is known as the classical propagator. Since the classical propagator cannot be evaluated analytically for any but the simplest of systems, it would seem that Eq. (17) is little better than a formal device. In fact, Eq. (17) is the starting point for the derivation of practically useful numerical integrators. In order to use Eq. (17) in this way, it is necessary to introduce an approximation to the classical propagator. To begin, note that iL can be written in the form iL = iL 1 + iL 2

(18)

where iL 1 =

N pi i=1

mi

·

∂ ∂ri

iL 2 =

N

Fi ·

i=1

∂ ∂pi

(19)

Although these two operators do not commute, the propagator exp(iLt) can be factorized according to the Trotter theorem: eiLt = lim

M→∞

eiL 2 t /2M eiL 1 t /M eiL 2 t /2M

M

(20)

where M is an integer. As will be seen shortly, each of the operators in brackets can be evaluated analytically. Thus, the exact propagator could be evaluated by dividing the time t into an infinite number of “steps” of length t/M and evaluating the operator in brackets for each of these steps. While this is obviously not possible in practice, if we approximate M as a finite number, a practical scheme emerges. For finite M, Eq. (20) becomes

eiLt ≈ eiL 2 t /2M eiL 1 t /M eiL 2 t /2M

M

+ O(t 3 /M 2 )

eiLt /M ≈ eiL 2 t /2M eiL 1 t /M eiL 2 t /2M + O(t 3 /M 3 ) eiLt ≈ eiL 2 t /2 eiL 1 t eiL 2 t /2 + O(t 3 )

(21)

where, in the second line, the 1/M power of both sides is taken, and, in the third line, the identification t = t/M is made. The error terms in each line illustrate the difference between the global error in the long-time limit and the error in a single short time step. While the latter is t 3 , the former is t 3 /M 2 = tt 2 , indicating that the error in a long trajectory generated by repeated application of the approximate propagator in Eq. (21) is actually t 2 , despite the fact that the error in the approximate short-time propagator is t 3 . In order to illustrate how to evaluate the action of the approximate propagator in Eq. (21), consider a single particle moving in one dimension. Let q and p be the coordinate and conjugate momentum of the particle. The equations of motion are simply q˙ = p/m and p˙ = F(q). Thus, the approximate propagator becomes ∂ ∂ p ∂ t t F(q) F(q) exp t exp (22) exp[iLt] = exp 2 ∂p m ∂q 2 ∂p

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In order to evaluate the action of each of the operators, we only need the operator identity

exp c

∂ ∂x

f (x) = f (x + c)

(23)

where c is independent of x. This identity can be proved by expanding the exponential of the operator in a Taylor series. This type of operator is called a shift or translation operator because it has the effect of shifting x by an amount c. Applying the operator to the phase space vector (q, p) gives

q(t) p(t)

∂ ∂ p ∂ t t F(q) F(q) = exp exp t exp 2 ∂p m ∂q 2 ∂p 

p ∂  ∂ t = exp exp t F(q) 2 ∂p m ∂q p+

= exp

t ∂  F(q)  2 ∂p p+



q+



= 

p+

t 2

p+

t 2

p+

F(q) + F q + q+



t m

t 2

t m

p+

F(q) + F q +

F q+

t 2 t m

t 2 2m

t mp

p+

p+

F(q)



t 2

  

F(q) 

F(q)

t m



 

F(q)



q + t mp



=



q p



q t 2

t 2 2m

 

(24)

F(q)

Since the last line is just (q(t), p(t)) staring from the initial condition (q, p), the algorithm becomes, after substituting in (q(0), p(0)) for the initial condition: q(t) = q(0) + tv(0) + v(t) = v(0) +

t 2 F(q(0)) 2m

t F(q(0)) + F(q(t)) 2m

(25)

where the momentum has been replaced by the velocity v = p/m. Equation (25) is the well known velocity Verlet algorithm. However, it has been derived in a very powerful way starting from the classical propagator. In fact, the real power of the operator approach is that it can eliminate the need to derive a set of explicit finite difference equations. To see this, note that the velocity Verlet


597

algorithm can be written in the following equivalent way t F(q(0)) 2m q(t) = q(0) + tv(t/2) t v(t) = v(t/2) + F(q(t)) 2m

v(t/2) = v(0) +

(26)

Written in this way, it becomes clear that the three assignments in Eq. (26) correspond to the three operators in Eq. (22), i.e., a shift by an amount (t/2m) F(q(0)) applied to the velocity v(0), followed by a shift of the coordinate q(0) by tv(t/2), followed by a shift of v(t/2) by an amount (t/2m) F(q(t)). Note that the input to each operation is just the output of the previous operation. This fact suggests that one can simply look at an operator such as that of Eq. (22) and directly write the instructions in code corresponding to each operator, only keeping in mind that when the coordinate changes, the force needs to be recalculated. We call this technique of translating the operators in a given factorization scheme directly into instructions in code the direct translation method [8]. Applying this approach to Eq. (22), the following pseudocode could be written down immediately just by looking at the operator expression: v ←− v + t ∗ F/m q ←− q + t ∗ v Call GetNewForce(q, F) v ←− v + t ∗ F/m

!! Shift the velocity !! Shift the coordinate !! Evaluate force at new coordinate !! Shift the velocity

(27)

The velocity Verlet method is an example of a symplectic integrator as can be shown by computing the Jacobian of the transformation (q(0), p(0) → (q(t), p(t)). One could also factorize the propagator according to

exp[iLt] = exp

∂ t p ∂ t p ∂ exp t F(q) exp 2 m ∂q ∂p 2 m ∂q

(28)

and obtain yet another symplectic integrator known as the position Verlet method [9]. The use of the Liouville operator formalism also allows for easy development of integrators capable of exploiting the natural separation of time scales in many complex systems to yield more efficient algorithms [9]. Having seen how to devise numerical integration algorithms for the microcanonical ensemble, we now take up the issue of generating other ensembles. The next case we will consider is that of a system interacting with its surroundings via exchange of thermal energy. If the temperature of the surroundings is T , then, in equilibrium, the system will also have this temperature, and its internal energy will fluctuate. However, since only thermal energy is exchanged with the surroundings, the number of particles N and volume V of the system

598

M.E. Tuckerman

are trivially conserved. Thus, in this case, we have an ensemble whose thermodynamic control variables are N , V and T , known as the canonical ensemble. In this ensemble, the average of any quantity A(p, r) is given by A =

CN Q(N, V, T )

dp

dr A(p, r)e−β H (p,r)

(29)

D(V )

where C N = 1/(N !h 3N ), β = 1/kT , and Q(N, V, T ) is the canonical partition function Q(N, V, T ) = C N

dp

dr e−β H (p,r)

(30)

D(V )

Thermodynamic quantities in the canonical ensemble are given in terms of the partition function as follows: The Helmholtz free energy is A(N, V, T ) = −

1 ln Q(N, V, T ) β

(31)

The pressure, internal energy, chemical potential, and heat capacity at constant volume are given by

∂ ln Q(N, V, T ) P = kT ∂V N,T ∂ ln Q(N, V, T ) E =− ∂β N,V ∂ ln Q(N, V, T ) µ = −kT ∂N V ,T

C V = kβ

2

∂ 2 ln Q(N, V, T ) ∂β 2

(32) N,V

In the canonical ensemble, the surroundings act as a heat bath coupled to the system. Thus, unless we treat explicitly the surroundings that might be present in an actual constant temperature experiment, we cannot determine how this coupling will affect the dynamics of the system. Since this is clearly out of the question, the only alternative is to mimic the effect of the surroundings in a simple way so as to ensure that the system will be driven to generate a canonical distribution. There is no unique way to accomplish this, a fact that has lead practitioners of MD to propose a variety of methods. One class of methods that has become increasingly popular since their introduction are the so called extended phase space methods, originally pioneered by Andersen [10]. In this class of methods, the physical position and momentum variables of the particles in the system are supplemented by additional phase space variables that mimic the effect of the surroundings by controlling the fluctuations in certain quantities in such a way that their averages are


599

consistent with the desired ensemble. For example, in the canonical ensemble, additional variables are used to control the fluctuation in the instantaneous kinetic energy i p2i /2m i such that its average is 3N kT /2. Extended phase space methods based on both Hamiltonian and non-Hamiltonian dynamical systems have been proposed. The former include the original formulation by Nosé [11], and the more recent Nosé-Poincaré method [12]. The latter include the well known Nosé–Hoover [13] and Nosé–Hoover chain approaches [13] as well as the more recent generalized Gaussian moment method [14]. It is not possible to discuss all of these methods here, so we will focus on the Nosé– Hoover and Nosé–Hoover chain approaches, which are among the most widely used. Since these methods are of the non-Hamiltonian variety, it is necessary to review some of the basic statistical mechanics of non-Hamiltonian systems [15, 16]. Consider a non-Hamiltonian system with a generic smooth evolution equation x˙ = ξ(x)

(33)

where ξ(x) is a vector function. A clear signature of a non-Hamiltonian system will be a non-vanishing compressibility, κ(x), although non-Hamiltonian systems with vanishing compressibility exist as well. The consequence of nonzero compressibility is that the Jacobian of the transformation x0 −→ xt is no longer 1, and the Liouville theorem of Eq. (13) does not hold. However, for a large class of non-Hamiltonian systems described by Eq. (33), a generalization of Liouville’s theorem can be derived [15, 16]. This generalization states that a metric-weighted volume element is conserved, i.e.,

g(xt , t)dxt =

g(x0 , 0)dx0

where the metric factor

√

g(xt , t) = e−w(xt ,t )

(34)

g(xt , t) is given by (35)

where the function w(x) is related to the compressibility by κ(xt )=dw(xt , t)/dt. Equation (34) shows that for non-Hamiltonian systems, phase space integrals should use e−w(x,t )dx as the integration measure rather than just dx. This will be an important point in the analysis of the dynamical systems we will be considering. Finally, although Eq. (34) allows for time-dependent metrics, the systems we will be considering all have time-independent metric factors. Suppose the non-Hamiltonian in Eq. (33) has a time-independent metric factor and a set of Nc conservation laws k (x) = Ck , k = 1, . . . , Nc , where k is a function on the phase space and Ck is a constant. Then,if the system is ergodic, it Nc δ(k (x) − Ck ), which will generate a microcanonical distribution f (x) = k=1

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M.E. Tuckerman

satisfies a non-Hamiltonian generalization of the Liouville equation [15, 16]. The corresponding partition function is =

dx e−w(x)

Nc

δ(k (x) − Ck )

(36)

k=1

The first non-Hamiltonian system we will consider for generating the canonical distribution are the Nosé–Hoover equations (NH) [17]. In the Nosé–Hoover system, an additional variable η and its corresponding momentum pη and “mass” Q (so designated because Q actually has units of energy × time2 ) are introduced into a Hamiltonian system as follows: pi r˙ i = mi pη p˙ i = Fi − pi Q pη (37) η˙ = Q N p2i − 3N kT p˙η = mi i=1 The physics embodied in Eqs. (37) is based on the fact that the term −( pη /Q)pi in the momentum equation acts as a kind of dynamic frictional force. Although the average pη = 0, instantaneously, pη can be positive or negative and, therefore, act to damp or boost the momentum. According to the equation for pη , if the kinetic energy is larger than 3N kT /2, pη will increase and have a greater damping effect on the momenta, while if the kinetic energy is less than 3N kT /2, pη will decrease and have a greater boosting effect on the mometa. In this way, the NH system acts as a “thermostat” regulating the kinetic energy so that its average is the correct canonical value. Equations (37) have the conserved energy H =

N p2i i=1

2m i

+ U (r1 , . . . , r N ) +

= H (p, r) +

pη2 + 3N kTη 2Q

pη2 + 3N kT 2Q

(38)

where H (p, r) is the Hamiltonian of the physical system. Moreover, the compressibility of Eqs. (37) is κ(x) =

N ∂

∂pi pη = −3N Q = −3N η˙ i=1

· p˙ i +

∂ p˙η ∂ ∂ η˙ + · r˙ i + ∂ri ∂η ∂ pη

(39)


601

√ This implies that w(x) = −3N η, and the metric factor is g(x) = exp(3N η). If Eq. (38) is the only conservation law, then the partition function generated by Eqs. (37) can be written down as =

dp D(V )

pη2 + 3N kTη − E dr dη d pη e3Nη δ H (p, r) + 2Q

(40)

Performing the integrals over the variables η and pη yields the partition function of the physical subsystem

pη2 1 1 dp dr d pη exp E − H (p, r) − = 3N kT kT 2Q D(V ) √ 2π QkT e E/ kT = dp dr e−H (p,r)/ kT 3N kT

(41)

D(V )

which shows that the partition function for the physical system is canonical apart from the prefactors. Although this analysis would suggest that the NH equations should always produce a canonical distribution, it turns out that if even a single additional conservation law is obeyed by the system, Eqs. (37) will fail [16]. Figure 1 shows that for a simple harmonic oscillator coupled to the NH thermostat, the physical phase space and position and momentum distribution are not those of the canonical ensemble. Note that in N -particle systems, a common additional conservation law is conservation of N total momentum i=1 pi = K, where K is a constant vector. This conservation N Fi = law is obeyed by systems on which no external forces act, so that i=1 0. Conservation of total momentum is an example of a common conservation law in N -particle systems that can cause the NH equations to fail rather spectacularly [16]. A solution to this problem was devised by Martyna et al. [13] in the form of the Nosé–Hoover chain equations. In this scheme, the heat bath variables, themselves, are connected to a heat bath, which, in turn is connected to a heat bath, until a “chain” of M heat baths is generated. The equations of motion are r˙ i =

pi mi

p˙ i = Fi − η˙ k =

pηk Qk

p˙ηk = G k − p˙η M = G M

pη1 pi Q1 k = 1, . . . , M pηk+1 pη Q k+1 k

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M.E. Tuckerman

Figure 1. Simple harmonic oscillator with momentum p, coordinate q, mass m = 1, frequency ω = 1 and temperature kT = 1. Top left: Poincaré section ( pq plane) of the oscillator when coupled to the Nosé–Hoover thermostat with Q = 1 and q(0) = 0, p(0) = 1, η(0) = 0, pη (0) = 1. Middle left: The position distribution function of the oscillator. The solid line is the distribution function generated by the NH dynamics while the dashed line is the analytical result for a canonical ensemble. Bottom left: Same for the momentum distribution. Top right: Poincaré section for the Nosé-Hoover chain scheme with M = 4, q(0) = 0, p(0) = 1, ηk (0) = 0, pηk (0) = (−1)k . Middle right: The position distribution function. The solid line is the distribution function generated by the NHC dynamics while the dashed line is the analytical result. Bottom right: Same for the momentum distribution. In all simulations, the equations of motion were integrated for 5×106 steps using a time step of 0.01 and a fifth-order SY decomposition with n c = 5.

where the heat-bath forces have been introduced and are given by G1 =

N p2i i=1

mi

− 3N kT

Gk =

pη2k−1 Q k−1

− kT

(42)

Equations (42) have the conserved energy H = H (p, r) +

M pη2k k=1

2Q k

+ d N kT η1 + kT

M k=2

ηk

(43)


603

and a compressibility κ(x) = −3N η˙1 −

M

η˙ k

(44)

k=2

By allowing the “length” of the chain to be arbitrarily long, the problem of unexpected conservation laws is avoided. In Fig. 1, the physical phase space and momentum and position distributions for a harmonic oscillator coupled to a thermostat chain of length M = 4 is shown. It can be seen that the correct canonical distribution is obtained. The general proof that the canonical distribution is generated by Eqs. (42) follows the same pattern as for the NH equations. However, if additional conservation laws, such as conservation of total momentum, are obeyed, the NHC equations will still generate the correct distribution [16]. The NHC scheme can be used in a flexible manner to enhance the equilibration of a system. For example, rather than using a single global NHC thermostat, it is also possible to couple many NHCs to a system, one to each of a small number of degrees of freedom. In fact, coupling one NHC to each degree of freedom has been shown to lead to a highly effective method for studying quantum systems via the Feynman path integral using molecular dynamics [18]. In order to develop a numerical integration algorithm for the NHC equations, it is important to keep in mind the modified Liouville theorem, Eq. (34). The complexity of the NHC equation is such that a Taylor series approach cannot be employed to derive a satisfactory integrator, i.e., one that does not lead to substantial drifts in the conserved energy [19]. Thus, the NHC system is an example of a problem on which the power of the Liouville operator method can be brought to bear. We begin by writing the total Liouville operator for Eqs. (42) as iL = iL 1 + iL 2 + iL T

(45)

where iL 1 and iL 2 are given by Eq. (19) and iL T =

M k=1

N M−1 pη ∂ pηk ∂ ∂ pη1 ∂ k+1 + Gk − pi · − pηk Q k ∂ηk ∂ pηk Q1 ∂pi Q k+1 ∂ pηk i=1 k=1

(46) The propagator is now factorized in a manner very similar to the velocity Verlet algorithm

eiLt = eiL T t /2eiL 2 t /2eiL 1 t eiL 2 t /2eiL T t /2 + O t 3

(47)

The only new feature in this scheme is the operator exp(iL T t/2). Application of this operator to the phase space requires some care. Clearly, the operator needs to be further factorized into individual operators that can be applied

604

M.E. Tuckerman

analytically. However, the NHC equations constitute a stiff set of differential equations and, therefore, a simple O(t 3 ) factorization scheme will not be accurate enough. Thus, for this operator, a higher-order factorization is needed. Note that the overall integrator will still be O(t 3 ) despite the use of a higherorder method on the thermostat operator. The higher order method we choose is the Suzuki–Yoshida (SY) scheme [20, 21], which involves the introduction of weighted time steps, w j t, j = 1, . . . , n sy , the value of n sy determines the n order of the method. The weights w j are required to satisfy j sy=1 w j = 1 and are chosen so as to cancel out the lower order error terms. Applying the SY scheme, the operator exp(iL T t/2) becomes eiL T t /2 =

n sy

eiL T w j t /2

(48)

j =1

In order to avoid needed to choose n sy too high, another device can be introduced, namely, simply cutting the time step by a factor of n c and applying the operator in Eq. (48) n c times, i.e., e

iL T t /2

=

n sy nc

eiL Tw j t /2nc

(49)

i=1 j =1

In this way, both n c and n sy can be adjusted so as to minimize the number of operations needed for satisfactory performance of the overall integrator. Having introduced the above scheme, it only remains to specify a particular factorization of the operator exp(iL T w j t/2n c ). Defining δ j = w j t/n c , we choose the following factorization

δj δj ∂ GM = exp exp iL T 2 4 ∂ pη M

δj ∂ Gk × exp 4 ∂ pηk

N

1 k=M−1

δ j pηk+1 ∂ exp − pηk 8 Q k+1 ∂ pηk

δ j pηk+1 ∂ exp − pηk 8 Q k+1 ∂ pηk

δ j pη1 ∂ × exp − pi · 2 Q1 ∂pi i=1 ×

M−1 k=1

M

δ j pηk ∂ exp − 2 Q k ∂ηk k=1

δ j pηk+1 ∂ δj ∂ Gk exp − pηk exp 8 Q k+1 ∂ pηk 4 ∂ pηk

δ j pηk+1 ∂ × exp − pη 8 Q k+1 k ∂ pηk

δj ∂ GM exp 4 ∂ pη M

(50)

Although the overall scheme may seem complicated, the use of the direct translation technique simplifies considerably the job of coding the algorithm.


605

All of the operators appearing in Eq. (50) are either translation operators or operators of the form exp(cx∂/∂ x), the action of which is

exp cx

∂ x = xec ∂x

(51)

We call such operators scaling operators, because the effect is to multiply x by an x-independent factor ec . The examples of Fig. 1 were generated using the above scheme. The last ensemble we will discuss corresponds to a system that interacts with its surroundings through exchange of thermal energy and via a mechanical piston that adjusts the volume of the system until its internal pressure is equal to the external pressure of the surroundings. Such an ensemble will be characterized by constant particle number, N , internal pressure P, and temperature T and is known as the isothermal-isobaric ensemble. In this ensemble, it is necessary to consider all possible values of the volume. Thus, the average of any quantity A(p, r) is given by DN A = (N, P, T )

∞

dV e

−β P V

dp

dr A(p, r)e−β H (p,r)

(52)

D(V )

0

where D N = 1/(N !h 3N V0 ), with V0 being a reference volume, and where the partition function (N, P, T ) is given by (N, P, T ) = D N

∞

dV e

−β P V

dp

dr e−β H (p,r)

(53)

D(V )

0

The thermodynamic quantities defined in this ensemble are the Gibbs free energy, given by G(N, P, T ) = −

1 ln (N, P, T ) β

(54)

and the average volume, average enthalpy, chemical potential, and constantpressure heat capacity, given, respectively, by

∂ ln (N, P, T ) ∂P N,T ∂ ln (N, P, T ) H=− ∂β N,P ∂ ln (N, P, T ) µ = −kT ∂N P,T

V = −kT

C P = kβ

2

∂ 2 ln (N, P, T ) ∂β 2

N,P

(55)

606

M.E. Tuckerman

As with the canonical ensemble,there is no unique way to generate the correct volume fluctuations. Nevertheless, among the various algorithms that have been proposed for constant pressure MD, it can be shown [16] that they do not all generate the correct isothermal-isobaric distribution. We shall, therefore, focus on the Martyna–Tobias–Klein (MTK) algorithm [22], which has been shown to give both the correct phase space and volume distributions. The MTK approach uses both a set of thermostat variables to control the kinetic energy fluctuations as well as a barostat to control the fluctuations in the instantaneous pressure. The latter is given by the virial expression

N N 1 ∂U p2i + ri · Fi − 3V Pint = 3V i=1 m i ∂V i=1

(56)

Finally, the volume V is also treated as a dynamical variable. Thus, the equations of motion take the form pi p + ri r˙ i = m i W 1 p pη pi − 1 pi p˙ i = Fi − 1 + N W Q1 3V p V˙ = W N 1 pξ p2i p˙ = (Pint − P) + − 1 p N i=1 m i Q1 pηk (57) η˙ k = k = 1, . . . , M Qk pη p˙ηk = G k − k+1 pηk Q k+1 p˙η M = G M pξ ξ˙k = k k = 1, . . . , M Qk pξ p˙ξk = G k − k+1 pξk Q k+1 p˙ξ M = G M In Eqs. (57), the variable p with mass parameter W (having units of energy × time2 ) corresponds to the barostat, coupling both to the positions and the momenta. If the system is subject to a set of holonomic constraints, leaving only N f degrees of freedom, then the 1/N factors appearing in Eq. (57) must be replaced by 3/N f in three spatial dimensions. Moreover, note that two Nosé– Hoover chains are coupled to the system, one to the particles and the other to the barostat. This device is particularly important, as the barostat tends to evolve on a much slower time scale than the particles. The heat-bath forces G k are defined by G 1 =

p 2 − kT W

G k =

pξ2k−1

Q k−1

− kT

(58)


607

The MTK equations have the conserved energy M p2 H = H (p, r) + + P V + 2W k=1

+ kT

M

ηk + kT

k=2

M

pη2k pξ2k + 2Q k 2Q k

ξk

+ dN kT η1 (59)

k=1

and a phase space metric factor

g(x) = exp dN η1 +

M

ηk +

k=2

M

ξk

(60)

k=1

In order to prove that the MTK equations generate a correct isothermalisobaric distribution, one needs to substitute Eqs. (60) and (59) into Eq. (36) and perform the integrals over all of the heat bath variables and p following the same procedure as was done for the canonical ensemble. Moreover, since Nosé-Hoover chain thermostats are employed in the MTK scheme, the correct distribution will also be generated even if additional conservation laws, such as total momentum, are obeyed by the system. Integrating the MTK equations is only slightly more difficult than integrating the NHC equations and builds on the technology already developed. We begin by introducing the variable = (1/3) ln(V / V0 ) and writing the total Liouville operator as iL = iL 1 + iL 2 + iL ,1 + iL ,2 + iL T−baro + iL T−part

(61)

where iL 1 =

N pi i=1

N

p ∂ + ri · mi W ∂ri

p ∂ iL 2 = Fi − α pi · W ∂pi i=1 p ∂ iL ,1 = W ∂ ∂ iL ,2 = G ∂ p

(62)

and iL T−part and iL T−baro are defined in an analogous manner to Eq. (46). In Eq. (62), α = 1 + 1/N , and G = α

p2 i i

mi

+

N i=1

ri · Fi − 3V

∂φ − PV ∂V

(63)

608

M.E. Tuckerman

The propagator is factorized in a manner that bears a very close resemblance to that of the NHC equations, namely

t t t exp iL T−part exp iL ,2 exp(i Lt) = exp iL T−baro 2 2 2 t × exp iL 2 exp iL ,1 t exp (iL 1 t) 2 t t t exp iL ,2 exp iL T−part × exp iL 2 2 2 2 t × exp iL T−baro + O(t 3 ) 2

(64)

In evaluating the action of this propagator, the Suzuki–Yoshida decomposition already developed for the NHC equations is applied to the operators exp(iL T−baro t/2) and exp(iL T−part t/2). The operators exp(iL ,1 t) and exp(iL ,2 t/2) are simple translation operators. The operators exp(iL 1 t) and exp(iL 2 t/2) are somewhat more complicated than their microcanonical or canonical ensemble counterparts due to the barostat coupling. The action of the operator exp(iL 1 t) can be determined by solving the differential equation r˙ i = vi + v ri

(65)

for constant vi =pi /m i and constant v = p /W for an arbitrary initial condition ri (0) and evaluating the solution at t = t. This yields the evolution ri (t) = ri (0)ev t + tvi (0)ev t /2

sinh(v t/2) v t/2

(66)

Similarly, the action of exp(i L 2 t/2) can be determined by solving the differential equation v˙ i =

Fi − αv vi mi

(67)

for an arbitrary initial condition vi (0) and evaluating the solution at t = t. This yields the evolution vi (t/2) = vi (0)e−αv t /2 +

t sinh(αv t/4) Fi (0)e−αv t /4 2m i αv t/4

(68)

In practice, the factor sinh(x)/x should be evaluated by a power series for x small to avoid numerical instabilities. These equations together with the Suzuki–Yoshida factorization of the thermostat operators completely define an integrator for the isothermal-isobaric ensemble that can be shown to satisfy Eq. (34). The integrator can be easily coded using the direct translation technique. As an example, the MTK algorithm is applied to the problem of a


609

particle moving in a one-dimensional potential

2π q mω2 V 2 1 − cos (69) 2 4π V where V is the one-dimensional “volume” or box length. The system is coupled to the MTK thermostat/barostat and subject to periodic boundary conditions. Figure 2 shows the position and volume distributions generated together with the analytical results. It can be seen that the method is capable of generating correct distributions of both the phase space and of the volume. We conclude this contribution with a few closing remarks. First, the MTK equations can be generalized [22] to treat anisotropic pressure fluctuations as the Parrinello-Rahman scheme [23]. In this case, one considers the full 3 × 3 φ(q, V ) =

1.5

f(q)

1

0.5

0

0

1

2

3

4

6

8

q 0.5 0.4

f(V)

0.3 0.2 0.1 0 0

2

4

V Figure 2. Top: The position distribution of the system described by the periodic potential of Eq. (69) in the isothermal-isobaric ensemble. The numerical and analytical distributions are shown as the solid and dashed lines, respectively. Bottom: Same for the volume distribution. Nosé–Hoover chain lengths of 4 were coupled to the particle and to the barostat. The mass m and frequency ω were both taken to be 1, W = 18, kT = 1, P = 1, Q k = 1, Q k = 9. The time step was taken to be 0.005, and the equations of motion were integrated for 5×107 steps using a seventh-order SY scheme with n c = 6.

610

M.E. Tuckerman

cell matrix h = (a, b, c), where a, b, and c, which form the columns of h, are the three cell vectors. The partition function for this ensemble is (N, P, T ) =

1 dh e−β Pdet(h) [det(h)]2

dp

dr e−β H (p,r)

(70)

D(h)

Although we will not discuss the equations of motion here, we remark that it is important to generate the correct factors of det(h) (recall det(h) = V ) in the distribution. The generalized MTK algorithm has been shown to achieve this. Next, the reader may have noticed the glaringly obvious absence of a pure MD based approach to the grand canonical ensemble. Although a number of important proposals for generating this ensemble via MD have appeared in the literature, there is no standard, widely adopted approach to this problem, as is the case for the canonical and isothermal-isobaric ensembles, and the development of such a method for the grand canonical ensemble remains an open question. The main problem with the grand canonical ensemble comes from the need to treat the fluctuations in a discrete variable, N . Here, adiabatic dynamics techniques adopted to allow slow insertion and deletion of particles in the system at constant chemical potential might be useful. Finally, although we encourage the use of the Liouville operator approach in developing integrators for new sets of equations of motion, this method is not foolproof and must be used with some degree of caution, particularly for nonHamiltonian systems. Not every factorization scheme applied to the propagator of a non-Hamiltonian system is guaranteed to preserve the phase space volume as Eq. (34) requires. Although significant attempts have been made to develop a general procedure for devising such factorization schemes, not enough is known at this point about the phase space structure of non-Hamiltonian systems for a truly general theory of numerical integration, so that this, too, remains an open area. An advantage, however, of the Liouville operator approach is that it renders the problem of combining the NHC and MTK schemes with multiple time scale methods [9] and constraints [24] relatively transparent.

References [1] G.M. Torrie and J.P. Valleau, “Nonphysical sampling distributions in Monte Carlo free energy estimation: umbrella sampling,” J. Comp. Phys., 23, 187, 1977. [2] E.A. Carter, G. Ciccotti, J.T. Hynes, and R. Kapral, “Constrained reaction coordinate dynamics for the simulation of rare events,” Chem. Phys. Lett., 156, 472, 1989. [3] M. Sprik and G. Ciccotti, “Free energy from constrained molecular dynamics,” J. Chem. Phys., 109, 7737, 1998. [4] Z. Zhu, M.E. Tuckerman, S.O. Samuelson, and G.J. Martyna, “Using novel variable transformations to enhance conformational sampling in molecular dynamics,” Phys. Rev. Lett., 88, 100201, 2002.


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[5] J.I. Siepmann and D. Frenkel, “Configurational bias Monte Carlo – a new sampling scheme for flexible chains,” Mol. Phys., 75, 59, 1992. [6] S. Duane, A.D. Kennedy, B.J. Pendleton, and D. Roweth, “Hybrid Monte Carlo,” Phys. Lett. B, 195, 216, 1987. [7] S. Plimpton, “Fast parallel algorithms for short-range molecular dynamics,” J. Comput. Phys., 117, 1, 1995. [8] G.J. Martyna, M.E. Tuckerman, D.J. Tobias, and M.L. Klein, “Explicit reversible integrators for extended systems dynamics,” Mol. Phys., 87, 1117, 1996. [9] M.E. Tuckerman, G.J. Martyna, and B.J. Berne, “Reversible multiple time scale molecular dynamics,” J. Chem. Phys., 97, 1990, 1992. [10] H. Andersen, “Molecular dynamics at constant temperature and/or pressure,” J. Chem. Phys., 72, 2384, 1980. [11] S. Nosé, “A unified formulation of the constant temperature molecular dynamics methods,” J. Chem. Phys., 81, 511, 1984. [12] S.D. Bond, B.J. Leimkuhler, and B.B. Laird, “The nosé–poincaré method for constant temperature molecular dynamics,” J. Comput. Phys., 151, 114, 1999. [13] G.J. Martyna, M.E. Tuckerman, and M.L. Klein, “Nosé–Hoover chains: the canonical ensemble via continuous dynamics,” J. Chem. Phys., 97, 2635, 1992. [14] Y. Liu and M.E. Tuckerman, “Generalized Gaussian moment thermostatting: a new continuous dynamical approach to the canonical ensemble,” J. Chem. Phys., 112, 1685, 2000. [15] M.E. Tuckerman, C.J. Mundy, and G.J. Martyna, “On the classical statistical mechanics of non-Hamiltonian systems,” Europhys. Lett., 45, 149, 1999. [16] M.E. Tuckerman, Y. Liu, G. Ciccotti, and G.J. Martyna, “Non-Hamiltonian molecular dynamics: Generalizing Hamiltonian phase space principles to non-Hamiltonian systems,” J. Chem. Phys., 115, 1678, 2001. [17] W.G. Hoover, “Canonical dynamics – equilibrium phase space distributions,” Phys. Rev. A, 31, 1695, 1985. [18] M.E. Tuckerman, B.J. Berne, G.J. Martyna, and M.L. Klein, “Efficient molecular dynamics and hybrid Monte Carlo algorithms for path integrals,” J. Chem. Phys., 99, 2796, 1993. [19] M.E. Tuckerman and G.J. Martyna, Comment on “Simple reversible molecular dynamics algorithms for No´se–Hoover chain dynamics,” J. Chem. Phys., 110, 3623, 1999. [20] H. Yoshida, “Construction of higher-order symplectic integrators,” Phys. Lett. A, 150, 262, 1990. [21] M. Suzuki, “General-theory of fractal path-integrals with applications to many-body theories and statistical physics,” J. Math. Phys., 32, 400, 1991. [22] G.J. Martyna, D.J. Tobias, and M.L. Klein, “Constant-pressure molecular-dynamics algorithms,” J. Chem. Phys., 101, 4177, 1994. [23] M. Parrinello and A. Rahman, “Crystal-structure and pair potentials – a moleculardynamics study,” Phys. Rev. Lett., 45, 1196, 1980. [24] J.P. Ryckaert, G. Ciccotti, and H.J.C. Berendsen, “Numerical-integration of cartesian equations of motion of a system with constraints – molecular-dynamics of n-alkanes,” J. Comput. Phys., 23, 327, 1977.

2.10 BASIC MONTE CARLO MODELS: EQUILIBRIUM AND KINETICS George Gilmer1 and Sidney Yip2 1 Lawrence Livermore National Laboratory, P.O. box 808, Livermore, CA 94550 USA 2

Department of Nuclear Science and Engineering and Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

1.

Monte Carlo Simulations in Statistical Physics

Monte Carlo (MC) is a very general computational technique that can be used to carry out sampling of distributions. Random numbers are employed in the sampling, and often in other parts of the code. One definition of MC based on common usage in the literature is, any calculation that involves significant applications of random numbers. Historical accounts place the naming of this method in March 1947, when Metropolis suggested it for his method of evaluating the equilibrium properties of atomic systems, and this is the application that we will discuss in this section [1]. An important sampling technique is the one named after Metropolis, which we will describe below. There are several areas of computation besides the statistical mechanics of atomic systems where MC is used. An efficient method for the numerical evaluation of many-dimensional integrals is to apply random sampling techniques on the integrand [2]. A second application is the simulation of random walk diffusion processes in statistical mechanics and condensed matter physics [3]. Tracking particles and radiation (neutrons, photons, charged particles) during transport in non-equilibrium systems is another important area [4–7]. Models for crystal growth, ion implantation, radiation damage and other nonequilibrium systems often make use of random numbers. For example, in most of the MC models of ion implantation, the positions where the ions impinge on the surface of the target are selected using random numbers, whereas the trajectories of ions and target atoms are calculated deterministically using atomic collision theory. In models with diffusion, such as crystal growth and the annealing of radiation damage, the decision on which direction to move a particle or defect performing a random walk will be determined by random numbers. 613 S. Yip (ed.), Handbook of Materials Modeling, 613–628. c 2005 Springer. Printed in the Netherlands.

614

1.1.

G. Gilmer and S. Yip

Metropolis Sampling

In statistical physics one can find the average of a property A({r }) that is a function of the coordinates {r} of N particles, in a system that is in thermodynamic equilibrium,

A =

d3N r A({r }) exp[−U ({r})/kT ] . d3N r exp[−U ({r})/kT ]

(1)

The calculation involves averaging the dynamical variable of interest, A, which depends on the positions of all the particles in the system, over an appropriate thermodynamic ensemble. Often the canonical ensemble is chosen; one with a fixed number of particles, volume and temperature, N , V , and T . In this case the configurations are weighted by the Boltzmann factor exp[−U ({r})/kT ], where U is the potential energy of the system, and k the Boltzmann constant. Integration is over the positions of all particles (3N coordinates). The denominator in Eq. (1) is needed for normalization, and is an important quantity in its own right, because the Helmholtz free energy can be obtained from it (for a system with the independent variables, N , V , and T ). We consider two ways to perform the indicated integral. It is clearly overkill to integrate over all of configuration phase space, because the number of integrals is 3N , where N may have values of thousands or millions. The selection of some representative points seems like a reasonable alternative. One approach is to sample distinct configurations randomly and then obtain A by approximating Eq. (1) by a sum over a set of configurations A =

i=1

A({r }i ) exp[−U ({r}i )/kT ] . i=1 exp[−U ({r}i )/kT ]

(2)

The configurations could be selected by use of a random number generator. One could obtain coordinates to assign to the N atoms with which to fill the cell with N atoms using a sequence of numbers ξ i that are uniform in the range (0, 1), and scaling 3N values of ξ by the edge lengths of the rectangular computational cell. However this procedure would also be grossly inefficient. In a solid or liquid system, many of the atoms in such a random configuration would be overlapping, giving a huge potential energy, and hence a negligible weight, exp[−U ({r})/kT ], in the sampling procedure. The net result is that only a small fraction of “low energy” configurations would determine the value of A, and even these configurations would likely have potential energies much larger than the actual value U . To get around this difficulty, a second approach may be used, where the sampled configurations are picked in a way that is biased by the probability that they will appear in the equilibrium ensemble, i.e., using the factor exp[−U ({r})/kT ]. Then A is determined by weighing the contribution from

Basic monte carlo models: equilibrium and kinetics

615

each configuration equally, since the bias in the selection of configurations accounts for the Boltzmann weighting factor, A =

,Cn i=1

A({r i })

,Cn i=1

δii

,

(3)

where {r}i are configurations sampled from the biased distribution, as indicated by Cn above the summation sign. (The denominator is simply the number of states summed over, or .) How does one do this biased summation? One way is to adopt a procedure developed by Metropolis et al. in 1953 [8]. This procedure is an example of the concept of importance sampling in MC methods [9].

1.2.

Metropolis Sampling

One option for obtaining a set of configurations biased by exp[−U ({r})/ kT ] is to take small excursions from an initial configuration that has a low energy U ({r}). The initial coordinates could be the coordinates of N atoms in a perfect crystalline lattice structure at 0 K. Then, an atom is picked at random, and given a displacement that is small enough that the atom will not approach a neighbor too closely, and yet long enough to produce a significant displacement or change in the system energy. Let the initial position of the particle be (x, y, z). Imagine now displacing the particle from its initial position to a trial position (x + αξx , y + αξ y , z + αξz ), where α is a constant, and ξx , ξ y , and ξz are uniform in the interval (−1, 1). The value of α for obtaining the optimum sampling of phase space depends on the conditions, including density and T , among others. This could be determined from a preliminary run, or optimized as the simulation proceeds. With this move the system goes from configuration {r} j → {r} j +1 . The Metropolis procedure now consists of four steps. 1. Move system in the way just described. 2. Calculate U = U (final) − U (initial) = Uj +1 − Uj , i.e., U is the energy change resulting from the move. 3. If U < 0, accept the move. This means leaving the particle in its new position. 4. If, U > 0, accept the move provided ξ <exp[−U/kT ], where ξ is a fourth random number in the interval (0, 1). The Metropolis sampling technique generates a series of configurations, each of which is closely related to the previous one. This is true because of the small change on the total configuration affected by the displacement of only one atom. The series is, however, a Markov chain, since it satisfies the condition that the new configuration is derived from the previous one, without

616


i i⫹1 Markov chain (Time)

i⫹2

i⫹3

i⫹4

Figure 1. Illustration of the chain of states created by the Metropolis algorithm for a model of group of adatoms on a crystal surface. Each state differs from the one preceding it by the displacement of one atom to a neighboring lattice site.

taking into account the history of states before it. This is very different from molecular dynamics (MD) simulations, where the momentum of the particles plays an important role in determining the configuration of the next iteration. Figure 1 shows a schematic of the states generated by the Metropolis algorithm for a lattice gas, modeling a group of adatoms on the (100) face of a crystal. The elementary move in this model is a diffusion hop of an atom to a neighboring lattice site, and clearly the four hops in this series left much of the system unchanged. We see that it is the uphill moves of step 2 that account for the effect of temperature on the distribution of the system over the energy states. High temperature increases the magnitude of the Boltzmann factor, and therefore the probability of acceptance of moves that increase the energy of the system. If not for step 2, step 3 would only allow the system to go downhill in energy U , which would mean that the system of atoms would lose potential energy systematically and end up in a local energy minimum.

2.

Proof that Metropolis Sampling Results in a Canonical Ensemble

One can show that the Metropolis procedure allows one to sample the distribution of states biased by exp[−U/kT ]. Consider two states (configurations) of the system, i and j , and let Ui > U j . According to the Metropolis procedure, the probability of an (i → j ) transition is Pi ν ij , where Pi is the probability that the system is in state i, and ν ij is the transition probability that a system in state i will go to state j . Similarly, the probability of a ( j → i) transition is P j ν ij exp[−(Ui − U j )/kT ], where we have used the fact that ν j i= ν ij exp[−(Ui − U j )/kT ] according to the Metropolis procedure described above. At equilibrium the two transitions must have equal probabilities, otherwise the populations of some states in the ensemble could be increasing in probability, others decreasing, and the system would not be in equilibrium. This is the principle of microscopic reversibility, or detailed balance. Figure 2 shows


617

Vij Vji

state i

state j

Vij exp (⫺Ui/kT) ⫽ Vji exp (⫺Uj/kT) Figure 2. The microscopic reversibility condition on the transition rates (or probabilities) between two states. This condition is necessary to insure that there is an equilibrium state for the system.

an example of this for a lattice model of an atomic system. Thus, equating the probability of an (i → j ) transition to that for the reverse transition, we find: Pi = P j exp[−(Ui − U j )/kT ] or Pi = C exp[−Ui /kT ] and P j = C exp[−U j /kT ],

(4)

where C is a normalization constant. Whereas (4) relates the probability of finding the ensemble in state i to that for state j , based on the direct transitions between the two states, it also applies to states without direct transitions. Of course, a system can reach internal equilibrium only if there is a sequence of states, connected by direct transitions, between any two states in the system. That is, all of the states are interconnected. Any model that does not satisfy this condition will have isolated pockets of states in phase space that will not equilibrate with each other. But, a system of states that are interconnected in this way will have all states satisfying Eq. (4), which is the canonical ensemble. This completes the proof of the Metropolis sampling method. Stated again, the Metropolis method is an efficient way to sample states of the system with a bias equal to the Boltzmann factor, and that has the same form as the canonical distribution in thermodynamics. It is worthwhile to note that this method can be used in optimization problems, where one is interested in finding the global minimum of multidimensional parameters. One example is to calculate the optimum arrangement of the components of a silicon device to minimize the path length of electrical interconnect lines. The analog of energy is the total length of the conducting lines. The method is better than the standard energy minimization methods such as the conjugate gradient procedure, because it allows the system energy (length of interconnect lines) to increase occasionally in the search for the global minimum. This feature allows it to surmount energy

618


barriers and visit more than one global minimum. The approach to optimization problems is similar to that used to find the global minimum in the energy of an atomic system. A large initial “annealing temperature” is chosen, since this allows the system to pass between global minima. The “temperature” is then reduced in steps for annealing until eventually reaching zero temperature and a minimum, hopefully the global minimum, and the desired optimum value. This is the basis of the “simulated annealing” algorithm used for optimization problems [10].

3.

Free Energy Calculations

As mentioned earlier, the Helmholtz free energy of an atomic system can be obtained from an integration of the Boltzmann factor over phase space, and this is given by

F = −kT · ln V−N

d3N r exp[−U ({r})/kT ] ,

or in an unbiased sample equivalent to Eq. (2), F = −kT · ln

i=1

exp[−U ({r}i )/kT ] . i=1 δii

(5)

This result is not very helpful for obtaining F, however, for the same reason that Eq. (2) is not a useful way to average properties in a canonical ensemble. Again, the major contributions to the sum in (5) occur in well-ordered configurations with atoms avoiding close encounters with their neighbors, whereas the random sampling approach will yield very few such low potential energy configurations indeed. The Metropolis algorithm does not help either. In the biased sample derived from the Metropolis technique, the equivalent of (5) includes a term that cancels the bias in the sum in the numerator and denominator. For purposes of understanding we can assume that we sum over the same states, but that the number of times a given state is included in the Metropolis series, or its degeneracy, is proportional to the Boltzmann factor. That is, each state in the canonical ensemble has, in effect, been multiplied by exp[−U ({r}i )/kT ] because of the preferential choice of states with low potential energy. Therefore to obtain the equivalent of Eq. (5) in the canonical ensemble sums, we simply multiply each term of the sums by exp[U ({r}i )/kT ], giving

F = −kT · ln

,Cn ,Cn

i=1

δii

exp[U ({r}i )/kT ] 1 = −kT · ln . exp[U ({r}i )/kT]Cn

i=1

(6)


619

Although the evaluation of exp[U ({r}i )/kT ]Cn by the Metropolis method is a valid way to obtain the free energy, it is also totally impractical. The sum in the denominator of the middle expression in Eq. (6) will not be evaluated accurately, since it is large when bias factor is small, and vice versa. Therefore all states are equally important for evaluating the average exp[U ({r}i )/kT ]Cn , with the bias factor canceling the exponential in each term of the sum. Importance sampling fails, because each term is equally important, even states that have essentially zero probability of appearing in the ensemble, because these terms are multiplied by the huge exponential, exp[U ({r}i )/kT ]. One approach to calculating the free energy of a system of atoms is to relate it to a known reference system, i.e., a set of Einstein oscillators. If we define a potential energy U ({r}i ) = λU1 ({r}i ) + (1−λ)U0 ({r}i ), then when λ goes from 0 to 1, the potential energy goes from that corresponding to the interatomic potential for U0 ({r}i ) to that for U1 ({r}i ). Differentiating Eq. (5) with respect to λ, using our definition of U ({r}i ), we obtain

(U1 ({r }i ) − U0 ({r}i )) exp[−U ({r}i )/kT ] ∂F , = i=1 ∂λ i=1 exp[−U ({r}i )/kT ∂F = U1 ({r}i ) − U0 ({r i })Cn , ∂λ

or

(7) (8)

where the sampling in Eq. (8) is over an ensemble weighted with the Boltzmann factor exp[−{λU1{r }i + (1 − λ)U0{r }i }/kT ]. Integration of the derivative of F with respect to λ then gives the change in F between the reference state and the state with the desired configuration. Another method known as “umbrella sampling” has been used in situations where is it desired to compare two systems with almost identical interatomic potentials, or with slightly different temperatures [11,12]. If the interatomic potential is changed only a small amount, U ({r}i ) = U ({r}i ) − U0 ({r}i ), then it may be possible to make accurate calculations of the differences in the free energies or other properties A in a single Metropolis MC run. Then one chooses an “unphysical” bias potential, exp[−UUMB ({r}i )/kT , that will, ideally, reproduce the minimum values of both U0 ({r}i ) and of U ({r}i ). Then A0 is given by A0 =

,CnUMB i=1

A({r }i ) exp[−U0 + UUMB ]/kT

,CnUMB i=1

δii exp[−U0 + UUMB ]/kT

,

(9)

as discussed in Ref. [8]. Comparing Eq. (9) with Eq. (3) and the discussion following it, we see that the modified Metropolis method generates only one set of configurations, based on the bias potential, but that the average value of A must be calculated from these configurations weighted by the appropriate exponential, as shown in (9). An analogous expression holds for A for the interatomic potential giving U ({r}i ). Accurate results are only obtained for

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small differences in the potential, and if the size of the atomic configuration is less than several hundred atoms. The choice of bias functions is also crucial for accurate results. But the selection of these functions usually requires some laborious trial-and-error runs. A more complete discussion on methods to obtain free energy differences is given in Chapter 2.15 by de Koning and Reinhardt. MC methods have a number of advantages over MD for obtaining free energies and other equilibrium properties. The ability to bias the sampling process and transition rates while retaining the conditions for an equilibrium ensemble provides some powerful methodologies. One of these applies to the evaluation of the properties of metastable and other defects such as dislocations, surfaces, and interfaces. Because of the small number of atoms involved compared to the total number in the system, statistical noise from the fluctuations in the bulk system will interfere with the measurement of the relatively small impact of the defect on the properties of the atomic system. MC methods allow the concentration of events on the region around the defect being investigated, while retaining the essential condition of microscopic reversibility. In this way, slowly relaxing regions can be allowed to approach a metastable equilibrium without spending most of the computer time on a less important part of the system. Slow structural rearrangements can be accommodated at the interface, without spending computer power simulating the uninteresting parts of the system as they perform their equilibrium fluctuations. MD simulations tend to be more efficient computationally than MC in the case where a system of atoms is being equilibrated at a new temperature or some other change in its conditions is implemented. The advantage for MD results from the fact that the displacements of the atoms during an MD time step are quite different from those discussed earlier for the MC methods. With classical MC, a displacement of a particle has nothing to do with the environment of the particle, but is chosen by random numbers along the three orthogonal coordinate axes. A particle that is close to a neighbor and therefore in a strong repulsive force field may be given a displacement moving it even closer. Such a move will likely cause a large increase in energy and be rejected, but the cost of generating the random numbers for the unsuccessful move affects the efficiency of the process. Furthermore, coordinated moves of a number of particles such as those moving into a region of reduced pressure are not possible with Metropolis MC, whereas their presence in MD allows fast relaxation of a pressure pulse or recovery from artificial initial conditions. Force bias MC was developed to speed up MC relaxation of atomic systems [13]. In this technique, atomic displacements with a large component in the same direction as the force on an atom are selected preferentially to those that are mainly in a direction orthogonal to the force. To maintain microscopic reversibility, atoms moving against the force must also be given a larger selection probability, but


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since they are likely to be moving uphill in energy and to have their move rejected, the result is that more atoms move in the desired direction. This technique is found to be effective and to increase the speed of relaxation in many MC systems. But the calculation of the forces requires extra computer time, so that some applications are still faster if done by basic MC methods [13]. In cases where the flexibility of the MC technique provides strong advantages, it is likely to be advantageous to implement the force-bias algorithm.

4.

Kinetic Interpretation of MC [6]

The Metropolis algorithm was developed primarily for obtaining equilibrium properties of a physical system. Strictly speaking, however, the method never reaches complete equilibrium condition; that is, states whose appearance in an ensemble occurs with the probability Pi = C exp[−Ui /kT ]. Consider the behavior of an infinite ensemble, i.e., an infinite number of identical computational cells, and all starting in the same state, but run with different random number sequences. Calculate the ensemble average properties Ai at each MC step i, starting with the initial state i = 0. In other words, we obtain the average of the system property A by averaging over the computational cells composing the ensemble after each MC event. This differs from the usual procedure, where A is averaged over the successive states of a single computational cell generated by the Metropolis method. The ensemble average Ai will initially have properties similar to the initial state, A0 , since most of the atoms will be in the same position as the starting state. Unless the initial state has very unusual properties, Ai will change its value as i increases, and eventually approach an asymptotic value corresponding to equilibrium, with Pi = C exp[−Ui /kT ]. The approach to the equilibrium ensemble is a property of the system “kinetics,” and depends strongly on the probabilities for transitions between states, ν ij . The ν ij can be thought of as transition rates, in which case the approach to the equilibrium ensemble can be plotted as a function of time instead of MC event number i. A transition with U < 0 has the highest probability ν ij , and would correspond to the highest transition rate. However, transition rates proportional to the Metropolis transition probabilities are unphysical, and would not yield the kinetics of any real system. For this purpose, it is necessary to obtain rate constants for atomic diffusion, chemical reactions, and other unit mechanisms that are relevant for the physical system being studied. These may be obtained by the use of interatomic potentials in molecular dynamics simulations as discussed in preceding chapters, or from molecular dynamics or saddle point evaluations using density functional theory as discussed in Chapter 1.

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Kinetic Monte Carlo (KMC) is similar to equilibrium MC, but with transition rates appropriate for real systems. It can be applied both to equilibrium conditions and to conditions where the system is out of equilibrium. In order to distinguish KMC from equilibrium MC, we will use different terminology. Let P(x, t) be the probability that the system configuration is x at time t. Note that the configuration previously represented by {r}i is now simply x. Then P(x, t) satisfies the equation dP(x, t) W (x → x )P(x, t) + W (x → x)P(x , t), =− dt x x

(10)

where W (x → x ) is the transition probability per unit time of going from x to x (W is analogous to ν ij in the Metropolis method above). Equation (10) is called the Master equation. For the system to be able to reach equilibrium, as discussed above, the transition probabilities must satisfy the condition of microscopic reversibility, (cf. Eq. (4)). Peq (x)W (x → x ) = Peq (x )W (x → x).

(11)

At equilibrium, P(x, t) = Peq (x) and dP(x, t)/dt = 0. Since the probability of occupying state x is Peq (x) =

1 exp[−U (x)/kT ], Z

(12)

where Z is the partition function, Z = i exp[−U ({r}i )/kT ], and (11) gives the basic condition that must be satisfied by the transition probabilities imposed by microscopic reversibility, we have W (x → x ) = exp[{U (x) − U (x )}/kT ]. W (x → x)

(13)

Equation (13) is satisfied by the Metropolis procedure, but other transition rates also satisfy this condition. As we noted above, the Metropolis procedure is unphysical, but real systems also have equilibrium states when the transition rates that satisfy Eq. (13).

5.

Lattice MC: Crystal Growth

Kinetic Monte Carlo models of thin film and crystal growth are often based on the simplification of the lattice model, where atoms are confined to lattice sites on a perfect crystal lattice. We introduced a simple case in Fig. 1, where we discussed a model of a group of atoms diffusing on a crystal surface, and the model consisted of moving the atoms between lattice sites corresponding to a square array of binding sites on a fcc(100) substrate.


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The potential energies of the KMC lattice gas model (KMC LG) can be obtained from empirical interatomic potentials developed for MD simulations, or from simple bond-counting methods if the properties of the model are not required to match experiments. Usually the interactions are limited to nearest neighbors, although the embedded atom potentials have an effective range that is greater than the cut-off value because of indirect interactions through the embedding function. Thus, a potential that has an embedding function and pair interaction limited to first neighbors actually has interactions extending to second or third neighbors. Most KMC LG models do not account for stress fields, and as a result the potential energies U (x) ¯ take on discrete values. The Boltzmann factors for the allowed displacements can then be easily tabulated for computational efficiency. The efficiency of KMC LG models depends on the disparity of the different atomic displacement rates. The example of vapor deposition onto a crystal surface illustrates the possible effects of a large disparity. In the case of Al, the diffusion of an adatom to an adjacent site on a (111) surface requires crossing a potential energy barrier of less than 0.1 eV, according to first principles calculations, implying a rate of approximately of 1010 hops/s at room temperature. On the other hand, the deposition of atoms by sputtering gives an accumulation rate of only about 4 nm/s for the deposited material, or a rate of 20 atoms/s impinging on every surface site. Since the models are usually designed to measure film growth processes and morphologies, it is apparent that the simulations require runs corresponding to real deposition times on the order of a second or more. But it is also necessary to include all of the diffusion hops, which require spending a large fraction of the computer time on moving adatoms around on the surface. However, the capability for performing such simulations has been increasing dramatically, both as a result of cheaper computational power, and because of new algorithms that dramatically speed up the simulations. Techniques are being developed to model random walk diffusion processes, without the necessity of simulating explicitly each of the millions of diffusion hops, by making use of the known properties of random walk diffusion processes [14]. In addition, there are several methods that handle highly disparate events without the inefficiency of spending computer time calculating moves that subsequently get rejected, as in the case of the Metropolis algorithm [15–17]. Methods to treat systems with long-range correlations efficiently have also been developed [18].

6.

Off-lattice KMC: Ion Implantation and Radiation Damage

The implantation of dopant ions into silicon wafers is the primary means to insert the electrically active atoms during the manufacturing of silicon

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devices. Atomistic models of this process are receiving much attention recently because of the decreasing size of silicon device components. Atomistic effects are becoming important since fluctuations in dopant atoms may degrade uniformity in device properties, and control of the distribution of the dopant atoms is becoming more critical. Two distinct models are required for the simulations. First, a model describing the entry of the energetic ions into the crystal, together with the damage resulting from silicon atoms displaced from their lattice sites. Although these models, for example, MARLOWE [19], involve some use of random numbers as mentioned above, most of the computer time is involved with calculating the collisions of the energetic particles with the silicon atoms. After the ions are implanted, the wafer is usually annealed to reduce the damage and improve the electrical properties of the device. This requires the simulation of several types of defects and dopant atoms diffusing through the crystal. Vacancies and interstitials are the two main defects, although the diffusion of complexes such as interstitial-dopant and vacancy-dopant pairs, interstitial dimers, divacancies, and larger clusters can have a significant influence on the redistribution and clustering of dopant atoms. Rather complex set of events can be simulated by the KMC OL method. In these simulations, the defects and clusters diffuse through a complex path of saddle points and potential energy minima; only the vacancy spends most of its time on lattice sites. Furthermore, the exact path of the diffusing species as a function of time is not particularly important for the KMC OL simulation, although they are essential for the more detailed first principles calculations used to calculate overall diffusion rates. The crucial parameters for KMC OL are the binding energies between defects and dopant atoms and their mobilities, the defect–defect binding energies, cross-sections for capture, and the recombination cross section for vacancies and interstitials. Fortunately, there have been a number of first principles calculations for these parameters, at least for the smaller clusters and defects. As in the case of surface diffusion, the disparity of diffusion rates is quite large, and it is essential to employ efficient algorithms for the simulations. An example of the complexity of the simulations, is given in Fig. 3, where we show model calculations of the relatively simple case of the implantation of silicon ions into a silicon target using the DADOS simulator [20]. Silicon ions (5 keV of kinetic energy) are implanted into perfect crystalline silicon dislodging some silicon atoms from their lattice sites creating vacancies (dark spheres) and interstitials (grey spheres). Figure 3(a) shows the high concentration of defects after implantation at room temperature, with many vacancy-interstitial pairs created by the energetic ions. After a few seconds of annealing, Fig. 3(b), a large number of point defects have recombined, leaving an excess of interstitials corresponding to the implanted ions. The excess interstitials gradually aggregate and form {311} defects, Fig. 3(c) and (d). Note that


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Figure 3. Kinetic Monte Carlo results showing point defects in crystalline silicon after implantation of Si ions into perfect crystalline Si at room temperature, and during subsequent annealing at 800◦ C [19]. Grey spheres represent interstititals, and dark ones vacancies; only the defects are shown. (a) corresponds to the defects after implantation at room temperature, (b) 1 s anneal, (c) 40 s anneal, and (d) 250 s anneal.

simulation does not predict the structure of the interstitial clusters, because of the off-lattice nature of the model. The structure {311} of the defects is inserted into the model since it is important for the point-defect cluster interactions and cross-sections. As the defects diffuse and recombine in the initial stages and, later, as the {311} defects emit and absorb interstitials during the ripening phase, a very large number of diffusion hops take place demanding long KMC simulations. Eventually the interstitial clusters dissolve as the interstitial excess equilibrates with the surface.

7.

Simulation of Particle and Radiation Transport

MC is quite extensively used to track the individual particles as each moves through the medium of interest, streaming and colliding with the atomic constituents of the medium. To give a simple illustration, we consider the trajectory of a neutron as it enters a medium, as depicted in Fig. 4. Suppose the first interaction of this neutron is a scattering collision at point 1. After the scattering the neutron moves to point 2 where it is absorbed, causing a fission reaction which emits two neutrons and a photon. One of the neutrons streams to point 3 where it suffers a capture reaction with the emission of a photon, which in turn leaves the medium at point 6. The other neutron and the photon from the fission event both escape from the medium, to points 4 and 7, respectively, without undergoing any further collisions. By sampling a trajectory we mean that process in which one determines the position of point 1 where the scattering occurs, the outgoing neutron direction and its energy, the position of point 2 where fission occurs, the outgoing directions and energies of the two fission neutrons and the photon, etc. After tracking many such trajectories one can estimate the probability of a neutron penetrating the medium and the amount of energy deposited in the medium as a result of the reactions induced along the path of each trajectory. This is the kind of information

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Figure 4. Schematic of a typical particle trajectory simulated by Monte Carlo. By repeating the simulation many times one obtains sufficient statistics to estimate the probability of radiation penetration in the case of shielding calculations, or the probability of energy deposition in the case of dosimetry problems.

that one needs in shielding calculations, where one wants to know how much material is needed to prevent the radiation (particles) from getting across the medium (a biological shield), or in dosimetry calculations where one wants to know how much energy is deposited in the medium (human tissue) by the radiation.

8.

Comparison of MC with MD

As discussed in several of the sections of Chapter 2, MD is a technique to generate the atomic trajectories of a system of N particles by direct numerical integration of the Newtons equations of motion. In a similar spirit, we say that the purpose of MC is to generate an ensemble of atomic configurations by stochastic sampling. In both cases we have a system of N particles interacting through the same interatomic potential. In MD, the system evolves in time by following the Newtons equations of motion where particles move in response to forces created by their neighbors. The particles therefore follow the correct dynamics according to classical mechanics. In contrast, in MC the particles move by sampling a distribution such as the canonical distribution. The dynamics thus generated is stochastic or probabilistic rather than deterministic which is the case for MD. The difference is, dynamics becomes important in problems where we wish to simulate the system over a long period of time. Because MD is constrained to real dynamics, the time scale of the simulation is fixed by such factors as the interatomic potential, and the mass of the particle. This time scale is of the order of picoseconds (10−12 ). If one wants to observe a phenomenon on a longer scale such as microseconds, it would require extensive computer resources to simulate it directly by MD. On the other hand, the time scale of MC is not fixed in the same way. KMC models


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often are able to simulate many of the same phenomena as MD, but on a much longer time scale by using a simplified description of the motion. If we consider the system of atoms on a crystal surface represented in Fig. 1, the MD simulation would consist of a substrate that provides a potential consisting of a square array of binding sites. Mobile atoms on the substrate would vibrate around the potential energy minimum of the binding site, and occasionally surmount the barrier and hop to a neighboring site. The vibrations of the atoms around the binding site may not be of importance for many applications, but the diffusion hops to neighboring sites and the aggregation into larger clusters on the substrate could be important for studying thin film structures during annealing, as discussed earlier. A KMC model could be developed where the elementary move is a diffusion hop to a neighboring site, ignoring the vibrations. Information from the MD model on the hop rate to neighboring sites, together with the effect of neighboring atoms on the hop rate, is often used to develop the KMC model. Because of the greatly reduced frequency of the diffusion events compared to the vibrations, the simulation can cover much larger time and length scales, and yet provide the needed information on the atomic diffusion and clustering. Another way to characterize the difference between MC and MD is to consider each as a technique to sample the degrees of freedom of the system. Since we follow the particle positions and velocities in MD, we are sampling the evolution of the system of N particles in its phase space, the 6-N dimensional space of the positions and velocities of the N particles. In MC we generate a set of particle positions in the system of N particles, thus the sampling is carried out in the 3-N configurational space of the system. In both cases, the sampling generates a trajectory in the respective spaces, as shown in Fig. 5. Such trajectories then allow properties of the system to be calculated as averages over these trajectories. In MD one performs a time average whereas in MC one

Figure 5. Schematic depicting the evolution of the same N-particle system in the 3-N dimensional configurational space (µ) as sampled by MC, and in the 6-N dimensional phase space (γ ) sampled by MD. In each case, the sampling results in a trajectory in the appropriate space, which is the necessary information that allows average system properties to be calculated. For MC, the trajectory is that of a random walk (Markov chain) governed by stochastic dynamics, whereas for MD the trajectory is what we believe to be the correct dynamics as given by the Newton’s equations of motion in classical mechanics. The same interatomic potential is used in the two simulations.

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performs an average over discrete states. Under appropriate conditions MC and MD give the same results for equilibrium properties, a consequence of the so called ergodic hypothesis (ensemble average = time average); however, dynamical properties calculated using the two methods in general will not be the same.

References [1] N. Metropolis, “The beginning of the Monte Carlo method,” Los Alamos Sci., Special Issue, 125, 1987. [2] E.J. Janse van Rensburg and G.M. Torrie “Estimation of multidimensional integrals: is Monte Carlo the best method?” J. Phys. A: Math. Gen., 26, 943–953, 1993. [3] A.R. Kansal and S. Torquato, “Prediction of trapping rates in mixtures of partially absorbing spheres,” J. Chem. Phys., 116, 10589, 2002. [4] H. Gould and J. Tobochnik, An Introduction to Computer Simulation Methods, Part 2, Chaps 10–12, 14, 15, Addison-Wesley, Reading, 1988. [5] D.W. Hermann, Computer Simulation Methods, 2nd edn., Chap 4, Springer-Verlag, Berlin, 1990. [6] K. Binder and D.W. Hermann, Monte Carlo Simulation in Statistical Physics, An Introduction, Springer-Verlag, Berlin, 1988. [7] E.E. Lewis and W.F. Miller, Computational Methods of Neutron Transport, Chap 7, American Nuclear Society, La Grange Park, IL, 1993. [8] N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. Teller, “Equation of state calculations by fast computing machines,” J. Chem. Phys., 21, 1087, 1953. [9] M.H. Kalos and P.A. Whitlock, Monte Carlo Methods , Wiley, New York, 1986. [10] S. Kirkpatrick, C.D. Gelatt, and M.P. Vecchi, “Optimization by simulated annealing,” Science, 220, 671, 1983. [11] G.M. Torrie and J.P. Valleau, “Non-physical sampling distributions in Monte Carlo free energy estimation – umbrella sampling,” J. Comput. Phys., 23, 187, 1977. [12] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Oxford University Press, Oxford, 1987. [13] M. Rao, C. Pangali, and B.J. Berne, “On the force bias Monte Carlo simulation of water: methodology, optimization and comparison with molecular dynamics,” Mol. Phys., 37, 1773, 1979. [14] J. Dalla Torre, C.-C. Fu, F. Willaime, and J.-L. Bocquet, Simulations multi-echelles des experiences de recuit de resistivite isochrone dans le Fer-ultra pur irradie aux electrons: premiers resultants, CEA Annuel Rapport, p. 94, 2003. [15] D. T. Gillespie, “General method for numerically simulating stochastic time evolution of coupled chemical-reactions,” Comp. Phys., 22, 403–434, 1976. [16] A.B. Bortz, M.H. Kalos, and J. L. Lebowitz, J. Comput. Phys., 17, 10, 1975. [17] G. H. Gilmer, “Growth on imperfect crystal faces,” J. Cryst, Growth, 36, 15, 1976. [18] R.H. Swendsen and J.S. Wang, “Replica Monte Carlo simulation of spin-glasses,” Phys. Rev. Lett., 57, 2607, 1986. [19] M.T. Robinson, “The binary collision approximation: background and introduction, Rad. Eff. Defects Sol., 130–131, 3, 1994. [20] M.E. Law, G.H. Gilmer, and M. Jaraiz, “Simulation of defects and diffusion phenomena in silicon,” MRS Bull., 25, 45, 2000.

2.11 ACCELERATED MOLECULAR DYNAMICS METHODS Blas P. Uberuaga1, Francesco Montalenti2 , Timothy C. Germann3, and Arthur F. Voter4 1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 2

INFM, L-NESS, and Dipartimento di Scienza dei Materiali, Università degli Studi di Milano-Bicocca, Via Cozzi 53, I-20125 Milan, Italy 3 Applied Physics Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA 4 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Molecular dynamics (MD) simulation, in which atom positions are evolved by integrating the classical equations of motion in time, is now a well established and powerful method in materials research. An appealing feature of MD is that it follows the actual dynamical evolution of the system, making no assumptions beyond those in the interatomic potential, which can, in principle, be made as accurate as desired. However, the limitation in the accessible simulation time represents a substantial obstacle in making useful predictions with MD. Resolving individual atomic vibrations – a necessity for maintaining accuracy in the integration – requires time steps on the order of femtoseconds, so that reaching even one microsecond is very difficult on today’s fastest processors. Because this integration is inherently sequential in nature, direct, spatial parallelization does not help significantly; it just allows simulations of nanoseconds on much larger systems. Beginning in the late 1990s, methods based on a new concept have been developed for circumventing this time scale problem. For systems in which the long-time dynamical evolution is characterized by a sequence of activated events, these “accelerated molecular dynamics” methods [1] can extend the accessible time scale by orders of magnitude relative to direct MD, while retaining full atomistic detail. These methods – hyperdynamics, parallel-replica dynamics, and temperature accelerated dynamics (TAD) – have already been demonstrated on problems in surface and bulk diffusion and surface growth. With more development they will become useful for a broad range of key materials problems, such as pipe diffusion along a dislocation core, impurity clustering, grain 629 S. Yip (ed.), Handbook of Materials Modeling, 629–648. c 2005 Springer. Printed in the Netherlands.

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growth, dislocation climb and dislocation kink nucleation. Here we give an introduction to these methods, discuss their current strengths and limitations, and predict how their capabilities may develop in the next few years.

1. 1.1.

Background Infrequent Event Systems

We begin by defining an “infrequent-event” system, as this is the type of system we will focus on in this article. The dynamical evolution of such a system is characterized by the occasional activated event that takes the system from basin to basin, events that are separated by possibly millions of thermal vibrations within one basin. A simple example of an infrequent-event system is an adatom on a metal surface at a temperature that is low relative to the diffusive jump barrier. We will exclusively consider thermal systems, characterized by a temperature T , a fixed number of atoms N , and a fixed volume V ; i.e., the canonical ensemble. Typically, there is a large number of possible paths for escape from any given basin. As a trajectory in the 3N -dimensional coordinate space in which the system resides passes from one basin to another, it crosses a (3N –1)dimensional “dividing surface” at the ridgetop separating the two basins. While on average these crossings are infrequent, successive crossings can sometimes occur within just a few vibrational periods; these are termed “correlated dynamical events” [2–4]. An example would be a double jump of the adatom on the surface. For this discussion it is sufficient, but important, to realize that such events can occur. In most of the methods presented below, we will assume that these correlated events do not occur – this is the primary assumption of transition state theory – which is actually a very good approximation for many solid-state diffusive processes. We define the “correlation time” (τcorr ) of the system as the duration of the system memory. A trajectory that has resided in a particular basin for longer than τcorr has no memory of its history and, consequently, how it got to that basin, in the sense that when it later escapes from the basin, the probability for escape is independent of how it entered the state. The relative probability for escape to a given adjacent state is proportional to the rate constant for that escape path, which we will define below. An infrequent event system, then, is one in which the residence time in a state (τrxn ) is much longer than the correlation time (τcorr ). We will focus here on systems with energetic barriers to escape, but the infrequent-event concept applies equally well to entropic bottlenecks.1 The key to the accelerated

1 For systems with entropic bottlenecks, the parallel-replica dynamics method can be applied very

effectively [1].

Accelerated molecular dynamics methods

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dynamics methods described here is recognizing that to obtain the right sequence of state-to-state transitions, we need not evolve the vibrational dynamics perfectly, as long as the relative probability of finding each of the possible escape paths is preserved.

1.2.

Transition State Theory

Transition state theory (TST) [5–9] is the formalism underpinning all of the accelerated dynamics methods, directly or indirectly. In the TST approximation, the classical rate constant for escape from state A to some adjacent state B is taken to be the equilibrium flux through the dividing surface between A and B (Fig. 1). If there are no correlated dynamical events, the TST rate is the exact rate constant for the system to move from state A to state B. The power of TST comes from the fact that this flux is an equilibrium property of the system. Thus, we can compute the TST rate without ever propagating a trajectory. The appropriate ensemble average for the rate constant for escape from A, k TST A→ , is k TST A→ = |dx/dt | δ(x − q) A ,

(1)

where x ∈ r is the reaction coordinate and x = q the dividing surface bounding state A. The angular brackets indicate the ratio of Boltzmann-weighted integrals over 6N -dimensional phase space (configuration space r and momentum space p). That is, for some property P(r, p),

P =

P(r, p)exp[−H (r, p)/kB T ] dr dp , exp[−H (r, p)/kB T ] dr dp

(2)

A

Ea

B

Figure 1. A two-state system illustrating the definition of the transition state theory rate constant as the outgoing flux through the dividing surface bounding state A.

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where kB is the Boltzmann constant and H (r, p) is the total energy of the system, kinetic plus potential. The subscript A in Eq. (1) indicates the configuration space integrals are restricted to the space belonging to state A. If the effective mass (m) of the reaction coordinate is constant over the dividing surface, Eq. (1) reduces to a simpler ensemble average over configuration space only [10], k TST A→ =

2kB T /π m δ(x − q) A .

(3)

The essence of this expression, and of TST, is that the Dirac delta function picks out the probability of the system being at the dividing surface, relative to everywhere else it can be in state A. Note that there is no dependence on the nature of the final state B. In a system with correlated events, not every dividing surface crossing corresponds to a reactive event, so that, in general, the TST rate is an upper bound on the exact rate. For diffusive events in materials at moderate temperatures, these correlated dynamical events typically do not cause a large change in the rate constants, so TST is often an excellent approximation. This is a key point; this behavior is markedly different than in some chemical systems, such as molecular reactions in solution or the gas phase, where TST is just a starting point and dynamical corrections can lower the rate significantly [11]. While in the traditional use of TST, rate constants are computed after the dividing surface is specified, in the accelerated dynamics methods we exploit the TST formalism to design approaches that do not require knowing in advance where the dividing surfaces will be, or even what product states might exist.

1.3.

Harmonic Transition State Theory

If we have identified a saddle point on the potential energy surface for the reaction pathway between A and B, we can use a further approximation to TST. We assume that the potential energy near the basin minimum is well described, out to displacements sampled thermally, with a second-order energy expansion – i.e., that the vibrational modes are harmonic – and that the same is true for the modes perpendicular to the reaction coordinate at the saddle point. Under these conditions, the TST rate constant becomes simply −E a / kB T , k HTST A→B = ν0 e

(4)

where 3N

min i νi ν0 = 3N−1 . νisad i

(5)


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Here E a is the static barrier height, or activation energy (the difference in energy between the saddle point and the minimum of state A (Fig. 1)), {νimin } are the normal mode frequencies at the minimum of A, and {νisad } are the nonimaginary normal mode frequencies at the saddle separating A from B. This is often referred to as the Vineyard [12] equation. The analytic integration of Eq. (1) over the whole phase space thus leaves a very simple Arrhenius temperature dependence.2 To the extent that there are no recrossings and the modes are truly harmonic, this is an exact expression for the rate. This harmonic TST expression is employed in the temperature accelerated dynamics method (without requiring calculation of the prefactor ν0 ).

1.4.

Complex Infrequent Event Systems

The motivation for developing accelerated molecular dynamics methods becomes particularly clear when we try to understand the dynamical evolution of what we will term complex infrequent event systems. In these systems, we simply cannot guess where the state-to-state evolution might lead. The underlying mechanisms may be too numerous, too complicated, and/or have an interplay whose consequences cannot be predicted by considering them individually. In very simple systems we can raise the temperature to make diffusive transitions occur on an MD-accessible time scale. However, as systems become more complex, changing the temperature causes corresponding changes in the relative probability of competing mechanisms. Thus, this strategy will cause the system to select a different sequence of state-to-state dynamics, ultimately leading to a completely different evolution of the system, and making it impossible to address the questions that the simulation was attempting to answer. Many, if not most, materials problems are characterized by such complex infrequent events. We may want to know what happens on the time scale of milliseconds, seconds or longer, while with MD we can barely reach one microsecond. Running at higher T or trying to guess what the underlying atomic processes are can mislead us about how the system really behaves. Often for these systems, if we could get a glimpse of what happens at these longer times, even if we could only afford to run a single trajectory for that long, our understanding of the system would improve substantially. This, in essence, is the primary motivation for the development of the methods described here.

2 Note that although the exponent in Eq. (4) depends only on the static barrier height E , in this HTST a

approximation there is no assumption that trajectory passes exactly through the saddle point.

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1.5.


Dividing Surfaces and Transition Detection

We have implied that the ridgetops between basins are the appropriate dividing surfaces in these systems. For a system that obeys TST, these ridgetops are the optimal dividing surfaces; recrossings will occur for any other choice of dividing surface. A ridgetop can be defined in terms of steepest-descent paths – it is the 3N –1-dimensional boundary surface that separates those points connected by steepest descent paths to the minimum of one basin from those that are connected to the minimum of an adjacent basin. This definition also leads to a simple way to detect transitions as a simulation proceeds, a requirement of parallel-replica dynamics and temperature accelerated dynamics. Intermittently, the trajectory is interrupted and minimized through steepest descent. If this minimization leads to a basin minimum that is distinguishable from the minimum of the previous basin, a transition has occurred. An appealing feature of this approach is that it requires virtually no knowledge of the type of transition that might occur. Often only a few steepest descent steps are required to determine that no transition has occurred. While this is a fairly robust detection algorithm, and the one used for the simulations presented below, more efficient approaches can be tailored to the system being studied.

2.

Parallel-Replica Dynamics

The parallel-replica method [13] is the simplest and most accurate of the accelerated dynamics techniques, with the only assumption being that the infrequent events obey first-order kinetics (exponential decay); i.e., for any time t > τcorr after entering a state, the probability distribution function for the time of the next escape is given by p(t) = ktot e−ktot t ,

(6)

where ktot is the rate constant for escape from the state. For example, Eq. (6) arises naturally for ergodic, chaotic exploration of an energy basin. Parallelreplica allows for the parallelization of the state-to-state dynamics of such a system on M processors. We sketch the derivation here for equal-speed processors. For a state in which the rate to escape is ktot , on M processors the effective escape rate will be Mktot , as the state is being explored M times faster. Also, if the time accumulated on one processor is t1 , on the M processors a total time of tsum = Mt1 will be accumulated. Thus, we find that p(t1 ) dt1 = Mktot e−Mktot t1 dt1 p(t1 ) dt1 = ktot e−ktot tsum dtsum p(t1 ) dt1 = p(tsum ) dtsum

(7a) (7b) (7c)


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and the probability to leave the state per unit time, expressed in tsum units, is the same whether it is run on one or M processors. A variation on this derivation shows that the M processors need not run at the same speed, allowing the method to be used on a heterogeneous or distributed computer; see Ref. [13]. The algorithm is schematically shown in Fig. 2. Starting with an N -atom system in a particular state (basin), the entire system is replicated on each of M available parallel or distributed processors. After a short dephasing stage during which each replica is evolved forward with independent noise for a time tdeph ≥ τcorr to eliminate correlations between replicas, each processor carries out an independent constant-temperature MD trajectory for the entire N -atom system, thus exploring phase space within the particular basin M times faster than a single trajectory would. Whenever a transition is detected on any processor, all processors are alerted to stop. The simulation clock is advanced by the accumulated trajectory time summed over all replicas, i.e., the total time τrxn spent exploring phase space within the basin until the transition occurred. The parallel-replica method also correctly accounts for correlated dynamical events (i.e., there is no requirement that the system obeys TST), unlike the other accelerated dynamics methods. This is accomplished by allowing the trajectory that made the transition to continue on its processor for a further amount of time tcorr ≥ τcorr , during which recrossings or follow-on events may occur. The simulation clock is then advanced by tcorr , the final state is replicated on all processors, and the whole process is repeated. Parallelreplica dynamics then gives exact state-to-state dynamical evolution, because the escape times obey the correct probability distribution, nothing about the procedure corrupts the relative probabilities of the possible escape paths, and the correlated dynamical events are properly accounted for.

A

B

C

D

A

Figure 2. Schematic illustration of the parallel-replica method (after Ref. [1]). The four steps, described in the text, are (A) replication of the system into M copies, (B) dephasing of the replicas, (C) evolution of independent trajectories until a transition is detected in any of the replicas, and (D) brief continuation of the transitioning trajectory to allow for correlated events such as recrossings or follow-on transitions to other states. The resulting configuration is then replicated, beginning the process again.

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The efficiency of the method is limited by both the dephasing stage, which does not advance the system clock, and the correlated event stage, during which only one processor accumulates time. (This is illustrated schematically in Fig. 2, where dashed line trajectories advance the simulation clock but dotted line trajectories do not.) Thus, the overall efficiency will be high when τrxn /M tdeph + tcorr .

(8)

Some tricks can further reduce this requirement. For example, whenever the system revisits a state, on all but one processor the interrupted trajectory from the previous visit can be immediately restarted, eliminating the dephasing stage. Also, the correlation stage (which only involves one processor) can be overlapped with the subsequent dephasing stage for the new state on the other processors, in the hope that there are no correlated crossings that lead to a different state. Figure 3 shows an example of a parallel-replica simulation; an Ag(111) island-on-island structure decays over a period of 1 µs at T = 400 K. Many of the transitions involve concerted mechanisms. Parallel-replica dynamics has the advantage of being fairly simple to program, with very few “knobs” to adjust – tdeph and tcorr , which can be conservatively set at a few ps for most systems. As multiprocessing environments become more ubiquitous, with more processors within a node or even on a chip, and loosely linked Beowulf clusters of such nodes, parallel-replica dynamics will become an increasingly important simulation tool. Recently, parallel-replica dynamics has been extended to driven systems, such as systems with some externally applied strain rate. The requirement here is that the drive rate is slow enough that at any given time the rates for the processes in the system depend only on the instantaneous configuration of the system.

3.

Hyperdynamics

Hyperdynamics builds on the basic concept of importance sampling [14, 15], extending it into the time domain. In the hyperdynamics approach [16], the potential surface V (r) of the system is modified by adding to it a nonnegative bias potential Vb (r). The dynamics of the system is then evolved on this biased potential surface, V (r) + Vb (r). A schematic illustration is shown in Fig. 4. The derivation of the method requires that the system obeys TST – that there are no correlated events. There are also important requirements on the form of the bias potential. It must be zero at all the dividing surfaces, and the system must still obey TST for dynamics on the modified potential surface. If such a bias potential can be constructed, a challenging


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t = 0.00 µs

t = 0.15 µs

t = 0.25 µs

t = 0.39 µs

t = 0.41 µs

t = 0.42 µs

t = 0.44 µs

t = 0.45 µs

t = 1.00 µs

Figure 3. Snapshots from a parallel-replica simulation of an island on top of an island on the Ag(111) surface at T = 400 K (after Ref. [1]). On a microsecond time scale, the upper island gives up all its atoms to the lower island, filling vacancies and kink sites as it does so. This simulation took 5 days to reach 1 µs on 32 1 GHz Pentium III processors.

task in itself, we can substitute the modified potential V (r) + Vb (r) into Eq. (1) to find k TST A→ =

|v A | δ(x − q)Ab , eβVb (r) Ab

(9)

where β = 1/kB T and the state Ab is the same as state A but with the bias potential Vb applied. This leads to a very appealing result: a trajectory on this modified surface, while relatively meaningless on vibrational time scales,

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C A

B

Figure 4. Schematic illustration of the hyperdynamics method. A bias potential (V (r)), is added to the original potential (V (r), solid line). Provided that V (r) meets certain conditions, primarily that it be zero at the dividing surfaces between states, a trajectory on the biased potential surface (V (r) + V (r), dashed line) escapes more rapidly from each state without corrupting the relative escape probabilities. The accelerated time is estimated as the simulation proceeds.

evolves correctly from state to state at an accelerated pace. That is, the relative rates of events leaving A are preserved: k TST k TST Ab →B A→B = . TST k TST k Ab →C A→C

(10)

This is because these relative probabilities depend only on the numerator of Eq. (9) which is unchanged by the introduction of Vb since, by construction, Vb = 0 at the dividing surface. Moreover, the accelerated time is easily estimated as the simulation proceeds. For a regular MD trajectory, the time advances at each integration step by tMD , the MD time step (often on the order of 1 fs). In hyperdynamics, the time advance at each step is tMD multiplied by an instantaneous boost factor, the inverse Boltzmann factor for the bias potential at that point, so that the total time after n integration steps is thyper =

n

tMD eV (r(t j ))/ kB T.

(11)

j =1

Time thus takes on a statistical nature, advancing monotonically but nonlinearly. In the long-time limit, it converges on the correct value for the


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accelerated time with vanishing relative error. The overall computational speedup is then given by the average boost factor,

boost(hyperdynamics) = thyper/tMD = eV (r)/ kB T

Ab ,

(12)

divided by the extra computational cost of calculating the bias potential and its forces. If all the visited states are equivalent (e.g., this is common in calculations to test or demonstrate a particular bias potential), Eq. (12) takes on the meaning of a true ensemble average. The rate at which the trajectory escapes from a state is enhanced because the positive bias potential within the well lowers the effective barrier. Note, however, that the shape of the bottom of the well after biasing is irrelevant; no assumption of harmonicity is made. Figure 5 illustrates an application of hyperdynamics for a two-dimensional, periodic model potential using a Hessian-based bias potential [16]. The hopping diffusion rate was compared against MD at high temperature, where the two calculations agreed very well. At lower temperatures where the MD calculations would be too costly, it is compared against the result computed ⫺5

⫺10

In(D)

47 200

⫺15

⫺20

3435 8682

⫺25 4

6

8 1/kBT

10

12

Figure 5. Arrhenius plot of the diffusion coefficients for a model potential, showing a comparison of direct MD (), hyperdynamics (•), and TST + dynamical corrections (+). The symbols are sized for clarity. The line is the full harmonic TST approximation, and is indistinguishable from a least-square line through the TST points (not shown). Also shown are the boost factors, relative to direct MD, for each hyperdynamics result. The boost increases dramatically as the temperature is lowered (after Ref. [16]).

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using TST plus dynamical corrections. As the temperature is lowered, the effective boost gained by using hyperdynamics increased to the point that, at kB T = 0.09, the boost factor was over 8500. See Ref. [16] for details. The ideal bias potential should give a large boost factor, have low computational overhead (though more overhead is acceptable if the boost factor is very high), and, to a good approximation, meet the requirements stated above. This is very challenging, since we want, as much as possible, to avoid utilizing any prior knowledge of the dividing surfaces or the available escape paths. To date, proposed bias potentials typically have either been computationally intensive, have been tailored to very specific systems, have assumed localized transitions, or have been limited to low-dimensional systems. But the potential boost factor available from hyperdynamics is tantalizing, so developing bias potentials capable of treating realistic many-dimensional systems remains a subject of ongoing research by several groups. See Ref. [1] for a detailed discussion on bias potentials and results generated using various forms.

4.

Temperature Accelerated Dynamics

In the temperature accelerated dynamics (TAD) method [17], the idea is to speed up the transitions by increasing the temperature, while filtering out the transitions that should not have occurred at the original temperature. This filtering is critical, since without it the state-to-state dynamics will be inappropriately guided by entropically favored higher-barrier transitions. The TAD method is more approximate than the previous two methods, as it relies on harmonic TST, but for many applications this additional approximation is acceptable, and the TAD method often gives substantially more boost than hyperdynamics or parallel-replica dynamics. Consistent with the accelerated dynamics concept, the trajectory in TAD is allowed to wander on its own to find each escape path, so that no prior information is required about the nature of the reaction mechanisms. In each basin, the system is evolved at a high temperature Thigh (while the temperature of interest is some lower temperature Tlow ). Whenever a transition out of the basin is detected, the saddle point for the transition is found. The trajectory is then reflected back into the basin and continued. This “basin constrained molecular dynamics” (BCMD) procedure generates a list of escape paths and attempted escape times for the high-temperature system. Assuming that TST holds and that the system is chaotic and ergodic, the probability distribution for the first-escape time for each mechanism is an exponential (Eq. (6)). Because harmonic TST gives an Arrhenius dependence of the rate on temperature (Eq. (4)), depending only on the static barrier height, we can then extrapolate each escape time observed at Thigh to obtain a corresponding escape time at Tlow that is drawn correctly from the exponential distribution at Tlow . This extrapolation, which requires knowledge of the saddle point energy, but not the preexponential factor, can be illustrated graphically in an


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Arrhenius-style plot (ln(1/t) vs. 1/T ), as shown in Fig. 6. The time for each event seen at Thigh extrapolated to Tlow is then tlow = thigh e Ea (βlow −βhigh ) ,

(13)

Tlow time

Thigh time

In(νmin)

ln(1/t)

In(ν*min)

low

ln(1/t short ) ln(1/tstop)

1/Thigh

1/Tlow 1/T

Figure 6. Schematic illustration of the temperature accelerated dynamics method. Progress of the high-temperature trajectory can be thought of as moving down the vertical time line at 1/Thigh . For each transition detected during the run, the trajectory is reflected back into the basin, the saddle point is found, and the time of the transition (solid dot on left time line) is transformed (arrow) into a time on the low-temperature time line. Plotted in this Arrhenius-like form, this transformation is a simple extrapolation along a line whose slope is the negative of the barrier height for the event. The dashed termination line connects the shortest-time transition recorded so far on the low temperature time line with the confidence-modified minimum =ν preexponential (νmin min /ln(1/δ)) on the y-axis. The intersection of this line with the highT time line gives the time (tstop , open circle) at which the trajectory can be terminated. With confidence 1-δ, we can say that any transition observed after tstop could only extrapolate to a shorter time on the low-T time line if it had a preexponential lower than νmin .

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where, again, β = 1/kB T . The event with the shortest time at low temperature is the correct transition for escape from this basin. Because the extrapolation can in general cause a reordering of the escape times, a new shorter-time event may be discovered as the BCMD is continued at Thigh. If we make the additional assumption that there is a minimum preexponential factor, νmin , which bounds from below all the preexponential factors in the system, we can define a time at which the BCMD trajectory can be stopped, knowing that the probability that any transition observed after that time would replace the first transition at Tlow is less than δ. This “stop” time is given by

ln(1/δ) νmin tlow,short thigh,stop ≡ νmin ln (1/δ)

Tlow /Thigh

,

(14)

where tlow,short is the shortest transition time at Tlow . Once this stop time is reached, the system clock is advanced by tlow,short, the transition corresponding to tlow,short is accepted, and the TAD procedure is started again in the new basin. The average boost in TAD can be dramatic when barriers are high and Thigh/Tlow is large. However, any anharmonicity error at Thigh transfers to Tlow ; a rate that is twice the Vineyard harmonic rate due to anharmonicity at Thigh will cause the transition times at Thigh for that pathway to be 50% shorter, which in turn extrapolate to transition times that are 50% shorter at Tlow . If the Vineyard approximation is perfect at Tlow , these events will occur at twice the rate they should. This anharmonicity error can be controlled by choosing a Thigh that is not too high. As in the other methods, the boost is limited by the lowest barrier, although this effect can be mitigated somewhat by treating repeated transitions in a “synthetic” mode [17]. This is in essence a kinetic Monte Carlo treatment of the low-barrier transitions, in which the rate is estimated accurately from the observed transitions at Thigh , and the subsequent low-barrier escapes observed during BCMD are excluded from the extrapolation analysis. Temperature accelerated dynamics is particularly useful for simulating vapor-deposited crystal growth, where the typical time scale can exceed minutes. Figure 7 shows an example of TAD applied to such a problem. Vapor deposited growth of a Cu(100) surface was simulated at a deposition rate of one monolayer per 15 s and a temperature T = 77 K, exactly matching (except for the system size) the experimental conditions of Ref. [18]. Each deposition event was simulated using direct MD for 2 ps, long enough for the atom to collide with the surface and settle into a binding site. A TAD simulation with Thigh = 550 K then propagated the system for the remaining time until the next deposition event was required, on average 0.3 s later. The overall boost factor was ∼ 107 . A key feature of this simulation was that, even at this low temperature, many events accepted during the growth process


1 ML

2 ML

3 ML

4 ML

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5 ML Figure 7. Snapshots from a TAD simulation of the deposition of five monolayers (ML) of Cu onto Cu(100) at 0.067 ML/s and T =77 K, matching the experimental conditions of Egelhoff and Jacob [18]. Deposition of each new atom was performed using direct molecular dynamics for 2 ps, while the intervening time (0.3 s on average for this 50 atom/layer simulation cell) was simulated using the TAD method. The boost factor for this simulation was ∼107 over direct MD (after Ref. [1]).

involved concerted mechanisms, such as the concerted sliding of an eight-atom cluster [1]. This MD/TAD procedure for simulating film growth has been applied also to Ag/Ag(100) at low temperatures [19] and Cu/Ag(100) [20]. Heteroepitaxial systems are especially hard to treat with techniques such as kinetic Monte Carlo because of the increased tendency for the system to go off lattice due

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to mismatch strain, and because the rate table needs to be considerably larger when neighboring atoms can have multiple types. Recently, enhancements to TAD, beyond the “synthetic mode” mentioned above, have been developed that can increase the efficiency of the simulation. For systems that revisit states, the time required to accept an event can be reduced for each revisit by taking advantage of the time accumulated in previous visits [21]. This procedure is exact; no assumptions beyond the ones required by the original TAD method are needed. After many visits, the procedure converges. The minimum barrier for escape from that state (E min ) is then known to within uncertainty δ. In this converged mode (ETAD), the average time at Thigh required to accept an event no longer depends on δ, and the average boost factor becomes simply

t low,short boost(ETAD) = = exp E min t high,stop

1 1 − kB Tlow kB Thigh

(15)

for that state. The additional boost (when converged) compared to the original TAD can be an order of magnitude or more. For systems that seldom (or never) revisit the same state, it is still possible to exploit this extra boost by running in ETAD mode with E min supplied externally. One way of doing this is to combine TAD with the dimer method [22]. In this combined dimer-TAD approach, first proposed by Montalenti and Voter [21], upon entering a new state, a number of dimer searches are used to find the minimum barrier for escape, after which ETAD is employed to quickly find a dynamically appropriate escape path. This exploits the power of the dimer method to quickly find low-barrier pathways, while eliminating the danger associated with the possibility that it might miss important escape paths. Although the dimer method might fail to find the lowest barrier correctly, this is a much weaker demand on the dimer method than trying to find all relevant barriers. In addition, the ETAD phase has some chance of correcting the simulation during the BCMD if the dimer searches did not find E min .

5.

Outlook

As these accelerated dynamics methods become more widely used and further developed (including the possible emergence of new methods), their application to important problems in materials science will continue to grow. We conclude this article by comparing and contrasting the three methods presented here, with some guidelines for deciding which method may be most appropriate for a given problem. We point out some important limitations of the methods, areas in which further development may significantly increase their usefulness. Finally, we discuss the prospects for these methods in the immediate future.


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The key feature of all of the accelerated dynamics methods is that they collapse the waiting time between successive transitions from its natural time (τrxn ) to (at best) a small number of vibrational periods. Each method accomplishes this in a different way. TAD exploits the enhanced rate at higher temperature, hyperdynamics effectively lowers the barriers to escape by filling in the basin, and parallel-replica dynamics spreads the work across many processors. The choice of which accelerated dynamics method to apply to a problem will typically depend on three factors. The first is the desired level of accuracy in following the exact dynamics of the system. As described previously, parallel-replica is the most exact of the three methods; the only assumption is that the kinetics are first order. Not even TST is assumed, as correlated dynamical events are treated correctly in the method. This is not true with hyperdynamics, which does rely upon the assumptions of TST, in particular the absence of correlated events. Finally, temperature accelerated dynamics makes the further assumptions inherent in the harmonic approximation to TST, and is thus the most approximate of the three methods. If complete accuracy is the main goal of the simulation, parallel-replica is the superior choice. The second consideration is the potential gain in accessible time scales that the accelerated dynamics method can achieve for the system. Typically, TAD is the method of choice when considering this factor. While in all three methods the boost for escaping from each state will be limited by the smallest barrier, if the barriers are high relative to the temperature of interest, TAD will typically achieve the largest boost factor. In principle, hyperdynamics can also achieve very significant boosts, but, in practice, existing bias potentials either have a very simple form which generally provide limited boosts for complex many-atom systems, or more sophisticated (e.g., Hessian-based) forms whose overhead reduces the boosts actually attainable. It may be possible, using prior knowledge about particular systems, to construct a computationally inexpensive bias potential which simultaneously offers large boosts, in which case hyperdynamics could be competitive with TAD. Finally, parallel-replica dynamics usually offers the smallest boost given the typical access to parallel computing today (e.g., tens of processors or fewer per user for continuous use), since the maximum possible boost is exactly the number of processors. For some systems, the overhead of, for example, finding saddle points in TAD may be so great that parallel-replica can give more overall boost. However, in general, the price of the increased accuracy of parallel-replica dynamics will be shorter achievable time scales. It should be emphasized that the limitations of parallel-replica in terms of accessible time scales are not inherent in the method, but rather are a consequence of the currently limited computing power which is available. As massively parallel processing becomes commonplace for individual users, and any number can be used in the study of a given problem, parallel-replica should become just as efficient as the other methods. If enough processors are available

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so that the amount of simulation time each processor has to do for each transition is on the order of ps, parallel-replica will be just as efficient as TAD or hyperdynamics. This analysis may be complicated by issues of communication between processors, but the future of parallel-replica is very promising. The last main factor determining which method is best suited to a problem is the shape of the potential energy surface (PES). Both TAD and hyperdynamics require that the PES be relatively smooth. In the case of TAD, this is because saddle points must be found and standard techniques for finding them often perform poorly for rough landscapes. The same is true for the hyperdynamics bias potentials that require information about the shape of the PES. Parallel-replica, however, only requires a method for detecting transitions. No further analysis of the potential energy surface is needed. Thus, if the PES describing the system of interest is relatively rough, parallel-replica dynamics may be the only method that can be applied effectively. The temperature dependence of the boost in hyperdynamics and TAD gives rise to an interesting prediction about their power and utility in the future. Sometimes, even accelerating the dynamics may not make the activated processes occur frequently enough to study a particular process. A common trick is to raise the temperature just enough that at least some events will occur in the available computer time, hoping, of course, that the behavior of interest is still representative of the lower-T system. When faster computers become available, the same system can be studied at a lower, more desirable, temperature. This in turn increases the boost factor (e.g., see Eqs. (12) and (14)), so that, effectively, there is a superlinear increase in the power of accelerated dynamics with increasing computer speed. Thus, the accelerated dynamics approaches will become increasingly more powerful in future years simply because computers keep getting faster. A particularly appealing prospect is that of accelerated electronic structurebased molecular dynamics simulations (e.g., by combining density functional theory (DFT) or quantum chemistry with the methods discussed here), since accessible electronic structure time scales are even shorter, currently on the order of ps. However, because of the additional expense involved in these techniques, the converse of the argument given in the previous paragraph indicates that, for example, accelerated DFT dynamics simulations will not give much useful boost on current computers (i.e., using DFT to calculate the forces is like having a very slow computer). DFT hyperdynamics may be a powerful tool in 5–10 years, when breakeven (boost = overhead) is reached, and this could happen sooner with the development of less expensive bias potentials. TAD is probably close to being viable for combination with DFT, while parallel-replica dynamics and dimer-TAD could probably be used on today’s computers for electronic structure studies on some systems. Currently, these methods are very efficient when applied to systems in which the barriers are much higher than the temperature of interest. This is often true


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for systems such as ordered solids, but there are many important systems that do not so cleanly fall into this class, a prime example being glasses. Such systems are characterized by either a continuum of barrier heights, or a set of low barriers that describe uninteresting events, like conformational changes in a molecule. Low barriers typically degrade the boost of all of the accelerated dynamics methods, as well as the efficiency of standard kinetic Monte Carlo. However, even these systems will be amenable to study through accelerated dynamics methods as progress is made on this low-barrier problem. A final note should be made about the computational scaling of these methods with system size. While the exact scaling depends on the type of system and many aspects of the implementation, a few general points can be made. In the case of TAD, if the work of finding saddles and detecting transitions can be localized, it can be shown that the scaling goes as N 2−Tlow /Thigh [21] for the simple case of a system that has been enlarged by replication. This is improved greatly with ETAD, which scales as O(N ), the same as regular MD. Real systems are more complicated and, typically, lower barrier processes will arise as the system size is increased. Thus, even hyperdynamics with a bias potential requiring no overhead might scale worse than N . The accelerated dynamics methods, as a whole, are still in their infancy. Even so, they are currently powerful enough to study a wide range of materials problems that were previously intractable. As these methods continue to mature, their applicability, and the physical insights gained by their use, can be expected to grow.

Acknowledgments We gratefully acknowledge vital discussions with Graeme Henkelman. This work was supported by the United States Department of Energy (DOE), Office of Basic Energy Sciences, under DOE Contract No. W-7405-ENG-36.

References [1] A.F. Voter, F. Montalenti, and T.C. Germann, “Extending the time scale in atomistic simulation of materials,” Annu. Rev. Mater. Res., 32, 321–346, 2002. [2] D. Chandler, “Statistical-mechanics of isomerization dynamics in liquids and transition-state approximation,” J. Chem. Phys., 68, 2959–2970, 1978. [3] A.F. Voter and J.D. Doll, “Dynamical corrections to transition state theory for multistate systems: surface self-diffusion in the rare-event regime,” J. Chem. Phys., 82, 80–92, 1985. [4] C.H. Bennett, “Molecular dynamics and transition state theory: simulation of infrequent events,” ACS Symp. Ser., 63–97, 1977. [5] R. Marcelin, “Contribution a` l’étude de la cinétique physico-chimique,” Ann. Physique, 3, 120–231, 1915.

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B.P. Uberuaga et al. [6] E.P. Wigner, “On the penetration of potential barriers in chemical reactions,” Z. Phys. Chemie B, 19, 203, 1932. [7] H. Eyring, “The activated complex in chemical reactions,” J. Chem. Phys., 3, 107–115, 1935. [8] P. Pechukas, “Transition state theory,” Ann. Rev. Phys. Chem., 32, 159–177, 1981. [9] D.G. Truhlar, B.C. Garrett, and S.J. Klippenstein, “Current status of transition state theory,” J. Phys. Chem., 100, 12771–12800, 1996. [10] A.F. Voter and J.D. Doll, “Transition state theory description of surface selfdiffusion: comparison with classical trajectory results,” J. Chem. Phys., 80, 5832– 5838, 1984. [11] B.J. Berne, M. Borkovec, and J.E. Straub, “Classical and modern methods in reaction-rate theory,” J. Phys. Chem., 92, 3711–3725, 1988. [12] G.H. Vineyard, “Frequency factors and isotope effects in solid state rate processes,” J. Phys. Chem. Solids, 3, 121–127, 1957. [13] A.F. Voter, “Parallel-replica method for dynamics of infrequent events,” Phys. Rev. B, 57, 13985–13988, 1998. [14] J.P. Valleau and S.G. Whittington, “A guide to Monte Carlo for statistical mechanics: 1. highways,” In: B.J. Berne (ed.), Statistical Mechanics. A. A Modern Theoretical Chemistry, vol. 5, Plenum, New York, pp. 137–168, 1977. [15] B.J. Berne, G. Ciccotti, and D.F. Coker (eds.), Classical and Quantum Dynamics in Condensed Phase Simulations, World Scientific, Singapore, 1998. [16] A.F. Voter, “A method for accelerating the molecular dynamics simulation of infrequent events,” J. Chem. Phys., 106, 4665–4677, 1997. [17] M.R. Sørensen and A.F. Voter, “Temperature-accelerated dynamics for simulation of infrequent events,” J. Chem. Phys., 112, 9599–9606, 2000. [18] W.F. Egelhoff, Jr. and I. Jacob, “Reflection high-energy electron-diffraction (RHEED) oscillations at 77K,” Phys. Rev. Lett., 62, 921–924, 1989. [19] F. Montalenti, M.R. Sørensen, and A.F. Voter, “Closing the gap between experiment and theory: crystal growth by temperature accelerated dynamics,” Phys. Rev. Lett., 87, 126101, 2001. [20] J.A. Sprague, F. Montalenti, B.P. Uberuaga, J.D. Kress, and A.F. Voter, “Simulation of growth of Cu on Ag(001) at experimental deposition rates” Phys. Rev. B, 66, 205415, 2002. [21] F. Montalenti and A.F. Voter, “Exploiting past visits or minimum-barrier knowledge to gain further boost in the temperature-accelerated dynamics method,” J. Chem. Phys., 116, 4819–4828, 2002. [22] G. Henkelman and H. Jónsson, “A dimer method for finding saddle points on high dimensional potential surfaces using only first derivatives,” J. Chem. Phys., 111, 7010–7022, 1999.

2.12 CONCURRENT MULTISCALE SIMULATION AT FINITE TEMPERATURE: COARSE-GRAINED MOLECULAR DYNAMICS Robert E. Rudd Lawrence Livermore National Laboratory, University of California, L-045 Livermore, CA 94551, USA

1.

Embedded Nanomechanics and Computer Simulation

With the advent of nanotechnology, predictive simulations of nanoscale systems have become in great demand. In some cases, nanoscale systems can be simulated directly at the level of atoms. The atomistic techniques used range from models based on a quantum mechanical treatment of the electronic bonds to those based on more empirical descriptions of the interatomic forces. In many cases, however, even nanoscale systems are too big for a purely atomistic approach, typically because the nanoscale device is coupled to its surroundings, and it is necessary to simulate the entire system comprising billions of atoms. A well-known example is the growth of nanoscale epitaxial quantum dots in which the size, shape and location of the dot is affected by the elastic strain developed in a large volume of the substrate as well as the local atomic bonding. The natural solution is to model the surroundings with a more coarse-grained (CG) description, suitable for the intrinsically longer length scale. The challenge then is to develop the computational methodology suitable for this kind of concurrent multiscale modeling, one in which the simulated length scale can be changed smoothly and seamlessly from one region of the simulation to another while maintaining the fidelity of the relevant mechanics, dynamics and thermodynamics. The realization that Nature has different relevant length scales goes back at least as far as Democritus. Some 24 centuries ago he put forward the idea that solid matter is comprised ultimately at small scales by a fundamental constituent that he termed as atom. Implicit in his philosophy was the idea that an 649 S. Yip (ed.), Handbook of Materials Modeling, 649–661. c 2005 Springer. Printed in the Netherlands.

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understanding of the atom would lead to a more robust understanding of the macroscopic world around us. In the intervening period, of course, not only has the science of this atomistic picture been put on a sound footing through the inventions of chemistry, the discovery of the nucleus and the development of quantum mechanics and modern condensed matter physics, but a host of additional length scales with their own relevant physics has been uncovered. A great deal of scientific innovation has gone into the development of physical models to describe the phenomena observed at these individual length scales. In the past decade a growing effort has been devoted in understanding how physics at different length scales works in concert to give rise to the observed behavior of solid materials. The use of models at multiple length scales, especially computer models optimized in this way, has been known as multiscale modeling. An example of multiscale modeling that we will consider in some detail is the modeling of the elastic deformation of solids at the atomistic and continuum levels. Clearly one kind of multiscale model would be to calculate the mass density and elastic constants within an atomistic model, and to use those data to parameterize a continuum model to describe large-scale elastic deformation. Such a parameter-passing, hierarchical approach has been used extensively to study a variety of systems [1]. Its success relies on the occurrence of well-separated length scales. We shall refer to such an approach as sequential multiscale modeling. In some systems, it is not clear how to separate the various length scales. An example would be turbulence, in which vortex structures are generated at many length scales and hierarchical models have to date only worked in very special cases [2]. Alternatively, the system of interest may be inhomogeneous and have regions in which small-scale physics dominates embedded in regions governed by large-scale physics. Examples would include fracture [3, 4], various nucleation phenomena [5], nanoscale moving mechanical components on computer chips (NEMS) [6], ion implantation and radiation damage events [7], epitaxial quantum dot growth [8] and so on. In either case hierarchical approach is not ideal, and concurrent multiscale modeling is preferred [9]. Here we focus on the inhomogeneous systems, and in particular on systems like those mentioned above in which the most interesting behavior involves the mechanics of a nanoscale region, but the overall behavior also depends on how the nanoscale region is coupled to its large-scale surroundings. This embedded nanomechanics may be studied effectively with concurrent multiscale modeling, where regions dominated by different length scales are treated with different models, either explicitly through a hybrid approach or effectively through a derivative approach. Here we focus on the methodology of coarse-grained molecular dynamics (CGMD) [9–12], one example of a concurrent multiscale model. CGMD describes the dynamic behavior of solids concurrently at the atomistic level and at more coarse-grained levels. The CG description is similar to finite element

Concurrent multiscale simulation at finite temperature

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modeling (FEM) of continuum elasticity, with several important distinctions. CGMD is derived directly from the atomistic model without recourse to a continuum description. This approach is important because it allows a more seamless coupling of the atomistic and coarse-grained models. The other important distinction is that CGMD is designed for finite temperature, and the coarse-graining procedure makes use of the techniques of statistical mechanics to ensure that the model provides a robust description of the thermodynamics. Several other concurrent multiscale models for solids have been proposed and used [13–18]. The Quasicontinuum technique is of particular note in this context, because it is also derived entirely from the underlying atomistic model [14]. CGMD was the first concurrent multiscale model designed for finite temperature simulations [10]. Recently, another finite temperature concurrent multiscale model has been developed using renormalization group techniques, including time renormalization [17]. This model is very interesting, although to date its formulation is based on bond decimation procedures that is limited to simple models with pair-wise nearest-neighbor interactions. The formulation of CGMD is more flexible, making it compatible with most classical interatomic potentials. It has been applied to realistic potentials in 3D whose range extends beyond nearest neighbors.

2.

Formulation of CGMD

Coarse-grained molecular dynamics provides a model whose minimum length scale may vary from one location to another in the system. The CGMD formulation begins with a specification of a mesh that defines the length scales that will be represented in each region (see Fig. 1). As in finite element modeling [19], the mesh is unstructured, and it comes with a set of shape functions that define how fields are continuously interpolated on the mesh. For example, the displacement field is the most basic field in CGMD, and it is approximated as u(x) ≈

u j N j (x),

(1)

j

where N j (x) is the value of the j th shape function evaluated at the point x in the undeformed (reference) configuration. It is often useful to let N j (x) have support at node j so that the coefficient u j represents the displacement at node j , but it need not be so for the derivation of CGMD. We will refer to u j as nodal displacements, bearing in mind that the coarse-grained fields could be more general. Ultimately the usual criteria to ensure well-behaved numerics will apply, such as the cells should not have high aspect ratios and the mesh size should not change too abruptly; for the purposes of the formulation, the only requirement we impose is that if a region of the mesh is at the atomic

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R.E. Rudd Micron Resonator

CG

MD

Figure 1. Schematic diagram of a concurrent multiscale simulation of a NEMS silicon microresonator [4–6] to illustrate how a system may be decomposed into atomistic (MD) and coarse-grained (CG) regions. The CG region comprises most of the volume, but the MD region contains most of the simulated degrees of freedom. Note that the CG mesh is refined to the atomic scale where it joins with the MD lattice.

scale, the positions of the nodes coincide with equilibrium lattice sites. This is not required for coarser regions of the mesh. To the first approximation, CGMD is governed by mass and stiffness matrices. They are derived from the underlying atomistic physics, described by a molecular dynamics (MD) model [20]. Define the discrete shape functions by evaluating the shape function N j (x) at the equilibrium lattice site x0µ of atom µ: N jµ = N j (x0µ ).

(2)

The discrete shape functions allow us to approximate the atomic displace ments uµ ≈ j u j N jµ . If we were to make this a strict equality, we would be on the path to the Quasicontinuum technique. Instead, we consider this a constraint on the system, and allow all of the unconstrained degrees of freedom in the system to fluctuate in thermal equilibrium. In particular, we demand that the interpolating fields be best fits to the underlying atomistic degrees of freedom of the system. In the case of the displacement field this requirement means that the nodal displacements minimize the chi-squared error of the fit: 2 2 u j N j µ . χ = uµ − µ j

(3)


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The minimum of χ 2 is given by u j = (N N T )−1 j k Nkµ uµ ≡ f jµ uµ ,

(4)

where repeated indices are summed and the inverse is a matrix inverse. We have introduced the weighting function expressed in terms of the discrete shape function as f jµ = (N N T )−1 j k Nkµ . Equation (4) provides the needed correspondence between the coarse and fine degrees of freedom. Once the weighting function f jµ is defined, the CGMD energy is defined as an average energy over the ensemble of systems in different points in phase space satisfying the correspondence relation (4). Mathematically, this is expressed as E(uk , u˙ k ) = Z

−1

dxµ dpµ HMD e−β HMD ,

(5)

where Z is the constrained partition function (the same integral without the HMD pre-exponential factor). The integral runs over the full 6Natom -dimensional MD phase space. The inverse temperature is given by β = 1/kT . The factor HMD is the MD Hamiltonian, the sum of the atomistic kinetic and potential energies. The potential energy is determined by an interatomic potential, a generalization of the well-known Lennard–Jones potential that typically includes non-linear many-body interactions [20]. The factor is a product of delta functions enforcing the constraint, =

j

δ uj −

µ

uµ f jµ δ u˙ j −

pµ f j µ µ

mµ

.

(6)

Once the energy (5) is determined, the equations of motion are derived as the corresponding Euler–Lagrange equations. The CGMD energy (5) consists of kinetic and potential terms. The CGMD kinetic energy can be computed exactly using analytic techniques for any system; the CGMD potential energy can also be calculated exactly, provided the MD interatomic potential is harmonic. Anharmonic corrections may be computed in perturbation theory. The details are given in Ref. [11]. Here we focus on the harmonic case, in which the potential energy is quadratic in the atomic displacements, and the coefficient of the quadratic term (times 2) is known as the dynamical matrix, Dµν . The result for harmonic CGMD is that E(uk , u˙ k ) = Uint + 12 (M j k u˙ j · u˙ k + u j · K j k uk ), Uint = Natom E + 3(Natom − Nnode )kT, Mij = m Niµ N jµ , −1 f j ν )−1 K ij = ( f iµ Dµν × ˜ −1 × Dµν D j ν , = Niµ Dµν N j ν − Diµ coh

(7) (8) (9) (10) (11)

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where Mij is the mass matrix and K ij is the stiffness matrix. Here again and throughout this Article a sum is implied whenever indices are repeated on one side of an equation unless otherwise noted. The internal energy Uint includes the total cohesive energy of the system, Natom E coh , as well as the internal energy of a collection of (Natom − Nnode ) harmonic oscillators at finite temperature. The form of the mass matrix (9) assumes a monatomic lattice. A more general form is given in Ref. [11]. The two forms of the stiffness matrix are equivalent in principle, although in practice numerical considerations have favored one form or the other for particular applications. The first form (10) was used for the early CGMD applications. It is most suited for applications in which the nodal index may be Fourier transformed, such as the computation of phonon spectra. The second form (11) is better suited for real space applications. It depends on an off-diagonal block of the dynamical matrix

D ×jµ = δµρ − N jµ f jρ Dρν N j ν

(12)

−1 for the internal and a regularized form of the lattice Green function D˜ µν degrees of freedom that is defined in Ref. [11]. Note that the mass matrix and the compliance matrix (the inverse of the stiffness matrix) are weighted averages of the corresponding MD quantities, the MD mass and MD lattice Green function, respectively. The CGMD equations of motion are derived from the CGMD Hamiltonian (5) using the Euler–Lagrange procedure

M j k u¨ k = −K j k uk + Fext j ,

(13)

where we have included the possibility of an external body force on node j given by Fext j . The anharmonic corrections to these equations of motion form an infinite Taylor series in powers of uk [11]. In regions of the mesh refined to the atomic level, it has been shown that the infinite series sums up to the MD interatomic forces; i.e., the original MD equations of motion are recovered in regions of the mesh refined to the atomic scale [10]. In the case of a harmonic system, the recovery of the MD equations of motion in the atomic limit should be clear from the equations for the mass and stiffness matrices. In this limit Niµ = δiµ and f iµ = δiµ , so Mij = mδij and K ij = Dij from Eqs. (9) and (10), respectively. In practice, we define two complementary regions of the simulation. In the CG region, the harmonic CGMD equations of motion (13) are used, whereas in the region of the mesh refined to the atomic level, called the MD region, the anharmonic terms are restored through the use of the full MD equations of motion. In a CGMD simulation the mass and stiffness matrices are calculated once at the beginning of the simulation. The reciprocal space (Fourier transform) representation of the dynamical matrix is used in order to make the calculation of the stiffness matrix tractable. This representation implicitly assumes that the solid in the form of a crystal lattice free from defects in the CG region.


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The CGMD mass matrix involves couplings between nearest neighbor nodes in the CG region, just as the distributed mass matrix of finite element modeling does. The fact that the mass matrix is not diagonal is inconvenient, since a system of equations must be solved in order to determine the nodal accelerations. The system of equations is sparse, but this step introduces some computational overhead, and it is desirable to eliminate it. In FEM, the distributed mass matrix is often replaced by a diagonal approximation, the lumped mass matrix [19]. In CGMD, the lumped mass approximation, lump

Mij

= m δij

Niµ

(no sum on i)

(14)

µ

has proven useful in the same way [9]. This definition assumes that the shape functions form a partition of unity, so that i Niµ = 1 for all µ. In principle, the determination of the equations of motion together with the relevant initial and boundary conditions completely specifies the problem. In practice, we have typically used a thermalized initial state and a mixture of periodic and free boundary conditions suitable for the problem of interest. The equations of motion are integrated in time using a velocity Verlet time integrator [20] with the conventional MD time step used throughout the simulation. The natural time scale of the CG nodes is longer due to the greater mass and greater compliance of the larger cells, and it would be natural to use a longer time step in the CG region. We have found little motivation to explore this possibility, however, since the computational cost of our simulations is typically dominated by the MD region, so there is little to gain by speeding up the computation in the CG region. We now turn to the question of how CGMD simulations are analyzed. Much of the analysis of CGMD simulations is accomplished using standard MD techniques. The simulations are typically constructed such that the most interesting phenomena occur in the MD region, and here most of the usual MD tools may be brought to bear. Thermodynamic quantities are calculated in the usual way, and the identification and tracking of crystal lattice defects may be accomplished with conventional techniques. In some cases it may be of interest to analyze the simulation in the CG region, as well. For example, it may be of interest to plot the temperature throughout the simulation in order to verify that the behavior at the MD/CG interface is reasonable. In MD the temperature is directly related to the mean ˙ 2 , where the brackets indicate the kinetic energy of the atoms: kT = 13 m|u| average [20]. In CGMD, a similar expression holds [11] kT = 13 |u˙ i |2 /Mii−1

(no sum on i),

(15)

where Mii−1 is the diagonal component corresponding to node i of the inverse of the mass matrix. This analysis of the temperature and thermal oscillations is

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closely tied to the kinetic energy in the CG region. Similar tools are available to analyze the potential energy and the related quantites such as deformation, pressure and stress [11].

3.

Validation

Validation of concurrent multiscale models is a challenge in its own right, and the development of quantitative tools and performance measures to analyze models like CGMD has taken place at the same time as the development of the first models. CGMD has been tested in several ways to see how it compares with a full MD simulation of a test system, as well as other concurrent multiscale simulations. The first test was the calculation of the spectrum of elastic waves or phonons. The techniques to calculate these spectra in atomistic systems have been developed long ago in the field of lattice dynamics [21]. In general the phonon spectrum is comprised of D acoustic mode branches (where D is the number of dimensions) together with D(Nunit −1) optical branches (where Nunit is the number of atoms in the elementary unit cell of the crystal lattice) [22]. The acoustic modes are distinguished by the fact that their frequency goes to zero as their wavelength becomes large. The infinite wavelength corresponds to uniform translation of the system, a process that costs no energy and hence corresponds to zero frequency. Elastic wave spectra are an interesting test of CGMD and other concurrent multiscale techniques because they represent a test of dynamics and because elastic waves have a natural length scale associated with them: the wavelength. When a CG mesh is introduced, the shortest wavelengths are excluded. These modes are eliminated because they are irrelevant in the CG region, and their elimination increases the efficiency of the simulation. The test then is to see how well the model describes those longer wavelength modes that are represented in the CG region. The elastic wave spectra for solid argon were computed in CGMD on a uniform mesh for various mesh sizes, and compared to the MD spectra and spectra computed using a FEM model based on continuum elasticity [9, 11]. The bonds between argon atoms were modeled with a Lennard–Jones potential cut off at the fifth shell of neighboring atoms. Several interesting results were found. First, both CGMD and FEM agreed with the MD spectrum at long wavelengths. This is to be expected, since for wavelengths much longer than the mesh spacing, the waveform should be well represented on the mesh. Also, at long wavelengths the FEM assumption of a continuous medium is justified, and the slope of the spectrum gives the sound velocity, c = ω/k for k → 0. Here ω is the (angular) frequency and k is the wave number. The error in ω(k) was found to be of order O(k 2 ) for FEM, as expected. It goes to zero in the long wavelength limit, k → 0. One nice feature of CGMD was a reduced


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error of order O(k 4 ) [10]. Moreover, CGMD provides a better approximation of the elastic wave spectra for all wavelengths supported on the mesh. Of course, CGMD also has the important feature that the elastic wave spectra are reproduced exactly when the mesh is refined to the atomic level, a property that FEM does not possess. Interatomic forces are not merely FEM elasticity on an atomic sized grid. Solid argon forms a face-centered cubic crystal lattice and hence has only three acoustic wave branches in its phonon spectrum. For crystals with optical phonon branches, there is more than one way to implement the coarsegraining, depending on the physics that is of interest, but the general CGMD framework continues to work well [23]. The other validation of CGMD has been the study of the reflection of elastic waves from the MD/CG interface. For applications such as crack propagation, it has proven important to control this unphysical reflection. The reflected waves can propagate back into the heart of the MD simulation and interfere with the processes of interest. In the case of crack propagation, a noticeable anomaly in the crack speed occurs at the point in time when the reflected waves reach the crack tip [24]. The reflection coefficient, a measure of the amount of elastic wave energy reflected at a given wavelength, has been calculated for CGMD and FEM based on continuum elasticity [10, 11]. Typical results are shown in Fig. 2. Long wavelength elastic waves are transmitted into the CG region, whereas short wavelength modes are reflected. The short wavelengths cannot be supported on the mesh, and since energy is conserved, they must go somewhere and they are reflected. The transmission threshold is expected to occur at a wave number k0 = π/(Nmax a). The CGMD threshold occurs precisely at 1 CGMD lump FEM dist FEM

0.8

Reflection Coefficient


1

0.6 0.4 0.2

⫺5

10

⫺10

10

CGMD lumped mass FEM distributed mass FEM

⫺15

10

⫺20

0

10 0

0.2

0.4

0.6

0.8

Wave number k/k0

1

1.2

1.4

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Wave number k/k0

Figure 2. A comparison of the reflection of elastic waves from a CG region in three cases: CGMD and two varieties of FEM. Note that the reflection coefficient is plotted on a log scale. A similar graph plotted on a linear scale is shown in Ref. [10]. The dashed line marks the natural cutoff [k0 = π/(Nmax a)], where Nmax is the number of atoms in the largest cells. The bumps in the curves are scattering resonances. Note that at long wavelengths CGMD offers significantly suppressed scattering.

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R.E. Rudd

this wave number, while the threshold for transmission in distributed mass and lumped mass FEM models occurs somewhat above and below this value, respectively. The scattering in the long wavelength limit shows a generalized Rayleigh scattering behavior. In conventional Rayleigh scattering the scattering crosssection goes like σ ∼ k 4 , which is the behavior exhibited by scattering here in FEM. For CGMD, the scattering drops off more quickly at long wavelengths, with the reflection coefficient approximately proportional to k 8 [11]. One aspect of concurrent multiscale modeling that remains poorly understood is the requirements for a suitable mesh. Certainly, many of the desired properties are clear either from the nature of the problem or from experience with FEM. For example, the mesh needs to be refined to the atomic level in the MD region, so here the mesh nodes should coincide with equilibrium crystal lattice sites. In the periphery large cells are desirable since the gain in efficiency is proportional to the cell size. From FEM it is well known that the aspect ratio of the cells should not be too large. Beyond these basic criteria, one is left with the task of generating a mesh that interpolates between the atomic-sized cells in the MD region to the large cells in the periphery without introducing high aspect ratio cells. One question we have investigated is whether the abruptness of this transition matters, and indeed it does matter. Figure 3 shows the reflection coefficient as a function of the wave number for two meshes that go between an MD region and a CG region with a maximum cell size of 20 lattice spacings. In one case, the transition is made gradually, whereas in the other case it is made abruptly. The mesh with the

1 CGMD smooth mesh

0.8



1

CGMD abrupt mesh 0.6 0.4 0.2 0

⫺5

10

⫺10

10

CGMD smooth mesh ⫺15

CGMD abrupt mesh

10

⫺20

0

0.2

0.4

0.6

0.8

Wave number k/k0

1

1.2

1.4

10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Wave number k/k0

Figure 3. A comparison of the reflection of elastic waves from a CG region whose mesh varies smoothly in cell size and one with an abrupt change in cell size, both computed in CGMD. In both cases the reflection coefficient is plotted as a function of the wave number in units of the natural cutoff [k0 = π/(Nmax a)], where a is the lattice constant and Nmax a = 20a is the maximum linear cell size in the mesh. The pronounced series of scattering resonances in the case of the abruptly changing mesh is undesirable. The second panel is a log-linear plot of the same data in order to show how the series of scattering resonances continues at decreasing amplitudes to long wavelengths.


659

abrupt transition exhibits markedly increased scattering, including a series of strong scattering resonances. Note that the envelope of the scattering curve is well defined in the case of the abrupt mesh, a property used to calculate the scaling of the reflection coefficient, R ∼ k 8 .

4.

Outlook

CGMD provides a formalism for concurrent multiscale modeling at finite temperature. The initial tests have been very encouraging, but there are still many ways in which CGMD can be developed. One area of active research is numerical algorithms to make CGMD more efficient for large simulations. The calculation of the stiffness matrix involves the inverse of a large matrix whose size grows with the number of nodes in the CG region, NCGnode . The 3 calculation of the inverse scales like NCGnode and the matrix storage scales 2 like NCGnode , for the exact matrix without any cutoff. Even though the calculation of the stiffness matrix need only be done once during the simulation, the calculation has proven sufficiently onerous to prevent the application of CGMD to the large-scale simulations for which it was originally intended. Only now are linear scaling CGMD algorithms starting to become available. There are several directions in which CGMD has begun to be extended for specific applications. The implementation of CGMD described in this Article conserves energy. It implicitly makes the assumption that the only thermal fluctuations that are relevant to the problem are those supported on the mesh. Fluctuations of the degrees of freedom that have been integrated out are neglected. Those fluctuations can be physically relevant in several ways [12]. First, they exert random and dissipative forces on the coarse-grained degrees of freedom in a process that is analogous to the forces in Brownian motion exerted on a large particle by the atoms in the surrounding liquid. Second, they also act as a heat bath that is able to exchange and transport thermal energy. Finally, they can transport energy in non-equilibrium processes, such as the waves generated by a propagating crack discussed above. A careful treatment of the CG system leads to a generalization of the CGMD equations of motion presented above [12]. In addition to the conservative forces, there are random and dissipative forces that form a generalized Langevin equation. The dissipative forces involve a memory function in time and space that acts to absorb waves that cannot be supported in the CG region. The memory kernel is similar to those that have been discussed in the context of absorbing boundary conditions for MD simulations [25, 26], except that in CGMD the range of the kernel is shorter because the long wavelength modes are able to propagate into the CG region and do not need to be absorbed. Interestingly, in the case of a CG region surrounded by MD regions, the memory kernel also contains propagators that recreate the absorbed waves on the far

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side of the CG region after the appropriate propagation delay [12]. Of course, use of the generalized Langevin incurs additional computational expenses both in terms of run time and memory. There are many other ways in which CGMD could be extended. Additional CG fields could be introduced to model various material phenomena such as electrical polarization, defect concentrations and local temperature. Fluxes such as heat flow and defect diffusion can be included through the technique of coarse-graining the atomistic conservation equations. CGMD provides a powerful framework in which to formulate finite temperature multiscale models for a variety of applications.

Acknowledgments This article was prepared under the auspices of the US Department of Energy by University of California, Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.

References [1] J.A. Moriarty, J.F. Belak, R.E. Rudd, P. Soderlind, F.H. Streitz, and L.H. Yang, “Quantum-based atomistic simulation of materials properties in transition metals,” J. Phys.: Condens. Matter, 14, 2825–2857, 2002. [2] A.A. Townsend, The Structure of Turbulent Shear Flow, 2nd edition, Cambridge University Press, Cambridge, 1976. [3] F.F. Abraham, J.Q. Broughton, E. Kaxiras, and N. Bernstein, “Spanning the length scales in dynamic simulation,” Comput. in Phys., 12, 538–546, 1998. [4] F.F. Abraham, R. Walkup, H. Gao, M. Duchaineau, T. Diaz de la Rubia, and M. Seager, “Simulating materials failure by using up to one billion atoms and the world’s fastest computer: work-hardening,” Proc. Natl. Acad. Sci. USA, 99, 5783–5787, 2002. [5] D.R. Mason, R.E. Rudd, and A.P. Sutton, “Atomistic modelling of diffusional phase transformations with elastic strain,” J. Phys.: Condens. Matter, 16, S2679–S2697, 2004. [6] R.E. Rudd and J.Q. Broughton, “Atomistic simulation of MEMS resonators through the coupling of length scales,” J. Model. Simul. Microsys., 1, 29–38, 1999. [7] R.S. Averback and T. Diaz de la Rubia, “Fundamental studies of radiation effects in solids,” Solid State Phys., 51, 281–402, 1998. [8] R.E. Rudd, G.A.D. Briggs, A.P. Sutton, G. Medieros-Ribiero, and R.S. Williams, “Equilibrium model of bimodal distributions of epitaxial island growth,” Phys. Rev. Lett., 90, 146101, 2003. [9] R.E. Rudd and J.Q. Broughton, “Concurrent multiscale simulation of solid state systems,” Phys. Stat. Sol. (b), 217, 251–291, 2000. [10] R.E. Rudd and J.Q. Broughton, “Coarse-grained molecular dynamics and the atomic limit of finite elements,” Phys. Rev. B, 58, R5893–R5896, 1998.


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[11] R.E. Rudd and J.Q. Broughton, “Coarse-grained molecular dynamics: non-linear finite elements and finite temperature,” Phys. Rev. B, 2004 (unpublished). [12] R.E. Rudd, Coarse-grained molecular dynamics: Dissipation due to internal modes. Mater. Res. Soc. Symp. Proc., 695, T10.2, 2002. [13] S. Kohlhoff, P. Gumbsch, and H.F. Fischmeister, “Crack-propagation in bcc crystals studied with a combined finite-element and atomistic model,” Philos. Mag. A, 64, 851–878, 1991. [14] E.B. Tadmor, M. Ortiz, and R. Phillips, “Quasicontinuum analysis of defects in solids,” Philos. Mag. A, 73, 1529–1563, 1996. [15] J.Q. Broughton, F.F. Abraham, N. Bernstein, and E. Kaxiras, “Concurrent coupling of length scales: methodology and application,” Phys. Rev. B, 60, 2391–2403, 1999. [16] L.E. Shilkrot, R.E. Miller, and W.A. Curtin, “Coupled atomistic and discrete dislocation plasticity,” Phys. Rev. Lett., 89, 025501, 2002. [17] S. Curtarolo and G. Ceder, “Dynamics of an inhomogeneously coarse grained multiscale system,” Phys. Rev. Lett., 88, 255504, 2002. [18] W.A. Curtin and R.E. Miller, “Atomistic/continuum coupling in computational materials science,” Modell. Simul. Mater. Sci. Eng., 11, R33–R68, 2003. [19] T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover, Mineola, 2000. [20] M.P. Allen and D.J. Tildesley, Computer Simulation of Liquids, Clarendon Press, Oxford, 1987. [21] M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon Press, Oxford, 1954. [22] N.W. Ashcroft and N.D. Mermin, Solid State Physics, Saunders College Press, Philadelphia, 1976. [23] B. Kraczek, private communication, 2003. [24] B.L. Holian and R. Ravelo, “Fracture simulations using large-scale moleculardynamics,” Phys. Rev. B, 51, 11275–11288, 1995. [25] W. Cai, M. de Koning, V.V. Bulatov, and S. Yip, “Minimizing boundary reflections in coupled-domain simulations,” Phys. Rev. Lett., 85, 3213–3216, 2000. [26] W.E and Z. Huang, “Matching conditions in atomistic-continuum modeling of materials,” Phys. Rev. Lett., 87, 135501, 2001.

2.13 THE THEORY AND IMPLEMENTATION OF THE QUASICONTINUUM METHOD E.B. Tadmor1 and R.E. Miller2 1 Technion–Israel Institute of Technology, Haifa, Israel 2

Carleton University, Ottawa, ON, Canada

While atomistic simulations have provided great insight into the basic mechanisms of processes like plasticity, diffusion and phase transformations in solids, there is an important limitation to these methods. Specifically, the large number of atoms in any realistic macroscopic structure is typically much too large for direct simulation. Consider that the current benchmark for largescale fully atomistic simulations is on the order of 109 atoms, using massively paralleled computer facilities with hundreds or thousands of CPUs. This represents 1/10 000 of the number of atoms in a typical grain of aluminum, and 1/1 000 000 of the atoms in a typical micro-electro-mechanical systems (MEMS) device. Further, it is apparent that with such a large number of atoms, substantial regions of a problem of interest are essentially behaving like a continuum. Clearly, while fully atomistic calculations are essential to our understanding of the basic “unit” mechanisms of deformation, they will never replace continuum models altogether. The goal for many researchers, then, has been to develop techniques that retain a largely continuum mechanics framework, but impart on that framework enough atomistic information to be relevant to modeling a problem of interest. In many examples, this means that a certain, relatively small, fraction of a problem require full atomistic detail while the rest can be modeled using the assumptions of continuum mechanics. The quasicontinuum method (QC) has been developed as a framework for such mixed atomistic/continuum modeling. The QC philosophy is to consider the atomistic description as the “exact” model of material behaviour, but at the same time acknowledge that the sheer number of atoms make most problems intractable in a fully atomistic framework. Then, the QC uses continuum assumptions to reduce the degrees of freedom and computational demand without losing atomistic detail in regions where it is required. 663 S. Yip (ed.), Handbook of Materials Modeling, 663–682. c 2005 Springer. Printed in the Netherlands.

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The purpose of this article is to provide an overview of the theoretical underpinnings of the QC method, and to shed light on practical issues involved in its implementation. The focus of the article will be on the specific implementation of the QC method as put forward in Refs. [1–4]. Variations on this implementation, enhancements, and details of specific applications will not be presented. For the interested reader, these additional topics can be found in several QC review articles [5–8] and of course in the original references. The most recent of the QC reviews [5] provides an extensive literature survey, detailing many different implementations, extensions and applications of the QC. Also included in that review are several other coupled methods that are either direct descendants of the QC or are similar alternatives developed independently. For a detailed comparison between several coupled atomistic/continuum methods including the QC, the reader may find the review by Curtin and Miller [9] of interest. A QC website designed to serve as a clearinghouse for information on the QC method has been established at www.qcmethod.com. The site includes information on QC research, links to researchers, downloadable QC code and documentation. The downloadable code is freely available and corresponds to the QC implementation discussed in this paper.

1.

Atomistic Modeling of Crystalline Solids

In the QC, the point-of-view which is adopted is that there is an underlying atomistic model of the material which is the “correct” description of the material behaviour. This could, in principle, be a quantum-mechanically based description such as density functional theory (DFT), but in practice the focus has been primarily on atomistic models based on semi-empirical interatomic potentials. A review of such methods can be found, for example, in [10]. Here, we present only the features of such models which are essential for our discussion. We focus on lattice statics solutions, i.e., we are looking for equilibrium atomic configurations for a given model geometry and externally imposed forces or displacements, because most applications of the QC have used a static implementation. Recent work to extend QC to finite temperature and dynamic simulations shows promise, and can be found in Ref. [11]. We assume that there is some reference configuration of N atomic nuclei, confined to a lattice. Thus, the reference position of the ith atom in the model X i is found from an integer combination of lattice vectors and a reference (origin) atom position, X 0 X i = X 0 + li A1 + m i A2 + n i A3 ,

(1)

The theory and implementation of the quasicontinuum method

665

where (li , m i , n i ) are integers, A j is the j th Bravais lattice vector.1 The deformed position of the ith atom x i , is then found from a unique displacement vector ui for each atom. x i = X i + ui .

(2)

The displacements ui , while only having physical meaning on the atomic sites, can be treated as a continuous field u(X) throughout the body with the property that u(X i ) ≡ ui . This approach, while not the conventional one in atomistic models, is useful in effecting the connection to continuum mechanics. Note that for brevity we will often refer to the field u to represent the set of all atomic displacements {u1 , u2 , . . . , u N } where N is the number of atoms in the body. In standard lattice statics approaches using semi-empirical potentials, there is a well defined total energy function E tot that is determined from the relative positions of all the atoms in the problem. In many semi-empirical models, this energy can be written as a sum over the energy of each individual atom. Specifically, E tot =

N

E i (u),

(3)

i=1

where E i is the site energy of atom i, which depends on the displacements u through the relative positions of all the atoms in the deformed configuration. For example, within the embedded atom method (EAM) [13, 14] atomistic model, this site energy is given by E i = Ui (ρ¯i ) +

1 Vi j (ri j ), 2 j =/ i

(4)

where Ui can be interpreted as an electron-density dependent embedding energy, Vi j is a pair potential between atom i and its neighbor j and ri j = (x i − x j ) · (x i − x j ) is the interatomic distance. The electron density at the position of atom i, ρ¯i , is the superposition of spherically averaged density contributions from each of the neighbors, ρ j : ρ¯i =

ρ j (ri j ).

(5)

j= /i

A similar site energy can be identified for other empirical atomistic models, such as those of the Stillinger–Weber type [15], for instance. 1 We omit a discussion of complex lattices with more than one atom at each Bravais lattice site. This topic is discussed in Refs. [5, 12].

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In addition to the potential energy of the atoms, there may be energy due to external loads applied to atoms. Thus, the total potential energy of the system (atoms plus external loads) can be written as (u) = E tot(u) −

N

f i ui ,

(6)

i=1

where − f i ui is the potential energy of the applied load f i on atom i. In lattice statics, we seek the displacements u such that this potential energy is minimized.

2.

The QC Method

The goal of the static QC method is to find the atomic displacements that minimize Eq. (6) by approximating the total energy of Eq. (3) such that: 1. the number of degrees of freedom is substantially reduced from 3N , but the full atomistic description is retained in certain “critical” regions, 2. the computation of the energy in Eq. (3) is accurately approximated without the need to explicitly compute the site energy of all the atoms, 3. the fully atomistic, critical regions can evolve with the deformation, during the simulation. In this section, the details of how the QC achieves each of these goals are presented.

2.1.

Removing Degrees of Freedom

A key measure of a displacement field is the deformation gradient F. A body deforms from reference state X to deformed state x = X + u(X), from which we define F(X) ≡

∂x ∂u =I+ , ∂X ∂X

(7)

where I is the identity tensor. If the deformation gradient changes gradually on the atomic scale, then it is not necessary to explicitly track the displacement of every atom in the region. Instead, the displacements of a small fraction of the atoms (called representative atoms or “repatoms”) can be treated explicitly, with the displacements of the remaining atoms approximately found through interpolation. In this way, the degrees of freedom are reduced to only the coordinates of the repatoms.


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The QC incorporates such a scheme by recourse to the interpolation functions of the finite element method (FEM) (see, for example, [16]). Figure 1 illustrates the approach in two-dimensions in the vicinity of a dislocation core. The filled atoms are the selected repatoms, which are meshed by a space-filling set of linear triangular finite elements. Any atom not chosen as a repatom, like the one labeled “A”, is subsequently constrained to move according to the interpolated displacements of the element in which is resides. The density of repatoms is chosen to vary in space according to the needs of the problem of interest. In regions where full atomistic detail is required, all atoms are chosen as repatoms, with correspondingly fewer in regions of more slowly varying deformation gradient. This is illustrated in Fig. 1, where all the atoms around the dislocation core are chosen as repatoms. Further away, where the crystal experiences only the linear elastic strains due to the dislocation, the density of repatoms is reduced. This first approximation of the QC, then, is to replace the energy E tot by tot,h E : E

tot,h

=

N

E i (uh ).

(8)

i=1

In this equation the atomic displacements are now found through the interpolation functions and take the form h

u =

Nrep

Sα uα ,

(9)

α=1

where Sα is the interpolation (shape) function associated with repatom α, and Nrep is the number of repatoms, Nrep N . Note that the formal summation over the shape functions in Eq. (9) is in practice much simpler due to the compact support of the finite element shape functions. Specifically, shape functions are identically zero in every element not immediately adjacent to a specific repatom. Referring back to Fig. 1, this means that the displacement of atom A is determined entirely from the sum over the three repatoms B, C and D defining the element containing A: uh (X A ) = SB (X A )uB + SC (X A )uC + SD (X A )uD .

(10)

Introducing this kinematic constraint on most of the atoms in the body will achieve the goal of reducing the number of degrees of freedom in the problem, but notice that for the purpose of energy minimization we must still compute the energy and forces on the degrees of freedom by explicitly visiting every atom – not just the repatoms – and building its neighbor environment from the interpolated displacement fields. Next, we discuss how these calculations are approximated and made computationally tractable.

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E.B. Tadmor and R.E. Miller (a)

A

(b)

D

B

A

C

Figure 1. Selection of repatoms from all the atoms near a dislocation core are shown in (a), which are then meshed by linear triangular elements in (b). The density of the repatoms varies according to the severity of the variation in the deformation gradient. After Ref. [5]. Reproduced with permission.


2.2.

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Efficient Energy Calculations: The Local QC

In addition to the degree of freedom reduction described in Section 2.1, the QC requires an efficient means of computing the energy and forces without the need to visit every atom in the problem as implied by Eq. (8). The first way to accomplish this is by recourse to the so-called Cauchy–Born (CB) rule (see Ref. [17] and references therein), resulting in what is referred to as the local formulation of the QC.1 The use of linear shape functions to interpolate the displacement field means that within each element, the deformation gradient will be uniform. The Cauchy–Born rule assumes that a uniform deformation gradient at the macro-scale can be mapped directly to the same uniform deformation on the micro-scale. For crystalline solids with a simple lattice structure,2 this means that every atom in a region subject to a uniform deformation gradient will be energetically equivalent. Thus, the energy within an element can be estimated by computing the energy of one atom in the deformed state and multiplying by the number of atoms in the element. In practice, the calculation of the CB energy is done separately from the model in a “black box,” where for a given deformation gradient F, a unit cell with periodic boundary conditions is deformed appropriately and its energy is computed. The strain energy density in the element is then given by E(F) =

E 0 (F) , 0

(11)

where 0 is the unit cell volume (in the reference configuration) and E 0 is the energy of the unit cell when its lattice vectors are distorted according to F. Now the total energy of an element is simply this energy density times the element volume, and the total energy of the problem is simply the sum of element energies: E

tot,h

≈E

tot,h

=

N element

e E(F e ),

(12)

e=1

where e is the volume of element e. The important computational saving made here is that a sum over all the atoms in the body has been replaced by a sum over all the elements, each one requiring an explicit energy calculation for only one atom. Since the number of elements is typically several orders of magnitude smaller than the total number of atoms, the computational 1 The term “local” refers to the fact that use of the CB rule implies that the energy at each point in the

continuum will only be a function of the deformation at that point and not on its surroundings. 2 A simple lattice structure is one for which there is only one atom at each Bravais lattice site. In a complex lattice with two or more atoms per site, the Cauchy–Born rule must be generalized to permit shuffling of the off-site atoms. See Ref. [12].

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savings is substantial. The number of elements scales linearly with the number of repatoms, and so the local QC scales as O(Nrep ). Note, however, that even in the case where the deformation is uniform within each element, the local prescription for the energy in the element is only approximate. This is because in the constrained displacement field uh , the deformation gradient varies from one element to the next. At element boundaries and free surfaces, atoms can have energies that differ significantly from that of an atom in a bulk, uniformly deformed lattice. Figure 2 illustrates this schematically for an initially square lattice deformed according to two different deformation gradients in two neighboring regions. The energy of the atom labeled as a “bulk atom” can be accurately computed from the CB rule; its neighbor environment is uniform even though some of its neighbors occupy other elements. However, the “interface atom” and “surface atom” are not accurately described by the CB rule, which assumes that these atoms see uniformly deformed bulk environments. In situations where the deformation is varying slowly from one element to the next and where surface energetics are not important, the local approximation is a good one. Using the CB rule as in Eq. (11), the QC can be thought of as a purely continuum formulation, but with a constitutive law that is based on

Reference

Deformed

interface atom

surface atom

bulk atom

Figure 2. On the left, the reference configuration of a square lattice meshed by triangular elements. On the right, the deformed mesh shows a bulk atom, for which the CB rule is exactly correct, and two other atoms for which the CB rule will give the wrong energy due to its inability to describe surfaces or changes in the deformation gradient. After Ref. [5]. Reproduced with permission.


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atomistics rather than on an assumed phenomenological form. The CB constitutive law automatically ensures that the correct anisotropic crystal elasticity response will be recovered for small deformations. It is non-linear elastic (as dictated by the underlying atomistic potentials) for intermediate strains and includes lattice invariance for large deformations; for example, a shear deformation that corresponds to the twinning of the lattice will lead to a rotated crystal structure with zero strain energy density. An advantage of the local QC formulation is that it allows the use of quantum-mechanical atomistic models that cannot be written as a sum over individual atom energies such as tight binding (TB) and DFT. In these models only the total energy of a collection of atoms can be obtained. However, for a lattice undergoing a uniform deformation it is possible to compute the energy density E(F) from a single unit cell with periodic boundary conditions. Incorporation of quantum-mechanical information into the atomic model generally ensures that the description is more transferable, i.e., it provides a better description of the energy of atomic configurations away from the reference structure to which empirical potentials are fitted. This allows truly firstprinciples simulations of some macroscopic processes such as homogeneous phase transformations.

2.3.

More Accurate Calculations: Mixed Local/Non-Local QC

The local QC formulation successfully enhances the continuum FEM framework with atomistic properties such as nonlinearity, crystal symmetry and lattice invariance. The latter property means that dislocations may exist in the local QC. However, the core structure and energy of these dislocations will only be coarsely represented due to the CB approximation of the energy. The same is true for other defects such as surfaces and interfaces, where the deformation of the crystal is non-uniform over distances shorter than the cutoff radius of the interatomic potentials. For example, to correctly account for the energy of the interface shown in Fig. 2, the non-uniform environment of the atoms along the interface must be correctly accounted for. While the local QC can support deformations (such as twinning) which may lead to microstructures containing such interfaces, it will not account for the energy cost of the interface itself. In order to correctly capture these details, the QC must be made non-local in certain regions. The energy of Eq. (8), which in the local QC was approximated by Eq. (12), must instead be approximated in a way that is sensitive to non-uniform deformation and free surfaces, especially in the limit where full atomistic detail is required.

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We now make the ansatz that the energy of Eq. (8) can be approximated by computing only the energy of the repatoms, but we will identify each repatom as being either local or non-local depending on its deformation environment. Thus, the repatoms are divided into Nloc local repatoms and Nnl non-local repatoms (Nloc + Nnl = Nrep ). The energy expression is then approximated as E

tot,h

≈

Nnl

n α E α (uh ) +

α=1

Nloc

n α E α (uh ).

(13)

α=1

The important difference between Eq. (8) and Eq. (13) is that the sum on all the atoms in the problem has been replaced with a sum on only the repatoms. The function n α is a weight assigned to repatom α, which will be high for repatoms in regions of low repatom density and vice versa. For consistency, the weight functions must be chosen so that Nrep

n α = N,

(14)

α=1

which further implies (through the consideration of a special case where every atom in a problem is made a repatom) that in atomically-refined regions, all n α = 1. From Eq. (14), the weight functions can be physically interpreted as the number of atoms represented by each repatom α. The weight n α for each repatom (local or non-local) is determined from a tessellation that divides the body into cells around each repatom. One physically sensible tessellation is Voronoi cells [18], but an approximate Voronoi diagram can be used instead due to the high computational overhead of the Voronoi construction. In practice, the coupled QC formulation makes use of a simple tessellation based on the existing finite element mesh, partitioning each element equally between each of its nodes. The volume of the tessellation cell for a given repatom, divided by the volume of a single atom (the Wigner–Seitz volume) provides n α for the repatom. In typical QC simulations, non-local regions are fully refined down to the atomic scale, and so the weight of the non-local repatoms is one. To compute the energy of a local repatom α, we recognize that of the n α atoms it represents, n eα reside in each element e adjacent to the repatom. The weighted energy contribution of the repatom is then found by applying the CB rule within each element adjacent to α such that Eα =

M n eα e=1

nα

0 E(F e ),

nα =

M

n eα ,

(15)

e=1

where E(F e ) is the energy density in element e by the CB rule, 0 is the Wigner–Seitz volume of a single atom and e runs over all elements adjacent to α.


673

Note that this description of the local repatoms is exactly equivalent to the element-by-element summation of the local QC in Eq. (12); it is only the way that the energy partitioning is written that is different. In a mesh containing only local repatoms, the two formulations are the same, but the summations have been rearranged from one over elements in Eq. (12) to one over the repatoms here. The energy of each non-local repatom is computed from the deformed neighbor environment dictated from the current interpolated displacements in the elements. In essence, every atom in the vicinity of a non-local repatom is displaced to the deformed configuration, the energy of each non-local repatom in this configuration is computed from Eq. (4), and the total energy is the sum of these repatom energies weighted by n α . For example, the energy of the repatom identified as an “interface atom” in Fig. 2 requires that the neighbor environment be generated by displacing each neighbor according to the element in which it resides. Thus, the energy of each non-local repatom is exactly as it should be under the displacement field uh , while the local approximation is used in regions where the deformation is uniform on the atomic scale. From this starting point, the forces on all the repatoms can be obtained as the appropriate derivatives of Eq. (13), and energy minimization can proceed. When making use of the mixed formulation described in Eq. (13), it now becomes necessary to decide whether a given repatom should be local or non-local. This is achieved automatically in the QC using a non-locality criterion. Note that simply having a large deformation in a region does not in itself require a non-local repatom, as the CB rule of the local formulation will exactly describe the energy of any uniform deformation, regardless of the severity. The key feature that should trigger a non-local treatment of a repatom is a significant variation in the deformation gradient on the atomic scale in the repatom’s proximity. Thus, the non-locality criterion in implemented as follows. A cut-off, rnl , is empirically chosen to be between two and three times the cut-off radius of the interatomic potentials. The deformation gradients in every element within this cut-off of a given representative atom are compared, by looking at the differences between their eigenvalues. The criterion is then: max |λak − λbk | < ,

(16)

a,b;k

where λak is the kth eigenvalue of the right stretch tensor U a = F aT F a in element a, k = 1 · · · 3, and the indices a and b run over all elements within rnl of a given repatom. The repatom will be made local if this inequality is satisfied, and non-local otherwise. In practice, the tolerance is determined empirically. A value of 0.1 has been used in a number of tests and found to give good results. The effect of this criterion is clusters of non-local atoms in regions of rapidly varying deformation.

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The fact that the non-local repatoms tend to cluster into atomistically refined regions surrounded by local regions leads to non-local/local interfaces in the QC. As in all attempts to couple a non-local atomistic region to a local continuum region found in the literature, this will lead to spurious forces near the interface. These forces, dubbed “ghost-forces” in the QC literature, arise due to the fact that there is an inherent mismatch between the local (continuum) and non-local (atomistic) regions in the problem. In short, the finite range of interaction in the non-local region mean that the motion of repatoms in the local region will effect the energy of non-local repatoms, while the converse may not be true. Upon differentiating Eq. (13), forces on repatoms in the vicinity of the interface may include a non-physical contribution due to this asymmetry. Note that these ghost forces are a consequence of differentiating an approximate energy functional, and therefore they still are “real” forces in the sense that they come from a well-defined potential. The problem is that the mixed local/non-local energy functional of Eq. (13) is approximate, and the error in this approximation is most apparent at the interface. A consequence of this is that a perfect, undistorted crystal containing an artificial local/nonlocal interface will be able to lower its energy below the ground-state energy by rearranging the atoms in the vicinity of the interface. This is clearly a non-physical result. In Ref. [3], a solution to the ghost forces was proposed whereby corrective forces were added as dead loads to the interface region. In this way, there is a well-defined contribution of the corrective forces to the total energy functional (since the dead loads are constant) and the minimization of the modified energy can proceed using standard conjugate gradient or Newton–Raphson techniques. The procedure can be iterated to self-consistency.

2.4.

Evolving Microstructure: Automatic Mesh Adaption

The QC approach outlined in the previous sections can only be successfully applied to general problems in crystalline deformation if it is possible to ensure that the fine structure in the deformation field will be captured. Without a priori knowledge of where the deformation field will require fine-scale resolution, it is necessary that the method have an automatic way to adapt the finite element mesh through the addition or removal of repatoms. To this end, the QC makes use of the finite element literature, where considerable attention has been given to adaptive meshing techniques for many years. Typically in finite element techniques, a scalar measure is defined to quantify the error introduced into the solution by the current density of nodes (or repatoms in the QC). Elements in which this error estimator is higher than some prescribed tolerance are targeted for adaption, while at the same time


675

the error estimator can be used to remove unnecessary nodes from the model. The error estimator of Zienkiewicz and Zhu [19], originally posed in terms of errors in the stresses, is re-cast for the QC in terms of the deformation gradient. Specifically, we define the error estimator to be 

 1

εe = 

e

1/2 

¯ − F e ) :( F ¯ − F e )d (F

,

(17)

e

where e is the volume of element e, F e is the QC solution for the deformation ¯ is the L 2 -projection of the QC solution for F, gradient in element e, and F given by ¯ = SF avg . (18) F Here, S is the shape function array, and F avg is the array of nodal values ¯ Because the deformation gradients of the projected deformation gradient F. are constant within the linear elements used in the QC , the nodal values F avg are simply computed by averaging the deformation gradients found in each element touching a given repatom. This is then interpolated throughout the elements using the shape functions, providing an estimate to the discretized field solution that would be obtained if higher order elements were used. The error, then, is defined as the difference between the actual solution and this estimate of the higher order solution. If this error is small, it implies that the higher order solution is well represented by the lower order elements in the region, and thus no refinement is required. The integral in Eq. (17) can be computed quickly and accurately using Gaussian quadrature. Elements for which the error εe is greater than some prescribed error tolerance are targeted for refinement. Refinement then proceeds by adding three new repatoms at the atomic sites closest to the mid-sides of the targeted elements. Notice that since repatoms must fall on actual atomic sites in the reference lattice, there is a natural lower limit to element size; if the nearest atomic sites to the mid-sides of the elements are the atoms at the element corners, the region is fully refined and no new repatoms can be added. The same error estimator is used in the QC to remove unnecessary repatoms from the mesh. In this process, a repatom is temporarily removed from the mesh and the surrounding region is locally remeshed. If the all of the elements produced by this remeshing process have a value of the error estimator below the threshold, the repatom can be eliminated.

3.

Practical Issues in QC Simulations

In this section, we will use a specific, simple example to highlight the practical issues surrounding solutions using the QC method. The example to be

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discussed is also provided with the QC download at qcmethod.com, and it is discussed in even greater detail in the documentation that accompanies that code.

3.1.

Problem Definition

Consider the problem of a twin boundary in face-centered cubic (FCC) aluminum. The boundary is perfect but for a small step. A question of interest may be “how does this stepped boundary respond to mechanical load?” In this example, we probe this question by using the QC method to solve the problem shown in Fig. 3(a), where two crystals, joined by a stepped twin boundary, are sheared until the boundary begins to migrate due to the load. The result will elucidate the mechanism of this migration. The implementation of the QC method used to solve this problem has been described as “two and a half” dimensional to emphasize that, while it is not a fully 3D model it is also not simply 2D. Specifically, the reference crystal structure is 3D, and all the underlying atomistic calculations (both local and non-local) consider the full, 3D environment of each atom. However, the deformation of the crystal is constrained such that the three components of displacement, u x , u y and u z are functions only of two coordinates x and y. This allows, for example, both edge and screw dislocations, but forces the line direction of the dislocations to be along z. For the reader who is familiar with purely atomistic simulations, this is equivalent to imposing periodic boundary conditions along the z direction, and then using a periodic cell with the

(a)

(b) 200

200

150

150 fcc Al

100

50

Stepped twin boundary

0

Y

Y

50

100

⫺50

0

⫺50

⫺100

⫺100

fcc Al

⫺150

⫺150

⫺200

⫺200 ⫺200 ⫺100

x

0

100

200

⫺200 ⫺100

x

0

100

200

Figure 3. (a) Initial coarse mesh used to define the simulation volume and (b) the final mesh after the automatic adaption.


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minimum possible thickness along z to produce the correct crystal structure. We sometimes refer to this as a “2D” implementation for brevity, but ask that the reader bears in mind the true nature of the model. The use of a 2D implementation of the QC to study this problem is appropriate given its geometry. However, fully 3D implementations of the QC exist and these must be used for many problems of interest (see examples in Ref. [5]). The starting point for a QC simulation is a crystal lattice, defined by an origin atom and a set of Bravais vectors as in Eq. (1). To allow the QC method to model polycrystals, it is necessary to define a unique crystal structure within each grain. The shape of each grain is defined by a simple polygon in 2D. Physically, it makes sense that the polygons defining each grain do not overlap, although it may be possible to have holes between the grains. In our example, it is easy to see how the shape of the two grains could be defined to include the grain boundary step. Mathematically, the line defining the boundary should be shared identically by the two grains, but this can lead to numerical complications; for example in checking whether two grains overlap. Fortunately, realistic atomistic models are unlikely to encounter atoms that are less than an Angström or so apart, and so there exists a natural “tolerance” in the definition of these polygons. For example, a gap between grains of 0.1 Å will usually provide sufficient numerical resolution between the grains without any atoms falling “in the gap” and therefore being omitting from the model. In the QC implementation, the definition of the grains is separate from the definition of the actual volume of material to be simulated. This simulation volume is defined by a finite element mesh between an initial set of repatoms. Each element in this mesh must lie within one or more of the grain polygons described above, but the finite element mesh need not fill the entire volume of the defined grains. It is useful to think of the actual model (the mesh) being “cut-out” from the previously defined grain structure. For our problem, a sensible choice for the initial mesh is shown in Fig. 3(a), where the grain boundary lies approximately (to within the height of the step) along the line y = 0. Elements whose centroid lie above or below the grain boundary are assumed to contain material oriented according to the lattice of the upper or lower grain, respectively. Since our interest here is atomic scale processes along the grain boundary, it is clear that the model shown in Fig. 3(a), with elements approximately 50 Å in width, will not provide the necessary accuracy. Thus, we can make use of the QC’s automatic adaption to increase the resolution near the grain boundary. The main adaption criterion, as outlined earlier, is based on error in the finite element interpolation of the deformation gradient. However, there will initially be no deformation near the grain boundary and thus no reason for automatic adaption to be triggered. It is therefore necessary to force the model to adapt in regions that are inhomogeneous at the atomic scale for reasons other than deformation. To this end, we can identify certain segments of the

678


grain boundary as “active” segments. Any repatom within a prescribed distance of an active segment will be made non-local. This further implies that the elements touching this repatom will be targetted for refinement, since we require that n α = 1 for all non-local repatoms. The effect of such a technique is shown in Fig. 3(b), where the segment of the boundary between x = −100 and 100 Å was defined to be active. The result is that the grain boundary structure is correctly captured in the vicinity of the step, as well as for some distance on either side of the step.

3.2.

Solution Procedure

In the static QC implementation, the solution procedure amounts to minimization of the total energy (elastic energy plus the potential energy of the applied loads, see Eq. (6)) for a given set of boundary conditions (applied displacements or forces on certain repatoms). However, problems solved using the QC method are typically highly nonlinear, and as such their energy functional typically includes many local minima. In order to find a physically realistic solution, it is necessary to use a quasi-static loading approach, whereby boundary conditions are gradually incremented, the energy is minimized, and the minimum energy configuration is used in generating an initial guess to the solution after the subsequent load increment. Again, we can refer to the specific example of the stepped twin boundary to make this more clear. Our desire, in this example, is to study the effect of applying a shear strain to the stepped twin boundary. Specifically, we may be interested in knowing the critical shear strain at which the boundary begins to migrate and to understand the mechanism of this migration. We begin by choosing a sensible strain increment to apply, such that the incremental deformation will not be too severe between minimization steps. For this example, the initial guess, un+1 0 , used to solve for the relaxed displacement, un+1 , of load step n + 1 is given by un+1 = un + F X, 0

(19)

where un is the relaxed, minimum energy displacement field from load step n, u0 = 0, and the matrix F corresponding to pure shear along the y direction is 

1 γ F =  0 1 0 0



0 0 . 1

(20)

Thus, a shear strain increment of γ is applied, the outer repatoms are held fixed to the resulting displacements, and all inner repatoms are relaxed until the


679

energy reaches a minimum. Then, another strain increment is superimposed on these relaxed displacements and the process repeated. After n load steps, a total macroscopic shear strain of γ = n γ has been applied to the outer boundary of the bi-crystal. The energy minimization can be performed using several standard approaches, such as the conjugate gradient (CG) or the Newton–Raphson (NR) methods (both of which are described, for example, in Ref. [20]). The CG method has the advantage over the NR technique in that it requires only the energy functional and its first derivatives with respect to the repatom positions (i.e., the forces). The NR method requires a second derivative, or “stiffness matrix” that is not straightforward to derive or to code in an efficient manner. Once correctly implemented, however, the NR method has the advantage of quadratic convergence (compared to linear convergence for the CG method) once the system is close to the energy minimizing configuration. By monitoring the applied force (measured as the sum of forces in the y-direction applied to the top surface of the bi-crystal) versus the accumulated shear strain, γ , it can be observed that there is an essentially linear response for the first six load steps, and then a sudden load drop from step six to seven. This jump corresponds to the first inelastic behaviour of the boundary, the mechanism of which is shown in Fig. 4. In Fig. 4(a), a close-up of the relaxed step at an applied strain of γ = 0.03 is shown, while Fig. 4(b) shows the relaxed configuration after the next strain increment at γ = 0.035. The mechanism of this boundary motion is the motion of two Shockley partial dislocations from the corners of the step along the boundary. This can be seen clearly by observing the finite element mesh between the repatoms in Fig. 4(c). Because the mesh is triangulated in the reference configuration, the effect of plastic slip is the shearing of a row of elements in the wake of the moving dislocations. One challenge in modeling dislocation motion in crystals at the atomic scale is evident in this simulation. In crystals with a low Peierls resistance like the FCC crystal modelled here, dislocations will move long distances under small applied stresses. In this simulation, the Shockley partials which nucleated at the step move to the ends of the region of atomic-scale refinement. In order to rigorously compute the equilibrium position of the dislocations, it would be necessary to further adapt the model. The presence of the dislocation in close proximity to the larger elements to the left of the fully refined region will trigger the adaption criterion, as well as increase the number of repatoms that are non-local according to the non-locality criterion defined earlier. This will allow the dislocations to move somewhat further upon subsequent relaxation. In principle, this process of iteratively adapting and relaxing can be repeated until the dislocations come to its true equilibrium, which in this example would be at the left and right free surfaces of the bi-crystal.

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(a)

Initial Boundary Location (b)

Boundary Migration (c)

Slip of Shockley partials Figure 4. Mechanism of migration of the twin boundary under shear. (a) Before migration. (b) After migration (c) Deformed mesh showing the motion of Shockley partial dislocations.

In practice, however, we may not be interested in the full details of where this dislocation comes to rest, if we are willing to accept some degree of error in the simulation. Specifically, the fact that the dislocation is held artificially close to the step may effect the critical load level at which subsequent migration events occur. The compromise is made for the sake of computational speed, which will be significantly compromised if we were to iteratively adapt and relax many times for each load step.


4.

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Summary

This review has summarized the theory and practical implementation of the QC method. Rather than provide an exhaustive review of the QC literature (which can already be found, for example, in Ref. [5]), the intent has been to provide a simple overview for someone interested in understanding one implementation of the QC method. More specific details, including free, opensource code and documentation, can be found at www.qcmethod.com.

References [1] E.B. Tadmor, M. Ortiz, and R. Phillips, “Quasicontinuum analysis of defects in solids,” Phil. Mag. A, 73, 1529–1563, 1996a. [2] E.B. Tadmor, R. Phillips, and M. Ortiz, “Mixed atomistic and continuum models of deformation in solids,” Langmuir, 12, 4529–4534, 1996b. [3] V.B. Shenoy, R. Miller, E. Tadmor, D. Rodney, R. Phillips, and M. Ortiz, “An adaptive methodology for atomic scale mechanics: the quasicontinuum method,” J. Mech. Phys. Sol., 47, 611–642, 1998a. [4] V.B. Shenoy, R. Miller, E.B. Tadmor, R. Phillips, and M. Ortiz, “Quasicontinuum models of interfacial structure and deformation,” Phys. Rev. Lett., 80, 742–745, 1998b. [5] R.E. Miller and E.B. Tadmor, “The quasicontinuum method: overview, applications and current directions,” J. of Computer-Aided Mater. Design, 9(3), 203–231, 2002. [6] M. Ortiz, A.M. Cuitino, J. Knap, and M. Koslowski, “Mixed atomistic continuum models of material behavior: the art of transcending atomistics and informing continua,” MRS Bull., 26, 216–221, 2001. [7] D. Rodney, “Mixed atomistic/continuum methods: static and dynamic quasicontinuum methods,” In: A. Finel, D. Maziere, and M. Veron (eds.), NATO Science Series II, Vol. 108, “Thermodynamics, Microstructures and Plasticity,” Kluwer Academic Publishers, Dordrecht, 265–274, 2003. [8] M. Ortiz and R. Phillips, “Nanomechanics of defects in solids,” Adv. Appl. Mech., 36, 1–79, 1999. [9] W.A. Curtin and R.E. Miller, “Atomistic/continuum coupling methods in multi-scale materials modeling,” Model. Simul. Mater. Sci. Eng., Vol. 11(3), R33–R68, 2003. [10] A. Carlsson, “Beyond pair potentials in elemental transition metals and semiconductors,” Sol. Stat. Phys., 43, 1–91, 1990. [11] V. Shenoy, V. Shenoy, and R. Phillips, “Finite temperature quasicontinuum methods,” Mater. Res. Soc. Symp. Proc., 538, 465–471, 1999. [12] E. Tadmor, G. Smith, N. Bernstein, and E. Kaxiras, “Mixed finite element and atomistic formulation for complex crystals,” Phys. Rev. B, 59, 235–245, 1999. [13] M. Daw and M. Baskes, “Embedded-atom method: derivation and application to impurities, surfaces, and other defects in metals,” Phys. Rev. B, 29, 6443–6453, 1984. [14] J. Norskøv and N. Lang, “Effective-medium theory of chemical binding: application to chemisorption,” Phys. Rev. B, 21, 2131–2136, 1980. [15] F. Stillinger and T. Weber, “Computer-simulation of local order in condensed phases of silicon,” Phys. Rev. B, 31, 5262–5271, 1985.

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E.B. Tadmor and R.E. Miller [16] O.C. Zienkiewicz, The Finite Element Method, vols. 1–2, 4th edn. McGraw-Hill, London, 1991. [17] J. Ericksen, In: M. Gurtin (ed.), Phase Transformations and Material Instabilities in Solids, Academic Press, New York. [18] A. Okabe, Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, Wiley, Chichester, England, 1992. [19] O.C. Zienkiewicz and J. Z. Zhu, “A simple error estimator and adaptive procedure for practical engineering analysis,” Int. J. Numer. Meth. Eng., 24, 337–357, 1987. [20] W.H. Press, S.A. Teukolsky, W.T. Vetterling, and B.P. Flannery, Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge, 1992.

2.14 PERSPECTIVE: FREE ENERGIES AND PHASE EQUILIBRIA David A. Kofke1 and Daan Frenkel2 1 University at Buffalo, The State University of New York, Buffalo, New York, USA 2

FOM Institute for Atomic and Molecular Physics, Amsterdam, The Netherlands

Analysis of the free energy is required to understand and predict the equilibrium behavior of thermodynamic systems, which is to say, systems in which temperature has some influence on the equilibrium condition. In practice, all processes in the world around us proceed at a finite temperature, so any application of molecular simulation that aims to evaluate the equilibrium behavior must consider the free energy. There are many such phenomena to which simulation has been applied for this purpose. Examples include chemical-reaction equilibrium, protein-ligand affinity, solubility, melting and boiling. Some of these are examples of phase equilibria, which are an especially important and practical class of thermodynamic phenomena. Phase transformations are characterized by some macroscopically observable change signifying a wholesale rearrangement or restructuring occurring at the molecular level. Typically this change occurs at a specific value of some thermodynamic variable such as the temperature or pressure. At the exact point where the transition occurs, both phases are equally stable – have equal free energy – and we find a condition of phase equilibrium or coexistence [1].

1.

Free-Energy Measurement

Free-energy calculations are among the most difficult but most important encountered in molecular simulation. A key “feature” of these calculations is their tendency to be inaccurate, yielding highly reproducible results that are nevertheless wrong, despite the calculation being performed in a way that is technically correct. Often seemingly innocuous changes in the way the calculation is performed can introduce (or eliminate) significant inaccuracies. So it 683 S. Yip (ed.), Handbook of Materials Modeling, 683–705. c 2005 Springer. Printed in the Netherlands.

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is important when performing these calculations to have a strong sense of how they can go awry, and proceed in a way that avoids their pitfalls. The aim of any free-energy calculation is to evaluate the difference in free energy between two systems. “System” is used here in a very general sense. The systems may differ in thermodynamic state (temperature, pressure, chemical composition), in the presence or absence of a constraint, or most generally in their Hamiltonian. Often the free energy of one system is known, either because it is sufficiently simple to permit evaluation analytically (e.g., an ideal gas or a harmonic crystal), or because its free energy was established by a separate calculation. In many cases the free-energy difference is itself the principal quantity of interest. The important point here is that free-energy calculations always involve two (or more) systems. We will label these systems A and B in our subsequent discussion, and their free energy difference will be defined F = FB − FA . Once the systems of interest have been identified, a large variety of methods are available to evaluate F. At first glance the methods seem to be very diverse and unrelated, but they nevertheless can be grouped into two broad categories: (a) methods based on measurement of density of states and (b) methods based on work calculations. Implicit in both approaches is the idea of a path joining the two systems, and one way that specific methods differ is in how this path is defined. As free energy is a state function, the free-energy difference of course does not depend on the path, but the performance of a method can depend greatly on this choice (and other details). It is always possible to define a parameter λ that locates a position on the path, such that one value λ A corresponds to system A and another value λ B indicates system B. The parameter λ may be continuous or discrete (in fact, it is not uncommon that it have only two values, λ A and λ B ), and may represent a single variable or a set of variables, depending on the choice of the path. Moreover, for a given path, the parameter λ can be viewed as a state variable, such that a free energy F(λ) can be associated with each value of λ. Thus F = F(λ B ) − F(λ A ). The term “Landau free energy” is sometimes used in connection with this dependence.

1.1.

Density-of-States Methods

If a system is given complete freedom to move back and forth across the path joining A and B, it will explore all possible values of the path variable λ, but it will (in general) not spend equal time at each value. The probability p(λ) that the system is observed to be at a particular point λ on the path is related to the value of the free energy there p(λ) ∝ exp (−F(λ)/kT ) ,

(1)

Perspective: free energies and phase equilibria

685

where T is the absolute temperature and k is Boltzmann’s constant. This relation is the basic idea behind the density-of-states methods. The specific way in which λ samples values depends on how the simulation is implemented. Typically density-of-states calculations are performed as part of Monte Carlo (MC) simulations. In this case sampling includes trial moves in which λ is perturbed to a new value, and a decision to accept the trial is taken in the usual MC fashion. It is possible also to have λ vary as part of a molecular dynamics (MD) simulation. In such a situation λ must couple to the equations of motion of the system, usually via an extended-Lagrangian formalism [2]. Then λ follows a deterministic dynamical trajectory akin to the way that the particles’ coordinates do. In almost all cases of practical interest, conventional Boltzmann sampling will probe only a small fraction of all possible λ-values. The variation of the free energy F(λ) can be many times kT when considered over all λ values of interest, and consequently the probability p(λ) can vary over many orders of magnitude. Extra measures must therefore be taken to ensure that sufficient information is gathered over all λ to evaluate the desired free-energy difference, and one of the features distinguishing different density-of-states methods is the way that they take these measures. Almost always an artificial bias φ(λ) must be imposed to force the system to examine values of λ where the free energy is unfavorable, Usually the aim is to formulate the bias to lead to a uniform sampling over λ, which is achieved if φ(λ) = −F(λ). Of course, inasmuch as the aim is to evaluate F(λ) it is necessary to set up a scheme in which the free energy can be estimated either through preliminary simulations or as part of a systematic process of iteration. The greatest difficulty is found if the free energy change is extensive, meaning that λ affects the entire system and not just a small part of it (e.g., a path that results in a change in the thermodynamic phase, versus a path in which a single molecule is added to the system). In such cases F(λ) scales with the system size and is likely to vary by very large amounts with λ. The practical consequence is that the bias must be tuned very precisely to ensure that good sampling over all λ is accomplished. A robust solution to the problem is the use of windowing, in which the problem of evaluating the full free energy profile F(λ) is broken into smaller problems, each involving only a small range of all λ of interest. Separate simulations are performed over each λ range, and the composite data are assembled to yield the full profile. Even here there are different ways that one can proceed, and a popular approach to this end uses the histogram-reweighting method, which optimally combines the data in a way that accounts for their relative precision. Histogram reweighting is discussed in another chapter of this volume. Within the framework outlined above, the most obvious way to measure the probability distribution p(λ) is to use a visited-states approach: MC or MD sampling of λ values is performed, perhaps in the presence of the bias φ, and

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a histogram is recorded of the frequency with which each value (or bin of values) of λ is occupied. The Wang-Landau method [3, 4] (and its extensions) is the most prominent such technique today. Another approach of this type applies a history-dependent bias using a Gaussian basis [5]. An alternative to visited-states has recently emerged in the form of transition-matrix methods [6–10]. In such an approach one does not tabulate the occupancy of each λ value; rather one tallies statistics about the attempts to transition from one λ to another in a MC simulation. The movement among different λs forms a Markov process, and knowledge of the transition probabilities is sufficient to derive the limiting distribution p(λ). Interestingly, even rejected MC trials contribute information to the transition matrix, so it seems that this approach is gathering information that is discarded in visited-states methods. The transition-matrix approach has several other appealing features. The method can accommodate the use of a bias to flatten the sampling, but the bias does not enter into the transition matrix, so if the bias is updated as part of a scheme to achieve a flat distribution the previously recorded transition probabilities do not have to be discarded, as they must be in visited-states methods (at least in its simpler formulations). Moreover, if windowing is applied to obtain uniform samples across λ, it is easy to join data from different windows. It is not even required that adjacent windows overlap, just that they attempt trials (without necessarily accepting) into each other’s domain. Details of the transition-matrix methods are still being refined, and the versatility of the approach is currently being explored through its application to different problems. Additionally, there are efforts now to combine visited-states and transition-matrix approaches, exploiting the relatively fast (but rough) convergence of the former while relying on the more complete data collection abilities of the latter to obtain the best precision [11].

1.2.

Work-Based Methods

Classical thermodynamics relates the difference in free energy between two systems to the work associated with a reversible process that takes one into the other. A straightforward application of this idea leads to the thermodynamic integration (TI) free-energy method, which has a long history and has seen widespread application. The TI method is but one of several approaches in a class based on the connection between F and the work involved in transforming a system from A to B. A very important development in this area occurred recently, when Jarzynski showed that F could be related to work associated with any such process, not just a reversible one [12–15]. Jarzynski’s non-equilibrium work (NEW) approach requires evaluation of an ensemble of


687

work values, and thus involves repeated transformation from A to B, evaluating the work each time. The connection to the free energy is then exp(−F/kT ) = exp(−W/kT ),

(2)

where W is the total work, and the overbar on the right-hand side indicates an average taken over many realizations of the path from A to B, always starting from an equilibrium A condition. For an equilibrium (reversible) path, the repeated work measurements will each yield exactly the same value (within the precision of the calculations), while for an arbitrary non-equilibrium transformation a distribution of work values will be observed. It is remarkable that these non-equilibrium transformations can be analyzed to yield a quantity related to the equilibrium states. The instantaneous work w involved in the transformation λ → λ + λ will in general depend upon the detailed molecular configuration of the system at the instant of the change. Assuming that there is no process of heat transfer accompanying the transformation, this work is given simply by the change in the total energy of the system w = E(r N ; λ + λ) − E(r N ; λ).

(3)

For sufficiently small λ, this difference can be given in terms of the derivative dE(λ) λ, (4) w= dλ r N which can be interpreted in terms of a force acting on the parameter λ. The derivative relation is the natural formulation for use in MD simulations, in which the work is evaluated by integrating the product of this force times the displacement in λ over the complete path. The former expression (Eq. (3)) is more appropriate for MC simulation, in which larger steps in λ are typically taken across the path from A to B. Thermodynamic integration is perhaps the first method by which free energies were calculated by molecular simulation. Thermodynamic integration methods are usually derived from classical thermodynamics [1], with molecular simulation appearing simply to measure the integrand. As indicated above, TI also derives as a special (reversible) case of Jarzynski’s NEW formalism, whereby F =W rev for the reversible path. The total work W rev is in turn given by integration of Eq. (4), leading to: F =

λ B

w(λ) dλ.

(5)

λA

Equilibrium values of w are measured in separate simulations at a few discrete λ points along the path. It is then assumed that w is a smooth function

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of λ, and simple quadrature formulas (e.g., trapezoid rule) can be applied. The primary mechanism for the failure of TI is the occurrence of a phase transition, and therefore a discontinuity in w, along the path. Otherwise TI has been successfully applied to a very wide variety of systems, dating to the earliest simulations. Its primary disadvantage is that it does not provide direct measurement of the free energy, and if one is not interested in behavior for points along the integration path then another approach might be preferred. TI approximates a reversible path by smoothing equilibrium, ensembleaveraged, “forces” measured discretely along the path. Alternatively, one can access a reversible path by mimicking a truly reversible process, i.e., by attempting to traverse the path via a slow, continuous transition. In this manner the simulation constantly evolves from system A to system B, such that every MC or MD move is accompanied by a tiny step in λ (or some variation of this protocol). The differential work associated with these changes is accumulated to yield the total work W , which then approximates the free-energy difference. The process may proceed isothermally or adiabatically, the latter being the so-called adiabatic-switch method (and which instead yields the entropy difference between A and B) [16]. The weakness of these methods is in the uncertainty on whether the evolution of the system is sufficiently slow to be considered reversible. Such concerns can be allayed by implementing the calculation using the Jarzynski free-energy formula, Eq. (9); however this remedy then requires averaging of repeated realizations of the transition. One is then led to ask whether it is better to average, say, ten NEW passes, or to perform a single switch ten times more slowly. Free-energy perturbation (FEP) is obtained as the special case of the NEW method in which the transformation from A to B is taken in a single step. Free-energy perturbation is a well established and widely used method. Its principal advantage is that it permits F to be given as an ensemble average over configurations of the A system, removing the complication and expense of defining and traversing a path. The working formula emphasizes this feature exp(−βF) = exp [−β(E B − E A )]A .

(6)

A given NEW calculation can in principle be performed in either direction, starting from A and transforming to B, or vice versa. In practice the calculation will give different results when applied in one or the other direction; moreover these results will bracket the correct value of F. The results differ because they are inaccurate, and the fact that they bracket the correct value makes it tempting to take their average as the “best” result. But this practice is not a good idea, because the magnitude of the inaccuracies is in general not the same for the two directions [17,18]. In fact, it is not uncommon for one direction to provide the right result while the other yields an inaccurate one. But it is also not uncommon in other cases for the average to give a better estimate than either direction individually. The point is that one often does not know what


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is the best way to interpret the results. The more careful practitioners will apply sufficient calculation (and perhaps use sufficient stages) until a point is reached in which the results from each direction match each other. However, this practice can be wasteful. To understand the problem and its remedy it is helpful to consider the systems A and B from the perspective of configuration space.

1.3.

Configuration Space

Configuration space is a high-dimensional space of all molecular configurations, such that any particular arrangement of the N atoms in real space is represented by a single point in 3N -dimensional configuration space (more generally we may consider 6N -dimensional phase space, which includes also the momenta) [19]. An arbitrary point in configuration space will typically describe a configuration that is unrealistic and unimportant, in the sense that one would not expect ever to observe the configuration to arise spontaneously in course of the system’s natural dynamics. For example, it might be a configuration in which two atoms occupy overlapping positions. Configuration space will of course contain points that do represent realistic, or important configurations, ones that are in fact observed in the system. It is helpful to consider the set * of all such configurations, as we do schematically in Fig. 1. The enclosing square represents the high-dimensional configuration space, and the ovals drawn within it represent (in a highly simplified manner) the set of all important configurations for the systems. The concept of “important configurations” is relevant to free-energy calculations because the ease with which a reliable (accurate) free-energy difference can be measured depends largely on the relation between the * regions of the two systems defining the free-energy difference. There are five general possibilities [20], summarized in Fig. 1. In a FEP calculation perturbing from A to B, the simulation samples the region labeled ∗A and at intervals it examines its present configuration and gauges its importance to the B system. Three general outcomes are possible for the difference E B − E A seen in Eq. (6): (a) it is a large positive number and the contribution to the FEP average is small; this occurs if the point is in ∗A but not in ∗B ; (b) it is a number of order unity, and a significant contribution is made to the FEP average; this occurs if the point is in ∗A and in ∗B ; or (c) it is a large negative number, and an enormous contribution is made to the FEP average; this occurs if the point is not in ∗A but is in ∗B . The third case will arise rarely if ever, because the sampling is by definition largely confined to the region ∗A . This contradiction (a large contribution made by a configuration that is never sampled) is the source of the inaccuracy in FEP calculation, and it arises if any part of ∗B lies outside of ∗A .

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(a)

Γ

(b)

(c) Γ*B

Γ*B Γ*A

Γ*A (d)

Γ*A

Γ*B

(e) Γ*B

Γ*A ⫻

Γ*A Γ*B

Figure 1. Schematic depiction of types of structures that can occur for the region of important configurations involving two systems. The square region represents all of phase space, and the filled regions are the important configurations ∗A and ∗B for the systems “A” and “B”, as indicated. (a) simple case in which ∗A and ∗B are roughly coincident, and there is no significant region of one that lies outside the other; (b) case in which the important configurations of A and B have no overlap, and energetic barriers prevent each from sampling the other; (c) case in which one system’s important configurations are a wholly contained, not-very-small subset of the others; (d) case in which ∗B is a very small subset of ∗A ; (e) case in which ∗A and ∗B overlap, but neither wholly contains the other.

This observation leads us to the most important rule for the reliable application of FEP: the reference and target systems must obey a configuration-space subset relation. That is, the important configuration space of the target system (B) must be wholly contained within the important configuration space of the system governing the sampling (A). Failure to adhere to this requirement will lead to an inaccurate result. Note the asymmetry of the relation “is a subset of” is directly related to the asymmetry of the FEP calculation. Exchange of the roles of A and B as target or reference can make or break the accuracy of the calculation. For example, consider the free energy change associated with the addition of a molecule to the system. In this case, F equals the excess chemical potential. The A system is one in which the “test” molecule has no interaction with the others, and the B system is one in which it interacts as all the other molecules do. Any configuration in which the test molecule overlaps another molecule is not important to B but is (potentially) important to A – the B system may be a subset of A, while A is most certainly not a subset of B. Whether all of ∗B is within ∗A cannot be stated for the general case. In more complex


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systems (e.g., water) it is likely that there are configurations sampled by B that would not be important to A, while in simpler systems (a Lennard–Jones fluid at moderate density) the subset relation is satisfied. This black-and-white picture, in which the * regions are well defined with crisp boundaries, presents only a conceptual illustration of the nature of the calculations. In reality the “importance” of a given configuration (point in ) is not so clear-cut, and the * regions for the A and B systems may overlap in shades of gray (i.e., degrees of importance). The discussion here is given in the context of a FEP calculation, but the same ideas are relevant to the more general NEW calculation. Each increment of work performed in a NEW calculation must adhere to the subset relation too. The difference with NEW is that if the change is made sufficiently slowly (approaching reversibility), then the important phase spaces at each step will differ by only small amounts (cf. Fig. 1(a)), and the subset relation will be satisfied. To the extent that a NEW calculation is performed irreversibly, the issue of inaccuracy and asymmetry becomes increasingly important.

1.4.

Staging Strategies

In practice one is confronted with pair of systems for which F is desired, and there is no control over whether their * regions satisfy a subset relation. Yet FEP and NEW cannot be safely applied unless this condition is met. Two remedies are possible. Phase space can be redefined, such that a given point in it can represent different configurations for the A and B systems [21–23]. This approach has been applied to evaluate free energy differences between crystal structures (e.g., fcc vs. bcc) of a given model system. The phase-space points are defined to represent deviations from a perfect-crystal configuration, and the reference crystal is defined differently for the two systems. The switch from A to B entails swapping the definition of the reference crystal while keeping the deviations (i.e., the redefined phase-space point) fixed. With this transformation, two systems having disjoint * regions are redefined such that their * at least have significant overlap, and perhaps obey the subset requirement. Multiple staging is a more general approach to deal with systems that do not satisfy the subset relation [24–26]. Here the desired free energy difference is expressed in terms of the free energy of one or more intermediate systems, typically defined only to facilitate the free-energy calculation. Thus, F = (FB − FM ) + (FM − FA ),

(7)

where M indicates the intermediate. Free-energy methods are then brought to evaluate separately the two differences, between the M and B and M and A systems, respectively. The M system should be defined such that a subset relation can be formed between it and both the A and B systems. There are

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several options to this end, depending on the * relation in place for the A and B systems. Figure 2 summarizes the possibilities, and the cases are named as follows: • Umbrella sampling. Here M is formulated to contain both A and B, and sampling is performed from it into each [27]. • Funnel sampling. This is possible only if B is already a subset of A. Then M is defined as a subset of A and superset of B, and each perturbation stage is performed accordingly [20, 25, 28]. • Overlap sampling. Here M is formulated to be a subset of both A and B, and sampling is performed on each with pesrturbation into M [29]. General ways to define M to satisfy these requirements are summarized in Table 1, which also lists the general working equations for each multistage scheme. Umbrella sampling is a well-established method but is has only recently been viewed from the perspective given here. Bennett’s acceptanceratio method is a particular type of overlap sampling in which an optimal

Figure 2. Schematic depiction of types of structures that can occur for the region of important configurations involving two systems and a weight system formulated for multistage sampling. The square region represents all of phase space, and the filled regions are the important configurations ∗A , ∗B , and ∗M for the systems A and B, and M as indicated. (a) well formulated umbrella potential defines important configuration that have both ∗A and ∗B as subsets; (b) safely formulated funnel potential needed to focus sampling on tiny set of configurations ∗B while still representing all configurations important to A; (c) well formulated overlap potential, with important configurations formed as a subset of both the A and B systems. Table 1. Summary of staging methods for free-energy perturbation calculations Method

Formula for e−β(FB −F A )

Preferred staging potential, e−β E M

−β(E −E ) B M e Umbrella sampling −β(E −E ) M e

A

e

M

M

Funnel sampling

−1

e−β(E A −F A ) + e−β(E B −FB )

M

−β(E −E ) M A e Overlap sampling −β(E −E ) A B

B

e+β(E A −F A ) + e+β(E B −FB )

e−β(E M −E A ) A e−β(E B −E M ) M No general formulation


693

M is selected to minimize the variance of F; it is a highly effective and underappreciated method. The funnel-sampling multistage scheme is new, and a general, effective formulation for an M system appropriate to it has not yet been identified. Overlap sampling and umbrella sampling are not particularly helpful if A and B already satisfy the subset relation – they do not give much better precision than a simple single-stage FEP calculation taken in the appropriate direction. However, if implemented correctly they do provide some measure of safety against problems of inaccuracy, which is useful because in most cases one usually does not know clearly the nature of the phase-space relation for the A and B systems, and whether (and which way) a single-stage calculation is safe to perform between them.

2.

Methods for Evaluation of Phase Coexistence

Our perspective now shifts to the calculation of phase coexistence by molecular simulation, for which free-energy methods play a major role. Applications in this area have exploded over the past decade or so, owing to fundamental advances in algorithms, hardware, and molecular models. Some of the methods and concepts surveyed here have been discussed in more detail in recent reviews [30, 31].

2.1.

What is a Phase?

An order parameter is a statistic for a configuration. It is a number (or perhaps a vector, tensor, or some other set of numbers) that can be calculated or measured for a system in a particular configuration, and that in some sense quantifies the configuration. Examples include the density, the mole fraction in a mixture, the magnetic moment of a ferromagnet, and so on. Some molecular order parameters are formulated as expansion coefficients of an appropriate distribution function rendered in a suitable basis set. For example, a natural choice for crystalline translation order parameters is the value of the structure factor for an appropriate wave vector k. Orientational order parameters are widely used in the field of liquid crystals, and a common choice is based on expansion of the orientation distribution in Legendre polynomials. Usually an order parameter is defined such that it has a physical manifestation that can be observed experimentally. A thermodynamic phase is the set of all configurations that have (or are near) a given value of an order parameter. Phases are important because a system will spontaneously change its phase in response to some external perturbation.

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In doing so, the configurations exhibited by the system change from those associated with one value of the order parameter to those of another. Usually such a large shift in the predominant configurations will cause the system’s physical properties (mechanical, electrical, optical, etc.) to change in ways that might be very useful. A well known example is the boiling of a liquid to form a vapor. In response to a small change in temperature, the observed configurations of the system go from those corresponding to a large density to those for a much smaller density. In both cases the system (being at fixed pressure) is free to adopt any desired density. In changing phase it overwhelmingly selects configurations for one density over another. This phenomenon, and its many variants, has a multitude of practical applications. Clearly, there is a close connection between this molecular picture of a phase transformation, and the ideas presented above about the important phase space for a system. When a system changes phase, it is actually changing its important phase space, and the * region for the system before and after the change can relate in any of the ways described in Fig. 1. Analysis of the free energy is required to identify the location of the phase change quantitatively. Often the order parameter describing the phase change serves as the path parameter λ when performing this analysis.

2.2.

Conditions for Phase Equilibria

In a typical phase-equilibrium problem one is interested in the two (or more) phases involved in the transformation. At the exact condition at which one becomes favored over the other, both are equally stable. Molecular simulation is applied to locate this point of phase equilibrium and to characterize the coexisting phases. Formally, the thermodynamic conditions of coexistence can be identified as those minimizing an appropriate free energy, or equivalently by finding the states in which the intensive “field” variables of temperature, pressure, and chemical potential (and perhaps others) are equal among the candidate phases. Most methods for evaluation of phase equilibria by molecular simulation are based on identifying the conditions that satisfy the thermodynamic phase-coexistence criteria, and consequently they require evaluation of free energies or a free-energy difference. Still there is a lot of variability in the approaches, because really there are two problems involved in the calculation. The first is the measurement of the thermodynamic properties, particularly the free energy, while the second is the numerical “root-finding” problem of locating the coexistence conditions. Methods differ largely in the way they combine these two numerical problems, and the most effective and popular methods synthesize these calculations in elegant ways.


2.3.

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Direct Contact of Phases, Spontaneous Transformations

Before turning to the free-energy based approaches for evaluating phase coexistence, it is worthwhile to consider the more intuitive approaches that mimic the way phase transitions are studied experimentally. By this we mean methods in which a system is simulated and the phase it spontaneously adopts is identified as the stable thermodynamic phase. Two general approaches can be taken, depending on the types of variables that are fixed in the simulation (i.e., the governing ensemble). In the first case, only one size variable is imposed (typically the number of molecules), and the remaining variables are fields (temperature, pressure, chemical potential difference). Then a scan is made of one or more of the fields (e.g., the temperature is increased), and one looks for the condition at which the phase changes spontaneously (e.g., the system undergoes a sudden expansion). For example, the temperature at which this happens, and the conditions of the phases before and after the transition, characterizes the coexistence point. In practice this method is effective only for producing a coarse description of the phase behavior. It is very easy for a system to remain in a metastable condition as the field variable moves through the transition point, and the spontaneous transformation may occur at a point well beyond the true value. The reverse process is susceptible to the same problem, so the transformation process exhibits hysteresis when the field is cycled back and forth through the transition value. In the second case, two or more extensive variables are imposed (i.e., the number of molecules and the volume), and the system is simulated at a condition inside the two-phase region. A macroscopic system in this situation would separate into the two phases, and both would coexist in the given volume. In principle, this too happens in a molecular simulation, but usually the system size is not sufficiently large to wash out effects due to the presence of the interface. In effect, neither bulk phase is simulated. Nevertheless, the directcontact method does work in some situations. Solid-fluid phase behavior has been studied this way. The interface is slow to equilibrate in this system, so one must be careful to ensure that the simulation begins with a well equilibrated solid. Vapor-liquid equilibria have also been examined using direct contact of the phases. Of course, this approach cannot be applied when too close to the critical point. Often such systems are examined because the interfacial properties are themselves of direct interest. Spontaneous formation of phases has been used recently to examine the behaviors of models that exhibit complex morphologies. Glotzer et al. have examined the mesophases formed by a wide variety of model nanoparticles, including hard particles with tethers, and particles with sticky patches [32].

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The systems have been observed to spontaneously form many complex structures, including columns, lamella, micelles, sheets, double layers, gyroid phases, and so on. The question remains of the absolute stability of the observed structures, but their spontaneous formation is a strong indicator that they are certainly relevant, and could likely be the most stable of all possible phases at the simulated conditions. The phase behaviors of other types of mesoscale models are also studied through the direct-observation methods. Systems modeled using dissipative particle dynamics [2, 33] are good candidates for this treatment, because they have a very soft repulsion and particles can in effect pass through each other; and as a consequence they equilibrate very quickly.

2.4.

Methods Based on Solution of Thermodynamic Equalities

A well worn approach to the free-energy based evaluation of phase equilibria focuses on satisfying the coexistence conditions given in terms of equality of the field parameters. In this approach each phase is studied separately, and state conditions are varied systematically until the coexistence conditions are met. An effective way to attack this problem is to combine the search for the coexistence point with the evaluation of the free energy through thermodynamic integration. For example, to evaluate a vapor-liquid coexistence point, one can start with a subcooled liquid of known chemical potential (evaluated using any of the methods reviewed above), and proceed with a series of isothermal-isobaric simulations following a line of decreasing pressure. At each point the chemical potential can be evaluated through the thermodynamic integration using the measured density µ(P) = µ(P0 ) +

P

d p/ρ( p).

(8)

P0

A similar series of simulations can be performed in the vapor separately, at the same temperature as the liquid simulations, but increasing the pressure toward the point of saturation (alternatively, an equation of state might be applied to characterize the vapor). Once the liquid and vapor simulations reach an overlapping range of pressures, the chemical potentials computed according to Eq. (8) can be examined at each pressure, until the point is found at which chemical potential is equal across the two phases for a given pressure. This general approach can be somewhat tedious to implement, but it is perhaps the most robust of all methods. It is likely to provide a good result for almost all types of coexistence. It has been applied to many types of phase equilibria, including those involving solids [34], liquid crystals [35], plastic


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crystals, as well as fluids. The search for the coexistence condition can be applied using almost any order parameter (density was used in this example), although one must perhaps put some effort toward developing the appropriate formalism defining a field to couple to the parameter, and implementing a simulation in which this field is applied. Complications arise if many field parameters are relevant. For example, if one is studying a mixture, then a separate field parameter (chemical potential) is needed to couple to each molefraction variable. The problem can be simplified by fixing all but one of the field variables in the two phases, but often this leads to a statement of the coexistence problem that is at odds with the problem of real interest (e.g., one might want to know the composition of the incipient phase arising from another phase of given composition, which in the context of vapor-liquid equilibria is known as a bubble-point or a dew-point calculation). For mixtures, this formulation is expressed by the semigrand ensemble [36]. This method, like many others, will suffer when applied to characterize a weak phase transition, that is, one that is accompanied by only a small change in the relevant order parameter. The order parameter is related to the slope of the line that is being mapped in this calculation, and consequently for a weak transition the slopes of these lines for the two phases will not be very different from each other. It can be difficult to locate precisely the intersection of two nearly parallel lines – any errors in the position of the lines will have a greatly magnified effect on the error in the point of intersection. Therefore the application of this method to a weak transition can fail if the relevant ensemble averages and the free energies for the initial points of the integration are not measured with high precision and accuracy.

2.5.

Gibbs Ensemble

A breakthrough in technique for the evaluation of phase coexistence by molecular simulation arrived in 1987 with the advent of the Gibbs ensemble [37]. This method presents a very clever synthesis of the problem of locating the conditions of coexistence and measuring the free energy in the candidate phases. It accomplishes this through the simulation of both phases simultaneously, each occupying its own simulation volume. Although the phases are not in “physical” contact, they are in contact thermodynamically. This means that they are capable of exchanging volume and mass in response to the thermodynamic driving forces of pressure and chemical potential difference, respectively. The systems evolve in this way, increasing or decreasing in density with the mass and volume exchanges, until the point of coexistence is found. Upon reaching this condition the systems will fluctuate in density about the values appropriate for the equilibrium state, which can then be measured as a simple

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ensemble average. Details of the method are available in several reviews and texts [2, 37, 38]. The Gibbs ensemble is the method of choice for straightforward evaluation of vapor–liquid and liquid–liquid equilibria. It does not suffer any particular complications when applied to mixtures, and it has been applied with great success to many phase coexistence calculations. However, there are several ways in which it can fail. First, an essential element of the technique is the exchange of molecules at random between the coexisting phases. If trials of this type are not accepted with sufficient frequency, the systems will not equilibrate and a poor result is obtained. This problem arises in applications to large, complex molecules, and/or at low temperatures and high densities. It can be overcome to a useful degree through the application of special sampling techniques, such as configurational bias. Second, in its basic form the Gibbs ensemble is not applicable to equilibria involving solids, or to lattice models. The problem is only partially due to the difficulty of inserting a molecule into a solid. The “mass balance” is the more insidious obstacle. The number of molecules present in each phase at equilibrium is set by the initial number of molecules and the volume of the composite system of both phases (as well as the values of the coexistence densities). A defect-free crystal can be set up in a periodic system using only a particular number of molecules. For example an fcc lattice in cubic periodic boundaries can be set up using 32, 108, 256, 500, and so on molecules (i.e., 4n 3 where n is an integer). When beginning a Gibbs ensemble calculation there is no simple way to ensure this condition will be met in the equilibrium system. Tilwani and Wu [39] have treated these problems with an alternative approach in which an atom is added to the unit box of the solid and this new unit box is used to fill up (tile) space. In this way, particles can be added or removed from the system, while the crystal structure is maintained. The Gibbs ensemble fails also upon approach to the critical point. As this condition is reached, contributions to the averages increase for densities in the region between the two phases. It then becomes possible, even likely, that the simulated phases will swap their roles as the liquid and vapor phases. This is not a fatal flaw, but it presents a complication to the method, and it is an indicator that the general approach is beginning to fail. Thus the consensus today is that in this region of the phase envelope density-of-states methods are more suitable for characterizing the coexistence behavior. More generally, the Gibbs ensemble can encounter difficulty when applied to any weak phase transition, if only because it is necessary to configure the composite system so that it lies in the two phase region – this can be difficult to do if this region is very narrow. Interestingly enough, the Gibbs ensemble can fail also if it is applied using very large system sizes. In this situation an interface is increasingly likely to form in one or both phases, and the result is that a clean separation of phases between the volumes is no longer in place – instead both


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simulation volumes each end up representing both phases. Typically the Gibbs ensemble is applied for its simplicity and ability to provide quick results, so the large systems needed to raise this problem are not usually encountered.

2.6.

Gibbs–Duhem Integration

The Gibbs–Duhem integration (GDI) method [40] applies thermodynamic integration to both parts of the combined problem of evaluating the free energy and locating the point of transition. In particular, the path of integration is constructed to follow the line of coexistence. All of this is neatly packaged by the Clapeyron differential equation for the coexistence line, which in the pressure–temperature plane is [1]

dP dT

= σ

H , T V

(9)

where H and V are the differences in molar enthalpy and molar volume, respectively, between the two phases; the σ subscript indicates a path along the coexistence line. The GDI procedure treats Eq. (9) as a numerical problem of integrating an ordinary differential equation. The complication, of course, is that the right-hand side must be evaluated through molecular simulation at the temperature and pressure specified by the integration procedure, and moreover separate simulations are required to characterize both phases involved in the difference. A simple iterative process is applied to refine the pressure according to Eq. (9) after a step in temperature is taken, using preliminary results for the ensemble averages from the simulations. Predictor-corrector methods are effective in performing the integration, and inasmuch as the primary error in the calculation arises from the imprecision of the ensemble averages, a low-order integration scheme suffices for the purpose. The GDI method applies much more broadly than indicated in this description. Any type of field variables can be used in the role held by pressure and temperature in Eq. (9), with appropriate modification to the right-hand side. For example, integrations have been performed along paths of varying composition, polydispersity, orientational order, and interparticle-potential softness, rigidity, or shape [36]. The method applies equally well to equilibria involving fluids or solids, or other types of phases. It has been used to follow three-phase coexistence lines too. In this application one must integrate two differential equations similar to Eq. (9), involving three field variables. In all cases there are a number of practical implementation issues to consider, such as how the integration is started, and the proper selection of the functional form of the field variables (e.g., integration in ln(P) vs. 1/T has advantages for tracing

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vapor–liquid coexistence lines). These issues have been discussed in some detail in recent reviews [36, 41]. The GDI method has some limitations. It does require an initial point of coexistence in order to begin the integration procedure. Concerns are often expressed that errors in this initial point will propagate throughout the integration, but this problem is not as bad as one might think. A stability analysis shows that any such errors will be attenuated if the integration is performed in a direction from a weaker to a stronger transition (e.g., away from the liquid– vapor critical point toward lower temperatures). On the other hand, if the integration is performed in the opposite direction, initial and accumulated errors will be amplified. Regardless it seems that in practice any such problems do not arise. A related concern is the general difficulty in treating weak phase transitions. If the differences on the right-hand side of Eq. (9) are small, and thus may be formed using averages that have stochastic errors comparable to the differences themselves, then it is clear that the method will not work well. In such cases one might be better off employing a method that directly bridges the difference between the phases, such as by mapping the full density of states in this region. The basic idea of tracing coexistence lines has been further generalized for mapping of other classes of phase equilibria, such as tracing of azeotropes [42], and dew/bubble-point lines [41]. Escobedo has developed and applied a general framework for these approaches [30, 43–47].

2.7.

Mapping the Density of States

Density of states methods evaluate coexisting phases by calculating the full free-energy profile across the range of values of the order parameter between and including the two phases. It is only in the past few years that this method has come to be viewed as generally viable, and even a good choice for evaluating phase coexistence. The effort involved in collecting information for the intermediate points seems wasteful, although with the approach these data are needed to obtain the relative free energies of the real states of interest (i.e., the coexisting phases). The methods reviewed above are popular because they avoid this complication and are more efficient because of it. However, there is some advantage in having the system cycle through the uninteresting states. It helps to move the sampling through phase space. Thus, a simulated system might go from a liquid configuration, then to a vapor, and back to the liquid but in a very different configuration from which it started. This is particularly important for complex fluids such as polymers (in the context of other phase equilibria), in which it is otherwise difficult to escape from ergodic traps. Second, the intermediate states may be of interest in themselves; they can be used, for example, to evaluate the surface tension associated with contacting the two


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phases [10]. Third, it may be that the distance between the coexisting phases is not so large (i.e., the transition is weak), so covering the ground between them does not introduce so much expense; moreover in such a situation other methods do not work very well. Regardless, continuing improvements in computing hardware and algorithms (some reviewed above), particularly in parallel methods and architectures, have made the density-of-states strategy look much more appealing. We describe the basic approach in the context of vapor–liquid equilibria. Simulation can be performed in the grand-canonical potential with a chemical potential selected to be in the vicinity of the coexistence value. The density of states is mapped as a function of number of molecules at fixed volume; the transition-matrix method with a biasing potential in N has been found to be convenient and effective in this application. The resulting density of states will most likely exhibit two unequal peaks, representing the two nearly coexisting phases. Histogram reweighting is then applied to the density of states to determine the value of the chemical potential that makes the peaks equal in size. This is taken to be the coexistence value of the chemical potential, and the positions of the peaks give the molecule numbers (densities) of the coexisting phases. The coexistence pressure can be determined from the grand potential, which is available from the density of states. Additional details are presented by Errington [9].

3.

Outlook

The nature of the questions that we address with the help of computer simulations is changing. Increasingly, we wish to be able to predict the changes that will occur in a system when external conditions (e.g., temperature, pressure or the chemical potential of one or more species) are changed. In order to predict the stable phase of a many-body system, or the “native” conformation of a macromolecule, we need to know the accessible volume in phase space that corresponds to this state or, in other words, its free energy. Both the MC and the MD methods were created in effectively the form in which we use them today. However, the techniques used to compute free energy differences have expanded tremendously and have become much more powerful and much more general than they were only a decade ago. Yet, the roots of some of these techniques go back a long way. For instance, the density-of-states method was already considered in the late 1950s [48] and was first implemented in the 1960s [49]. The aim of the present chapter is to provide a (very concise) review of some of the major developments. As the developments are in a state of flux, this review provides nothing more than a snapshot.

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It is always risky to identify challenges for the future, but some seem clear. First of all, it would seem that there must be a quantum-mechanical counterpart to Jarzynski’s NEW method. However, it is not at all obvious that this would lead to a tractable computational scheme. A second challenge has to do with the very nature of free energy. In its most general (Landau) form, the free energy of a system is a measure of the available phase space compatible with one or more constraints. In the case of the Helmholtz free energy, the quantities that we constrain are simply the volume V and the number of particles N . However, when we consider the pathway by which a system transforms from one state to another, the constraint may correspond to a non-thermodynamic order parameter. In simple cases, we know this order parameter, but often we do not. We know the initial and final states of the system and hopefully the transformation between the two can be characterized by one, or a few, order parameters. If such a low-dimensional picture is correct, it is meaningful to speak of the “free-energy landscape” of the system. However, although methods exist to find pathways that connect initial and final states in a barriercrossing process [50], we still lack systematic ways to construct optimal low-dimensional order-parameters to characterize the transformation of the system. To date, most successful schemes to map free-energy landscapes assume that the true reaction coordinates are spanned by a relatively small set of supposedly relevant coordinates. However, is not obvious that it will always be possible to find such coordinates. Yet, without a physical picture of the constraint or reaction coordinate, free energy surfaces are hardly more informative than the high-dimensional potential-energy surface from which they are ultimately derived. Without this knowledge we can still compute the relative stability of initial and final state (provided we have a criterion to distinguish the two), but we will be unable to gain physical insight into the factors that affect the rate of transformation from the metastable to the stable state.

Acknowledgments DAK’s activity in this area is supported by the U.S. Department of Energy, Office of Basic Energy Sciences. The work of the FOM Institute is part of the research program of FOM and is made possible by financial support from the Netherlands organization for Scientific Research (NWO).

References [1] K. Denbigh, Principles of Chemical Equilibrium, Cambridge: Cambridge University, 1971. [2] D. Frenkel and B. Smit, Understanding Molecular Simulation: From Algorithms to Applications, Academic Press, San Diego, 2002.


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2.15 FREE-ENERGY CALCULATION USING NONEQUILIBRIUM SIMULATIONS Maurice de Koning1 and William P. Reinhardt2 1 University of São Paulo São Paulo, Brazil 2

University of Washington Seattle, Washington, USA

1.

Introduction

Stimulated by the progress of computer technology over the past decades, the field of computer simulation has evolved into a mature branch of modern scientific investigation. It has had a profound impact in many areas of research including condensed-matter physics, chemistry, materials and polymer science, as well as in biophysics and biochemistry. Many problems of interest in all of these areas involve complex many-body systems and analytical solutions are generally not available. In this light, atomistic simulations play a particularly important role, giving detailed insight into the fundamental microscopic processes that control the behavior of complex systems at the macroscopic level. They provide key and effective tools for providing ab initio predictions, interpreting complex experimental data, as well as conducting computational “experiments” that are difficult or impossible to realize in a laboratory. In this article, we will discuss one of the most fundamental and difficult applications of atomistic simulation techniques such as Monte Carlo (MC) [1] and molecular dynamics (MD) [2, 3]; the determination of those thermodynamic properties that require determination of the entropy. The entropy, the chemical potential, and the various free energies are examples of thermal thermodynamic properties. In contrast their mechanical counterparts such as the enthalpy, thermal quantities cannot be computed as simple time, or ensemble, averages of functions of the dynamical variables of the system and, therefore, are not directly accessible in MC or MD simulations. Yet, the free energies are often the most fundamental of all thermodynamic functions. Under appropriate constraints they control chemical and phase equilibria, and transition state estimates of the rates of chemical reactions. Examples of applications 707 S. Yip (ed.), Handbook of Materials Modeling, 707–727. c 2005 Springer. Printed in the Netherlands.

708

M. de Koning and W.P. Reinhardt

range from determination of the influence of crystal defects on the mechanical properties of materials, to the mechanisms of protein folding. The development of efficient and accurate techniques for their calculation has therefore attracted considerable attention during the past fifteen years, and is still a very active field of research [4]. As detailed in the previous chapter [4], the evaluation of free energies (or, more specifically free-energy differences) requires simulations that collect data along a sequence of states on a thermodynamic path linking two equilibrium states. If the system is at equilibrium at every point along such a path, the simulated process is quasistatic and reversible, and standard thermodynamic results may be used to interpret collected data and to estimate the free-energy difference between the initial and final equilibrium states. The present chapter generalizes this approach to the case where data is collected during nonequilibrium, and thus irreversible, processes. Several important themes will emerge, making clear why this generalization is of interest, and how nonequilibrium calculations may be set up to provide both upper and lower bounds (and thus systematic in addition to statistical error estimates) to the desired thermal quantities. Additionally, the irreversible process may be optimized in a variational sense so as to improve such bounds. The statistical–mechanical theory of nonequilibrium systems within the regime of linear response will prove particularly helpful in this endeavor. Finally, newly developed re-averaging techniques have appeared that, in some cases, allow quite precise estimates of equilibrium thermal quantities directly from nonequilibrium data. The combination of such techniques with near-optimal paths can give well converged results from relatively short computations. In the illustrations that follow, for sake of conciseness, we will limit ourselves to the application of nonequilibrium methods within the realm of the classical canonical ensemble. For this representative case the relevant thermodynamic variables are the number of particles N , the volume V , and the temperature T ; and the appropriate free energy is the Helmholtz free energy, A(N, V, T ) = E(N, V, T ) − T S(N, V, T ), E and S being the internal energy and entropy, respectively. However, appropriate generalizations of nonequilibrium methods to other classical ensembles, as well as to quantum systems, are readily available.

2.

Equilibrium Free-Energy Simulations

The calculation of thermodynamic quantities by means of atomistic simulation is rooted in the framework of equilibrium statistical mechanics [5], which provides the link between the microscopic details of a system and its macroscopic thermodynamic properties. Let us consider a system consisting

Free-energy calculation using nonequilibrium simulations

709

of N classical particles with masses m i . A microscopic configuration of the system is fully specified by the set of N particle momenta {pi } and positions {ri }, and its energy is described in terms of a potential-energy function U ({ri }). Statistical mechanics in the canonical ensemble then tells us that the distribution of the particle positions and momenta is given by ρ(Γ) =

1 exp(−β H (Γ)), Z (N, V, T )

(1)

where Γ ≡ ({p}, {r}) denotes a microstate of the system, β = 1/k B T (with k B Boltzmann’s constant) and H (Γ) is the classical Hamiltonian. The denominator in Eq. (1) is referred to as the canonical partition function, defined as Z (N, V, T ) =

dΓ exp[−β H (Γ)],

(2)

and guarantees proper normalization of the distribution function. The mechanical thermodynamic properties such as the internal energy, enthalpy and pressure, can be expressed as ensemble averages over the distribution function ρ(Γ). Here, the attribute “mechanical” means that the quantity of interest, X , is associated with a specific function X = X (Γ) of the microstate, Γ, of the system and can be written as X =

dΓρ(Γ)X (Γ).

(3)

Standard atomistic simulation techniques such as Metropolis MC [1] and MD [2, 3] provide powerful algorithms for generating sequences of microstates (Γ1 , Γ2 , . . . , Γ M ) that are distributed according the particular statistical– mechanical (e.g., canonical) distribution function of interest. In this manner, the average implied by Eq. (3) is easily estimated by averaging the function X (Γ) over a sequence, Γj , of microstates generated using MC or MD simulation, X = lim

M→∞

M 1 X (Γ j ). M j =1

(4)

Although the partition function Z , itself, is not known this does not present a problem in the case one is interested in any of the mechanical properties of the system; since Z is implicit in the generation of the sequence of microstates, Γi , it is not needed to perform the ensemble average of Eq. (3). The calculation of thermal quantities is not so straightforward, however. For example, the Helmholtz free energy A(N, V, T ) = −

1 1 ln Z (N, V, T ) = − ln β β

dΓ exp[−β H (Γ)] ,

(5)

710


is seen to be an explicit function of the partition function Z rather than an average of the type shown in Eq. 3. Therefore, as Z is not directly accessible in an MC or MD simulation, indirect strategies must be used. The most widely adopted strategy is to construct a real or artificial thermodynamic path that consists of a continuous sequence of equilibrium states linking two states of interest of the system and then attempt to calculate the free-energy difference between them. Should the free energy of one of these states be exactly known, the free energy of the other may then be put on an absolute basis. This approach provides the basis for the common thermodynamic integration (TI) method. Usually TI relies on the definition of a thermodynamic path in the space of system Hamiltonians. Typically, this involves the construction of an “artificial” Hamiltonian H (Γ , λ), which, aside from the usual dependence on the microstate Γ is also a function of some generalized coordinate or switching parameter λ. This generalized Hamiltonian is then constructed in such a way that it leads to a continuous transformation from the Hamiltonian of a system of interest to that of a reference system of which the free energy is known beforehand. Within the canonical ensemble, the Helmholtz free-energy difference between the initial and final states of the path, characterized by the switching coordinate values λ1 and λ2 , respectively, is then given by A ≡ A(λ2 ; N, V, T ) − A(λ1 ; N, V, T ) λ2

dλ

= λ1

∂ A(λ; N, V, T ) ∂λ

λ

λ2

= dλ λ1

∂ H (Γ, λ) ∂λ

λ

≡ Wrev ,

(6)

where A(λ; N, V, T ) is the Helmholtz free energy of the system as a function of the switching coordinate λ for fixed N , V , and T , and the brackets in the second integral denote an average evaluated for the canonical ensemble associated with the generalized coordinate value λ = λ . From a thermodynamic standpoint, Eq. (6) may be interpreted in the following way. The free-energy difference between the initial and final states is equal to the reversible work Wrev done by the generalized thermodynamic driving force ∂ H (Γ, λ)/∂λ along a quasistatic, or reversible process connecting both states. By quasistatic we mean that the process is carried out so slowly that the system remains in equilibrium at all times and the instantaneous driving force is equal to the associated equilibrium ensemble average. In this way, the TI method represents a numerical discretization of the quasistatic process; Wrev is estimated by computing the equilibrium ensemble averages of the driving force on a grid of λ-values on the interval [λ1 , λ2 ], after which the integration is carried out using standard numerical techniques. For further details of the TI method and its applications we refer to the chapter by Kofke and Frenkel [4].


3. 3.1.

711

Nonequilibrium Free-Energy Estimation Establishing Free-Energy Bounds: Systematic and Statistical Errors

Nonequilibrium free-energy estimation is an alternative approach to measuring the reversible work Wrev . Instead of discretizing the quasistatic process in terms of a sequence of independent equilibrium states, the reversible work is estimated by means of a single, dynamical sequence of nonequilibrium states, explored along an out-of-equilibrium simulation. This is achieved by introducing an explicit “time-dependent” element into the originally static sequence of states by making λ = λ(t) an explicit function of the simulation “time” t. Here we have used the quotes to emphasize that t should not always be interpreted as a real physical time. For instance, in contrast to MD simulations, typical displacement MC simulations do not involve a natural time scale, in case of which t is simply an index variable that orders the sequence of sampling operations, measured in simulation steps. Suppose we choose λ(t) such that λ(0)=λ1 and λ(tsim )=λ2 , so that λ varies between λ1 and λ2 in a time tsim . Accordingly, the Hamiltonian H (Γ, λ) = H (Γ, λ(t)) also becomes a function of t, and is driven from the initial system H1 to the final system H2 in the same time. The irreversible work Wirr done by the driving force along this switching process, defined as tsim

dt

Wirr = 0

dλ dt

t

∂H ∂λ

λ(t )

,

(7)

provides an estimator for the reversible work Wrev done along the corresponding quasistatic process. The point of this nonequilibrium procedure is that values of Wirr can be found, in principle, from a single simulation, because the integration in Eq. (7) involves instantaneous values of the function ∂ H/∂λ rather than ensemble averages. If efficient, this would be much less costly than the TI procedure in Eq. (6), which requires a series of independent equilibrium simulations. But there is, of course, a trade-off. While the TI method is inherently “exact” in that the errors are associated only with statistical sampling and the discreteness of the mesh used for the numerical integration, the irreversible work procedure provides a biased estimator for Wrev . That is, aside from statistical errors arising from different choices of initial configurations for calculation of Eq. (7), the irreversible estimator Wirr is subject to a systematic error Esyst. Both types of error are due to the inherently irreversible nature of the nonequilibrium process. The statistical errors originate from the fact that, for a fixed and finite simulation time tsim , the value of the integral in Eq. (7) depends on the initial

712


conditions of the nonequilibrium process. In other words, for different initial conditions, Γ j (t = 0), and a finite simulation time tsim , the value of Wirr in Eq. (7) is not unique. Instead, it is a stochastic quantity characterized by a distribution function with a finite variance, giving rise to statistical errors of the sort arising in any MC or MD simulation. The systematic error manifests itself in terms of a shift of the mean of the irreversible work distribution with respect to the value of the ideal quasistatic work Wrev . This shift is caused by the dissipative entropy production characteristic of irreversible processes [6]. Because the entropy always increases, the systematic error Ediss is always positive, regardless of the sign of the reversible work Wrev . In this way, the average value Wirr of many measurements of the irreversible work will yield an upper bound to the reversible work Wrev , provided the average is taken over an ensemble of equilibrated initial conditions j (t = 0) at the starting point, t = 0. The importance of satisfying the latter condition was demonstrated by Hunter et al. [7]. From a purely thermodynamic point of view, the bounding error is simply a consequence of the Helmholtz inequality. Starting from an equilibrium initial state, for instance at λ = λ1 , the irreversible work upon driving the system to λ = λ2 is always an upper bound to the actual free-energy change between the equilibrium states of initial and final systems, i.e., Wirr ≥ A = A(λ2 ; N, V, T ) − A(λ1 ; N, V, T ).

(8)

Only in the limit of an ideally quasistatic, or reversible process, represented by the tsim → ∞ limit, does the inequality in Eq. (8) become the equality, Wrev = A, as also manifested in Eq. (6). The preceding ideas are illustrated conceptually in Fig. 1(a) and (b), which show typical distribution functions of irreversible work measurements starting from an ensemble of equilibrated initial conditions. Figure 1(a) compares the results that might be obtained for irreversible work measurements for two different finite simulation times tsim = t1 and tsim = t2 , with t2 > t1 to the ideally reversible tsim → ∞ limit. Both finite-time results show distribution functions with a finite variance and whose mean values have been shifted with respect to the reversible work value by a positive systematic error. Both the variance and systematic error for tsim = t1 are larger than the corresponding values for tsim = t2 , given that the latter process proceeds in a slower manner, leading to smaller irreversibility. Figure 1(b) shows the irreversible work estimators obtained for the reversible work associated with a quasistatic process in which system 1 is transformed into system 2 as obtained in the forward (1 → 2) and backward (2 → 1) directions using the same simulation time tsim . Given that the systematic error is always positive, the forward and backward processes provide upper and lower bounds to the reversible work value, respectively. However, in general, the systematic and statistical errors need not be equal for both directions.

Free-energy calculation using nonequilibrium simulations (a)

713

(b) tsim → ∞

tsim t2 > t1

(2 → 1)

(1 → 2)

∆Ediss (t2)

tsim t1 ∆Ediss(t1)

Wrev (1 → 2)

Wrev Wirr

Wirr

Figure 1. Conceptual illustration of typical irreversible work distributions obtained from nonequilibrium simulations. (a) compares the results that might be obtained for irreversible work measurements for two different finite simulation times tsim = t1 and tsim = t2 , with t2 > t1 to the ideally reversible tsim → ∞ limit. (b) shows the irreversible work estimators obtained for the reversible work associated with a quasistatic process in which system 1 is transformed into system 2 as obtained in the forward (1 → 2) and backward (2 → 1) directions using the same simulation time tsim .

3.2.

Optimizing Free-Energy Bounds: Insight from Nonequilibrium Statistical Mechanics

A natural question that arises after considering the discussion in previous section is how one might tune the nonequilibrium process so as to minimize the systematic and statistical errors associated with the irreversibility for given initial and final equilibrium states and a given simulation time tsim . To answer this question, it is useful to investigate the microscopic origin of entropy production in nonequilibrium processes. For this purpose, it is particularly helpful to consider the particular class of close-to-equilibrium nonequilibrium processes for which the instantaneous distribution functions of nonequilibrium states do not deviate too much from the ideally quasistatic equilibrium distribution functions and where theory of linear response [5] is appropriate. As we will see later on, it is not too difficult to reach this condition in practical situations. As described by Onsager’s regression hypothesis [5], when a nonequilibrium state is not too far from equilibrium, the relaxation of any mechanical property can be described in terms of the proper equilibrium autocorrelation function. In other words, the hypothesis states that the relaxation of a nonequilibrium disturbance is governed by the same laws as the regression of spontaneous microscopic fluctuations in an equilibrium system.

714


Under the assumption of proximity to equilibrium, one can then derive the following expression for the mean dissipated energy, i.e., the systematic error Ediss(tsim ), for a series a irreversible work measurements obtained from nonequilibrium simulations of duration tsim [8–10]: 1 Ediss(tsim ) = kB T

tsim

dt 0

dλ dt

2 t

∂H τ [λ(t )] var ∂λ

.

λ(t )

(9)

Aside from the switching rate, the integrand in Eq. (9) contains both the correlation time as well as the equilibrium variance of the driving force ∂ H/∂λ. These two factors describe, respectively, how quickly the fluctuations in the driving force decay and how large these fluctuations are in the equilibrium state. It is clear that the integral is positive-definite, as it must be. Moreover, it indicates that, for near-equilibrium processes, the systematic error should be the same for forward and backward processes. This means that, in the linear–response regime, one can obtain an unbaised estimator for the reversible work Wrev by combining the results obtained from forward and backward processes. More specifically, in this regime we have Wirr (1 → 2) = Wrev (1 → 2) + Ediss ,

(10)

Wirr (2 → 1) = −Wrev (1 → 2) + Ediss ,

(11)

and

leading to the unbaised estimator (i.e., subject to statistical fluctuations only) Wrev (1 → 2) = 12 (Wirr (1 → 2) − Wirr (2 → 1) .

(12)

Concerning minimization of dissipation, Eq. (9) tells us that one should attempt to reduce both the magnitude of the fluctuations in the driving force as well as the associated correlation times. This involves both a static component, i.e., the magnitude of the equilibrium fluctuations, and a dynamic one, namely the typical decay time of equilibrium correlations. This shows that not only the choice of the path, H (λ), but also the simulation algorithm by which the system is propagated in “time” (i.e., MC or MD simulation) will affect the dissipation in the irreversible work measurements. Whereas the magnitude of the equilibrium fluctuations should be algorithm independent (as long as the algorithms sample the same equilibrium distribution function), the correlation time is certainly algorithm-dependent. In case of displacement MC simulation, as we will see below, the choice of the maximum displacement parameter affects the correlation time τ , and, consequently, the magnitude of the dissipation.


715

Finally, let us now assume that we have a prescribed path H (λ) and a simulation algorithm to sample the nonequilibrium process between the systems H (λ1 ) and H (λ2 ). How do we now choose the functional form of the time-dependent switching function λ(t) to minimize the dissipation? Equation (9) provides us with an explicit answer. To see this, we first perform a change of integration variable, setting x = t /tsim , obtaining Ediss(tsim ) =

1 tsim

Ediss[λ(x)],

(13)

with 1 Ediss[λ(x)] = kB T

1

dx 0

dλ dx

2 x

∂H τ (λ(x )) var ∂λ

λ(x )

.

(14)

Equation (14) is a functional of the common form [11] 1

S[λ(x)] =

dx F(λ (x), λ(x), x).

(15)

0

The minimization of the dissipation is thus equivalent to finding the function λ(x) that minimizes a functional of the type (15) subject to the boundary conditions λ(0)=λ1 and λ(1)=λ2 . Standard variational calculus then shows that the solution is obtained by solving the Euler–Lagrange equation [11] associated with the functional, d ∂F ∂F = , dx ∂λ ∂λ

(16)

subject to the mentioned boundary conditions.

4.

Applications of Nonequilibrium Free-Energy Estimation

To illustrate the discussion of the previous sections we will now discuss a number of applications of nonequilibrium free-energy estimation, demonstrating the bounding properties of irreversible-work measurements, as well as aspects of dissipation optimization.

4.1.

Harmonic Oscillators

In the first application we consider the problem of computing the free-energy difference between two systems consisting of 100 identical, independent,

716


one-dimensional harmonic oscillators of unit mass with different characteristic frequencies [9]. In particular we will consider the path defined by H (λ) =

100 1 [(1 − λ)ω12 + λω22 ] xi2 , 2 i=1

(17)

with ω1 = 4 and ω2 = 0.5 at a temperature k B T = 2. Note that we are considering only the potential energy of the oscillators here and have neglected any kinetic energy contributions. We can do this because the free-energy difference between two harmonic oscillators at a fixed temperature is determined only by the configurational part of the partition function. The value of the desired reversible work Wrev per oscillator associated with a quasistatic modification of the frequency from ω1 to ω2 is known analytically: ω1 = −4.15888. (18) Wrev (ω1 → ω2 ) = −k B T ln ω2 The simulation algorithm we utilize is standard Metropolis displacement MC with a fixed maximum trial displacement xmax = 0.3. First we consider the statistics of the irreversible work measurements as a function of the simulation “time” tsim , which here stands for the number of MC sweeps (one sweep corresponds to one trial displacement per oscillator) per process, for a linear switching function. The results are shown as the dashed line curves in Fig. 2(a) and (b), in which each data point represents the mean value of Wirr over 50 independent initial conditions. Figure 2(a) shows that the upper and lower

Upper/Lower bounds to Work

2 3 4 5 6

Analytical Linear function Optimized function

7 8 9 10

0

0.5

1

1.5

2

2.5

tsim ( 104 MC sweeps)

Average of forward and backward

(b)

(a)

4.0 4.5 5.0 Analytical Linear function Optimized function

5.5 6.0 6.5

0

0.5

1

1.5

2

2.5

tsim ( 104 MC sweeps)

Figure 2. Results of irreversible-work measurements per oscillator as a function of the switching time tsim for the linear (dashed lines) and optimal (solid lines) switching function. The analytical reversible work value is also shown (dot dashed line). (a) shows the results of the forward (upperbounds) and backward (lowerbounds) directions. (b) shows the values of the combined estimator of Eq. (12).


717

limit do converge toward the reversible value Wrev , although they do so quite slowly. The slow convergence becomes more apparent when we consider the behavior of the combined estimator of Eq. (12) in Fig. 2(b). If the process were sufficiently slow for linear–response theory to be accurate, the combined estimator should be unbiased and show no systematic deviation. It is clear that this is only the case for the slowest process, at tsim =2.56×104 MC sweeps. All shorter simulations show a systematic deviation, indicating that the associated processes remain quite far from equilibrium, hampering convergence. Next, we attempt to minimize dissipation in the simulation by using the switching function λ(x) that satisfies the Euler–Lagrange Eq. (16). For this purpose we first measured the equilibrium variance in the driving force and the characteristic correlation time of decay as a function of λ from a series of equilibrium simulations (i.e., fixed λ), after which we numerically solved Eq. (16), subject to the boundary conditions λ(0) = 0 and λ(1) = 1. The equilibrium variances, correlation times and the resulting optimal switching function are shown in Fig. 3(a)–(c), respectively. The results in Fig. 3(a) and (b) indicate that the main contribution to the dissipation originates from the region λ ≈ 1, where both the magnitude as well the characteristic decay time of the fluctuations in the driving force increase sharply. The optimal switching function in Fig. 3(c) captures this effect, prescribing a slow switching rate where one should and going faster where one can. The results obtained with this function for the irreversible work measurements are shown as the red lines in Fig. 2(a) and (b). The improvement compared to the linear switching function is quite significant. Figure 2(b), for instance, shows that for tsim as short as 3.2 × 103 MC sweeps, the nonequilibrium process has already reached the linear–response regime. The above optimization procedure is useful in cases where the thermodynamic path H (λ) is prescribed beforehand. This is the case, for instance, for

5

(a)

10

0 0

(c)

(b) 1.0

100

Correlation time

Variance

10 20

0.5

1

0.8

50

0.6 0.4

Linear Optimized

0.2 0

0.0 0

0.5

1

0

0.2 0.4 0.6 0.8 1.0

x

Figure 3. (a) The equilibrium variance (∂ H/∂λ), and (b) the correlation decay time (in MC sweeps) as a function of λ. (c) shows the optimal switching function, as determined by numerically solving Euler–Lagrange equation (16).

718


the reversible-scaling method [12], in which each state along the fixed path H (λ) = λV (V is the interatomic interaction potential) represents the physical system of interest in a different temperature state. In this manner, a single irreversible-work simulation along the scaling path provides a continuous series of estimators of the system’s free energy on a finite temperature interval. If one has some information about the behavior of the magnitude of the and correlation-decay times of the fluctuations of the driving force, one may use the variational method described above to optimize the switching function and minimize dissipation effects.

4.2.

Compression of Confined Lennard–Jones Particles

In the following application we consider a system consisting of 30 Lennard– Jones particles, constrained to move on the x-axis only. In addition, the particles are subject to an external field whose strength is controlled by an external parameter L. More specifically, we consider the path 

6  12 σ σ  2xi 26  − + , H (L) =

xi j

xi j

L

(19)

where xi describes the position of particle i on the x-axis and xi j ≡ |xi − x j | is the distance between particles i and j . The second term in Eq. (19) is the external field, which is a very steeply rising potential and has the effect of confining the particles through very strong interactions with the first and last particles, effectively causing the 30 particles to lie approximately evenly spaced between x = ±L/2. Now consider the compression process wherein L changes from L 0 = 30σ to L 1 = 26σ , forcing the line of particles to undergo a one-dimensional compression. As in the previous example, we will attempt to compute the reversible work associated with this process by measuring the irreversible work Wirr for both process directions. Once again we utilize the Metropolis MC algorithm, but instead of fixing the algorithm parameter xmax , describing the maximum trial displacement, we now consider the effects of changing the sampling algorithm on the convergence of the upper and lower bounds. Although the variance of the driving force var (∂ H/∂λ) will not be affected, the correlation time will certainly depend on the choice of xmax . This is illustrated in Fig. 4, which shows the convergence of the upper and lower bounds to the reversible work as obtained for 3 different values of max at a temperature k B T = 0.35 : xmax = 0.6σ , 0.1σ , and 0.04σ , respectively. Effectively, the variation of this algorithm parameter may be thought of as changing the strength of the coupling between the MC “thermostat” and the system of particles. We utilized the linear switching function which varies L linearly between L 0 and L 1 in tsim MC sweeps (each sweep


719

Figure 4. Results of forward (upperbound) and backward (lowerbound) irreversible-work measurements (in units of ) as a function of the switching time tsim for the linear switching function for three different values of the MC algorithm parameter xmax .

consisting of 30 MC single-particle trial moves). Each data point and corresponding error bar (±1 standard deviation) were obtained from a set of 21 irreversible work measurements initiated for independent, equilibrated initial conditions. It is also useful to note that it is not necessary to explicitly compute the work Wirr by using (7). All that is needed, through the first law of thermodynamics which applies equally to reversible and irreversible processes, is to calculate the work as Wirr = E − Q, where E is the difference in internal energies of the system between the first and last switching steps, and Q is the heat accumulated during the switching process. This heat, Q, is simply the sum of energies added to, or subtracted from, the system as MC configurations evolve during a simulation. Given that these energies, εi , are already calculated in determining whether moves for particle i are to be accepted or rejected according to the canonical exp(−εi /k B T ), no extra programming is needed to calculate Wirr . It is immediately seen that the strength of the system-thermostat coupling through the algorithm parameter max is indeed a variational parameter

720


for the free-energy computations. Accordingly, rather than selecting a pre-set acceptance ratio of trial moves, as is usually done in equilibrium MC simulations, xmax should be determined so as to minimize the difference between the upper and lower bounds to A. The results show that for all three values of xmax , the upper and lower bounds show convergence. Yet, the convergence properties are clearly different for the three parameter values, giving the best results for xmax = 0.1 and the worst for xmax = 0.04, indicating that the correlation decay time for the fluctuations in the driving force are the shortest for the former and the longest for the latter. Nevertheless, the convergence of the bounds is still quite slow, in that hundreds of thousands of MC sweeps are required to obtain convergence of to within a few percent. This is a consequence of the strong interactions between the particles, as their hard cores interact during the compression from the “ends” of the line of particles and such hard core density gradients are typically slow to work themselves out through single particle MC moves. Contrary to the simple harmonic oscillator problem discussed in the previous section, this problem will be ubiquitous in most atomic and molecular systems in the condensed phase, seemingly rendering the free-energy computations on realistic systems of interest problematic. The questions that now arise are as to whether we can estimate the systematic errors Ediss from data already in hand and use it to improve the estimates of Fig. 4; and/or if we can optimize the thermodynamic path to reduce dissipation and achieve better behavior at short switching times; or perhaps both?

4.3.

Estimating Equilibrium Work from Nonequilibrium Data

Recently, Jarzynski [13] has generalized the Gibbs–Feynman identity, A = A1 − A0 = −k B T lnexp[−(H1 − H0 )/k B T ]0

(20)

where · · · 0 denotes canonical averaging with respect to configurations generated by H0 , and which is the basis of thermodynamic perturbation theory [4], to finite-time processes. Equation (20) is an identity, but in practice it is useful only when the configurations generated by canonical sampling with respect to H0 strongly overlap those generated by H1 . For hard core fluids this would be unusual unless H1 and H0 are quite “close”, resulting in the perturbative use of Eq. (20). Jarzynski now allows H0 to dynamically approach H1 along a path, in analogy with the above discussions. The result, in the context discussed here, suggests that for a given set of N irreversible-work measurements Wi ≡ Wirr (i , t = 0), with i = 1, . . . , N , instead of estimating Wirr as the sim-


721

ple arithmetic mean of the Wi , one should calculate the Boltzmann weighted “Jarzynski” (or “Jz”) average W Jz =

M 1 exp(−Wi /k B T ), M i=1

(21)

and then estimate the free energy change as AJz ≡ −k B T lnW Jz .

(22)

In this way bounding is sacrificed, but a more accurate result is not precluded given that, in principle, the Jz-average is unbiased. This approach has been shown to be effective both in the analysis of simulation data as well as finite-time polymer extension experiments, which are of course irreversible. An immediate concern, however, is that, although in the limit of complete sampling as in the Gibbs–Feynman identity, the Jarzynski results are exact in the context of a dissipation-free system, incomplete MC sampling may result in unsatisfactory results.

Work of Compression 300 Forward arithmetic average Backward arithmetic average Forward Jarzynski average Backward Jarzynski average


250

200

150

100

50

0 100

1000

10

4

10

5

tsim (MC sweeps) Figure 5. Results of forward and backward irreversible-work averages (in units of ) for the 30-particle confined Lennard–Jones system as a function of the switching time tsim . The results show both the simple arithmetic averages as well as the Boltzmann-weighted Jarzynski averages.

722


This is illustrated in Fig. 5, where data used to generate the bounds to A in Fig. 4, are plotted over a much larger range of switching times tsim , and compared to the AJz estimates. Both the simple arithmetic as well as the Jarzynski averages for both directions were computed over the 21 independent initial conditions. It is evident that, although not giving bounds, the AJz estimates indeed improve the upper and lower bounds compared to those calculated as simple averages. However, the Jarzynski averages become useful when the convergence of the simple arithmetic averages has reached the order of less than 1 k B T per particle. In this fashion, although a promising computational asset, the Jarzynski procedure still requires systematic procedures for finding more reversible paths.

4.4.

Path Optimization through Scaling and Shifting of Coordinates

As we have seen in the harmonic oscillator and Lennard–Jones problems, the choice of the thermodynamic path and the used switching function is quite crucial to the success of nonequilibrium free-energy estimation. In the case of the harmonic oscillator problem it was relatively straightforward to find a good switching function by explicitly solving the variational problem in Eqs. (15) and (16), which lead to an optimized simulation that “spends the right amount of time along each segment” of the already defined path. Here it is important to note that this variational optimization should be carried out over an ensemble averaged Wirr , being identical for every member of the ensemble, independently of any specific i (t = 0). This is the reason why early attempts by Pearlman and Kollman [14] to determine paths “on the fly” by looking ahead and avoiding strong dissipative collisions in specific configurations may result in the unintentional introduction of a Maxwell demon [15], violating the second law of thermodynamics, which is of course the fundamental origin of the Helmholtz inequality. Compared to the simple harmonic oscillator problem, the optimization of the nonequilibrium simulation of the confined Lennard–Jones system is significantly more challenging because of the strong interactions between the particles as during the compression of the system. Given that this type of interaction is expected to occur in most interesting problems, it is of interest to design thermodynamic paths that are different from the ones in which one simply follows H (λ) as λ runs from an initial to a final value, like we did in the case of the harmonic oscillator problem. We now present two approaches that follow this idea and lead to thermodynamic paths that are significantly more reversible. Both the coordinate scaling [16] and coordinate shifting methods discussed below derive from


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the same fundamental thought: is there a (λ-dependent) coordinate system in which all particles are apparently at rest with relative to one another during the switching process? In such a coordinate system perhaps all particles will have little difficulty in remaining close to equilibrium during the whole switching process, with only the magnitude of their local fluctuations changing.

4.4.1. Coordinate scaling Figure 6 illustrates the possibilities of such an approach, when applied to the simple problem of compression discussed above. Here, in an admittedly simple example, all particles should be compressed “uniformly,” rather than by the nonuniform compression generated through the interactions of the confining potential with the particles at both ends of the line. This is accomplished by writing the coordinates as s(λ) xi , where s(λ) is a (common) scaling parameter, which may then be variationally optimized. The greatly improved bounds of Fig. 6 indicate that a better path has indeed been found. How does this fit the “at rest” criterion mentioned earlier? If one watches the MC dynamics in the unscaled “xi ” coordinates using an optimized s(λ), rather than in the actual physical coordinates, s(λ) xi , it appears that the equilibrium positions xi do not change during the switching, and thus, indeed, the only irreversibility arises from the changes in the RMS fluctuations about the equilibrium positions. It should be noted, however, that, as these scalings may be regarded as a change in the metric that affects the length and volumes definitions, one should include a entropic (calculable) correction to obtain the desired free-energy difference. Recently, there has been a variety of applications of the scaling approach [16–18], including the determination of the absolute free energy of Lennard– Jones clusters and a smooth metric scaling through a first order solid–solid phase transition, fcc to bcc, with no apparent hysteresis with its resulting irreversibility.

4.4.2. Coordinate shifting In the applications of metric scaling, thermodynamic paths are often easily determined when a clear symmetry is present. Another approach, namely coordinate shifting is more useful when such symmetries are absent. As an alternative to writing a moving coordinate using the scaling relation s(λ) xi , one can take xi = xifluct + xiref (λ). Here each particle moves in a concerted fashion along a λ-dependent reference path, chosen by symmetry, or by methods such as simulated annealing, to avoid strong hard core interactions or other

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likely causes of irreversibility. As λ evolves, only the fluctuation coordinates xifluct are subject to MC variations: should the physical environment of each particle remain at least roughly constant, one may hope that the fluctuations from the xiref (λ) do not depend strongly on λ. To the extent that this is the case, the fluctuation coordinates are always at equilibrium, and thus the path is reversible! Figure 7 illustrates the efficacy of this method for the linear compression problem. As opposed to coordinate scaling, coordinate shifting does not change the metric, dispensing the need for entropic corrections and paving the way for applications involving inhomogeneous systems where the possible absence of symmetries obscures the choice of an appropriate metric obvious and complicates the computation of scaling entropy corrections. As is also clear from the results shown in Figure 7, the finite-time upper and lower bounds converge sufficiently quickly for the Jarzynski averaging to actually markedly improve even the shortest-time results. More general “non-linear” combinations of scaling and shifting may also be used to advantage, as in [19].

Work of Compression: Optimized Scaling


225

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25 1

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2

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3

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4

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tsim (MC sweeps) Figure 6. Convergence of upper and lower bounds to the free-energy change associated with the compression of the confined Lennard–Jones system at k B T = 0.35 as a function of the switching time tsim . The outer pair of lines are from standard finite-time switching, whereas the inner pair represents the results from finite-time switching using linear metric scaling. The vertical bars represent the standard error in the mean of 100 replicas.


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Work of Compression: Optimized Shifting 71 Forward arithmetic average Backward arithmetic average Forward Jarzynski average Backward Jarzynski average


70 69 68 67 66 65 64 63 10

2

10

3

10

4

tsim (MC sweeps) Figure 7. Convergence of upper and lower bounds to the free-energy change associated with the compression of the confined Lennard–Jones system at k B T = 0.35 as a function of the switching time tsim as obtained by optimized coordinate shifting. The vertical bars represent the standard error in the mean of 21 replicas. The results obtained with Jarzynski averages are also shown.

5.

Outlook

One of the most fundamental and challenging applications of atomistic simulation techniques concerns the determination of those thermodynamic properties that require determination of the entropy, the chemical potential and the various free energies, which are all examples of thermal thermodynamic properties. In contrast to their mechanical counterparts (e.g., enthalpy, pressure) they cannot be computed as ensemble (or time) averages, and indirect strategies must be adopted. Here, we have discussed the basic aspects of a particular strategy, that of using nonequilibrium simulations to obtain estimators of reversible work between equilibrium states. The point of this approach is that, in contrast to equilibrium methods such as thermodynamic integration, the desired value can, in principle, be estimated from a single simulation. But there is a trade-off, in that the nonequilibrium estimators are subject to both systematic and statistical errors, caused by the inherently irreversible nature of nonequilibrium processes.

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Yet, the approach allows one to systematically obtain upper and lower bounds to the requested reversible result by exploring the nonequilibrium processes both in forward and backward directions. The bounds for a given process become tighter with decreasing process rates. But more importantly, it is possible to optimize the nonequilibrium process so as to minimize irreversibility and, for a given process time, decrease the bounds. We have discussed a number of methods by which to conduct this optimization task, including explicit functional optimization using standard variational calculus and techniques based on special coordinate transformations aimed at the reduction of irreversibility. These techniques have been quite successful so far, allowing accurate free-energy measurements using relatively short nonequilibrium simulations. In this light, the idea of using nonequilibrium simulations has now grown into a robust and efficient computational approach to the problem of computing thermal thermodynamic properties using atomistic simulation methods. Nevertheless, further development remains necessary, in particular toward improving/generalizing the existing optimization schemes.

References [1] G. Gilmer and S. Yip, Handbook of Materials Modeling, vol. I, chap. 2.14, Kluwer, 2004. [2] J. Li, Handbook of Materials Modeling, vol. I, chap. 2.8, Kluwer, 2004. [3] M.E. Tuckerman, Handbook of Materials Modeling, vol. I, chap. 2.9, Kluwer, 2004. [4] D.A. Kofke and D. Frenkel, Handbook of Materials Modeling, vol. I, chap. 2.14, Kluwer, 2004. [5] D. Chandler, Introduction to Modern Statistical Mechanics, Oxford University Press, Oxford, 1987. [6] L.D. Landau and E.M. Lifshitz, Statistical Physics, Part 1, 3rd edn., Pergamon Press, Oxford, 1980. [7] J.E. Hunter III, W.P. Reinhardt, and T.F. Davis, “A finite-time variational method for determining optimal paths and obtaining bounds on free energy changes from computer simulations,” J. Chem. Phys., 99, 6856, 1993. [8] L.W. Tsao, S.Y. Sheu, and C.Y. Mou, “Absolute entropy of simple point charge water by adiabatic switching processes,” J. Chem . Phys., 101, 2302, 1994. [9] M. de Koning and A. Antonelli, “Einstein crystal as a reference system in free energy estimation using adiabatic switching,” Phys. Rev. E, 53, 465, 1996. [10] M. de Koning and A. Antonelli, “Adiabatic switching applied to realistic crystalline solids: vacancy-formation free energy in copper,” Phys. Rev. B, 55, 735, 1997. [11] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1, Wiley, New York, 1953. [12] M. de Koning, A. Antonelli, and S. Yip, “Optimized free energy evaluation using a single reversible-scaling simulation,” Phys. Rev. Lett., 83, 3973, 1999. [13] C. Jarzynski, “Nonequilibrium equality for free energy differences,” Phys. Rev. Lett., 78, 2690, 1997.


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[14] D.A. Pearlman and P.A. Kollman, “The lag between the Hamiltonian and the system configuration in free energy perturbation calculations,” J. Chem. Phys., 91, 7831, 1989. [15] H.S. Leff and A.F. Rex, Maxwell’s Demon 2, Entropy, Classical and Quantum Information, Computing, Institute of Physics Publishing, Bristol, U.K, 2002. [16] M.A. Miller and W.P. Reinhardt, “Efficient free energy calculations by variationally optimized metric scaling: concepts and applications to the volume dependence of cluster free energies and to solid–solid phase transitions,” J. Chem. Phys., 113, 7035, 2000.

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M. de Koning and W.P. Reinhardt [17] L.M. Amon and W.P. Reinhardt, “Development of reference states for use in absolute free energy calculations of atomic clusters with application to 55-atom Lennard– Jones clusters in the solid and liquid states,” J. Chem. Phys., 113, 3573, 2000. [18] W.P. Reinhardt, M.A. Miller, and L.M. Amon, “Why is it so difficult to simulate entropies, free energies and their differences?” Accts. Chem. Res., 34, 607, 2001. [19] C. Jarzynski, “Targeted free energy perturbation,” Phys. Rev. E, 65, 046122, 2002.

2.16 ENSEMBLES AND COMPUTER SIMULATION CALCULATION OF RESPONSE FUNCTIONS John R. Ray 1190 Old Seneca Road, Central, South Carolina 29630, USA

1.

Statistical Ensembles and Computer Simulation

Calculation of thermodynamic quantities in molecular dynamics (MD) and Monte Carlo (MC) computer simulations is a useful, often employed tool [1–3]. In this procedure one chooses a particular statistical ensemble for the computer simulation. Historically, this was the microcanonical, or (EhN) ensemble for MD and the canonical, or (ThN) ensemble for MC, but there are several choices available for MD or MC. The notations, (EhN), (ThN) denote ensembles by the thermodynamic state variables that are constant in an equilibrium simulation; energy E, shape-size matrix h, particle number N and temperature T . (There could be other thermodynamic state variables, gi , i = 1, 2, . . . , such as electric or magnetic field applied to the system, and these additional variables would be in the defining brackets.) The shape-size matrix is made up of the three vectors defining the computational MD or MC cell. If the vectors defining the parallelepiped, containing the particles in the computational cell, are denoted (a, b, c) then the 3×3 shape-size matrix is defined by having its columns constructed from the three cell vectors, h = (a, b, c).The volume V of the computational cell is related to the h matrix by V = det(h). For simplicity, we assume that the atoms in the simulation are described by classical physics using an effective potential energy function to describe the inter-particle interactions. Unless explicitly stated otherwise we suppose that periodic boundary conditions are applied to the particles in the computational cell. The periodic boundary conditions have the effect of removing surface effects and, conveniently, making the calculated system properties approximately equal to those of bulk matter. We assume the system obeys 729 S. Yip (ed.), Handbook of Materials Modeling, 729–743. c 2005 Springer. Printed in the Netherlands.

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the Born–Oppenheimer approximation and can be described by a potential energy U using classical mechanic and classical statistical mechanics.

2.

Ensembles

For a single component system there are eight basic ensembles that are convenient to introduce. These ensembles and their connection to their reservoirs are shown in Fig. 1 [4]. Each ensemble represents a system in contact with different types of reservoirs. These eight systems are physically realizable and each can be employed in MD or MC simulations. The combined reservoir is a thermal reservoir, a tension (or stress) and pressure reservoir (the pressure reservoir in Fig. 1 represents a tension and pressure reservoir) and a chemical potential reservoir. The reservoirs are used to impose, respectively,

Figure 1. Shown are the eight ensembles for a single component system. The systems interact through a combined temperature, pressure and chemical potential reservoir. The ensembles on the left are adiabatically insulated from the reservoir while those on the right are in thermal contact with the reservoir. Pistons and porous walls allow for volume and particle exchange. Adiabatic walls are shown cross-hatched while dithermal walls are shown as solid lines. Ensembles on the same line like a and e are related by Laplace and inverse Laplace transformations. The pressure stands for the pressure and the tension.

Ensembles and computer simulation calculation

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constant temperature, tension and pressure, and chemical potential. The eight ensembles naturally divide into pairs of ensembles. The left-hand column in Fig. 1, a–d are constant energy ensembles while ensembles in the right hand column, e–h have constant temperature. These pairs of ensembles are connected to each other by direct and inverse Laplace transformations, a ↔ e, et cet. The energies that are associated with each ensemble are related to the internal energy E by Legendre transformations [4]. The eight ensembles may be defined using the state variables that are held constant in the ensemble ([5] pp. 293–304). The eight ensembles include the (EhN) and (ThN) ensembles introduced earlier. Another pair of ensembles is the (H t and P N) and (T t and P N) ensembles where H = E + Vo Tr(tε) + PV is the enthalpy, tij is the thermodynamic tension tensor, εij the strain tensor, P the pressure and Tr represents the trace operation. The thermodynamic tension is a modified stress tensor applied to the system that is introduced in the thermodynamics of anisotropic media. Due to definitions in the thermodynamic of non-linear elasticity we denote the tension and pressure separately. A third pair of ensembles is the (Lhµ) and (Thµ), where L is the Hill energy L = E−µN and µ the chemical potential for the one component system. The isothermal member of this latter pair of ensembles is Gibb’s grand canonical ensemble, (Thµ) ensemble. The final pair of ensembles is the (R t and Pµ) and (T t and Pµ) ensembles where R = E + Vo Tr(tε) + PV −µN is the R-energy. The latter member of this ensemble pair was introduced by Guggenheim [6] and is interesting since it has all intensive variables, T, P, µ, and these are all held fixed, but we know only two of these can be independent. Nevertheless, this ensemble can be used in simulations although its size will increase or decrease in the simulation. The (R t and P µ) ensemble allows variable particle number along with variable shape/size. These last four ensembles all have constant chemical potential and variable particle number. For multi-component systems there are a series of hybrid ensembles that are useful. As an example, for two component systems we can use the (T t and P µ1 N2 ) ensemble that is useful for studying the absorption of species 1 in species 2 as for example the absorption of hydrogen gas in a solid [7, 8]. Each of the eight ensembles, for a single component system, may be simulated using either MD or MC simulations. The probability distributions are exponentials for the isothermal ensembles and power laws for the adiabatic ensembles. For example, for the (TVN) ensemble the probability density has the Boltzmann form P(q; T VN ) = Ce−U (q)/(kB T ) with U (q) the potential energy and C a constant. For the (H t and PN) ensemble P(q;H, t,P,N) = CV N (H −Vo Tr(tε) −PV−U(q))(3N/2−1) . The trial MC moves involve particle moves and shape/size matrix moves [9]. For the (R t and Pµ) ensemble MC moves involve particle moves, shape/size matrix moves and attempted creation and destruction events [10]. For MC simulation of these ensembles one uses the probability density directly in the simulation, whereas for MD simulations

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ordinary differential equations of motion are solved for equations arising from Hamilton’s equations. An important advancement in using MD to simulate different ensembles was the extended variable approach introduced by Andersen [11]. In this approach, which some variation is used in all but the (EhN) ensemble, extra variables are introduced into the system to introduce the variation of the variable in the ensemble. Although these variations are fictitious it can be proven that the correct ensemble is generated using these extended variable schemes. In the original approach for the (H PN) ensemble Andersen introduced an equation of motion for the volume that responds to a force that is the difference between the internal microscopic pressure and an external constant pressure imposed by the reservoir. This leads to volume fluctuations that are appropriate to the (H PN) ensemble, see Fig. 1. Nose, thereafter, generalized MD to the isothermal ensembles by introducing a mass scaling variable that allows for energy fluctuations in the (ThN) and the other isothermal ensembles [12]. These energy fluctuations mimic the interaction of the system with the heat reservoir and allow MD to generate the probability densities of the isothermal ensembles. Which ensemble/ensembles to use, and whether to use MD or MC depends on user preference and the particular problem under consideration. For the variable particle number ensembles (those involving the chemical potential in their designation) one usually employs MC methods since simulations using these ensembles involve attempted creation and destruction of particles and this fits naturally with the stochastic nature of the MC method. However, MD simulations of these ensembles have been investigated and performed [13].

3.

Response Function Calculation

Response functions are thermodynamic properties of the system that are often measured, such as specific heats, heat capacities, expansion coefficients, and elastic constants to name a few. Response functions are associated with derivatives of the basic thermodynamic state variables like energy, pressure, entropy and include the basic thermodynamic state variables themselves. We do not include (non-equilibrium) transport properties, such as thermal conductivity, electrical conductivity, and viscosity, in our discussions since they fall under a different calculation schema that uses time correlation functions [14]. Formulas, that may be used to calculate response functions in simulations, may be derived by differentiation of quantities connecting thermodynamic state variables with integrals over functions of microscopic particle variables. These formulas are specific to each ensemble, and are standard statistical mechanics relations. Such a quantity, in the canonical ensemble, is the partition


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function Z (T, h, N), which for a N particle system in three-dimensions has the form Z (T, h, N ) =

1 N !(2π)3N

e−H (q, p,h)/ kB T d 3N qd 3N p,

(1)

where q and p denote the 6N -dimensional phase space canonical coordinates of the system, H the system Hamiltonian, kB Boltzmann’s constant, Plank’s constant, and dτ = d 3N qd3N p the phase space volume element. The integral in Eq. (1) is carried out over the entire phase space. Although we have indicated the Hamiltonian depends on the cell vectors, h, it would also depend on additional thermodynamic state variables gi . For liquids and gases the dependence on h is replaced by simple dependence on the volume V ; for discussions of elastic properties of solids it is important to include the dependence on the shape and size of the system through the shape size matrix h or some function of h. The Helmholtz free energy A(T, h, N ) is obtained from the canonical ensemble partition function A(T, h, N ) = −k B T ln Z (T, h, N ).

(2)

Average values of phase space functions may be calculated using the phase space probability, which for the canonical ensemble is the integrand in the partition function in Eq. (1). For example, the canonical ensemble average for the phase space function f(q,p,h)is f=

f e−H/k B T dτ e−H/k B T dτ .

(3)

In an MD or MC simulation the thermodynamic quantity f is calculated by using a simple average over the simulation configurations, for MD this is an average over time, whereas for MC it is an average over the Markov chain of configurations generated. If the value of f at each configuration (each value of q, p, h) is f n , n = 1, 2, 3, . . . , M. for M time-steps in MD or trials in MC, then the average of f for the simulation is M

fn

. (4) M In the simulation Eq. (4) is the approximation to the phase space average in Eq. (3). If, for example, H = f , then this average gives the thermodynamic energy E = H and the caloric equation of state E = E(T, h, N ). The assumption that Eq. (4) approximates the integral in Eq. (3) is often referred to in the literature by saying that MD or MC “generates the ensemble”. The approximate equality of these two results in MD is the quasi-ergodic hypothesis of statistical mechanics which states that ensemble averages, Eq. (3) and time averages, Eq. (4) are equal. This hypothesis has never been proven f=

n=1

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for realistic Hamiltonians but it is the pillar on which statistical mechanics rests. In what follows we shall assume that averages over simulation-generated configurations are equal to statistical mechanics ensemble averages. Thus, we use formulas from statistical mechanics but calculate the average values in simulations using Eq. (4) employing MD or MC. An important point to note is that for calculation of meaningful averages in a simulation we must “equilibrate” the system before collecting the values f n in Eq. (4). This is done by carrying out the simulation for a “long enough time” and then discarding these configurations and starting the simulation from that point. This removes transient behavior, associated with the particular initial conditions used to start the simulation, from overly influencing the average in Eq. (4). How long one must “equilibrate” the system depends on relaxation rates in the system, that are initially unknown. Tasks like the equilibration of the system, the estimate of the accuracy of calculated values, and so forth are part and parcel of the art of carrying out valid and, therefore, useful simulations and must be learned by actually carrying out simulations. In this aspect computer simulations have a similarity to experimental science, like gaining experience with the measuring apparatus, but, of course, they are theoretical calculations made possible by computers. From our discussion, so far, it might seem, to those who know thermodynamics, that the problem of calculating all response functions is finished, since if the Helmholtz free energy is known from Eq. (2) then all response functions may be calculated by differentiation of the Helmholtz free energy with respect to various variables. For example, the energy H may be found from H = kT 2

∂( A/kT ) . ∂T

(5)

Unfortunately, in MC or MD only average values like Eq. (3), that are ratios of phase space integrals, can be easily evaluated in simulations and not the 6N dimensional phase space integral itself, like Eq. (1). The reason for this is that in high-dimensions (dimensions greater than say, 10) the numerical methods used to accurately calculate integrals (e.g., Simpson’s rule) require computer resources beyond those presently available. For example, in 10 dimensions, for a grid of 100 intervals in each dimension, 1020 variables are required for the grid. Even with the most advanced computer, this number of variables is not easy to handle. In a typical simulation the dimension is typically hundreds or thousands, not ten. One might think that the high dimensional integrals could be calculated directly by MD or MC methods but this also does not work since the integrand in the high dimensional phase space is rapidly varying and one cannot sample for long enough to smooth out this rapid variation. The integral is determined by the value of the integrand in a few pockets (“equilibrium pockets”) in phase space that will only be sampled infrequently. For the ratio of high dimensional integrals, MD or MC methods have the


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effect of focusing the sampling on just those important regions. The difficulty, in high dimensions, of calculating quantities that require the evaluation of an integral as compared to the ratio of integrals leads to a classification of quantities to be calculated by computer simulation as thermal or mechanical properties. Thermal properties require the value of the partition function, or some other high-dimensional integral, for their evaluation whereas mechanical properties do not require the value of the partition function for their evaluation, but are a ratio of two high dimensional integrals. As examples, for the canonical ensemble the Helmholtz free energy is a thermal variable and the energy is a mechanical variable. Other thermal variables are the entropy, chemical potential, and Gibbs free energy. Other mechanical variables are temperature, pressure, enthalpy, thermal expansion coefficient, elastic constants, heat capacity, and so forth. Special methods must be developed for calculating thermal properties and the calculation of thermal properties is, in general, more difficult. We have developed novel methods to calculate thermal variables using different ensembles [15, 16] but shall not discuss them in detail in this contribution. As an example of the calculation of a mechanical response function, consider the fluctuation formula for the heat capacity in the canonical ensemble. Differentiation of the average energy H in Eq. (3) with respect to T while holding the cell vectors rigid leads to the heat capacity at constant shape-size CV CV =

1 ∂H = (H 2 − H 2 ). ∂T kB T 2

(6)

Recall that in the simulation the average values in Eq. (2) are approximated by simple averages of the quantity. Thus, in a single canonical ensemble simulation, MC or MD we may calculate the heat capacity of the system at the given thermodynamic state point by calculating the average value of the square of the energy, subtracting the average value of the energy squared and dividing by kB T 2 . The quantity, δ H 2 = H 2 − H 2 ,

(7)

the variance in probability theory, is called the fluctuation in the energy H. The fluctuation of quantities enters into the formulas for response functions for mechanical variables. It should be noted that a direct way of calculating the heat capacity CV is to calculate the thermal equation of state at a number of temperatures and then numerically differentiate H with respect to T . This requires a series of simulations and is not as convenient or as easy to determine an estimate of accuracy but is simple and is a useful check on the value obtained from the fluctuation formula, Eq. (6). We refer to this method of calculating response functions as the direct method. Any mechanical response function can, in principle, be calculated by the direct method.

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Thermodynamics of Anisotropic Media

For the present we choose the reference state to be the equilibrium state of the system with zero tension applied to the system. The h matrix for this reference state is h o while for an arbitrary state of tension we have h. The following formulation of the thermodynamics of aniostropic media is consistent with nonlinear or finite elasticity theory. In the following repeated indices are summed over. The elastic energy Uel is defined by Uel = Vo Tr(tε),

(8)

where Vo is the reference volume, t is the thermodynamic tension tensor, ε is the strain tensor and Tr implies trace. The h matrix maps the particle coordinates into fractional coordinates, sai , in the unit cube through the relation xai = h ij sa j . The strain of the system relative to the unstressed state is εij = 12 (h oT −1 Gh −1 0 − I )ij ,

(9)

where G = h T h is the metric tensor. Here h o is the reference value for measuring strain, that is, the value of h when the system is unstrained. This value can be obtained by carrying out a (H t and PN) simulation, MD or MC with the tension set to zero. Equation (9) can be derived by noting that the deformation gradient can be written in terms of the h matrices as ∂ xi /∂ xoj = h ik h −1 okj , and using this in the defining relation for the Lagrangian strain of the system. The thermodynamic tension tensor is defined so that the work done in an infintesimal distortion of the system is given by dW = V o Tr(tdε). The stress tensor, σ , is related to the thermodynamic tension by T −1 T σ = Vo hh −1 h / V. o th o

(10)

The thermodynamic law is T d S = dE + Vo Tr(t dε),

(11)

where T is the temperature, S the entropy and E the energy of the particles. Using the definition of the strain, Eq. (9), the thermodynamic law can be recast as T −1 T d S = dE + Vo Tr(h −1 dG)/2. o th o

(12)

From this latter we obtain T −1 (∂ E/∂ G kn ) S = −(Vo h −1 )kn /2. o th o

(13)

In the (EhN) ensemble we have the general relation (∂ E/∂ G kn ) S = ∂ H/∂ G kn ,

(14)


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where H is the particle Hamiltonian and the average is the (EhN) ensemble average. Combining the last two equations leads to T −1 )kn /2. ∂ H/G kn = −(Vo h −1 o th o

(15)

The particle Hamiltonian is transformed by the canonical transformation xai = h ij sa j, pai = h ijT −1 πa j , into H (sa , πa , h) =

N 1 πai G −1 ij πa j /m a + U (r12 , r13 , . . .), 2 a=1

(16)

where the distance between particles a and b is to be replaced by the relation2 = sabi G ij sabj and sabi is the fractional coordinate difference between a ship rab and b. The microscopic stress tensor ij may be obtained by differentiation of the particle Hamiltonian with respect to the h matrix while holding constant (sa , πa ) : ∂ H/∂h ij = ik Akj , where A is the area tensor A=VhT −1 . For the Hamiltonian, Eq. (16), the microscopic stress tensor is 1 ij = V

pai pa j /m a −

a

∂U a R), this is all that can be said from the point of view of aymptotics. Among others, we have established that the motion is described by a well-defined asymptotic solution. Hence after a very short time, details of the microscopic mechanisms leading to coalescence have been “forgotten”. In a numerical simulation, “surgery” done on a sufficiently small scale will meet a similar fate, and one soon ends up following the unique physical solution. Finally, if ν R, there is a region where the motion is almost inviscid. From a balance of surface tension forces with inertial forces at the meniscus one deduces [10] that

rm ∝

γR ρ

1/4

t 1/2 .

(12)

This behavior has been confirmed by recent numerical simulations of Ref. [11]. However, there is an unexpected complication: as the meniscus retracts, capillary waves grow ahead of it, whose amplitude finally equals the width of the channel. Thus the two sides of the drops touch, and a toroidal void is enclosed. This process repeats itself, leaving behind a self-similar succession of voids. In summary, one can often obtain analytical solutions to the equations of motion near a singularity, explaining some universal features of breakup and coalescence events. This is important for estimating errors introduced by a given numerical procedure used to describe topological transitions. Matching numerics to known analytical solutions can lead to considerable savings in numerical effort.

Breakup and coalescence of free surface flows (a)

(b)

0.040

y

0.040

0.030

0.030

0.020

y 0.020

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0.010

0.000 ⫺0.0010

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0.000 ⫺0.0010 ⫺0.0005

0.0010

x

0.0000

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(c)

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⫺0.005

0.000

0.005

0.010

x

Figure 7. A closeup of the point of contact during coalescence of two identical drops for the two cases of no outer fluid, (a), and two fluids of equal viscosity, ((b) and (c)). Part (a) is Hopper’s solution (no outer fluid) for rm /R = 10−3 , 10−2.5 , 10−2 , and 10−1.5 . Part (b) is a numerical simulation of the case where the inner and outer viscosities are the same, showing fluid that collects in a bubble at the meniscus. Note that the two axes are scaled differently, so the bubble is almost circular. For large values of rm , as shown in (c), the fluid finally escapes from the bubble.

References [1] R. Scardovelli and S. Zaleski, “Direct numerical simulation of free-surface and interfacial flow,” Annu. Rev. Fluid Mech., 31, 567–603, 1999. [2] J. Eggers, “Non-linear dynamic and breakup of free-surface flows,” Rev. Mod. Phys., 69, 865–929, 1997. [3] L.D. Landau and E.M. Lifshitz, Fluid Mechanics, Pergamon, Oxford, 1984. [4] A. Menchaca-Rocha et al., “Coalescence of liquid drops by surface tension,” Phys. Rev. E, 63, 046309, 1–5, 2001.

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[5] B. Ambravaneswaran, E.D. Wilkes, and O.A. Basaran, “Drop formation from a capillary tube: comparison of one-dimensional and two-dimensional analyses and occurence of satellite drops,” Phys. Fluids, 14, 2606–2621, 2002. [6] A.U. Chen, P.K. Notz, and O.A. Basaran, “Computational and experimental analysis of pinch-off and scaling,” Phys. Rev. Lett., 88, 174501, 1–4, 2002. [7] G.I. Barenblatt, Scaling, Self-Similarity, and Intermedeate Asymptotics, Cambridge, 1996. [8] S.P. Lin, Breakup of Liquid Sheets and Jets, Cambridge, 2003. [9] Y. Amarouchene, G. Cristobal, and H. Kellay, “Noncoalescing drops,” Phys. Rev. Lett., 87, 206104, 1–4, 2002. [10] J. Eggers, J.R. Lister, and H.A. Stone, “Coalescence of liquid drops,” J. Fluid Mech., 401, 293–310, 1999. [11] L. Duchemin, J. Eggers, and C. Josserand, “Inviscid coalescence of drops,” J. Fluid Mech., 487, 167–178, 2003.

4.10 CONFORMAL MAPPING METHODS FOR INTERFACIAL DYNAMICS Martin Z. Bazant1 and Darren Crowdy2 1

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA 2 Department of Mathematics, Imperial College, London, UK

Microstructural evolution is typically beyond the reach of mathematical analysis, but in two dimensions certain problems become tractable by complex analysis. Via the analogy between the geometry of the plane and the algebra of complex numbers, moving free boundary problems may be elegantly formulated in terms of conformal maps. For over half a century, conformal mapping has been applied to continuous interfacial dynamics, primarily in models of viscous fingering and solidification. Current developments in materials science include models of void electro-migration in metals, brittle fracture, and viscous sintering. Recently, conformal-map dynamics has also been formulated for stochastic problems, such as diffusion-limited aggregation and dielectric breakdown, which has re-invigorated the subject of fractal pattern formation. Although restricted to relatively simple models, conformal-map dynamics offers unique advantages over other numerical methods discussed in this chapter (such as the Level–Set Method) and in Chapter 9 (such as the phase field method). By absorbing all geometrical complexity into a time-dependent conformal map, it is possible to transform a moving free boundary problem to a simple, static domain, such as a circle or square, which obviates the need for front tracking. Conformal mapping also allows the exact representation of very complicated domains, which are not easily discretized, even by the most sophisticated adaptive meshes. Above all, however, conformal mapping offers analytical insights for otherwise intractable problems. After reviewing some elementary concepts from complex analysis in Section 1, we consider the classical application of conformal mapping methods to continuous-time interfacial free boundary problems in Section 2. This includes cases where the governing field equation is harmonic, biharmonic, or in a more general conformally invariant class. In Section 3, we discuss the 1417 S. Yip (ed.), Handbook of Materials Modeling, 1417–1451. c 2005 Springer. Printed in the Netherlands.

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M.Z. Bazant and D. Crowdy

recent use of random, iterated conformal maps to describe analogous discretetime phenomena of fractal growth. Although most of our examples involve planar domains, we note in Section 4 that interfacial dynamics can also be formulated on curved surfaces in terms of more general conformal maps, such as stereographic projections. We conclude in Section 5 with some open questions and an outlook for future research.

1.

Analytic Functions and Conformal Maps

We begin by reviewing some basic concepts from complex analysis found in textbooks such as Churchill and Brown [1]. For a fresh geometrical perspective, see Needham [2]. A general function of a complex variable depends on the real and imaginary parts, x and y, or, equivalently, on the linear combinations, z = x + i y and z¯ = x − i y. In contrast, an analytic function, which is differentiable in some domain, can be written simply as w = u + iv = f (z). The condition, ∂ f /∂ z¯ = 0, is equivalent to the Cauchy–Riemann equations, ∂u ∂v = ∂x ∂y

and

∂u ∂v =− , ∂y ∂x

(1)

which follow from the existence of a unique derivative, f =

∂ f ∂u ∂v ∂f ∂v ∂u = +i = = −i , ∂x ∂x ∂ x ∂(i y) ∂ y ∂y

(2)

whether taken in the real or imaginary direction. Geometrically, analytic functions correspond to special mappings of the complex plane. In the vicinity of any point where the derivative is nonzero, f (z) =/ 0, the mapping is locally linear, dw = f (z) dz. Therefore, an infinitesimal vector, dz, centered at z is transformed into another infinitesimal vector, dw, centered at w = f (z) by a simple complex multiplication. Recalling Euler’s formula, (r1 eiθ1 )(r2 eiθ2 ) = (r1r2 )ei(θ1 + θ2 ) , this means that the mapping causes a local stretch by | f (z)| and local rotation by arg f (z), regardless of the orientation of dz. As a result, an analytic function with a nonzero derivative describes a conformal mapping of the plane, which preserves the angle between any pair of intersecting curves. Intuitively, a conformal mapping smoothly warps one domain into another with no local distortion. Conformal mapping provides a very convenient representation of free boundary problems. The Riemann Mapping Theorem guarantees the existence of a unique conformal mapping between any two simply connected domains, but the challenge is to derive its dynamics for a given problem. The only constraint is that the conformal mapping be univalent, or one-to-one, so that physical fields remain single-valued in the evolving domain.

Conformal mapping methods for interfacial dynamics

2. 2.1.

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Continuous Interfacial Dynamics Harmonic Fields

Most applications of conformal mapping involve harmonic functions, which are solutions to Laplace’s equation, ∇ 2 φ = 0.

(3)

From Eq. (1), it is easy to show that the real and imaginary parts of an analytic function are harmonic, but the converse is also true: Every harmonic function is the real part of an analytic function, φ = Re , the complex potential. This connection easily produces new solutions to Laplace’s equation in different geometries. Suppose that we know the solution, φ(w) = Re (w), in a simply connected domain in the w-plane, w , which can be reached by conformal mapping, w = f (z, t), from another, possibly time-dependent domain in the z-plane, z (t). A solution in z (t) is then given by φ(z, t) = Re (w) = Re ( f (z, t))

(4)

because ( f (z)) is also analytic, with a harmonic real part. The only caveat is that the boundary conditions be invariant under the mapping, which holds for Dirichlet (φ = constant) or Neumann (nˆ · ∇ φ = 0) conditions. Most other boundary conditions invalidate Eq. (4) and thus complicate the analysis. The complex potential is also convenient for calculating the gradient of a harmonic function. Using Eqs. (1) and (2), we have ∇z φ =

∂φ ∂φ +i = , ∂x ∂y

(5)

where ∇z is the complex gradient operator, representing the vector gradient, ∇ , in the z-plane.

2.1.1. Viscous fingering and solidification The classical application of conformal-map dynamics is to Laplacian growth, where a free boundary, Bz (t), moves with a (normal) velocity, dz ∝ ∇ φ, (6) dt proportional to the gradient of a harmonic function, φ, which vanishes on the boundary [3]. Conformal mapping for Laplacian growth was introduced independently by Polubarinova–Kochina and Galin in 1945 in the context of ground-water flow, where φ is the pressure field and u = (k/η)∇ ∇ φ is the velocity of the fluid of viscosity, η, in a porous medium of permeability, k, according v=

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to Darcy’s law. Laplace’s equation follows from incompressibility, ∇ · u = 0. The free boundary represents an interface with a less viscous, immiscible fluid at constant pressure, which is being forced into the more viscous fluid. In physics, Laplacian growth is viewed as a fundamental model for pattern formation. It also describes viscous fingering in Hele–Shaw cells, where a bubble of fluid, such as air, displaces a more viscous fluid, such as oil, in the narrow gap between parallel flat plates. In that case, the depth averaged velocity satisfies Darcy’s law in two dimensions. Laplacian growth also describes dendritic solidification in the limit of low undercooling, where φ is the temperature in the melt [4]. To illustrate the derivation of conformal-map dynamics, let us consider viscous fingering in a channel with impenetrable walls, as shown in Fig. 1(a). The viscous fluid domain, z (t), lies in a periodic horizontal strip, to the right of the free boundary, Bz (t), where uniform flow of velocity, U , is assumed far ahead of the interface. It is convenient to solve for the conformal map, z = g(w, t), to this domain from a half strip, Re w > 0, where the pressure is simply linear, φ = Re Uw/µ. We also switch to dimensionless variables, where length is scaled to a characteristic size of the initial condition, L, pressure to UL/µ, and time to L/U . Since ∇w φ = 1 in the half strip, the pressure gradient at a point, z = g(w, t), on the physical interface is easily obtained from Eq. (30): ∂f = ∇z φ = ∂z

∂g ∂w

−1

(7)

(a)

(b)

3 4

2 1

2

0

0

1

2

2 4 3 4

3

2

1

0

1

2

3

2

0

2

4

6

Figure 1. Exact solutions for Laplacian growth, a simple model of viscous fingering: (a) a Saffman–Taylor finger translating down an infinite channel, showing iso-pressure curves (dashed) and streamlines (solid) in the viscous fluid, and (b) the evolution of a perturbed circular bubble leading to cusp singularities in finite time. (Courtesy of Jaehyuk Choi.)


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where w = f (z, t) is the inverse mapping (which exists as long as the mapping remains univalent). Now consider a Lagrangian marker, z(t), on the interface, whose pre-image, w(t), lies on the imaginary axis in the w-plane. Using the chain rule and Eq. (7), the kinematic condition, Eq. (6), becomes, ∂g dw dz ∂g = + = dt ∂t ∂w dt

∂g ∂w

−1

.

(8)

Multiplying by ∂g/∂w =/ 0, this becomes

∂g ∂g ∂g 2 dw + = 1. ∂w ∂t ∂w dt

(9)

Since the pre-image moves along the imaginary axis, Re(dw/dt) = 0, we arrive at the Polubarinova–Galin equation for the conformal map:

Re

∂g ∂g ∂w ∂t

= 1,

for Re w = 0.

(10)

From the solution to Eq. (10), the pressure is given by φ = Re f (z, t). Note that the interfacial dynamics is nonlinear, even though the quasi-steady equations for φ are linear. The best-known solutions are the Saffman–Taylor fingers, t (11) g(w, t) = + w + 2(1 − λ) log(1 + e−w ) λ which translate at a constant velocity, λ−1 , without changing their shape [5]. Note that (11) is a solution to the fingering problem for all choices of the parameter λ. This parameter specifies the finger width and can be chosen arbitrarily in the solution (11). In experiments however, it is found that the viscous fingers that form are well fit by a Saffman–Taylor finger filling precisely half of the channel, that is with λ = 1/2, as shown in Fig. 1(a). Why this happens is a basic problem in pattern selection, which has been the focus of much debate in the literature over the last 25 years. To understand this problem, note that the viscous finger solutions (11) do not include any of the effects of surface tension on the interface between the two fluids. The intriguing pattern selection of the λ = 1/2 finger has been attributed to a singular perturbation effect of small surface tension. Surface tension, γ , is a significant complication because it is described by a non-conformally-invariant boundary condition, φ = γ κ,

for z ∈ Bz (t)

(12)

where κ is the local interfacial curvature, entering via the Young–Laplace pressure. Small surface tension can be treated analytically as a singular perturbation to gain insights into pattern selection [6, 7]. Since surface tension

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effects are only significant at points of high curvature κ in the interface, and given that the finger in Fig. 1(a) is very smooth with no such points of high curvature, it is surprising that surface tension acts to select the finger width. Indeed, the viscous fingering problem has been shown to be full of surprises [8]. In a radial geometry, the univalent mapping is from the exterior of the unit circle, |w| = 1, to the exterior of a finite bubble penetrating an infinite viscous liquid. Bensimon and Shraiman [9] introduced a pole dynamics formulation, where the map is expressed in terms of its zeros and poles, which must lie inside the unit circle to preserve univalency. They showed that Laplacian growth in this geometry is ill-posed, in the sense that cusp-like singularities occur in finite time (as a zero hits the unit circle) for a broad class of initial conditions, as illustrated in Fig. 1(b). (See Howison [3] for a simple, general proof due to Hohlov.) This initiated a large body of work on how Laplacian growth is “regularized” by surface tension or other effects in real systems. Despite the analytical complications introduced by surface tension, several exact steady solutions with non-zero surface tension are known [10, 11]. Surface tension can also be incorporated into numerical simulations based on the same conformal-mapping formalism [12], which show how cusps are avoided by the formation of new fingers [13]. For example, consider a threefold perturbation of a circular bubble, whose exact dynamics without surface tension is shown in Fig. 1(b). With surface tension included, the evolution is very similar until the cusps begin to form, at which point the tips bulge outward and split into new fingers, as shown in Fig. 2. This process repeats itself to produce a complicated fractal pattern [14], which curiously resembles the diffusion-limited particle aggregates discussed below in Section 3.

2.1.2. Density-driven instabilities in fluids An important class of problems in fluid mechanics involves the nonlinear dynamics of an interface between two immiscible fluids of different densities. In the presence of gravity, there are some familiar cases. Deep-water waves involve finite disturbances (such as steady “Stokes waves”) in the interface between lighter fluid (air) over a heavier fluid (water). With an inverted density gradient, the Rayleigh–Taylor instability develops when a heavier fluid lies above a lighter fluid, leading to large plumes of the former sinking into the latter. Tanveer [15] has used conformal mapping to analyze the Rayleigh– Taylor instability and has provided evidence to associate the formation of plumes with the approach of various conformal mapping singularities to the unit circle. A related problem is the Richtmyer–Meshkov instability, which occurs when a shock wave passes through an interface between fluids of different


1423 4 3 2 1 0 ⫺1 ⫺2 ⫺3

4

3

2

1

0

⫺1

⫺2

⫺3

⫺4

⫺4

Figure 2. Numerical simulation of viscous fingering, starting from a three-fold perturbation of a circular bubble. The only difference with the Laplacian-growth dynamics in Fig. 1(b) is the inclusion of surface tension, which prevents the formation of cusp singularities. (Courtesy of Michael Siegel.)

densities. Behind the shock, narrow fingers of the heavier fluid penetrate into the lighter fluid. The shock wave usually passes so quickly that compressibility only affects the onset of the instability, while the nonlinear evolution occurs much faster than the development of viscous effects. Therefore, it is reasonable to assume a potential flow in each fluid region, with randomly perturbed initial velocities. Although real plumes roll up in three dimensions and bubbles can form, conformal mapping in two dimensions still provides some insights, with direct relevance for shock tubes of high aspect ratio. A simple conformal-mapping analysis is possible for the case of a large density contrast, where the lighter fluid is assumed to be at uniform pressure. The Richtmyer–Meshkov instability (zero-gravity limit) is then similar to the Saffman–Taylor instability, except that the total volume of each fluid is fixed. A periodic interface in the y direction, analogous to the channel geometry in Fig. 1, can be described by the univalent mapping, z = g(w, t), from the

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interior of the unit circle in the mathematical w plane to the interior of the heavy-fluid finger in the physical z-plane. Zakharov [16] introduced a Hamiltonian formulation of the interfacial dynamics in terms of this conformal map, taking into account kinetic and potential energy, but not surface tension. One way to derive equations of motion is to expand the map in a Taylor series, g(w, t) = log w +

∞

an (t)w n ,

|w| < 1.

(13)

n=0

(The log w term first maps the disk to a periodic half strip.) On the unit circle, w = eiθ , the pre-image of the interface, this is simply a complex Fourier series. The Taylor coefficients, an (t), act as generalized coordinates describing n-fold shape perturbations within each period, and their time derivatives, a˙ n (t), act as velocities or momenta. Unfortunately, truncating the Taylor series results in a poor description of strongly nonlinear dynamics because the conformal map begins to vary wildly near the unit circle. An alternate approach used by Yoshikawa and Balk [17] is to expand in terms resembling Saffman–Taylor fingers, g(w, t) = log w + b(t) −

N

bn (t) log(1 − λn (t)w),

(14)

n=1

which can be viewed as a re-summation of the Taylor series in Eq. (13). As shown in Fig. 3, exact solutions exist with only a finite number of terms in the finger expansion, as long as the new generalized coordinates, λn (t), stay inside the unit disk, |λn | < 1. This example illustrates the importance of the choice of shape functions in the expansion of the conformal map, e.g., w n vs. log(1 − λn w).

2.1.3. Void electro-migration in metals Micron-scale interconnects in modern integrated circuits, typically made of aluminum, sustain enormous currents and high temperatures. The intense electron wind drives solid-state mass diffusion, especially along dislocations and grain boundaries, where voids also nucleate and grow. In the narrowest and most fragile interconnects, grain boundaries are often well separated enough that isolated voids migrate in a fairly homogeneous environment due to surface diffusion, driven by the electron wind. Voids tend to deform into slits, which propagate across the interconnect, causing it to sever. A theory of void electro-migration is thus important for predicting reliability. In the simplest two-dimensional model [18], a cylindrical void is modeled as a deformable, insulating inclusion in a conducting matrix. Outside the void,


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2 1 0

⫺1 ⫺2 ⫺3 ⫺4 ⫺5 ⫺6 ⫺7 ⫺6

⫺4

⫺2

0

2

4

6

Figure 3. Conformal-map dynamics for the strongly nonlinear regime of the RichtmyerMeshkov instability [17]. (Courtesy of Toshio Yoshikawa and Alexander Balk.)

the electrostatic potential, φ, satisfies Laplace’s equation, which invites the use of conformal mapping. The electric field, E = −∇ ∇ φ, is taken to be uniform far away and wraps around the void surface, due to a Neumann boundary condition, nˆ · E = 0. The difference with Laplacian growth lies in the kinematic condition, which is considerably more complicated. In place of Eq. (6), the normal velocity of the void surface is given by the surface divergence of the surface current, j , which takes the dimensionless form, nˆ · v =

∂ 2φ ∂ 2κ ∂j =χ 2 + 2, ∂s ∂s ∂s

(15)

where s is the local arc-length coordinate and χ is a dimensionless parameter comparing surface currents due to the electron wind force (first term) and due to gradients in surface tension (second term). This moving free boundary problem somewhat resembles the viscous fingering problem with surface tension, and it admits analogous finger solutions, albeit of width 2/3, not 1/2 [19]. To describe the evolution of a singly connected void, we consider the conformal map, z = g(w, t), from the exterior of the unit circle to the exterior of

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the void. As long as the map remains univalent, it has a Laurent series of the form, g(w, t) = A1 (t)w + A0 (t) +

∞

A−n (t)w −n ,

for |w| > 1,

(16)

n=1

where the Laurent coefficients, An (t), are now the generalized coordinates. As in the case of viscous fingering [3], a hierarchy of nonlinear ordinary differential equations (ODEs) for these coordinates can be derived. For void electromigration, Wang et al. [18] start from a variational principle accounting for surface tension and surface diffusion, using a Galerkin procedure. They truncate the expansion after 17 coefficients, so their numerical method breaks down if the void deforms significantly, e.g., into a curved slit. Nevertheless, as shown in Fig. 4(a), the numerical method is able to capture essential features of the early stages of strongly nonlinear dynamics. In the same regime, it is also possible to incorporate anisotropic surface tension or surface mobility. The latter involves multiplying the surface current by a factor (1 + gd cos mα), where α is the surface orientation in the physical z-plane, given at z = g(eiθ , t), by α = θ + arg

∂g iθ (e , t). ∂w

(17)

Some results are shown in Fig. 4(b), where the void develops dynamical facets.

(a)

(b)

Figure 4. Numerical conformal-mapping simulations of the electromigration of voids in aluminum interconnects [18]. (a) A small shape perturbation of a cylindrical void decaying (above) or deforming into a curved slit (below), depending on a dimensionless group, χ, comparing the electron wind to surface-tension gradients. (b) A void evolving with anisotropic surface diffusivity (χ = 100, gd = 100, m = 3). (Courtesy of Zhigang Suo.)


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2.1.4. Quadrature domains We end this section by commenting on some of the mathematics underlying the existence of exact solutions to continuous-time Laplacian-growth problems. Significantly, much of this mathematics carries over to problems in which the governing field equation is not necessarily harmonic, as will be seen in the following section. The steadily-translating finger solution (11) of Saffman and Taylor turns out to be but one of an increasingly large number of known exact solutions to the standard Hele–Shaw problem. Saffman [20] himself identified a class of unsteady finger-like solutions. This solution was later generalized by Howison [21] to solutions involving multiple fingers exhibiting such phenomena as tip-splitting where a single finger splits into two (or more) fingers. It is even possible to find exact solutions to the more realistic case where there is a second interface further down the channel [22] which must always be the case in any experiment. Besides finger-like solutions which are characterized by time-evolving conformal mappings having logarithmic branch-point singularities, other exact solutions, where the conformal mappings are rational functions with timeevolving poles and zeros, were first identified by Polubarinova–Kochina and Galin in 1945. Richardson [23] later rediscovered the latter solutions while simultaneously presenting important new theoretical connections between the Hele–Shaw problem and a class of planar domains known as quadrature domains. The simplest example of a quadrature domain is a circular disc D of radius r centered at the origin which satisfies the identity

h(z) dx dy = πr 2 h(0),

(18)

D

where h(z) is any function analytic in the disc (and integrable over it). Equation (18), which is known as a quadrature identity since it holds for any analytic function h(z), is simply a statement of the well-known mean-value theorem of complex analysis [24]. A more general domain D, satisfying a generalized quadrature identity of the form

h(z) dx dy = D

N n k −1

c j k h ( j )(z k )

(19)

k=1 j =0

is known as a quadrature domain. Here, {z k ∈ C} is a set of points inside D and h ( j )(z) denotes the j th derivative of h(z). If one makes the choice h(z) = z n in (19) the resulting integral quantities have become known as the Richardson moments of the domain. Richardson showed that the dynamics of the Hele–Shaw problem is such as to preserve quadrature domains. That is, if the initial fluid domain in a Hele–Shaw cell is a quadrature domain at time

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t = 0, it remains a quadrature domain at later times (so long as the solution does not break down). This result is highly significant and provides a link with many other areas of mathematics including potential theory, the notion of balayage, algebraic curves, Schwarz functions and Cauchy transforms. Richardson [25] discusses many of these connections while Varchenko and Etingof [26] provide a more general overview of the various mathematical implications of Richardson’s result. Shapiro [27] gives more general background on quadrature domain theory. It is a well-known result in the theory of quadrature domains [27] that simply-connected quadrature domains can be parameterized by rational function conformal mappings from a unit circle. Given Richardson’s result on the preservation of quadrature domains, this explains why Polubarinova–Kochina and Galin were able to find time-evolving rational function conformal mapping solutions to the Hele–Shaw problem. It also underlies the pole dynamics results of Bensimon and Shraiman [9]. But Richardson’s result is not restricted to simply-connected domains; multiply-connected quadrature domains are also preserved by the dynamics. Physically this corresponds to time-evolving fluid domains containing multiple bubbles of air. Indeed, motivated by such matters, recent research has focused on the analytical construction of multiplyconnected quadrature domains using conformal mapping ideas [28, 29]. In the higher-connected case, the conformal mappings are no longer simply rational functions but are given by conformal maps that are automorphic functions (or, meromorphic functions on compact Riemann surfaces). The important point here is that understanding the physical problem from the more general perspective of quadrature domain theory has led the way to the unveiling of more sophisticated classes of exact conformal mapping solutions.

2.2.

Bi-Harmonic Fields

Although not as well known as conformal mapping involving harmonic functions, there is also a substantial literature on complex-variable methods to solve the bi-harmonic equation, ∇ 2 ∇ 2 ψ = 0,

(20)

which arises in two-dimensional elasticity [30] and fluid mechanics [31]. Unlike harmonic functions, which can be expressed in terms of a single analytic function (the complex potential), bi-harmonic functions can be expressed in terms of two analytic functions, f (z) and g(z), in Goursat form [24]: ψ(z, z¯ ) = Im [¯z f (z) + g(z)].

(21)

Note that ψ is no longer just the imaginary part of an analytic function g(z) but also contains the imaginary part of the non-analytic component z¯ f (z).


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A difficulty with bi-harmonic problems is that the typical boundary conditions (see below) are not conformally invariant, so conformal mapping does not usually generate new solutions by simply a change of variables, as in Eq. (4). Nevertheless, the Goursat form of the solution, Eq. (21), is a major simplification, which enables analytical progress.

2.1.5. Viscous sintering Sintering describes a process by which a granular compact of particles (e.g., metal or glass) is raised to a sufficiently large temperature that the individual particles become mobile and release surface energy in such a way as to produce inter-particulate bonds. At the start of a sinter process, any two particles which are initially touching develop a thin “neck” which, as time evolves, grows in size to form a more developed bond. In compacts in which the packing is such that particles have more than one touching neighbor, as the necks grow in size, the sinter body densifies and any enclosed pores between particles tend to close up. The macroscopic material properties of the compact at the end of the sinter process depend heavily on the degree of densification. In industrial application, it is crucial to be able to obtain accurate and reliable estimates of the time taken for pores to close (or reduce to a sufficiently small size) within any given initial sinter body in order that industrial sinter times are optimized without compromising the macroscopic properties of the final densified sinter body. The fluid is modeled as a region D(t) of very viscous, incompressible fluid, in which the velocity field, u = (u, v) = (ψ y , −ψx )

(22)

is given by the curl of an out-of-plane vector, whose magnitude is a stream function, ψ(x, y, t), which satisfies the bi-harmonic equation [31]. On the boundary ∂ D(t), the tangential stress must vanish and the normal stress must be balanced by the uniform surface tension effect, i.e., − pn i + 2µei j = T κn i ,

(23)

where p is the fluid pressure, µ is the viscosity, T is the surface tension parameter, κ is the boundary curvature, n i denotes components of the outward normal n to ∂ D(t) and ei j is the usual fluid rate-of-strain tensor. The boundary is time-evolved in a quasi-steady fashion with a normal velocity, Vn , determined by the same kinematic condition, Vn = u · n, as in viscous fingering. In terms of the Goursat functions in (21) – which are now generally time-evolving – the stress condition (23) takes the form i f (z, t) + z f (¯z , t) + g (¯z , t) = − z s , 2

(24)

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where again s denotes arc length. Once f (z, t) has been determined from (24), the kinematic condition Im[z t z¯ s ] = Im[−2 f (z, t)¯z s ] −

1 2

(25)

is used to time-advance the interface. A significant contribution was made by Hopper [32] who showed, using complex variable methods based on the decomposition (21), that the problem for the surface-tension driven coalescence of two equal circular blobs of viscous fluid can be reduced to the evolution of a rational function conformal map, from a unit w-circle, of the form g(w, t) =

R(t)w . w 2 − a 2 (t)

(26)

The two time-evolving parameters R(t) and a(t) satisfy two coupled nonlinear ODEs. Figure 5 shows a sequence of shapes of the two coalescing blobs computed using Hopper’s solution. At large times, the configuration equilibrates to a single circular blob. While Hopper’s coalescence problem provides insight into the growth of the inter-particle neck region, there are no pores in this configuration and it is natural to ask whether more general exact solutions exist. Crowdy [33] reappraised the viscous sintering problem and showed, in direct analogy with Richardson’s result on Hele–Shaw flows, that the dynamics of the sintering problem is also such as to preserve quadrature domains. As in the Hele– Shaw problem, this perspective paved the way for the identification of new exact solutions, generalizing (26), for the evolution of doubly-connected fluid regions. Figure 6 shows the shrinkage of a pore enclosed by a typical “unit” in a doubly-connected square packing of touching near-circular blobs of viscous fluid. This calculation employs a conformal mapping to the doubly-connected fluid region (which is no longer a rational function but a more general automorphic function) derived by Crowdy [34] and, in the same spirit as Hopper’s solution (26), requires only the integration of three coupled nonlinear ODEs. The fluid regions shown in Fig. 6 are all doubly-connected quadrature domains. Richardson [35] has also considered similar Stokes flow problems using a different conformal mapping approach.

Figure 5. Evolution of the solution of Hopper [32] for the coalescence of two equal blobs of fluid under the effects of surface tension.


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Figure 6. The coalescence of fluid blobs and collapse of cylindrical pores in a model of viscous sintering. This sequence of images shows an analytical solution by Crowdy [34] using complex-variable methods.

2.1.6. Pores in elastic solids Solid elasticity in two dimensions is also governed by a bi-harmonic function, the Airy stress function [30]. Therefore, the stress tensor, σi j , and the displacement field, u i , may be expressed in terms of two analytic functions, f (z) and g(z): σ22 + σ11 = f (z) + f (z), 2 σ22 − σ11 + iσ12 = z f (z) + g (z), 2 Y (u 1 + iu 2 ) = κ f (z) − z f (z) − g(z) 1+ν

(27) (28) (29)

where Y is Young’s modulus, ν is Poisson’s ratio, and κ = (3 − ν)/(1 + ν) for plane stress and κ = 3 − 4ν for plane strain. As with bubbles in viscous flow, the use of Goursat functions allows conformal mapping to be applied to bi-harmonic free boundary problems in elastic solids, without solving explicitly for bulk stresses and strains. For example, Wang and Suo [36] have simulated the dynamics of a singlyconnected pore by surface diffusion in an infinite stressed elastic solid. As in the case of void electromigration described above, they solve nonlinear ODEs for the Laurent coefficients of the conformal map from the exterior of the unit disk, Eq. (16). Under uniaxial tension, there is a competition between surface tension, which prefers a circular shape, and the applied stress, which drives elongation and eventually fracture in the transverse direction. The numerical

1432


method, based on the truncated Laurent expansion, is able to capture the transition from stable elliptical shapes at small applied stress to the unstable growth of transverse protrusions at large applied stress, although naturally it breaks down when cusps resembling crack tips begin to form.

2.3.

Non-Harmonic Conformally Invariant Fields

The vast majority of applications of conformal mapping fall into one of the two classes above, involving harmonic or bi-harmonic functions, where the connections with analytic functions, Eqs. (4) and (21), are cleverly exploited. It turns out, however, that conformal mapping can be applied just as easily to a broad class of problems involving non-harmonic fields, recently discovered by Bazant [37]. Of course, in planar geometry, the conformal map itself is described by an analytic function, but the fields need not be, as long as they transform in a simple way under conformal mapping. The most convenient fields satisfy conformally invariant partial differential equations (PDEs), whose forms are unaltered by a conformal change of variables. It is straightforward to transform PDEs under a conformal mapping of the plane, w = f (z), by expressing them in terms of complex gradient operator introduced above, ∇z =

∂ ∂ ∂ +i =2 , ∂x ∂y ∂z

(30)

which we have related to the z partial derivative using the Cauchy–Riemann equations, Eq. (1). In this form, it is clear that ∇z f = 0 if and only if f (z) is analytic, in which case ∇ z f = 2 f . Using the chain rule, also obtain the transformation rule for the gradient, ∇ z = f ∇w .

(31)

To apply this formalism, we write Laplace’s equation in the form, ∇ z2 φ = Re ∇z ∇ z φ = ∇z ∇ z φ = 0,

(32)

which assumes that mixed partial derivatives can be taken in either order. (Note that a · b = Re ab.) The conformal invariance of Laplace’s equation, ∇w ∇ w φ = 0, then follows from a simple calculation, ∇z ∇ z = (∇z f )∇ w + | f |2 ∇w ∇ w = | f |2 ∇w ∇ w ,

(33)

where ∇z f = 0 because f is also analytic. As a result of conformal invariance, any harmonic function in the w-plane, φ(w), remains harmonic in the


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z-plane, φ( f (z)), after the simple substitution, w = f (z). We came to the same conclusion above in Eq. (4), using the connection between harmonic and analytic functions, but the argument here is more general and also applies to other PDEs. The bi-harmonic equation is not conformally invariant, but some other equations – and systems of equations – are. The key observation is that any “product of two gradients” transforms in the same way under conformal mapping, not only the Laplacian, ∇ · ∇ φ, but also the term, ∇ φ1 · ∇ φ2 = Re(∇φ1 )∇φ2 , which involves two real functions, φ1 and φ2 : Re(∇z φ1 ) ∇ z φ2 = | f |2 Re(∇w φ1 ) ∇ w φ2 .

(34)

(Todd Squires has since noted that the operator, ∇ φ1 × ∇ φ2 = Im(∇φ1 )∇φ2 , also transforms in the same way.) These observations imply the conformal invariance of a broad class of systems of nonlinear PDEs: N

  ai ∇ 2 φi +

N

i =1

j =i

ai j ∇ φi · ∇ φ j +

N



bi j ∇ φi × ∇ φ j  = 0,

(35)

j = i+1

where the coefficients ai (φ), ai j (φ), and bi j (φ) may be nonlinear functions of the unknowns, φ = (φ1 , φ2 , . . . , φ N ), but not of the independent variables or any derivatives of the unknowns. The general solutions to these equations are not harmonic and thus depend on both z and z. Nevertheless, conformal mapping works in precisely the same way: A solution, φ(w, w), can be mapped to another solution, φ( f (z), f (z)), by a simple substitution, w = f (z). This allows the conformal mapping techniques above (and below) to be extended to new kinds of moving free boundary problems.

2.1.7. Transport-limited growth phenomena For physical applications, the conformally invariant class, Eq. (35), includes the important set of steady conservation laws for gradient-driven flux densities, ∂ci = ∇ · Fi = 0, ∂t

Fi = ci ui − Di (ci ) ∇ ci ,

ui ∝ ∇ φ,

(36)

where {ci } are scalar fields, such as chemical concentrations or temperature, {Di (ci )} are nonlinear diffusivities, {ui } are irrotational vector fields causing advection, and φ is a potential [37]. Physical examples include advectiondiffusion, where φ is the harmonic velocity potential, and electrochemical transport, where φ is the non-harmonic electrostatic potential, determined implicitly by electro-neutrality.

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By modifying the classical methods described above for Laplacian growth, conformal-map dynamics can thus be formulated for more general, transportlimited growth phenomena [38]. The natural generalization of the kinematic condition, Eq. (6), is that the free boundary moves in proportion to one of the gradient-driven fluxes with velocity, v ∝ F1 . For the growth of a finite filament, driven by prescribed fluxes and/or concentrations at infinity, one obtains a generalization of the Polubarinova–Galin equation for the conformal map, z = g(w, t), from the exterior of the unit disk to the exterior of growing object, Re(w g gt ) = σ (w, t) on |w| = 1,

(37)

where σ (w, t) is the non-constant, time-dependent normal flux, nˆ · F1 , on the unit circle in the mathematical plane.

2.1.8. Solidification in a fluid flow A special case of the conformally invariant Eq. (35) has been known for almost a century: steady advection-diffusion of a scalar field, c, in a potential flow, u. The dimensionless PDEs are Pe u · ∇ c = ∇ 2 c,

u = ∇ φ,

∇ 2 φ = 0,

(38)

where we have introduced the Péclet number, Pe = UL/D, in terms of a characteristic length, L, velocity, U , and diffusivity, D. In 1905, Boussinesq showed that Eq. (38) takes a simpler form in streamline coordinates, (φ, ψ), where = φ + iψ is the complex velocity potential: ∂c = Pe ∂φ

∂ 2c ∂ 2c + ∂φ 2 ∂ψ 2

(39)

because advection (the left hand side) is directed only along streamlines, while diffusion (the right hand side) also occurs in the transverse direction, along isopotential lines. From the general perspective above, we recognize this as the conformal mapping of an invariant system of PDEs of the form (36) to the complex plane, where the flow is uniform and any obstacles in the flow are mapped to horizontal slits. Streamline coordinates form the basis for Maksimov’s method for interfacial growth by steady advection-diffusion in a background potential flow, which has been applied to freezing in ground-water flow and vapor deposition on textile fibers [4, 39]. The growing interface is a streamline held at a fixed concentration (or temperature) relative to the flowing bulk fluid at infinity. This is arguably the simplest growth model with two competing transport processes, and yet open questions remain about the nonlinear dynamics, even without surface tension.


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Figure 7. The exact self-similar solution, Eq. (40), for continuous advection-diffusion-limited growth in a uniform background potential flow (yellow streamlines) at the dynamical fixed point (Pe = ∞). The concentration field (color contour plot) is shown for Pe = 100. (Courtesy of Jaehyuk Choi.)

The normal flux distribution to a finite absorber in a uniform background flow, σ (w, t) in Eq. (37) is well known, but rather complicated [40], so it is replaced by asymptotic approximations for analytical work, such as √ σ ∼ 2 Pe/π sin(θ/2) as Pe → ∞, which is the fixed point of the dynamics. In this important limit, Choi et al. [41] have found an exact similarity solution,

g(w, t) = A1 (t) w(w − 1), A1 (t) = t 2/3 (40) √ iθ to Eq. (37) with σ (e , t) = A1 (t) sin(θ/2) (since Pe(t) ∝ A1 (t) for a fixed background flow). As shown in Fig. 7, this corresponds to a constant shape, 2/3 ◦ whose linear size grows like √ t , with a 90 cusp at the rear stagnation point, where a branch point of w(w − 1) lies on the unit circle. For any finite, Pe(t), however, the cusp is smoothed, and the map remains univalent, although other singularities may form. Curiously, when mapped to the channel geometry with log z, the solution (40) becomes a Saffman–Taylor finger of width, λ = 3/4.

3.

Stochastic Interfacial Dynamics

The continuous dynamics of conformal maps is a mature subject, but much attention is now focusing on analogous problems with discrete, stochastic dynamics. The essential change is in the kinematic condition: The expression for the interfacial velocity, e.g., Eq. (6), is re-interpreted as the probability

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density (per unit arc length) for advancing the interface with a discrete “bump”, e.g., to model a depositing particle. Continuous conformal-map dynamics is then replaced by rules for constructing and composing the bumps. This method of iterated conformal maps was introduced by Hastings and Levitov [42] in the context of Laplacian growth. Stochastic Laplacian growth has been discussed since the early 1980s, but Hastings and Levitov [42] first showed how to implement it with conformal mapping. They proposed the following family of bump functions,

f λ,θ (w) = eiθ f λ e−iθ w , |w| ≥ 1 

f λ (w) = w 1−a 





(41)

a

1−λ   (1 + λ)(w + 1)  w+1+ w 2 +1−2w −1 2w 1+λ

(42) as elementary univalent mappings of the exterior of the unit disk used to advance the interface (0 < a ≤ 1). The function, f λ,θ (w), places a bump of (approximate) area, λ, on the unit circle, centered at angle, θ. Compared to analytic functions of the unit disk, the Hastings–Levitov function (42) generates a much more localized perturbation, focused on the region between two branch points, leaving the rest of the unit circle unaltered √ [43]. For a = 1, the map produces a strike, which is a line segment of length λ emanating normal to the circle. For a = 1/2, the map is an invertible composition of simple linear, Möbius and Joukowski transformations, which inserts a semi-circular bump on the unit circle. As shown in Fig. 8, this yields a good description of (a)

(b) 4

400

2

200

0

0

⫺2

⫺200

⫺4

⫺400 ⫺4

⫺2

0

2

4

⫺400

⫺200

0

200

400

Figure 8. Simulation of the aggregation of (a) 4 and (b) 10 000 particles using the Hastings– Levitov algorithm (a = 1/2). Color contours show the quasi-steady concentration (or probability) field for mobile particles arriving from infinity, and purple curves indicate lines of diffusive flux (or probability current). (Courtesy of Jaehyuk Choi and Benny Davidovitch.)


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aggregating particles, although other choices, like a = 2/3, have also been considered [43]. Quantifying the effect of the bump shape remains a basic open question. Once the bump function is chosen, the conformal map, z = gn (w), from the exterior of the unit disk to the evolving domain with n bumps is constructed by iteration,

gn (w) = gn−1 f λn ,θn (w)

(43)

starting from the initial interface, given by g0 (w). All of the physics is contained in the sequence of bump parameters, {(λn , θn )}, which can be generated in different ways (in the w plane) to model a variety of physical processes (in the z-plane). As shown in Fig. 8(b), the interface often develops a very complicated, fractal structure, which is given, quite remarkably, by an exact mapping of the unit circle. The great advantage of stochastic conformal mapping over atomistic or Monte Carlo simulation of interfacial growth lies in its mathematical insight. For example, given the sequence {(λn , θn )} from a simulation of some physical growth process, the Laurent coefficients, Ak (n), of the conformal map, gn (w), as defined in Eq. (16), can be calculated analytically. For the bump function (42), Davidovitch et al. [43] provide a hierarchy of recursion relations, yielding formulae such as A1 (n) =

n

(1 + λm )a ,

(44)

m=1

and explain how to interpret the Laurent coefficients. For example, A1 is the conformal radius of the cluster, a convenient measure of its linear extent. It is also the radius of a grounded disk with the same capacitance (with respect to infinity) as the cluster. The Koebe “1/4 theorem” on univalent functions [44] ensures that the cluster (image of the unit disk) is always contained in a disk of radius 4A1 . The next Laurent coefficient, A0 , is the center of a uniformly charged disk, which would have the same asymptotic electric field as the cluster (if also charged). Similarly, higher Laurent coefficients encode higher multipole moments of the cluster. Mapping the unit circle with a truncated Laurent expansion defines the web, which wraps around the growing tips and exhibits a sort of turbulent dynamics, endlessly forming and smoothing cusp-like protrusions [42, 45]. The stochastic dynamics, however, does not suffer from finite-time singularities because the iterated map, by construction, remains univalent. In some sense, discreteness plays the role of surface tension, as another regularization of ill-posed continuum models like Laplacian growth.

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3.1.


Diffusion-Limited Aggregation (DLA)

The stochastic analog of Laplacian growth is the DLA model of Witten and Sander [46], illustrated in Fig. 8, in which particles perform random walks one-by-one from infinity until they stick irreversibly to a cluster, which grows from a seed at the origin. DLA and its variants (see below) provide simple models for many fractal patterns in nature, such as colloidal aggregates, dendritic electro-deposits, snowflakes, lightning strikes, mineral deposits, and surface patterns in ion-beam microscopy [14]. In spite of decades of research, however, DLA still presents theoretical mysteries, which are just beginning to unravel [47]. The Hastings–Levitov algorithm for DLA prescribes the bump parameters, {(λn , θn )}, as follows. As in Laplacian growth, the harmonic function for the concentration (or probability density) of the next random walker approaching an n-particle cluster is simply, φn (z) = A Re log gn−1 (z),

(45)

according to Eq. (4), since φ(w) = A Re log w = A log|w| is the (trivial) solution to Laplace’s equation in the mathematical w plane with φ = 0 on the unit disk with a circularly symmetric flux density, A, prescribed at infinity. Using the transformation rule, Eq. (31), we then find that the evolving harmonic measure, pn (z)|dz|, for the nth growth event corresponds to a uniform probability measure, Pn (θ) dθ, for angles, θn , on the unit circle, w = eiθ : ∇ φ dθ w pn (z)|dz| = |∇z φ||dz| = |gn−1 dw| = |∇w φ||dw| = = Pn (θ) dθ, g 2π n−1

(46) where we set A = 1/2π for normalization, which implicitly sets the time scale. The conformal invariance of the harmonic measure is well known in mathematics, but the surprising result of Hastings and Levitov [42] is that all the complexity of DLA is slaved to a sequence of independent, uniform random variables. Where the complexity resides is in the bump area, λn , which depends nontrivially on current cluster geometry and thus on the entire history of random angles, {θm | m ≤ n}. For DLA, the bump area in the mathematical w plane should be chosen such that it has a fixed value, λ0 , in the physical z-plane, equal to the aggregating particle area. As long as the new bump is sufficiently small, it is natural to try to correct only for the Jacobian factor Jn (w) = |gn (w)|2

(47)


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of the previous conformal map at the center of the new bump, λn =

λ0 , Jn−1 (eiθn )

(48)

although it is not clear a priori that such a local approximation is valid. Note at least that gn → ∞, and thus λn → 0, as the cluster grows, so this has a chance of working. Numerical simulations with the Hastings–Levitov algorithm do indeed produce nearly constant bump areas, as in Fig. 8. Nevertheless, much larger “particles”, which fill deep fjords in the cluster, occasionally occur where the map varies too wildly, as shown in Fig. 9(a). It is possible (but somewhat unsatisfying) to reject particles outside an “area acceptance window” to produce rather realistic DLA clusters, as shown in Fig. 9(b). It seems that the rejected large bumps are so rare that they do not much influence statistical scaling properties of the clusters [48], although this issue is by no means rigorously resolved.

3.2.

Fractal Geometry

Fractal patterns abound in nature, and DLA provides the most common way to understand them [14]. The fractal scaling of DLA has been debated for decades, but conformal dynamics is shedding new light on the problem. Simulations show that the conformal radius (44) exhibits fractal scaling, A1 (n) ∝ n 1/D f , where the fractal dimension, D f = 1.71, agrees with the accepted value from Monte Carlo (random walk) simulations of DLA, although the prefactor seems to depend on the bump function [43]. A perturbative renormalizationgroup analysis of the conformal dynamics by Hastings [45] gives a similar result, D f = 2 − 1/2 + 1/5 = 1.7. The multifractal spectrum of the harmonic measure has also been studied [49, 50]. Perhaps the most basic question is whether DLA clusters are truly fractal – statistically self-similar and free of any length scale. This long-standing question requires accurate statistics and very large simulations, to erase the surprisingly long memory of the initial conditions. Conformal dynamics provides exact formulae for cluster moments, but simulations are limited to at most 105 particles by poor O(n 2 ) scaling, caused by the history-dependent Jacobian in Eq. (48). In contrast, efficient random-walk simulations can aggregate many millions of particles. Therefore, Somfai et al. [51] developed a hybrid method relying only upon the existence of the conformal map, but not the Hastings–Levitov algorithm to construct it. Large clusters by Monte Carlo simulation, and approximate Laurent coefficients are computed, purely for their morphological information, as follows. For a given cluster of size N , M random walkers are launched

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(a)

(b)

(c)

(d)

(e)

(f)

Figure 9. Simulations of fractal aggregates by Stepanov and Levitov [48]: (a) Superimposed time series of the boundary, showing the aggregation of particles, represented by iterated conformal maps; (b) a larger simulation with a particle-area acceptance window; (c) the result of anisotropic growth probability with square symmetry; (d) square-anisotropic growth with noise control via flat particles; (e) triangular-anisotropic growth with noise control; (f) isotropic growth with noise control, which resembles radial viscous fingering. (Courtesy of Leonid Levitov.)

from far away, and the positions, z m , where they would first touch the cluster, are recorded. If the conformal map, z = gn (eiθ ), were known, the points z m would correspond to M angles θm on the unit circle. Since these must sample a uniform distribution, one assumes θm = 2π m/M for large M. From Eq. (16),


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the Laurent coefficientsare simply the Fourier coefficients of the discretely sampled function, z m = Ak eiθm k . Using this method, all Laurent coefficients appear to scale with the same fractal dimension,

|Ak (n)|2 ∝ n 2/D f

(49)

although the first few coefficients crossover extremely slowly to the asymptotic scaling.

3.3.

Snowflakes and Viscous Fingers

In conventional Monte Carlo simulations, many variants of DLA have been proposed to model real patterns found in nature [14]. For example, clusters closely resembling snowflakes can be grown by a combination of noise control (requiring multiple hits before attachment) and anisotropy (on a lattice). Conformal dynamics offers the same flexibility, as shown in Fig. 9, while allowing anisotropy and noise to be controlled independently [48]. Anisotropy can be introduced in the growth probability with a weight factor, 1 + c cos mαn , where αn is the surface orientation angle in the physical plane given by Eq. (17), or by simply rejecting angles outside some tolerance from the desired symmetry directions. Noise can be controlled by flattening the aspect ratio of the bumps. Without anisotropy, this produces smooth fluid-like patterns (Fig. 9(f)), reminiscent of viscous fingers (Fig. 2). The possible relation between DLA and viscous fingering is a tantalizing open question in pattern formation. Many authors have argued that the regularization of finite-time singularities in Laplacian growth by discreteness is somehow analogous to surface tension. Indeed, the average DLA cluster in a channel, grown by conformal mapping, is similar (but not identical) to a Saffman–Taylor finger of width 1/2 [52], and the instantaneous expected growth rate of a cluster can be related to the Polubarinova–Galin (or “Shraiman– Bensimon”) equation [42]. Conformal dynamics with many bumps grown simultaneously suggests that Laplacian growth and DLA are in different universality classes, due to the basic difference of layer-by-layer vs. one-byone growth, respectively [53]. Another multiple-bump algorithm with complete surface coverage, however, seems to yield the opposite conclusion [54].

3.4.

Dielectric Breakdown

In their original paper, Hastings and Levitov [42] allowed for the size of the bump in the physical plane to vary with an exponent, α, by replacing Jn−1

1442


with ( Jn−1 )α/2 in Eq. (48). In DLA (α = 2), the bump size is roughly constant, but for 0 < α < 2 the bump size grows with the local gradient of the Laplacian field. This is a simple model for dielectric breakdown, where the stochastic growth of an electric discharge penetrating a material is nonlinearly enhanced by the local electric field. One could use strikes (a = 0) rather than bumps (a = 1/2) to better reproduce the string-like branched patterns seen in laboratory experiments [14] and more familiar lightning strikes. The model displays a “stable-to-turbulent” phase transition: The relative surface roughness decreases with time for 0 ≤ α < 1 and grows for α > 1. The original Dielectric Breakdown Model (DBM) of Niemeyer et al. [55] has a more complicated conformal-dynamics representation. As usual, the growth is driven by the gradient of a harmonic function, φ (the electrostatic potential) on an iso-potential surface (the discharge region). Unlike the αmodel above, however, DBM growth events are assumed to have constant size, so the bump size in the mathematical plane is still chosen according to Eq. (48). The difference lies in the growth measure, which does not obey Eq. (46). Instead, the generalized harmonic measure in the physical z-plane is given by p(z) ∝ |∇z φ|η ,

(50)

where η is an exponent interpolating between the Eden model (η = 0), DLA (η = 1), and nonlinear dielectric breakdown (η > 1). For η =/ 1, the fortuitous cancellation in Eq. (46) does not occur. Instead, a similar calculation using Eq. (45) yields a non-uniform probability measure for the nth angle on the unit circle in the mathematical plane, (eiθn )|1−η , Pn (θn ) = |gn−1

(51)

which is complicated and depends on the entire history of the simulation. Nevertheless, conformal mapping can be applied fruitfully to DBM, because not solving Laplace’s equation around the cluster outweighs the difficulty of sampling the angle measure. Surmounting the latter with a Monte Carlo algorithm, Hastings [56] has performed DBM simulations of 104 growth events, an order of magnitude beyond standard methods solving Laplace’s equation on a lattice. The results, illustrated in Fig. 10, support the theoretical conjecture that DBM clusters become one-dimensional, and thus non-fractal, for η ≥ 4. Using the conformal-mapping formalism, efforts are also underway to develop a unified scaling theory of the η-model for the growth probability from DBM combined with the α-model above for the bump size [50].

Conformal mapping methods for interfacial dynamics (a)

1443

(b)

Figure 10. Conformal-mapping simulations by Hastings [56] of the Dielectric Breakdown Model with (a) η = 2 and (b) η = 3.5. (Courtesy of Matt Hastings.)

3.5.

Brittle Fracture

Modeling the stochastic dynamics of fracture is a daunting problem, especially in heterogeneous materials [14, 57]. The basic equations and boundary conditions are still the subject of debate, and even the simplest models are difficult to solve. In two dimensions, stochastic conformal mapping provides an elegant, new alternative to discrete-lattice and finite-element models. In brittle fracture, the bulk material is assumed to obey Lamé’s equation of linear elasticity, ∂ 2u = (λ + µ)∇ ∇ (∇ ∇ · u) + µ∇ ∇ 2 u, (52) ∂t 2 where u is the displacement field, ρ is the density, and µ and λ are Lamé’s constants. For conformal mapping, it is crucial to assume (i) two-dimensional symmetry of the fracture pattern and (ii) quasi-steady elasticity, which sets the left hand side to zero to obtain equations of the type described above. For Mode III fracture, where a constant out-of-plane shear stress is applied at infinity, we have ∇ · u = 0, so the steady Lamé equation reduces to Laplace’s equation for the out-of-plane displacement, ∇ 2 u z = 0, which allows the use of complex potentials. For Modes I and II, where a uniaxial, in-plane tensile stress is applied at infinity, the steady Lamé equation must be solved. As discussed above, this is equivalent to the bi-harmonic equation for the Airy stress function, which allows the use of Goursat functions. For all three modes, the method of iterated conformal maps can be adapted to produce fracture patterns for a variety of physical assumptions about crack dynamics [58]. For Modes I and II fracture, these models provide the first ρ

1444


examples of stochastic bi-harmonic growth, which have interesting differences with stochastic Laplacian growth for Mode III fracture. The Hastings–Levitov formalism is used with constant-size bumps, as in DLA, to represent the fracture process zone, where elasticity does not apply. The growth measure a function of the excess tangential stress, beyond a critical yield stress, σc , characterizing the local strength of the material. Quenched disorder is easily included by making σc a random variable. In spite of its many assumptions, the method provides analytical insights, while obviating the need to solve Eq. (52) during fracture dynamics, so it merits further study.

3.6.

Advection-Diffusion-Limited Aggregation

Non-local fractal growth models typically involve a single bulk field driving the dynamics, such as the particle concentration in DLA, the electric field in DBM, or the strain field in brittle fracture, and as a result these models tend to yield statistically similar structures, apart from the effect of boundary conditions. Pattern formation in nature, however, is often fueled by multiple transport processes, such as diffusion, electromigration, and/or advection in a fluid flow. The effect of such dynamical competition on growth morphology is an open question, which would be difficult to address with lattice-based or finite-element methods, since many large fractal clusters must be generated to fully explore the space and time dependence. Once again, conformal mapping provides a convenient means to formulate stochastic analogs of the non-Laplacian transport-limited growth models from Section 2.3 (in two dimensions). It is straightforward to adapt the Hastings– Levitov algorithm to construct stochastic dynamics driven by bulk fields satisfying the conformally invariant system of Eq. (35). A class of such models has recently been formulated by Bazant et al. [38]. Perhaps the simplest case involving two transport processes, illustrated in Fig. 11, is Advection-Diffusion-Limited Aggregation (ADLA), or “DLA in a flow”. Imagine a fluid carrying a dilute concentration of sticky particles flowing past a sticky object, which begins to collect a fractal aggregate. As the cluster grows, it causes the fluid to flow around it and changes the concentration field, which in turn alters the growth probability measure. Assuming a quasi-steady potential flow with a uniform speed far from the cluster, the dimensionless transport problem is Pe0 ∇ φ · ∇ c = ∇ 2 c, ∇ 2 φ = 0, c = 0, nˆ · ∇ φ = 0, σ = nˆ · ∇ c, c → 1, ∇ φ → xˆ ,

z ∈ z (t),

(53)

z ∈ ∂z (t),

(54)

|z| → ∞,

(55)


1445

Figure 11. A simulation of Advection-Diffusion-Limited Aggregation from Bazant et al. [38] In each row, the growth probabilities in the physical z-plane (on the right) are obtained by solving advection-diffusion in a potential flow past an absorbing cylinder in the mathematical w-plane (on the left), with the same time-dependent Péclet number.

where Pe0 is the initial Péclet number and σ is the diffusive flux to the surface, which drives the growth. The transport problem is solved in the mathematical w-plane, where it corresponds to a uniform potential flow of concentrated fluid past an absorbing circular cylinder. The normal diffusive flux on the cylinder, σ (θ, Pe), can be obtained from a tabulated numerical solution or an accurate analytical approximation [40]. Because the boundary condition on φ at infinity is not conformally invariant, the flow in the w-plane has a time-dependent Péclet number, Pe(t) = A1 (t)Pe0 , which grows with the conformal radius of the cluster. As a result, the

1446


probability of the nth growth event is given by a time-dependent, non-uniform measure for the angle on the unit circle, β Pn (θn ) = τn σ (eiθn , A1 (tn−1 )), (56) λ0 where β is a constant setting the mean growth rate. The waiting time between growth events is an exponential random variable with mean, τn , given by the current integrated flux to the object, λ0 = βτn

2π

σ (eiθ , A1 (tn−1 )) dθ.

(57)

0

Unlike DLA, the aggregation speeds up as the cluster grows, due to a larger cross section to catch new particles in the flow. As shown in Fig. 11, the model displays a universal dynamical crossover from DLA (the unstable fixed point) to an advection-dominated stable fixed point, since Pe(t) → ∞. Remarkably, the fractal dimension remains constant during the transition, equal to the value for DLA, in spite of dramatic changes in the growth rate and morphology (as indicated by higher Laurent coefficients). Moreover, the shape of the “average” ADLA cluster in the high-Pe regime of Fig. 11 is quite similar (but not identical) to the exact solution, Eq. (40), for the analogous continuous problem in Fig. 7. Much remains to be done to understand these kinds of models and apply them to materials problems.

4.

Curved Surfaces

Entov and Etingof (44) considered the generalized problem of Hele–Shaw flows in a non-planar cell having non-zero curvature. In such problems, the velocity of the viscous flow is still the (surface) gradient of a potential, φ, but this function is now a solution of the so-called Laplace–Beltrami equation on the curved surface. The Riemann mapping theorem extends to curved surfaces and says that any simply-connected smooth surface is conformally equivalent to the unit disk, the complex plane, or the Riemann sphere. A common example is the well-known stereographic projection of the surface of a sphere to the (compactified) complex plane. Under a conformal mapping, solutions of the Laplace–Beltrami equation map to solutions to Laplace’s equation and this combination of facts led Entov and Etingof (44) [59] to identify classes of explicit solutions to the continuous Hele–Shaw problem in a variety of non-planar cells. With very similar intent, Parisio et al. [60] have recently considered the evolution of Saffman–Taylor fingers on the surface of a sphere. By now, the reader may realize that most of the methods already considered in this article are, in principle, amenable to generalization to curved surfaces,


1447

which can be reached by conformal mapping of the plane. For example, Fig. 12 shows a simulation of a DLA cluster growing on the surface of a sphere, using a generalized Hastings–Levitov algorithm, which takes surface curvature into account. The key modification is to multiply the Jacobian in Eq. (47) by the Jacobian of the stereographic projection, 1 + |z/R|2 , where R is the radius of the sphere. It should also be clear that any continuous or discrete growth model driven by a conformally-invariant bulk field, such as ADLA, can be simulated on general curved surfaces by means of appropriate conformal projection to a complex plane. The reason is that the system of Eq. (35) is invariant under any conformal mapping, to a flat or curved surface, because each term transforms like the Laplacian, ∇ 2 φ → J ∇ 2 φ, where J is the Jacobian. The purpose of studying these models is not only to understand growth on a particular ideal shape, such as a sphere, but more generally to explore the effect of local surface curvature on pattern formation. For example, this could help interpret mineral deposit patterns in rough geological fracture surfaces, which form by the diffusion and advection of oxygen in slowly flowing water.

Figure 12. Conformal-mapping simulation of DLA on a sphere. Particles diffuse one by one from the North Pole and aggregate on a seed at a South Pole. (Courtesy of Jaehyuk Choi, Martin Bazant, and Darren Crowdy.)

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5.


Outlook

Although conformal mapping has been with us for centuries, new developments with applications continue to the present day. This appears to be the first pedagogical review of stochastic conformal-mapping methods for interfacial dynamics, which also covers the latest progress in continuum methods. Hopefully, this will encourage the further exchange of ideas (and people) between the two fields. Our focus has also been on materials problems, which provide many opportunities to apply and extend conformal mapping. Building on specific open questions scattered throughout the text, we close with a general outlook on directions for future research. A basic question for both stochastic and continuum methods is the effect of geometrical constraints, such as walls or curved surfaces, on interfacial dynamics. Most work to date has been for either radial or channel geometries, but it would be interesting to describe finite viscous fingers or DLA clusters growing near walls of various shapes, as is often the case in materials applications. The extension of conformal-map dynamics to multiply connected domains is another mathematically challenging area, which has received some attention recently but seems ripe for further development. Understanding the exact solution structure of Laplacian-growth problems using the mathematical abstraction of quadrature domain theory holds great potential, especially given that mathematicians have already begun to explore the extent to which the various mathematical concepts extend to higher-dimensions [27]. Describing multiply connected domains could pave the way for new mathematical theories of evolving material microstructures. Topology is the main difference between an isolated bubble and a dense sintering compact. Microstructural evolution in elastic solids may be an even more interesting, and challenging, direction for conformal-mapping methods. From a mathematical point of view, much remains to be done to place stochastic conformal-mapping methods for interfacial dynamics on more rigorous ground. This has recently been achieved in the simpler case of Stochastic Loewner evolution (SLE), which has a similar history to the interfacial problems discussed here [61]. Oded Schramm introduced SLE in 2000 as a stochastic version of the continuous Loewner evolution from univalent function theory, which grows a one-dimensional random filament from a disk or half plane. This important development in pure mathematics came a few years after the pioneering DLA papers of Hastings and Levitov in physics. A notable difference is that SLE has a rigorous mathematical theory based on stochastic calculus, which has enabled new proofs on the properties of percolation clusters and self-avoiding random walks (in two dimensions, of course). One hopes that someday DLA, DBM, ADLA, and other fractal-growth models will also be placed on such a rigorous footing.


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Returning to materials applications, it seems there are many new problems to be considered using conformal mapping. Relatively little work has been done so far on void electromigration, viscous sintering, solid pore evolution, brittle fracture, electrodeposition, and solidification in fluid flows. The reader is encouraged to explore these and other problems using a powerful mathematical tool, which deserves more attention in materials science.

References [1] R.V. Churchill and J.W. Brown, Complex Variables and Applications, 5th edn., McGraw-Hill, New York, 1990. [2] T. Needham, Visual Complex Analysis, Clarendon Press, Oxford, 1997. [3] S.D. Howison, “Complex variable methods in Hele-Shaw moving boundary problems,” Euro. J. Appl. Math., 3, 209–224, 1992. [4] L.M. Cummings, Y.E. Hohlov, S.D. Howison, and K. Kornev, “Two-dimensional soldification and melting in potential flows,” J. Fluid Mech., 378, 1–18, 1999. [5] P.G. Saffman and G.I. Taylor, “The penetration of a fluid into a porous medium or Hele–Shaw cell containing a more viscous liquid,” Proceedings of the Royal Society, London A, 245, 312–329, 1958. [6] M. Kruskal and H. Segur, “Asymptotics beyond all orders in a model of crystal growth,” Stud. Appl. Math., 85, 129, 1991. [7] S. Tanveer, “Evolution of Hele–Shaw interface for small surface tension,” Philosophical Transactions of the Royal Society of London A, 343, 155–204, 1993a. [8] S. Tanveer, “Surprises in viscous fingering,” J. Fluid Mech., 409, 273–308, 2000. [9] B. Bensimon and D. Shraiman, “Singularities in non-local interface dynamics,” Phys. Rev. A, 30, 2840–2842, 1984. [10] L.P. Kadanoff, “Exact solutions for the Saffman–Taylor problem with surface tension,” Phys. Rev. Lett., 65, 2986–2988, 1990. [11] D. Crowdy, “Hele–Shaw flows and water waves,” J. Fluid Mech., 409, 223–242, 2000. [12] J.W. Maclean and P.G Saffman, “The effect of surface tension on the shape of fingers in the Hele–Shaw cell,” J. Fluid Mech., 102, 455, 1981. [13] W.-S. Dai, L.P. Kadanoff, and S.-M. Zhou, “Interface dynamics and the motion of complex singularities,” Phys. Rev. A, 43, 6672–6682, 1991. [14] A. Bunde and S. Havlin (ed.), Fractals and Disordered Systems, 2nd edn., Springer, New York, 1996. [15] S. Tanveer, “Singularities in the classical Rayleigh–Taylor flow: formation and subsequent motion,” Proceedings of the Royal Society, A, 441, 501–525, 1993b. [16] V.E. Zakharov, “Stability of periodic waves of finite amplitude on the surface of deep fluid,” J. Appl. Mech. Tech. Phys., 2, 190, 1968. [17] T. Yoshikawa and A.M. Balk, “The growth of fingers and bubbles in the strongly nonlinear regime of the Richtmyer–Meshkov instability,” Phys. Lett. A, 251, 184– 190, 1999. [18] W. Wang, Z. Suo, and T.-H. Hao, “A simulation of electromigration-induced transgranular slits,” J. Appl. Phys., 79, 2394–2403, 1996. [19] M. Ben Amar, “Void electromigration as a moving free-boundary value problem,” Physica D, 134, 275–286, 1999.

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[20] P. Saffman, “Exact solutions for the growth of fingers from a flat interface between two fluids in a porous medium,” Q. J. Mech. Appl. Math., 12, 146–150, 1959. [21] S. Howison, “Fingering in Hele–Shaw cells,” J. Fluid Mech., 12, 439–453, 1986. [22] D. Crowdy and S. Tanveer, “The effect of finiteness in the Saffman–Taylor viscous fingering problem,” J. Stat. Phys., 114, 1501–1536, 2004. [23] S. Richardson, “Hele–Shaw flows with a free boundary produced by the injection of fluid into a narrow channel,” J. Fluid Mech., 56, 609–618, 1981. [24] G. Carrier, M. Krook, and C. Pearson, Functions of a Complex Variable, McGraw– Hill, New York, 1966. [25] S. Richardson, “Hele–Shaw flows with time-dependent free boundaries involving injection through slits,” Stud. Appl. Math., 87, 175–194, 1992. [26] A. Varchenko and P. Etingof, Why the Boundary of a Round Drop Becomes a Curve of Order Four, University Lecture Series, AMS, Providence, 1992. [27] H. Shapiro, The Schwarz Function and its Generalization to Higher dimension, Wiley, New York, 1992. [28] S. Richardson, “Hele–Shaw flows with time-dependent free boundaries involving a multiply-connected fluid region,” Eur. J. Appl. Math., 12, 571–599, 2001. [29] D. Crowdy and J. Marshall, “Constructing multiply-connected quadrature domains,” SIAM J. Appl. Math., 64, 1334–1359, 2004. [30] N. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, Holland, 1953. [31] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge University Press, 1967. [32] R. Hopper, “Plane stokes flow driven by capillarity on a free surface,” J. Fluid Mech., 213, 349–375, 1990. [33] D. Crowdy, “A note on viscous sintering and quadrature identities,” Eur. J. Appl. Math., 10, 623–634, 1999. [34] D.G. Crowdy, “Viscous sintering of unimodal and bimodal cylindrical packings with shrinking pores,” Eur. J. Appl. Math., 14, 421–445, 2003. [35] S. Richardson, “Plane stokes flow with time-dependent free boundaries in which the fluid occupies a doubly-connected region,” Eur. J. Appl. Math., 11, 249–269, 2000. [36] W. Wang and Z. Suo, “Shape change of a pore in a stressed solid via surface diffusion motivated by surface and elastic energy variations,” J. Mech. Phys. Solids, 45, 709– 729, 1997. [37] M.Z. Bazant, “Conformal mapping of some non-harmonic functions in transport theory,” Proceedings of the Royal Society, A, 460, 1433, 2004. [38] M.Z. Bazant, J. Choi, and B. Davidovitch, “Dynamics of conformal maps for a class of non-Laplacian growth phenomena,” Phys. Rev. Lett., 91, 045503, 2003. [39] K. Kornev and G. Mukhamadullina, “Mathematical theory of freezing for flow in porous media,” Proceedings of the Royal Society, London A, 447, 281–297, 1994. [40] J. Choi, D. Margetis, T.M. Squires, and M.Z. Bazant, “Steady advection-diffusion to finite absorbers in two-dimensional potential flows,” J. Fluid Mech., 2004b. [41] J. Choi, B. Davidovitch, and M.Z. Bazant, “Crossover and scaling of advectiondiffusion-limited aggregation,” In preparation, 2004a. [42] M.B. Hastings and L.S. Levitov, “Laplacian growth as one-dimensional turbulence,” Physica D, 116, 244–252, 1998. [43] B. Davidovitch, H.G.E. Hentschel, Z. Olami, I. Procaccia, L.M. Sander, and E. Somfai, “Diffusion-limited aggregation and iterated conformal maps,” Phys. Rev. E, 59, 1368–1378, 1999. [44] P.L. Duren, Univalent Functions, Springer-Verlag, New York, 1983.


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[45] M.B. Hastings, “Renormalization theory of stochastic growth,” Phys. Rev. E, 55, 135, 1997. [46] T.A. Witten and L.M. Sander, “Diffusion-limited aggregation: a kinetic critical phenomenon,” Phys. Rev. Lett., 47, 1400–1403, 1981. [47] T.C. Halsey, “Diffusion-limited aggregation: a model for pattern formation,” Phys. Today, 53, 36, 2000. [48] M.G. Stepanov and L.S. Levitov, “Laplacian growth with separately controlled noise and anisotropy,” Phys. Rev. E, 63, 061102, 2001. [49] M.H. Jensen, A. Levermann, J. Mathiesen, and I. Procaccia, “Multifractal structure of the harmonic measure of diffusion-limited aggregates,” Phys. Rev. E, 65, 046109, 2002. [50] R.C. Ball and E. Somfai, “Theory of diffusion controlled growth,” Phys. Rev. Lett., 89, 133503, 2002. [51] E. Somfai, L.M. Sander, and R.C. Ball, “Scaling and crossovers in diffusion limited aggregation,” Phys. Rev. Lett., 83, 5523, 1999. [52] E. Somfai, R.C. Ball, J.P. DeVita, and L.M. Sander, “Diffusion-limited aggregation in channel geometry,” Phys. Rev. E, 68, 020401, 2003. [53] F. Barra, B. Davidovitch, and I. Procaccia, “Iterated conformal dynamics and Laplacian growth,” Phys. Rev. E, 65, 046144, 2002a. [54] A. Levermann and I. Procaccia, “Algorithm for parallel laplacian growth by iterated conformal maps,” Phys. Rev. E, 69, 031401, 2004. [55] L. Niemeyer, L. Pietronero, and H.J. Wiesmann, “Fractal dimension of dielectric breakdown,” Phys. Rev. Lett., 52, 1033–1036, 1984. [56] M.B. Hastings, “Fractal to nonfractal phase transition in the dielectric breakdown model,” Phys. Rev. Lett., 87, 175502, 2001. [57] H.J. Hermann and S. Roux (eds.), Statistical Models for the Fracture of Disordered Media, North-Holland, Amsterdam, 1990. [58] F. Barra, A. Levermann, and I. Procaccia, “Quasistatic brittle fracture in inhomogeneous media and iterated conformal maps,” Phys. Rev. E, 66, 066122, 2002b. [59] V.M. Entov and P.I. Etingof, “Bubble contraction in Hele–Shaw cells,” Quart. J. Mech. Appl. Math., 507–535, 1991. [60] F. Parisio, F. Moreas, J.A. Miranda, and M. Widom, “Saffman–Taylor problem on a sphere,” Phys. Rev. E, 63, 036307, 2001. [61] W. Kager and B. Nienhuis, “A guide to stochastic loewner evolution and its applications,” J. Stat. Phys., 115, 1149–1229, 2004.

4.11 EQUATION-FREE MODELING FOR COMPLEX SYSTEMS Ioannis G. Kevrekidis1, C. William Gear1 , and Gerhard Hummer2 1 Princeton University, Princeton, NJ, USA 2

National Institutes of Health, Bethesda, MD, USA

A persistent feature of many complex systems is the emergence of macroscopic, coherent behavior from the interactions of microscopic “agents” – molecules, cells, individuals in a population – among themselves and with their environment. The implication is that macroscopic rules (a description of the system at a coarse-grained, high-level) can somehow be deduced from microscopic ones (a description at a much finer level). For laminar Newtonian fluid mechanics, a successful coarse-grained description (the Navier–Stokes equations) was known on a phenomenological basis long before its approximate derivation from kinetic theory [1]. Today we must frequently study systems for which the physics can be modeled at a microscopic, fine scale; yet it is practically impossible to explicitly derive a good macroscopic description from the microscopic rules. Hence, we look to the computer to explore the macroscopic behavior based on the microscopic description. It is difficult to define complexity in a precise, useful way. At the same time it pervades current modeling in engineering science, in the life and physical sciences, and beyond them (e.g., in economics) (see, e.g., Refs. [2, 3]). We may not typically think of a laminar Newtonian flow as complex, even though it involves interactions of enormous numbers of fluid molecules with themselves and with the boundaries of the flow. Such problems are considered simple because we have a good model, describing the behavior of the system at the level we need for practical purposes. If we are interested in pressure drops and flow rates over humanly relevant space/time scales, we do not need to know where each and every molecule is, or its individual velocity, at a given instant in time. Similarly, if a stirred chemical reactor can be modeled adequately, for design purposes, by a few ordinary differential equations (ODEs), the immense complexity of molecular interactions involved in flow, reaction and mixing in it goes unnoticed. The system is classified as simple, because 1453 S. Yip (ed.), Handbook of Materials Modeling, 1453–1475. c 2005 Springer. Printed in the Netherlands.

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a simple model of the behavior is adequate for practical purposes. This suggests that the scale of the observer, and the practical goals of the modeling, are crucial in classifying a system, its models, or its behavior as complex – or as simple. Macroscopic models of reaction and transport processes in our textbooks come in the form of conservation laws (species, mass, momentum, energy) closed through constitutive equations (reaction rates as a function of concentration, viscous stresses as functionals of velocity gradients). These models are written directly at the scale (alternatively, at the level of complexity) at which we are interested in practically modeling the system behavior. Because we observe the system at the level of concentrations or velocity fields,we sometimes forget that what is really evolving during an experiment is distributions of colliding and reacting molecules. We know, from experience with particular classes of problems, that it is possible to write predictive deterministic laws for the behavior observed at the level of concentrations or velocity fields – laws that are predictive over space and time scales relevant to engineering practice. Knowing the right level of observation at which we can be practically predictive, we attempt to write closed evolution equations for the system at this level. The closures may be based on experiment (e.g., through engineering correlations) or on mathematical modeling and approximation of what happens at more microscopic scales (e.g., the Chapman–Enskog expansion). In many problems of current modeling practice, ranging from materials science to ecology, and from engineering to computational chemistry, the physics are known at the microscopic/individual level, and the closures required to translate them to high-level, coarse-grained, macroscopic descriptions are not available. Sometimes we do not even know at what level of observation one can be practically predictive. Severe computational limitations arise in trying to bridge, through direct computer simulation, the enormous gap between the scale of the available description and the macroscopic, “system” scale at which the questions of interest are asked and the practical answers are required (see, e.g., Refs. [4, 5]). These computational limitations are a major stumbling block in current complex system modeling. Our objective is to describe a computational approach for dealing with any complex, multi-scale system whose collective, coarse-grained behavior is simple when we know in principle how to model such systems at a very fine scale (e.g., through molecular dynamics). We assume that we do not know how to write good simple model equations at the right coarse-grained, macroscopic scale for their collective, coarse-grained behavior. We will argue that, in many cases, the derivation of macroscopic equations can be circumvented; that by using short bursts of appropriately initialized microscopic simulation one can effectively solve the macroscopic equations without ever writing them down. A direct bridge can be built between microscopic simulation (e.g., kinetic Monte Carlo, agent-based modeling) and traditional continuum numerical

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analysis. It is possible to enable microscopic simulators to directly perform macroscopic, systems level tasks. The main idea is to consider the microscopic, fine-scale simulator as a (computational) experiment that one can set up, initialize, and run at will. The results of such appropriately designed, initialized and executed brief computational experiments allow us to estimate the same information that a macroscopic model would allow us to evaluate from explicit formulas. The heart of the approach can be conveyed through a simple example (see Fig. 1). Consider a single, autonomous ODE, dc = f (c). dt

(1)

Think of it as a model for the dynamics of a reactant concentration in a stirred reactor. Equations like this embody “practical determinism” as discussed above: given a finite amount of information (the state at the present time, c(t =0)) we can predict the state at a future time. Consider how this is done on the computer using – for illustration – the simplest numerical integration scheme, forward Euler: cn+1 ≡ c([n + 1]τ ) = cn + τ f (cn ).

(2)

Starting with the initial condition, c0 , we go to the equation and evaluate f (c0 ), the time derivative, or slope of the trajectory c(t); we use this value to make a prediction of the state of the system at the next time step, c1 . We then repeat the process: go to the equation with c1 to evaluate f (c1 ) and use the Euler scheme to predict c2 ; and so on. Forgetting for the moment accuracy and adaptive step size selection, consider how the equation is used: given the state we evaluate the time-derivative; and then, using mathematics (in particular, Taylor series and smoothness to create a local linear model of the process in time) we make a prediction of the state at the next time step. A numerical integration code will “ping” a sub-routine with the current state as input, and will obtain as output the time-derivative at this state. The code will then process this value, and use local Taylor series in order to make a prediction of the next state (the next value of c at which to call the sub-routine evaluating the function f ). Three simple things are important to notice. First, the task at hand (numerical integration) does not need a closed formula for f (c) – it only needs f (c) evaluated at a particular sequence of values cn . Whether the sub-routine evaluates f (c) from a single-line formula, uses a table lookup, or solves a large subsidiary problem, from the point of view of the integration code it is the same thing. Second, the sequence of values cn at which we need the time-derivative evaluated is not known a priori. It is generated as the task progresses, from processing results of previous function evaluations through the Euler formula. We know that protocols exist for designing experiments to

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t2

t

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(b) C C2 ~ f (C 1 )

C1 ~ f (C 0 )

C0

t0

t1

(c) C ⫺ Φτ (C)

(n) C (n) C ⫹ε

C (n⫹1)

Figure 1. (a) Forward Euler numerical integration, used (b) as a template for projective integration using the results of short experiments. (c) Fixed-point iteration for a timestepper.


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accomplish tasks such as parameter estimation [6]. In the same spirit, we can think of the Euler method, and of explicit numerical integrators in general, as protocols for specifying where to perform function evaluations based on the task we want to accomplish (computation of a temporal trajectory). Lastly, the form of the protocol (the Euler method here) is based on mathematics, particularly on smoothness and Taylor series. The trajectory is locally approximated as a linear function of time; the coefficients of this function are obtained from the model using function evaluations. Suppose now that we do not have the equation, but we have the experiment itself : we can fill up the stirred reactor with reactant at concentration c0 , run for some time, and record the time series of c(t). Using the results of a short run (over, say, 1 min) we can now estimate the slope, dc/dt at t = 0, and predict (using the Euler method) where the concentration will be in, say 10 min. Now, instead of waiting for 9 min for the reactor to get there, we stop the experiment and immediately start a new one: reinitialize the reactor at the predicted concentration; run for one more minute, and use forward Euler to predict what the concentration will be 20 min down the line. We are substituting short, appropriately initialized experiments, and estimation based on the experimental results, for the function evaluations that the sub-routine with the closed form f (c) would return. We are in effect doing forward Euler again; but the coefficients of the local linear model are obtained using experimentation “on demand ” [7] rather than function evaluations of an a priori available model. Many elements of this example are contrived; for example, the assumption that an Euler prediction with a 10 min step is reasonably accurate. It may also appear laughable that, instead of waiting nine more minutes for the reactor to get to the predicted concentration, we will initialize a fresh experiment at that concentration. It will probably take much more than 9 min to start a new experiment; there will be startup transients, and noise in the measurements. The point, however, remains: it is possible to do forward Euler integration using short bursts of appropriately initialized experiments if it is easy to initialize such experiments at will. An “outer” process (design of the next experiment, setting it up, measuring its results, processing them to design a new experiment) is wrapped around an “inner” process (the experiment). The outer wrapper is motivated by the task that we wish to perform (here, longtime integration) and is based on traditional, continuum numerical analysis. The inner layer is the process itself. It is clear that systems theory components (data acquisition and filtering, model identification, [8]) are vital in forming the connection between the outer layer and the inner layer (the task we want to accomplish and the system itself). Now we complete the argument: suppose that the inner layer is not a laboratory experiment, but a computational one, with a model at a different, much finer level of description (for the sake of the discussion, a lattice kinetic

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Monte Carlo, kMC, model of the reaction). Instead of running the kMC model for long times, and observing the evolution of the concentration, we can exploit the procedure described above, perform only short bursts of appropriately initialized microscopic simulation, and use their results to evolve the macroscopic behavior over hopefully much longer time scales. It is much easier to initialize a code at will – a computational experiment – as opposed to initializing a new laboratory experiment. Many new issues arise, notably noise, in the form of fluctuations, from the microscopic solver. The conceptual point, however, remains: even if we do not have the right macroscopic equation for the concentration, we can still perform its numerical integration without obtaining it in closed form. The skeleton of the wrapper (the integration algorithm) is the same one we would use if we had the macroscopic equation; but now function evaluations are substituted by short computational experiments with the microscopic simulator, whose results are appropriately processed for local macroscopic identification and estimation. If a large separation of time-scales exists between microscopic dynamics (here, the time we need to run kinetic Monte Carlo to estimate dc/dt) and the macroscopic evolution of the concentration, this procedure may be significantly more economical than direct simulation. Passing information between the microscopic and macroscopic scales at the beginning and the end of each computational experiment is a vitally important issue. It is accomplished through a lifting operator (macro- to micro-) and a restriction operator (micro- to macro-) as discussed below (see [9, 10] and references therein). Detailed, fine-level dynamics are typically given in terms of microscopically/stochastically evolving distributions of interacting “agents” (molecules, cells); the evolution rules could be molecular dynamics (classical, or Car–Parrinello [11]), MC or kMC, Brownian dynamics, etc. The macroscopic dynamics are described by closed evolution equations, typically ordinary (for macroscopically lumped) or partial differential/integrodifferential equations. The dependent variables in these equations are frequently a few, lower order moments of the evolving distributions (such as concentration, the zeroth moment). The proposed computational methodology consists of the following basic elements: (a) Choose the statistics of interest for describing the long-term behavior of the system and an appropriate representation for them. For example, in a gas simulation at the particle level, the statistics would probably be density and momentum (zeroth and first moment of the particle distribution over velocities) and we might choose to discretize them in a computational domain via finite elements. We call this the macroscopic description, u. These choices suggest possible restriction operators, M, from the microscopic-level description U, to the macroscopic description: u = MU;


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(b) Choose an appropriate lifting operator, µ from the macroscopic description, u, to one or more consistent microscopic descriptions, U. For example, in a gas simulation using pressure, etc. as the macroscopic-level variables, µ could make random particle assignments consistent with the macroscopic statistics. µM = I, i.e., lifting from the macroscopic to the microscopic and then restricting (projecting) down again should have no effect, except roundoff. (c) Start with a macroscopic condition (e.g., concentration profile) u(t0 ); (d) Transform it through lifting to one – or more – fine, consistent microscopic realizations U(t0 ) = µu(t0 ); (e) Evolve each realization using the microscopic simulator for the desired short macroscopic time T, generating the values U(t1 ) where t1 = t0 + T; (f) Obtain the restriction(s) u(t1 ) = MU(t1 ) (and average over them). This constitutes the coarse time-stepper, or coarse time-T map. If this map is accurate enough, we showed above how to use it in a two-tier procedure to perform Coarse Projective Integration [12–14]. • repeating steps (e–f) over several time steps and obtaining several U(ti ) as well as their restrictions u(ti ) = MU(ti ), i = 1, 2, . . . , k + 1 • using the chord approximating these successive time-stepper output points to estimate the derivative – the “right-hand-side” of the equations we do not have –, we can then • use this derivative in another, outer integrator scheme (such as forward Euler) to produce estimates of the macroscopic state much later in time u(tk+1+M ). • go back to step (d). The lifting step (creating microscopic distributions conditioned on a few of their lower moments, going back to Ehrenfest, [15]) is clearly not unique, and sometimes quite non-trivial: consider for example creating a distribution of particles on a lattice that has a prescribed average as well as a prescribed pair probability. A preparatory step (e.g., through simulated annealing) may be required to arrange the particles on the lattice consistently with the prescribed constraints. Through such appropriate preparation, one can even lift prescribed pair-correlation functions to consistent particle assemblies. Constrained dynamics algorithms, like SHAKE [16] can also be thought of as lifting procedures; see also Ref. [17]. An important point made in Fig. 2a is that an initial simulation interval must elapse before estimating the time-derivative of the macroscopic variables from the microscopic simulation. In the microscopic dynamics, every particle evolves while interacting with other particles, and all the moments of the distribution evolve in a coupled manner. It is therefore remarkable that practically predictive models are usually written in terms of only a few moments

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TI M E

Patch dynamics Lift µ

Project Restrict 2

Evolve 2

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Evolve 1 Interpolate

Lift µ

Interpolate Apply BC2

Boxes SPACE

(c)

Figure 2. Schematic illustrations of (a) coarse projectiveintegration; (b) patch dynamics; and (c) coarse-timestepper-based bifurcation computations (see text).


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of these evolving distributions. This is only possible because the remaining, higher-order moments quickly become functionals of the few, lower order, slow, “master” moments – our observation variables. This occurs over timescales that are short compared to the macroscopic observation time-scales. In this separation of time-scales (and concomitant space scales) lies the essential reduction step underpinning effective simplicity and practical determinism. The idea is that the long-term observable dynamics of the system evolve on a low-dimensional, strongly attracting, slow manifold in moments space; this is, effectively, a quasi-steady state approximation [18]. This manifold is parameterized by our observation variables (typically the lower distribution moments, like concentration) in terms of which we write macroscopic equations. The expected values of the remaining moments can be written as an (unspecified) function of the coarse variables; that is the graph of the manifold. A good example is the law of Newtonian viscosity: when one starts a molecular simulation, the stresses are not instantaneously proportional to velocity gradients – but for Newtonian fluids they become so within a few collision times, i.e., over times much shorter than the macroscopic observation times over which the Navier–Stokes equations become valid approximations. The coarse variables are therefore observation variables. If the fine-scale simulation, conditioned on values of the observation variables, is initialized “off manifold”, it only takes a fast (possibly constrained) initial transient to approach a neighborhood of this manifold. Through the restriction operator, we observe the dynamics on the hyperplane spanned by our chosen observation variables. After the system quickly relaxes to the manifold, we estimate the time-derivative of the observation variables, and use it in the projective integration scheme. The dynamics of the full system will then, after lifting and a short integration, spontaneously establish (by bringing us to the manifold) the missing closure: the effect of the full description on the observed dynamics. A direct conceptual analogy arises here with center manifolds in dynamical systems (parameterized using eigenvectors of the linearization at a steady state, see, e.g., Ref. [19]) or inertial manifolds for dissipative PDEs (parameterized using eigenfunctions of a linear dissipative operator, [20, 21]). Normal forms and (approximate) inertial forms are thus analogous to our macroscopic equations for the coarse observation variables. Low order moments have traditionally been the observation variables of choice in our textbooks. In principle, however, any set of variables that parameterizes this low-dimensional slow manifold can be used as observation variables with the appropriate lifting and restriction operators. Using more observation variables than necessary reduces computational efficiency; it is analogous to using a finer mesh than necessary for the accuracy required in solving a problem. Intelligently chosen order parameters usually provide a much more parsimonious basis set on which to observe the dynamics and apply our computational framework. There is a clear analogy here with

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empirical eigenfunctions [22] used for model reduction in the discretization of dissipative PDEs. The detection of good observables, capable of efficiently parameterizing this manifold, through statistical analysis of simulation results, is a crucial enabling technology for our computational framework. Using data mining techniques (e.g., see Ref. [23–25]) to find such observables can be thought of as the “variable-free” component of the equation-free modeling approach. In coarse projective integration we exploit the smoothness in time of the unavailable macroscopic equation in order to project (jump) to the future. In the case of macroscopically (spatially or otherwise) distributed systems, one can exploit smoothness of the unavailable macroscopic equation in space in order to perform the microscopic simulations only over small, but appropriately coupled, computational boxes (“teeth”). This is illustrated in Fig. 2b: (a) Coarse variable selection (same as above, but now the variable u(x) depends on “coarse space” x. We have chosen for simplicity to consider only one space dimension.) (b) Choice of lifting operator (same as above, but now we lift entire profiles of u(x, t) to profiles of U(y, t), where is microscopic space corresponding to the macroscopic space x. This lifting involves therefore not only the variables, but the space descriptions too. The basic idea is that a coarse point in x corresponds to an interval (a “box” or “tooth” in y). (c) Prescribe a macroscopic initial profile u(x, t 0 ) – the “coarse field”. In particular, consider the values u i (t0 ) at a number of macro-mesh points; the macroscopic profile arises from interpolation of these values of the coarse-field. (d) Lift the “mesh points” xi and the values u i (t0 ) to profiles Ui (yi , t0 ), in microscopic domains (“teeth”) yi corresponding to the coarse-mesh points xi . These profiles should be conditioned on the values u i , and it is a good idea that they are also conditioned on certain boundary conditions motivated by the coarse-field (e.g., be consistent with coarse slopes at the boundaries of the teeth that are computed from the coarse-field). (e) Evolve the microscopic dynamics in each of these boxes for a short time T based on the microscopic description, and through ensembles that enforce the coarsely inspired boundary conditions (see, e.g., Ref. [26]) – and thus generate Ui (yi , t1 ), where t1 = t0 + T. (f) Obtain the restriction from each patch to coarse variables u i (t1 ) = M Ui (yi , t1 ). (g) Interpolate between these to obtain the new coarse-field u(x, t1 ). Up to this point, we have the gaptooth scheme: a scheme that computes in small domains (the “teeth”) which communicate over the gaps between them


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through “coarse-field motivated” boundary conditions. We can now proceed by combining the gaptooth scheme with projective integration ideas to (h) Repeat the process (lift within the teeth, compute new boundary conditions, evolve microscopically, restrict to macroscopic variables and interpolate) for a few steps, and then (i) Project coarse-fields “long” into the future. For a projective forward Euler this would involve the chord between two successive coarse-fields to estimate the right-hand-side of the unavailable coarse equation, and then an Euler “projection” of the coarse-field long into the future. (j) Repeat the entire procedure starting with the lifting (d) above. This leads to patch dynamics: a computational framework in which simulations using the microscopic description over short times and small computational domains (“patches” in space-time) can be used to advance the macroscopic dynamics over long times and large computational domains [10, 27–29]. Initializing microscopic computations conditioned on macroscopic variables is an important component of coarse projective integration; similarly, imposing macroscopically motivated boundary conditions to microscopic computations is an important element of gaptooth and patch dynamics. The methods we discussed can, under appropriate conditions, drastically accelerate the direct simulation of the coarse-grained, macroscopic behavior of certain complex multi-scale systems. Direct simulation, however, is but the simplest computational task one can perform with a system model. It corresponds, in some sense, to physical experimentation: we set parameter values and initial conditions, let the system evolve on the computer and observe its behavior, just like performing a laboratory experiment. Depending on what we want to learn about the system, there exist much more interesting and efficient ways of using the model and the computer. Consider for example the location of steady states; fixed point algorithms, like the Netwon– Raphson, are a much more efficient way of finding steady states than direct integration (given a good initial guess). Such fixed point algorithms can locate both stable and unstable steady states (the latter would be extremely difficult or impossible to find with direct simulation). “The Jacobian of the solution is a treasure trove, not only for continuation, but also for analyzing stability of solutions, for detecting bifurcations of solution families, and for computing asymptotic estimates of the effects, on any solution, of small changes in parameters, boundary conditions and boundary shape” [30]. Beyond stability and sensitivity analysis, having the steady states and using Taylor series in their neighborhood (Jacobians, Hessians) one can design stabilizing controllers, observers, solve optimization problems, etc. There is a vast arsenal of algorithms (and codes implementing them) for the computer-aided analysis of system models, going much beyond direct simulation. Yet these algorithms

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are applicable to macroscopic equations: ODEs, Differential Algebraic Equations (DAEs), PDEs/PDAEs and their discretizations. Smoothness and Taylor series expansions (derivatives with respect to time, Frechet derivatives, partial derivatives with respect to parameters) are vital in formulating and implementing most of these algorithms. When the model comes in the form of microscopic/stochastic simulators at a much finer scale – without a closed formula for the equation, i.e., without a “right-hand side” for the time-derivative –, this arsenal of continuum numerical tools appears useless. Fortunately, the same coarse timestepping idea we used to accelerate direct simulation of an effectively simple multi-scale system can be used to enable its coarse-grained computer-assisted analysis even without explicit macroscopic equations. To illustrate this, we return to our simple scalar example in Fig. 1. We are given a black box timestepper for this equation: a code which, initialized with cn (t = nτ ) integrates the equation for time τ and returns the result cn+1 = c(t = [n + 1]τ ). We use the notation cn+1 = τ (cn ). If the task at hand is to find a steady state for the equation, this can be accomplished by calling the timestepper repeatedly (integrate forward in time) until the result does not change any more. Indeed a steady state of the equation is a fixed point for the timestepper, x ∗ =τ (x ∗ ). Yet this iteration will only find stable steady states, and the rate of convergence to them depends on the physical dynamics of the problem, becoming increasingly slow close to transition boundaries. The method of choice for finding a steady state (given a good initial guess) would be a Newton–Raphson iteration, which would converge quadratically to non-singular steady states.

df dc

c(n)

(c(n+1) − c(n) ) = − f (c(n) ).

Can we trick an integration code (the timestepper) into becoming a fixed point solver? In other words, if we do not have the equation for f (c), but can computationally evaluate the timestepper, can we still do Newton for the steady state? The answer is illustrated in Fig. 1c: we use the computationally evaluated timestepper to solve the fixed point problem G(c) ≡ c − (c) = 0. Calling the timestepper for an initial condition c(n) gives us (c(n) ) and the residual, G(c(n) ). Lacking a formula to compute the linearization, we call the timestepper with a nearby initial condition, c(n) + ε. This gives us (c(n) + ε), • ε. This estimate and the difference (using Taylor series) is approximately d dc of the action of the Jacobian can then be used in a secant method to compute the next iterate c(n + 1) of the steady-state search. Notice again the crucial issue of being able to initialize a simulator at will; after c(n+1) is estimated from


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the nearby integrations and the secant procedure, we can immediately call the timestepper with initial condition c(n + 1) and iterate the process. We have not done much more than estimating derivatives through differencing. Yet forward integration can now be used (through a computational superstructure, a “wrapper” that implements what we just described in words) to converge to unstable steady states, and eventually to compute bifurcation diagrams. We have enabled a simulation code to perform a task (fixed point computation) for which it had not been designed [31]. This procedure may initially appear hopeless in higher dimensions (e.g., for the large sets of ODEs arising in PDE discretizations). Fortunately, recent developments in large-scale computational linear algebra (the so-called matrix free solvers and eigensolvers) address precisely this point. Integrating with two nearby initial conditions (m-vectors, differing by the m-vector ε) and taking the difference of the timestepper results provides an estimate of DΦ · ε, the inner product of the m × m Jacobian matrix of the timestepper (which is not available in closed form) and the known m-vector ε. Matrix-free iterative algorithms (for example Newton–Krylov/GMRES methods based on the timestepper) can then be used to solve for the steady state (e.g., Refs. [32, 33]). Matrix-free eigensolvers (e.g., subspace iteration methods based on the timestepper) can be used to estimate the part of the spectrum of the linearization close to the imaginary axis, which is relevant for stability and bifurcation computations of the unavailable equation [34]. We see once more that the quantities necessary for computer-aided analysis (residuals, action of Jacobians) can be estimated by appropriately designed short calls to the timestepper and subsequent post-processing of the results, even if the equation is not available in closed form. Remarkably, and completely independently of complex/multi-scale computations, these software wrappers have the potential to enable legacy integration codes (large-scale, industrial dynamic simulators) to perform tasks such as stability/bifurcation and operability analysis, controller design and optimization. Our inspiration comes from precisely such a wrapper: the Recursive Projection Method of Ref. [35], which enables a class of large scale direct simulators (even slightly unstable ones) into becoming convergent fixed point solvers. Clearly, the same type of computational superstructure can turn coarsetimesteppers (lifting from macroscopic to consistent microscopic initial conditions, evolving with the fine-scale code, and restricting back to macroscopic variables) into coarse-fixed point algorithms, and, with appropriate augmentation, coarse bifurcation algorithms (Fig. 2c). Coarse residuals and the action of coarse slow Jacobians and Hessians can be estimated in a matrix-free context by systematic, judicious calls to the coarse timestepper. Coarse equation solvers and coarse eigensolvers can thus be implemented – many aspects of the computer-assisted analysis of the unavailable macroscopic equation can be

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performed without the equation. Motivated by the connection to matrix-free numerical analysis methods, we call the timestepper and coarse-timestepper based computer-assisted analysis equation free computation [10]. The scope of the approach is very general. Coarse projective integration and coarse bifurcation computations have been used to accelerate lattice kinetic Monte Carlo simulations of catalytic surface reactions ([36–39]); biased random walk kMC models of e-coli chemotaxis ([40]); kinetic theory-based, interacting particle simulations of hydrodynamic equations [28]; Brownian dynamics simulations of nematic liquid crystals [41]; lattice Boltzmann-BGK simulations of multi-phase, bubbly flows [31]; molecular dynamics simulations of the folding of a peptide fragment [42]; individual-based kMC models of evolving diseases such as influenza [43]; kMC models of dislocation movement in a lattice containing diffusing impurities [44]; molecular dynamics simulations of granular flows; and more. For some spatially distributed problems, this involved gaptooth and patch dynamics versions of the coarse-timestepper. As more experience is accumulated and the methods develop further, more problems may become accessible to equation-free computer aided analysis. Beyond simulation and stability/continuation computations, equation-free computation has been used to perform tasks such as linear stabilizing controller design for kMC, LB-BGK as well as Brownian Dynamics simulators [41, 45, 46]; case studies of coarse optimization [47] as well as coarse feedback linearization for kMC simulators [48, 49] have been performed; additional tasks like coarse reverse integration backward in time [50], and coarse dynamic renormalization [10, 51], for the equation-free computation of selfsimilar solutions are also possible. Wrappers for legacy codes have been designed (RPM has been wrapped around gPROMS to accelerate rapid pressure swing absorption computations, and coarse integration of an unavailable envelope equation has also been used for this purpose, [52]). Other problems can also be approached through the same basic scheme, including problems which we believe could be modeled by effective medium equations (such as flow in porous media, or reaction-diffusion over microcomposite catalysts). Here again, short bursts of detailed medium simulation can be used to estimate the timestepper of the effective medium equation without deriving this equation explicitly [53]. Similarly, the solution of effective continuum equations for spatially discrete problems (such as lattices of coupled neurons) can be attempted in an equation-free framework [54]. Most of the discussion so far was formulated in a deterministic context; yet many complex systems of interest are well-described by stochastic models. Every outcome of computations with such models is in principle different; noise destroys determinism at the level of a single experiment. Determinism is often restored, however, at a different level of observation: when one considers the distribution of the outcomes of several realizations. One can be deterministic (i.e., write predictive equations) about the expectation of a sufficiently


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large ensemble of experiments; possibly about the expectation and standard deviation of such an ensemble. Once again, higher order moments of a probability distribution (whose evolution is governed by a Fokker–Planck-type equation) get quickly slaved to lower order moments, and one can be practically predictive if one looks at an appropriately coarse-grained level. While, for example, we cannot know the fate of an individual after a year, we can be practically predictive about the evolution of a few basic statistics of the population of a country. For the right observables, the coarse-timestepper is then constructed by simulating a large enough ensemble of realizations of the stochastic problem. An important category of problems can be approximated by dynamics on low-dimensional free-energy surfaces, parametrized by a few well-chosen coarse variables (reaction coordinates). In the statistical mechanics of molecular systems the ability to be “practically predictive” with just a few meaningful reaction coordinates is intimately connected with separation of time scales. Formally, such coordinates could be defined with the help of the leading eigenfunctions of a Frobenius–Perron operator for the detailed problem [55]; yet this is practically unachievable. Instead, physical intuition, experience and data analysis is often used to suggest collective coordinates which hopefully provide dynamically relevant measures of the progress of a reaction. Projecting the full dynamics on such well-chosen reaction coordinates will then retain the macroscopically relevant features of the dynamics with only simplified representations of noise and memory [56, 57]. Short bursts of appropriately initialized molecular dynamics can again be used to estimate on demand the drift and the noise terms of effective Langevin or Fokker–Planck equations in these variables [58]; to find minima and saddles; to solve optimal path problems, and to construct approximate propagators for the density on this surface, without deriving or writing this effective equation in closed form. In our discussion we have endeavored to outline the new possibilities opened by such an equation-free framework. These possibilities are accompanied by many theoretical and practical difficulties. Some of these issues arise in algorithms of continuum numerical analysis themselves (stepsize selection in numerical integration, mesh-size selection in spatial discretizations, error monitoring and control in matrix-free iterative methods); some are particular to complex/multi-scale timesteppers (consistent initialization through lifting; estimation and filtering involved in restriction operators; imposition of macroscopically inspired boundary conditions); some arise from the coupling (choice of good observation variables). We will mention one special feature here. Adaptive step size selection is often performed by doing the computation with different step sizes and estimating the error a posteriori; similarly, adaptive mesh selection is based on computations performed at different mesh-sizes to estimate the error. To adaptively determine the level of coarse-graining at which we can be practically predictive, the coarse timestepper can be computed by

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conditioning the microscopic simulation at different observation levels, i.e., with different numbers of coarse variables (e.g., surface coverages only, vs. surface coverages and pair probabilities for lattice simulations of surface reactions). Matrix-free, timestepper-based eigensolvers can then be used to estimate the slow eigenvalues and corresponding eigenvectors for the timestepper, which should be tangent to the slow manifold (embodying the missing closure). Gaps in this spectrum, and the components of the corresponding eigenvectors can be used to probe the number and nature of coarse variables that should be used to observe the system dynamics (i.e., to locally parametrize the manifold). Handshaking between microscopic solvers and macroscopic continuum numerical analysis consists mainly of subjects traditionally studied in systems theory. System identification based on the results of computational experimentation with the fine-scale model is the most important component. Separation of time-scales underpins the low-dimensionality of the macroscopic dynamics. The dynamics of the hierarchy of distribution moments constitute a singularly perturbed system, and brief simulation is used to “cure off-manifold initial conditions” by bringing them back onto the manifold, healing the errors we commit when lifting. The dynamics themselves establish the missing closure; we can think of this as a “closure on demand” approach. Adaptive tabulation [59] can be used to economize in the design of experiments, and the importance of data assimilation/statistical analysis tools to identify non-linear correlations has already been stressed. The use of observer theory (e.g., [60, 61]) and realization balancing (e.g., Refs. [62, 63]) arises naturally: the microscopic system dynamics are observed on the macroscopic variables, but are realized through the microscopic simulator. Techniques for filtering [64] and variance reduction [65] will play an important role in determining how useful equation-free computations will ultimately be [66]. Timestepper-based methods are, in effect, alternative ensembles for performing microscopic (molecular dynamics, kMC, Brownian dynamics) simulations. These ensembles, however, are motivated by macroscopic numerical analysis, rather than statistical mechanical considerations. We are currently exploring the applicability of these “numerical analysis motivated” ensembles in accelerating equilibrium computations (grand canonical MC computations of micelle formation, [67, 68]). It is particularly interesting to consider ensembles motivated by the augmented systems arising in multi-parameter continuation. In such ensembles, like the pathostat [48, 49] based on pseudoarclength continuation, both the variables and the operating parameters themselves evolve, so that the system traces both stable and unstable parts of bifurcation diagrams. An increasing number of experimental systems appears in the literature for which finely spatially distributed actuation authority – coupled with sensing – is available; photosensitive chemical reactions addressed through a


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digital projector [69], laser-addressable catalytic reactions [70] and interfacial flows [71], colloidal particles manipulated through optical tweezers [72] or electric fields [73] are some such examples. When experiments can be initialized at will, the timestepper methods we discussed here can be applied to laboratory – rather than computational – experiments. Continuum numerical methods will then become experimental design protocols, tuned to the task we wish to perform. This way, mathematics might be performed directly on the physical system, and not on the (approximate) equations modeling it. Many of the mathematical and computational tools combined in this exposition (e.g., system identification, or inertial manifold theory) are wellestablished; we borrowed them, in our synthesis with tools developed in our group, as necessary. Innovative multi-scale/multi-level techniques proposed over the last decade include the quasi-continuum methods of Phillips and coworkers [74, 75]; the optimal prediction methods of Chorin and coworkers [76, 77]; the coupling of continuum fields with stochastic evolution in the work of Oettinger and coworkers [78, 79]; the kinetic-theory-based solvers proposed by Xu and Prendergast [80, 81], the modification of equation-free computation in the context of conservation laws by E and Engquist [82]; and the lattice coarse graining by Katsoulakis et al. [83] (see the review by Givon et al, [84] and the discussion in Ref. [10]. In the context of molecular dynamics simulations, the idea of using multiple, and possibly coupled replica runs to search conformation space (for systems with unmodified or artificially modified energy surfaces) forms the basis of approaches such as conformational flooding [85], parallel replica MD [86], SWARM-MD [87], coarse extended Lagrangian dynamics [88, 89], and simple averaging over multiple trajectories [90, 91]. It is fitting to close this perspective citing from a 1980 article entitled “Computer-aided analysis of nonlinear problems in transport phenomena” by Brown, Scriven and Silliman [30]: The nonlinear partial differential equations of mass, momentum, energy, species and charge transport, especially in two and three dimensions, can be solved in terms of functions of limited differentiability – no more than the physics warrants – rather than the analytical functions of classical analysis. . . . Organizing the polynomials in the so-called finite element basis functions facilitates generating and analyzing solutions by large, fast computers employing modern matrix techniques”. These sentences celebrate the transition from analytical solutions (of explicitly available equations) to computer-assisted solutions. The solutions are not analytically available for our class of complex/multiscale problems either; but now the equations themselves are not available, and they are solved in a computerassisted fashion using appropriate computational experiments at a different level of system description. The similarity of the list of important elements is remarkable: The right basis functions, dictated by the physics (discretizations of the right coarse observation variables); large, fast computers (now

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massively parallel clusters, each CPU computing one realization of trajectories for the same “coarse” initial condition); and modern matrix techniques (now matrix-free iterative linear algebra). The approach bridges traditional numerical analysis, computational experimentation with the microscopic simulator, and systems theory; its most vital element is the simple fact that a code can be initialized at will. If one has good macroscopic equations, one should use them. But when these equations are not available in closed form (and such cases arise with increasing frequency in contemporary modeling) the equation-free computational enabling technology we outlined here may hold the key to the engineering of effectively simple systems.

Acknowledgments This work was partially supported over the years by AFOSR, through an NSF/ITR grant, DARPA and Princeton University. A somewhat shortened version of this article has appeared as a Perspective in the July 2004 issue of the AIChE Journal.

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[47] A. Armaou and I.G. Kevrekidis, “Optimal switching policies using coarse timesteppers,” Proceedings of the 2003 CDC Conference, Hawaii, can be obtained as nlin.CG/0309024 at arXiv.org, 2003. [48] C.I. Siettos, N. Kazantzis, and I.G. Kevrekidis, “Coarse feedback linearization using timesteppers,” Submitted to Int. J. Bifurcations and Chaos, 2004. [49] C.I. Siettos, D. Maroudas, and I.G. Kevrekidis, “Coarse bifurcation diagrams via microscopic simulators: a state-feedback control-based approach,” Int. J. Bif. Chaos, 14(1), 207, 2004. [50] C.W. Gear and I.G. Kevrekidis, “Computing in the past with forward integration,” Phys. Lett. A, 321, 335, 2004. [51] L. Chen, P.G. Debenedetti, C.W. Gear, and I.G. Kevrekidis, “From molecular dynamics to coarse self-similar solutions: a simple example using equation-free computation,” J. Non-Newtonian Fluid Mech., 120, 215, 2004. [52] C.I. Siettos, C.C. Pantelides, and I.G. Kevrekidis, “Enabling dynamic process simulators to perform alternative tasks: a time-stepper based toolkit for computer-aided analysis,” Ind. Eng. Chem. Res., 42(26), 6795, 2003. [53] O. Runborg, I.G. Kevrekidis, and K. Theodoropoulos, “Effective stability and bifurcation analysis: a time stepper based approach,” Nonlinearity, 15, 491, 2002. [54] J. Moeller, O. Runborg, P.G. Kevrekidis, K. Lust, and I.G. Kevrekidis, “Effective equations for discrete systems: a time stepper based approach,” in press, Int. J. Bifurcations and Chaos, 2005. [55] C. Schuette, A. Fischer, W. Huisinga, and P. Deuflhard, “A direct approach to conformational dynamics based on hybrid Monte Carlo,” J. Comp. Phys., 151, 146, 1999. [56] R. Zwanzig, Nonequilibrium Statistical Mechanics, Oxford University Press, New York, 2001. [57] P. Haenggi, P. Talkner, and M. Borkovec, “Reaction-rate theory: 50 years After Kramers,” Rev. Mod. Phys., 62(2), 251, 1990. [58] R. Kupferman and A. Stuart, “Fitting SDE models to nonlinear Kac-Zwanzig heat bath models,” Phys. D, in press, 2005. [59] S. Pope, “Computationally efficient implementation of combustion chemistry using ins situ adaptive tabulation,” Comb. Theory Model., 1, 41, also Beam Technologies Inc, ISAT-CK Users’ Guide (Release 1.0), 1998. Beam Technologies Inc., Ithaca, NY, 1997. [60] D.G. Luenberger, “Observing the state of a linear system,” IEEE Trans. Military Electronics, 8, 74, 1964. [61] A.J. Krener, Nonlinear observers in control systems, robotics and automation. In: H. Unbehauen (ed.), Encyclopedie of Life Support Systems (EOLSS), Eolss Publishers, Oxford, 2003. [62] B.C. Moore, “Principal component analysis in linear systems: controllability, observability and model readuction,” IEEE Trans. Automatic Control, 26(1), 17, 1981. [63] S. Lall, J.E. Marsden, and S. Glavaski, “A subspace approach to balanced truncation for model reduction of nonlinear control systems,” Int. J. Robust Nonlinear Control, 12, 519, 2002. [64] R.E. Kalman and R.S. Bucy, “New results in linear filtering and prediction theory,” Trans. ASME, Part D, J. Basic Eng., 83, 95, 1961. [65] M. Melchior and H.C. Oettinger, “Variance reduced simulations of stochastic differential equations,” J. Chem. Phys., 103(21), 9506, 1995. [66] J. Li, P.G. Kevrekidis, C.W. Gear, and I.G. Kevrekidis, “Deciding the nature of the coarse equation through microscopic simulation,” SIAM MMS, 1(3), 391, 2003.

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[67] D. Kopelevich, A.Z. Panagiotopoulos, and I.G. Kevrekidis, “Coarse grained computations for a micellar system,” in press, 2005. [68] D. Kopelevich, A.Z. Panagiotopoulos, and I.G. Kevrekidis, “Coarse kinetic approach to rare events: application to micelle formation,” in press, J. Chem. Phys., 2005. [69] T. Sakurai, E. Mihaliuk, F. Chirila, and K. Showalter, “Design and control of wave propagation patterns in excitable media,” Science, 296 , 2009, 2002. [70] J. Wolff, A.G. Papathanasiou, I.G. Kevrekidis, H.H. Rotermund, and G. Ertl, “Spatiotemporal addressing of surface activity,” Science, 294, 134, 2001. [71] D. Semwogerere and M.F. Schatz, “Evolution of hexagonal patterns from controlled initial conditions in a Benard-Marangoni convection experiment,” Phys. Rev. Lett., 88, 054501, 2002. [72] D.G. Grier, “A revolution in optical manipulation,” Nature, 424, 810, 2003. [73] W.D. Ristenpart, I.A. Aksay, and D.A. Saville, “Electrically guided assembly of planar superlattices in binary colloidal suspensions,” Phys. Rev. Lett., 90, 12, 2003. [74] R. Phillips, Crystals, Defects and Microstructures, Cambridge University Press, Cambridge, 2001. [75] M. Ortiz and R. Phillips, “Nanomechanics of defects in solids,” Adv. Appl. Mech., 36, 1, 1999. [76] A. Chorin, A. Kast, and R. Kupferman, “Optimal prediction for underresolved dynamics,” Proc. Natl Acad. Sci. USA, 95, 4094, 1998. [77] A. Chorin, O. Hald, and R. Kupferman, “Optimal prediction and the Mori–Zwanzig representation of irreversible processes,” Proc. Natl Acad. Sci. USA, 97, 2968, 2000. [78] H.C. Oettinger, Stochastic Processes in Polymeric Fluids, Springer Verlag, New York, 1996. [79] M. Laso and H.-C. Oettinger, “Calculation of viscoelastic flow using molecular models: the CONNFFESSIT approach,” JNNFM, 47, 1, 1993. [80] K. Xu and K. Prendergast, “Numerical Navier–Stokes from gask kinetic theory,” J. Comp. Phys., 114, 9, 1994. [81] K. Xu, “A Gas-kinetic BGK scheme for the Navier–Stokes equations and its connection with artificial dissipation and the Godunov method,” J. Comp. Phys., 171, 289, 2001. [82] W.E. and B. Engquist, “The heterogeneous multiscale methods,” Commun. Math. Sci., 1, 87, 2003. [83] M.A. Katsoulakis, A.J. Majda, and D.G. Vlachos, “Coarse grained stochastic processes for microscopic lattice systems,” Proc. Natl. Acad. Sci. USA, 100(3), 782, 2003. [84] D. Givon, R. Kupferman, and A. Stuart, “Extracting macroscopic dynamics: model problems and algorithms,” Submitted to Nonlinearity, can be obtained as Warwick Preprint 11/2003, http://www.maths.warwick.ac.uk/ ∼stuart/extract.pdf, 2003. [85] H. Grubmueller, “Predicting slow structural transitions in macromolecular systems: conformational flooding,” Phys. Rev. E., 52(3), 2893, 1995. [86] A.F. Voter, “Parallel replica method for dynamics of infrequent events,” Phys. Rev. B, 57(22), R13985, 1998. [87] T. Huber and W.F. van Gunsteren, “SWARM-MD: searching conformational space by cooperative molecular dynamics,” J. Chem. Phys. A., 102(29), 5937, 1998. [88] M. Iannuzzi, A. Laio, and M. Parrinello, “Efficient exploration of reactive potential energy surfaces using Car-Parrinello molecular dynamics,” Phys. Rev. Lett., 90(23), 238302, 2003.


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[89] A. Laio and M. Parrinello, “Escaping free energy minima,” Proc. Natl Acad. Sci. USA, 99(20), 12562, 2002. [90] I.C. Yeh and G. Hummer, “Peptide loop-closure kinetics from microsecond molecular dynamics simulations in explicit solvent,” JACS, 124(23), 6563, 2002. [91] C.D. Snow, N. Nguyen, V.S. Pande, and M. Gruebele, “Absolute comparison of simulated and experimental protein folding,” Nature, 420(6911), 102, 2002.

4.12 MATHEMATICAL STRATEGIES FOR THE COARSE-GRAINING OF MICROSCOPIC MODELS Markos A. Katsoulakis1 and Dionisios G. Vlachos2 1

Department of Mathematics and Statistics, University of Massachusetts - Amherst, Amherst, MA 01002, USA 2 Department of Chemical Engineering, Center for Catalytic Science and Technology, University of Delaware, Newark, DE 19716, USA

1.

Introduction

Spatial inhomogeneity at some small length scale is the rule rather than the exception in most physicochemical processes ranging from advanced materials’ synthesis, to catalysis, to self-assembly, to atmospheric science, to molecular biology. These inhomogeneities arise from thermal fluctuations and complex interactions between microscopic mechanisms underlying conservation laws. While nanometer inhomogeneity and its corresponding ensemble average behavior can be studied via molecular simulation, such as molecular dynamics (MD) and Monte Carlo (MC) techniques, mesoscale inhomogeneity is beyond the realm of available molecular models and simulations. Mesoscopic inhomogeneities are encountered in self-assembly, pattern formation on surfaces and in solution, standing and traveling waves, as well as in systems exposed to an external field that varies spatially over micrometer to centimeter length scales. It is this class of problems that require “large scale” mesoscopic or coarse-grained molecular models and where the developments described herein are applicable. It is desirable that such mesoscopic or coarse-grained models meet the following needs: • They are derived from microscopic ones to retain microscopic mechanisms and interactions and enable a truly first principles multi-scale approach; • They reach large length and time scales, which are currently unattainable by micro scopic molecular models; 1477 S. Yip (ed.), Handbook of Materials Modeling, 1477–1490. c 2005 Springer. Printed in the Netherlands.

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• They give the correct statistical mechanics limits; • They describe equilibrium as well as dynamic properties accurately; • They retain the correct noise of molecular models to ensure that phenomena, such as nucleation, phase transitions, pattern formation, etc. at larger scales are properly modeled; • They are amenable to mathematical analysis in order to assess the errors introduced during coarse-graining and enable optimized coarse-graining strategies to be developed. Toward these goals, recent work in Refs. [1–3] focused on developing a novel stochastic modeling and computational framework, capable of describing efficiently much larger length and time scales than conventional microscopic models and simulations. Here, we did not directly attempt to speed up microscopic simulation algorithms such as MD or MC. Instead, our perspective was to derive a hierarchy of new coarse-grained stochastic models – referred to as Coarse-Grained MC (CGMC) – ordered by the magnitude of space/time scales. This new set of models involves a reduced set of observables compared to the original microscopic models, incorporating microscopic details and noise, as well as the interaction of the unresolved degrees of freedom. The outline of this approach can be summarized in the following heuristic steps: 1. Coarse-grid selection. We select a computational grid (lattice) Lc (see Fig. la) which will be referred to as the “coarse-grid”. The microscopic processes describe much smaller scales by explicitly simulating atoms or molecules–“particles”–and are defined at the subgrid level: for example in Ref. [1] they are defined on a “microscopic” grid L (see Fig. lb and Section 3 below). 2. Coarse-grained Monte Carlo methods. Using the microscopic stochastic model as a starting point, we derive by carrying out a "stochastic closure" a coarser stochastic model for a reduced number of observables, set on Lc (see Fig. la). These new stochastic processes define in essence Coarse Lattice LC 1

2

3

4

5

6

...

m

adsorption desorption diffusion Fine Lattice L 1 2 3 4 5 6 7 ...q

Figure 1.

Coarse and fine grids (lattices) with absorption/desorption and surface diffusion.

Mathematical strategies of microscopic models

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coarse-grained MC algorithms, which rather than describing dynamics of a single microscopic particle as conventional MC do, they model the evolution of a coarse observable on Lc . The CGMC models span a hierarchy of length scales starting from the microscopic to the mesoscopic scales, and involve Markovian birth–death and generalized exclusion processes. A key feature of our coarse-graining procedure is that the full hierarchy of our derived stochastic dynamics satisfies detailed balance relations and as a result not only yields self-consistent random fluctuation mechanisms, but which are also consistent with the underlying microscopic fluctuations. To demonstrate the basic ideas, we consider as our microscopic model an Ising-type system. This class of stochastic processes is employed in the modeling of adsorption, desorption, reaction and diffusion of interacting chemical species on surfaces or through nanopores of materials in numerous areas such as catalysis and microporous materials, growth of materials, biological molecules, magnetism, etc. The fundamental principle on which this type of modeling is based on is the following: when the binding of species on a surface or within a pore is relatively strong, these physical processes can be described as jump (hopping) processes from one site to another or to the gasphase (Fig. lb) with a transition probability that can be calculated, to varying degrees of rigor, from even smaller scales using quantum mechanical calculations and/or transition state theory, or from detailed experiments, see for instance [4].

2.

Microscopic Lattice Models

Ising-type systems are set on a periodic lattice L which is a discretization of the interval I = [0, 1]. We divide I in N (micro)cells and consider the microscopic grid L = 1/N Z ∩ I in Fig. lb. Throughout this discussion we concentrate on one-dimensional models, however, our results extend easily (and perform better!) in higher dimensions. At each lattice site ie x ∈ L the order parameter σ (x) is allowed to take the values 0 and 1 describing vacant and occupied sites, respectively. The energy H of the system, evaluated at the configuration σ = {σ (x) : x ∈ L} is given by the Hamiltonian, 1 J (x − y)σ (x)σ (y)+ hσ (x), (1) H (σ ) = − 2 x∈L y =/ x where h = h(x), x ∈ L, is the external field and J is the inter-particle potential. Equilibrium states of the Ising model are described by the Gibbs states at a prescribed temperature T , µL,β (dσ ) = Z L−1 exp (−β H (σ )) PN (dσ ),

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where β = 1/kT and k is the Boltzmann constant and Z L is the partition function. Furthermore the product Bernoulli distribution PN (σ ) with mean 1/2 is the prior distribution on L. The inter-particle potentials J account for interactions between occupied sites. We consider symmetric potentials with finite range interactions where by the integer L we denote the total number of interacting neighboring sites of a given point on L. The interaction potential can be written as J (x − y) =

1 V L

N (x − y) , L

x, y ∈ L,

(2)

where V (r) = V (−r), and V (r) = 0, |r| ≥ 1, accounting for possible finite range interactions. Note that for V summable, the choice of the scaling factor 1/L in (1) implies the summability of the potential J , even when N, L → ∞. An additional condition required in order to obtain error estimates for the coarse-graining procedure is that V is smooth away from 0 and R |∂r V (r)| dr < ∞. The derivation of the interaction potentials can be carried out either from quantum mechanics calculations (e.g., RKKY interactions in micromagnetics [5]) or experimentaly. Sometimes potentials involve only nearest neighbors since further interactions can be neglected, in which case we obtain the classical Ising model. However in many applications interactions are significant over a large but finite number of neighbors (see for instance the experimental results in Ref. [6]), or even involve true long range interactions such as electrostatics or the RKKY-type exchange energies mentioned earlier. The dynamics of Ising-type models considered in the literature consists of order parameter flips and/or exchanges that correspond to different physical processes. More specifically a flip at the site x ∈ L is a spontaneous change in the order parameter, 1 is converted to 0 and vice versa, while a spin exchange between the neighboring sites x, y ∈ L is a spontaneous exchange of the order parameters at the two locations, 1 is converted to 0 and vice versa. For instance, a spin flip can model the desorption of a particle from a surface described by the lattice to the gas phase above and conversely the adsorption of a particle from the gas phase to the surface, see Fig. lb. Such a model has also been proposed recently in the atmospheric sciences literature for describing certain unresolved features of tropical convection [7, 8]. On the other hand spin exchanges describe the diffusion of particles on a lattice; in this case the presence of interactions typically gives rise to a non-Fickian macroscopic behavior [9–11]. These mechanisms are set-up as follows: if σ is the configuration prior to a flip at x, then we denote the configuration after the flip by σ x . When the configuration is σ , a flip occurs at x with a rate c(x, σ ), i.e., the order parameter at x changes, during the time interval [t, t + t] with probability c(x, σ )t. The resulting stochastic process {σt }t ≥ 0 is defined as a continuous time jump Markov process with generator defined in terms of the


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rate c(x, σ ), [12]. The imposed condition of detailed balance implies that the dynamics leave the Gibbs measure invariant and is equivalent to c(x, σ ) exp(−β H (σ )) = c(x, σ x ) exp(−β H (σ x )). The simplest type of dynamics satisfying the detailed balance condition is the Metropolis-type dynamics [13] where the energy barrier for desorption or diffusion depends only on the energy difference between the initial and final states. This type of dynamics are usually employed as MC relaxational algorithms for sampling from the equilibrium canonical Gibbs measure. However, in the context of physicochemical applications involving non-equilibrium evolution of interacting chemical species on surfaces or through nanopores of materials, it is more appropriate to consider dynamics where the activation energy of desorption or diffusion is the energy barrier a species has to overcome in jumping from one lattice site to another or to the gas phase. This type of dynamics is called Arrhenius dynamics and can be derived from MD or transition state theory calculations (see for instance Ref. [4]), to varying degrees of rigor and approximation. The fundamental idea here is that when the binding of species on a surface or within a pore is relatively strong, desorption and diffusion can be modeled as a hopping process from one site to another or to the gas phase, with a transition probability that depends on the potential energy surface. The Arrhenius rate for the adsorption/desorption mechanism is: c(x, σ ) = d0 (1 − σ (x)) + d0 σ (x) exp[−βU (x, σ )], where U (x, σ ) =

(3)

J (x − z)σ (z) − h(x),

z= / x,z∈L

is the total energy contribution from the particle interactions with the particle located at the site x ∈ L, as well as the external field h. Typically an additional term corresponding to the energy associated with the surface binding of the particle at x, can be also included in the external field h in U ; finally d0 is a rate constant that mathematically can be chosen arbitrarily but physically is related to the pre-exponential of the microscopic processes. Similarly we can define an Arrhenius mechanism for diffusion; in both cases the dynamics satisfy detailed balance.

3.

Coarse-grained Stochastic Processes and CGMC Algorithms

First we construct the coarse grid Lc by dividing I = [0, 1] in m equal size coarse cells (see Fig. la); in turn, each coarse cell is subdivided into q

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(micro)cells. Hence I is divided in N = mq cells and L = 1/mq Z ∩ I is the microscopic lattice in Fig. lb. Each coarse cell is denoted by Dk , k = 1, . . . , m and the coarse lattice corresponding to the coarse cell partition (Fig. la) is defined as Lc = 1/m Z ∩ I. We consider the integers k = 1, . . . , m as the unsealed lattice points of Lc , the coarse-grained stochastic processes defined below are set on Lc while the Ising model is set on the microscopic lattice L. Next we define a coarse-grained observable on the coarse lattice Lc . One such intuitive choice motivated by renormalization theory [14] is the average over each coarse cell Dk :

F(σt )(k) : =

σt (y),

k = 1, . . . , m.

(4)

y∈Dk

Although F(σt ) is not a Markov process, our goal here is to derive a Markov process ηt , defined on the coarse lattice Lc , approximating the true microscopic average F(σ ). Computationally this new process η is advantageous over the underlying microscopic σ , since it has a substantially smaller state space than σ and can be simulated much more efficiently. We next derive with a direct calculation from the microscopic stochastic process the exact coarse-grained rates for adsoprtion and desorption for the microscopic average F(σt ) in coarse cell Dk ; these rates are, respectively c¯a (k) : =

c(x, σ ) (1 − σ (x)),

x∈Dk

c¯d (k) : =

c(x, σ )σ (x).

(5)

x∈Dk

In the case of Arrhenius diffusion the exact jump rate from cell Dk to Dl of the microscopic average (4) is given by c¯diff (k) : =

c(x, y, σ )σ (x)(1 − σ (y)).

(6)

x∈Dk, y ∈Dt

The main goal here is to express these exact coarse-grained rates, up to a controlled error, as functions of the “mesoscopic” random variable F(σ ), rather than the microscopic σ. This step yields a Markov process that will approximate in a probability metric the microscopic average (4). We refer to this procedure as a closure in analogy to closure arguments in kinetic theory and the derivation of coarse-grained deterministic PDE from interacting particle systems as hydrodynamic limits [12]. However, here we carry out a stochastic closure that retains fluctuations of the microscopic system. We demonstrate these arguments only in the case of Arrhenius dynamics; full details including other dynamics can be found in Refs. [1–3]. For the adsorption/desorption case we define the coarse-grained birth– death Markov process η = {η(k) : k ∈ Lc } approximating (4), where the random variable η(k) ∈ {0, 1, . . . , q} counts the number of particles in each coarse cell Dk . Using the rate calculations above we obtain the update rate with which the


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value η(k) ≈ F(σ ) is increased by 1 (adsorption rate of a single particle in the coarse cell Dk ) and decreased by 1 (desorption in Dk ), respectively: ca (k, η) = d0 [q − η(k)],

cd (k, η) = d0 η(k) exp[−β U¯ (k)],

(7)

¯ As we show in where U¯ (l) = k∈Lc k=1 J¯(l, k)η(k) + J¯(0, 0)(η(l) − 1) − h(l). Katsoulakis et al. 2003a this new rate can be obtained from (5) with an error of the order O(q/L), when replacing F(σ ) ≈ η. Finally, the coarse-grained potential J¯ is defined by including the average of all contributions of pairwise microscopic interactions between coarse cells and within the same coarse cell,

J¯(k, l) = m 2

J (r − s) dr ds,

(8)

Dl ×Dk

where the area of Dl × Dk is equal to 1/m 2 . The coarse-grained external field h¯ is defined accordingly. Wavelets with vanishing moments can also be used in the construction of the coarse-grained potential [11, 15]. Similarly, in the Arrhenius diffusion case we obtain [3] the new rate cdiff (k → l, η) = q1 η(k)(q − η(l)) exp[−β(U0 + U¯ (k, η))],

(9)

describing the migration of a particle from the coarse cell Dk to cell Dl if k, I are nearest neighbors, and cdiff (k → l, η) = 0 otherwise; the generator for the Markov process ηt is defined analogously. A crucial step, which is special for the diffusion case, in obtaining (9) from (6) is the approximation of the local function σ (x)(1 − σ )) in (6) as a function of the coarse-grained variable η. This last step is trivial in the spin flip dynamics since such local functions in (5) are linear. Here we make the closure assumption that the particles are at local equilibrium inside each coarse cell Dk , we thus can replace σ (x) by q −1 η(k) (resp. σ (y) by q −1 η(l)). This last substitution somewhat parallels the “Replacement Lemma” in the interacting particle systems literature, necessary to obtain deterministic PDE as hydrodynamic limits: relative entropy estimates describing local equilibration of interacting particles allow to approximately rewrite local functions as a function of the coarse grained variables, see Ref. [16]. This analogy becomes precise in the discussion in Section 6 of the relative entropy error estimates, discussed below (18), between the microscopic processes σ and coarse-grained η. The invariant measure for the coarse-grained process {ηt }t ≥0 is a canonical Gibbs measure related to the original microscopic dynamics {σt }t ≥0: µm,q,β (dη) =

1 Z m,q,β

exp(−β H¯ (η))Pm,q (dη),

(10)

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where the product binomial distribution Pm,q (η), is the prior distribution arising from the microscopic prior by including q independent sites. Furthermore, H¯ is the coarse-grained Hamiltonian derived from the microscopic H ,

1 H¯ (η) = − 2 l∈L +

c k∈Lc k=1

J¯(0, 0) J¯(k, l)η(k)η(l) − η(l)(η(l) − 1) 2 l∈L c

¯ hη(k)

(11)

k∈Lc

The same-cell interaction term η(l)(η(l) − 1), yields the global mean field theory when the coarse-graining is performed beyond the interaction parameter L, as well as at the other extreme of q = 1 it is consistent with the Ising case. As a result we obtain a complete hierarchy of MC models-termed coarsegrained MC-spanning from Ising (q = 1) to mean field statistical mechanics limits where the latter does not include detailed interactions but includes noise, unlike the usual ODE mean field theories. Finally it can be easily shown both in the adsorption/desorption and the diffusion case that the condition of detailed balance for η with respect to the measure µm,q,β holds. Thus, combined mechanisms of diffusion, adsorption and desorption, which typically coexist in physical systems [17], can be modeled and simulated consistently for every coarse-graining level q. Detailed balance guarantees the proper inclusion of fluctuations in the coarse-grained model as they arise from the microscopies. This is justified in part by the form of the prior in (10), it is tested numerically in Refs. [1, 3] and it is proved rigorously by the loss of information estimate (18) below.

4.

Coarse-grained Monte Carlo Algorithms

The implementation of coarse-grained MC (CGMC), based on (7) and (9), is essentially identical to the microscopic MC [18] with a few differences. First, the inter-particle potential J is coarse-grained at the beginning of a simulation to represent interactions between particles within each cell (a feature absent in microscopic MC) as well as interactions with neighboring cells. Second, the order parameter is still an integer but varies between zero and q, instead of zero and one which is typical for microscopic MC. Otherwise, microscopic and coarse-grained algorithms are basically the same. Finally, we should comment about the significant computational savings resulting from coarse graining. For CGMC the CPU time in kinetic MC simulation with global update, i.e., searching the entire lattice to identify the chosen site, scales approximately as O(m 3 ) vs. O(N 3 ) for a conventional MC algorithm. In addition, coarse-grained potentials J¯ are compressed through the wavelet expansion (4) and thus additional savings are made in the calculation of energetics.


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Overall in the case of adsorption/desorption processes the CPU time can decrease for the same real time with increasing q approximately as O(1/q 2 ). For example, even a very modest 10-fold reduction in the number of sites (q = 10) results in reduced CPU by a factor of 102 , yielding a significant enhancement in performance. Thus, while for macroscopic size systems in the millimeter length scale or larger, microscopic MC simulations are impractical on a single processor, the computational savings of CGMC make it a suitable tool capable of capturing large scale features, while retaining microscopic information on intermolecular forces and particle fluctuations. CGMC can capture mesoscale morphological features by incorporating the noise correctly, as well as simulating large length scales. For instance we refer to the standing wave example for adsorption/desorption computed by CGMC in Ref. [2] in this case we employed an exact analytic solution for the average coverage as a rigorous benchmark for the CGMC computations. A striking difference between diffusion and adsorption/desorption processes simulations is that in the case of diffusion we also have coarse-graining in time by a factor q 2 . This is certainly intuitively clear if one considers the additional space covered by a single coarsegrained jump, which would take q microscopic jumps. We refer to Ref. [3] for theory and simulations justifying and demonstrating precisely this coarse-graining in time effect. In turn, this approach contributes to improving the hydrodynamic slowdown effect in conservative MC and results in additional CPU savings. Overall, for long potentials CPU savings of up to q 4 , occur for continuous time KMC simulation.

5.

Connections to Stochastic Mesoscopic Models and Their Simulation

In this section we discuss connections of CGMC with coarse-grained models involving Stochastic PDE (SPDE) derived mainly in the physics and more recently in the mathematics communities. These approaches involve a heuristic and in some cases a rigorous passage to the infinite lattice limit in averaged quantities such as (4). Then, under suitable conditions, random fluctuations in the microscopic average (4) are suppressed in analogy to the Law of large numbers, but are accounted for as corrections similarly to the Central Limit Theorem. In the end the limit of (4) is expected to solve a SPDE. A classical example of such a SPDE is the stochastic Cahn–Hilliard–Cook model [19], which takes the abstract form:

ct − ∇ · µ[c]∇

δ E[c] δc

1 − √ ∇ · { 2µ[c]W˙ } = 0, N

(12)

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where W˙ = (W˙ 1 (x, t), . . . , W˙ d (x, t)) is a space/time white noise, δ E[c]/δc is the variational derivative of the free energy

|∇c| + βh

c(y) dy +

2

E[c] = D

F(c(y)) dy.

(13)

Here F(c) is a double-well potential and µ[c] is the mobility of the system. In the case of Cahn–Hilliard–Cook models the mobility is typically µ[c] = 1, or µ[c] = c(1 − c). In Ref. [10] we derived a stochastic PDE of the type (12) as a mesoscopic theory for diffusion of molecules interacting with a long range potential for microscopic dynamics by studying the asymptotics of (4), as the the number of interacting neighbors L → ∞. The free energy in this case is β E[c] = − 2 +

V (y − y )c(y)c(y ) dy dy + βh

c(y) dy

r(c(y)) dy .

(14)

where r(c) = c log c + (1 − c) log (1 − c), and the mobility depends explicitely on the choice of microscopic dynamics:

µ[c] =

βc(1 − c), βc(1 − c) exp(−βV ∗ c),

Metropolis-type, Arrhenius

(15)

where * denotes the convolution of two functions. Here the derivation of the noise is not based on a central limit theorem-type of scaling, which would linearize (12) and will not account for the expected hysteresis and metastability. Instead, the noise term is “designed” so that: (a) as expected (12) will satisfy a fluctuation–dissipation relation and (b) yield the same large deviation functional and rare events as the microscopic spin exchange process. We refer to Ref. [20] for an overview of mesoscopic PDE-based theories for both diffusion and adsorption and desorption processes. The connection of CGMC with SPDE such as (12) can be readily seen even with an equilibrium calculation: formally the Gibbs states associated with this Langevin-type stochastic equation is given by the free energy E[c]. On the other hand in Ref. [1] 2003a we derived an asymptotic formula for the coarse-grained Gibbs measure (10) as q → ∞: µm,qβ (η0 ) =

1 Z m,q,β

exp −qm(E m,q (η0 ) + oq (1)) ,

(16)

where E m,q [C] = −

β ¯ βh 1 Ck + r(Ck ), V (k, l)Ck Cl + m k∈L m k∈L 2m L¯ l k c c (17)


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and J¯ = 1/L V¯ and L¯ = L/q is the coarse-grained potential length of J¯; we also define the average coverage at k ∈ Lc , Ck = λk /q, where η0 = (λ1 , λ2 , . . . , λm ), 0 ≤ λi ≤ q, and r(c) = c log c + (1 − c) log (1 − c). It is now clear that when the coarse-grained potential V¯ is long ranged (17) is merely a discrete version of the free energy (14). On the other hand if V¯ is a nearest neighbor potential then (17) yields a discrete version of the Ginzburg–Landau energy (13). In passing we remark that (16) also implies that for large q and m fixed, the most probable equilibrium configurations of the coarse-grained process ηt are given by the minimizers of the discrete free energy (17). A notable advantage of the CGMC methods over numerically solving Cahn–Hilliard–Cook type equations is the explicit connection to the microscopic system. While the connection with the underlying microscopic system is clear for the stochastic mesoscopic equations (12), (15) their derivation from microscopies is valid for L 1, which is not a strict requirement for our coarse-grained systems, as the estimate (18) demonstrates. From a mathematical perspective, due to the singular nature of the noise term, such SPDEs are expected to have only distributional, at best, solutions in dimensions more than 1. As a result, although direct simulation of (12), (see (15)), may have the advantage that PDE-based spectral methods can be used to surpass the hydrodynamic slowdown of MC algorithms, see Horntrop et al. 2001, they, however, require the careful handling of the highly singular noise term so that the scheme satisfies the detailed balance condition. For detailed adsorption/desorption mechanisms, it is not even clear which is the stochastic mesoscopic analogue of (12) that still satisfies detailed balance. On the other hand, CGMC includes fluctuations consistently with the detailed balance principle, allowing for the mesoscopic modeling of multiple simultaneous mechanisms such as particle diffusion, adsorption, desorption and reaction and always including properly stochastic fluctuations.

6.

The Numerical Analysis of CGMC: An Information Theory Approach

In this section we discuss the error analysis between microscopic models and CGMC in a more traditional numerical analysis sense. The error here represents the loss of information in the transition from the microscopic probability measure to the coarse-grained one. Such relative entropy estimates give a first mathematical reasoning for the parameter regimes (e.g., degree of coarse-graining) for which CGMC is expected to give errors within a certain tolerance. In Refs. [1, 3] we rigorously and computationally demonstrated that coarse-grained and microscopic processes share the same asymptotic mean

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behavior, i.e., that averages of the microscopic and coarse-grained processes solve the same mesoscopic deterministic PDE in the long-range interactions limit L → ∞. In addition to comparing the asymptotic mean behavior of coarse-grained and microscopic systems, we would like to understand how well and in what regimes CGMC captures the fluctuations of the microscopic system. As a first step in this direction, in numerical simulations in Ref. [2] we observed almost pathwise agreement between CGMC and microscopic MC simulations in the adsorption/desorption case when the level of coarse graining q was substantially smaller than L, e.g., q/L ≈ .25 and L = 40 (we note that in two dimensions potentials with just three lattice units long interactions have L about 30). These simulations suggested that in order to understand questions beyond the agreement in average behavior, we would like to have a comparison of the entire probability measures of the microscopic and CG processes. Our principal idea in this direction is to obtain a quantitative measure of controlling the loss of information during coarse-graining from finer to coarser scales: we consider the exact coarse graining of the microscopic Gibbs measure, µL,β oF(η) : = µL,β ({σ : F(σ ) = η}), where F is the projection operator from fine-to-coarse variables (4), and compare it to the Gibbs measure in CGMC (10). The relative entropy between the two measures provides a first quantitative estimate of the loss of information, during the coarse-graining process from finer to coarser scales, [3]: R(µm,q,β |µL,β oF) : = N

−1

log

η

= O

q . L

µm,q,β (η) µm,q,β (η) µL,β ({σ : F(σ ) = η}) (18)

Notice that the estimate (18) is on the specific entropy which is the relative entropy normalized with the size N of the microscopic system; the loss of information – however, small in each coarse cell – grows linearly with size as we take into account a growing number of cells. Relation (18) gives some initial mathematical intuition, at least at equilibrium, on how to rationally design a “good” CGMC algorithm, i.e., decide how to select the extent of coarse-graining q, given a potential J with a total number of interacting neighbors L and a desired accuracy. In fact, (18) is essentially a numerical analysis estimate between the exact solution of the microscopic system σ and the approximating CGMC η. Such estimates for the solution of a PDE and a corresponding finite element approximation are usually done in an L p or Sobolev norm. Here the relative entropy provides the analogue of a norm, without strictly being one. Furthermore, due to the Pinsker inequality [22], the estimate (18) implies an estimate on the total variation norm of the probability measures.


7.

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Conclusions

Here we provided an overview of the first steps taken in deriving a mathematically founded framework for coarse-graining of stochastic processes and associated kinetic Monte Carlo simulations. We have shown that coarsegrained models and simulations can reach larger scales while retaining information about the microscopic mechanisms and interaction potentials and the correct noise. Information theory methods have been introduced to assess the errors (loos of information) during coarse-graining. We believe that these tools will be essential to providing strategies for optimized coarse-graining designs. Concluding, we remark that while our focus has been on simple Ising type of models, the concepts introduced here can be extended to more complex systems. One such application to atmospheric sciences arises in Ref. [8], where CGMC models, coupled with the macroscopic fluid and thermodynamic equations, are used to parametrize underresolved (subgrid) features of tropical convection. Furthermore, in recent years there is a great interest in the polymer science and biology literature in coarse-graining atomistic models of polymer chains; we refer to the review article on coarse-graining by Muller-Plathe Ref. [22], for further discussion. In this context, coarse-graining is typically achieved by collecting a number of atoms (on the order of 10–20) in a polymer chain into a “super-atom” and semi-empirically/analytically fit parameters to a known potential type U¯ , e.g., Lennard–Jones, to derive the coarse-grained potential for the super-atoms. Other coarse-graining techniques in the polymer science literature including the bond fluctuation model and its variants share the perspective of the CGMC: an atomistic chain model is mapped on a lattice, where a super-atom occupies a lattice cell (similarly to the coarse-cells Dk in Section 2). All these coarse-grained models have to varying degrees the drawback that they rely on parameterized coarse potentials. Hence at different conditions (e.g., temperature, density, composition) need to be re-parameterized [23]. Furthermore, since they are not directly derived from the atomistic dynamics, it is not clear if they reproduce transport and dynamic properties such as melt viscosities. We hope that our methods can eventually provide a new mathematical framework to these approaches and a more systematic – if not completely mathematical – way to construct coarse-grained dynamics and potentials for such complex systems.

References [1] M.A. Katsoulakis, A.J. Majda, and D.G. Vlachos, J. Comp. Phys., 186, 250, 2003. [2] M.A. Katsoulakis, A.J. Majda, and D.G. Vlachos, Proc. Natl. Acad. Sci. USA, 100, 782, 2003. [3] M.A. Katsoulakis and D.G. Vlachos, J. Chem. Phys., 112, 18, 2003.

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[4] S.M. Auerbach, Int. Rev. Phys. Chem., 19, 155, 2000. [5] R.C. O’Handley, Modern Magnetic Materials: Principles and Applications, Wiley, New York, 2000. [6] S. Renisch, R. Schuster, J. Wintterlin, and C. Ertl, Phys. Rev. Lett., 82, 3839, 1999. [7] A.J. Majda and B. Khouider, Proc. Natl. Acad. Sci. USA, 99, 1123, 2002. [8] B. Khouider, B. Majda, A. J. and M.A. Katsoulakis, Proc. Natl. Acad. Sci. USA, 100, 11941, 2003. [9] G. Giacomin and J.L. Lebowitz, J.L., J. Stat. Phys., 87, 37, 1997. [10] D.G. Vlachos and M.A. Katsoulakis, Phys. Rev. Lett., 85, 3898, 2000. [11] R. Lam, T. Basak, D.G. Vlachos, and M.A. Katsoulakis, J. Chem. Phys., 115, 11278, 2001. [12] C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, Springer, New York, 1999. [13] B. Gidas, Topics in Contemporary Probability and its Applications, J. Laurie Snell (ed.), CRC Press, Boca Raton, 1995. [14] N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group, vol. 85, Addison-Wesley, New York, 1992. [15] A.E. Ismail, G.C. Rutledge, and G. Stephanopoulos, J. Chem. Phys., 118, 4414, 2003. [16] H.T. Yau, Lett. Math. Phys., 22, 63, 1991. [17] M. Hildebrand and A.S. Mikhailov, J. Phys. Chem., 100, 19089, 1996. [18] D.P. Landau and K. Binder, A Guide to Monte Carlo Simulations in Statistical Physics, Cambridge University Press, London, 2000. [19] H.E. Cook, Acta Metall., 18, 297, 1970. [20] M.A. Katsoulakis and D.G. Vlachos, IMA Vol. Math. Appl., 136, 179, 2003. [21] D.J. Horntrop, M.A. Katsoulakis, and D.G. Vlachos, J. Comp. Phys., 173, 361, 2001. [22] T.M. Cover and J.A. Thomas, J.A., Elements of Information Theory, Wiley, New York, 1991. [23] F. Muller-Plathe, Chem. Phys. Chem., 3, 754, 2002. [24] G. Beylkin, R. Coifman, and V. Rokhlin, Commun Pure Appl. Math., 44, 141, 1991. [25] M. Hildebrand, A.S. Mikhailov, and G. Ertl, Phys. Rev. E, 58, 5483, 1998. [26] M. Seul and D. Andelman, Science, 267, 476, 1995. [27] A.F. Voter and J.D. Doll, J. Chem. Phys., 82, 80, 1985.

4.13 MULTISCALE MODELING OF CRYSTALLINE SOLIDS Weinan E and Xiantao Li Program in Applied and Computational Mathematics, Princeton University

1.

Introduction

Multiscale modeling and computation has recently become one of the most active research areas in applied science. With rapidly growing computing power, we are increasingly more capable of modeling the details of physical processes. Nevertheless, we still face the challenge that the phenomena of interest are oftentimes the result of strong interaction between multiple spatial and temporal scales, and the physical processes are described by radically different models at different scales. The mechanical behavior of solids is a typical example that exhibits such a multiscale characteristic. At the fundamental level, everything about the solid can be attributed to the electronic structures which obey the Schrödinger equation. Atomic interactions and crystal structures can be described at the atomistic scale using molecular dynamics. Mechanical properties at the scale of the material are often modeled using continuum mechanics for which one speaks of stresses and strains. In between there are carious levels of mesoscales where one deals with defects such as grain boundaries, dislocation dynamics, and dislocation bundles. What makes the problem challenging is that these different scales are often strongly coupled with each other. Continuum models usually offer an efficient way of studying material properties. But they suffer from inadequate accuracy and the lack of microstructural information that tells us the microscopic mechanisms for why the material responds in the way it does. Atomistic models, on the other hand, allow us to probe the detailed crystalline and defect structure. However, the length and time scales of our interest are often far beyond what a full atomistic computation can reach. This is where multiscale modeling comes into play. The idea is that by coupling microscopic models such as molecular dynamics (MD) 1491 S. Yip (ed.), Handbook of Materials Modeling, 1491–1506. c 2005 Springer. Printed in the Netherlands.

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with macroscopic models such as continuum mechanics, one might be able to develop numerical tools that have the accuracy that is comparable with the microscopic model and the efficiently that is comparable to the macroscopic model. In this article, we will review some of the strategies that have been proposed for this purpose. We will focus on the coupling between molecular dynamics and continuum mechanics, although some of the strategies can be formulated in a more general setting. In addition, for simplicity we will concentrate on concurrent coupling methods that link different scales “on the fly”. Broadly speaking, concurrent coupling methods can be divided into two main categories, those based on energetic formulations and those based on dynamic formulations. We will discuss them separately.

2.

Energy-based Methods

At the atomistic scale, the deformation of the solid is described by the (displaced) positions of atoms that make up the solid. At zero temperature, the positions of the atoms are obtained by minimizing the total energy of the system, which consists of the potential energy due to the interaction of the atoms and the energy due to applied forces: E tot = E(x1 , . . . , x N ) −

f (x j )

(1)

j

Here x j denotes the displaced position of the jth atom. We will use x0j to denote its reference position which is taken to be the equilibrium position. u j = x j − x0j is the displacement of the jth atom. At the continuum level, the deformation of the solid is described by the displacement field u which also minimizes the total energy of the system that consists of the elastic energy caused by the deformation and the energy due to external forces:

ε (∇u − fex u) dx

(2)

Here ε is the strain energy density. Numerically this problem is solved by finite element methods on an appropriate triangulation {α } of the domain that defines the solid. In both cases, dynamics can be generated using Hamilton’s equation for the corresponding energy functional. Clearly the continuum approach is more efficient once we know the strain energy density. The conventional approach in continuum mechanics is to model this empirically using a combination of experimental data and analytical reasoning. Recently developed multiscale approach, on the other hand, aims

Multiscale modeling of crystalline solids

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at computing the strain energy directly based on the atomistic model. Next we will discuss several methods that have been developed for this purpose. To begin with, let Q be an appropriately defined operator that maps the microscopic configuration u j of the atoms to the macroscopic displacement field u. Then consistency between (1) and (2) implies that the strain energy should be given in terms of the atomistic model by, e[u] = min

Q{u j } = u

E tot .

(3)

However, this formula is quite impractical for numerical purpose since the number of atoms involved is often too large, and one has to come up with appropriate approximation procedures.

2.1.

QC – Quasicontinuum Method

One remarkably successful approach is the (quasicontinuum QC) method [1, 2]. QC is a way of simulating the macroscale nonlinear deformation of crystalline solids using molecular mechanics. It consists of three main components. • A finite element method on an adaptively generated mesh, which is automatically refined to the atomistic level near defects. Away from the defects, the mesh is coarsened to reflect the slow variation of the displacement field. • A kinematic constraint by which a subset of atoms, called representative atoms, are selected. The deformation of the other atoms are expressed in terms of the deformation of the representative atoms. This reduces the number of degrees of freedom in the problem. • A summation rule that computes an approximation to the total energy of the system by visiting only a small subset of the atoms. A simple example of the summation rule is the Cauchy–Born rule which computes the local energy by assuming the deformation is locally uniform. We now discuss these components in some detail. Ideally, in order to calculate the total energy, one needs to visit all the atoms in the domain: E tot =

N

E i (x1 , x2 , . . . , x N ).

(4)

i=1

Here E i is the energy contribution from site xi . The analytical form of E i depends on the empirical potential models in use. In practice, the computation of E i only involves neighboring atoms. In the region where the displacement field is smooth, keeping track of each individual atom is unnecessary. After

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selecting some representative atoms (repatoms), the displacement of the rest of the atoms can be approximated via linear interpolation, uj =

Nrep

Sα x0j uα ,

α=1

where the subscript α identifies the representative atoms, Sα is an appropriate weight function, Nrep being the number of repatoms involved. This step reduces the number of degrees of freedom. But to compute the total energy, in principle we still need to visit every atom. To reduce the computational complexity involved in computing the total energy, several summation rules are introduced. The simplest of these is to assume that the deformation gradient A = ∂x/∂x0 is uniform within each element: namely that the Cauchy–Born rule holds. The strain energy in the element k can be approximately written as ε (Ak ) |k | in terms of the strain energy density ε (A). With these approximations, the evaluation of the total energy is reduced to a summation over the finite elements, E tot ≈

Ne

ε (Ak ) |k |

(5)

k=1

where Ne is the number of elements. This formulation is called the local version of QC. The advantage of local QC is the great reduction of the degrees of freedom since Nrep N . In the presence of defects, the deformation tends to be non-smooth. Therefore, the approximation made in local QC will be inaccurate. A nonlocal version of QC has been developed which proposes to compute the energy with the following ansatz: E tot ∼

Nrep

n α E α (uα )

(6)

α=1

Here the weight {n α } is related to atom density. The energy from each repatom {E α } is computed by visiting its neighboring atoms, which are generated using the local deformation. Near defects such as cracks or dislocations, the finite element mesh is also refined to atomic scale to reflect the local deformation more accurately. Practical implementations usually combine both local nonlocal version of the method, and a criterion has been suggested to identify the local/nonlocal regions so that the whole procedure can be applied adaptively. Another version of QC, which is based on the force calculation, has been put forward in Ref. [3]. The methods generate clusters around the repatoms and perform the force calculation using the atoms within the clusters, see Fig. 1.


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Figure 1. Schematic illustration of QC (courtesy of M. Ortiz). Only atoms in the small cluster need to be visited during the computation.

QC has been successfully applied to a number of problems∗ including dislocation structure, nanoindentation, crack propagation, deformation twinning, etc. The use of local QC to control the far-field region and thus create a continuum environment for material defects has become more and more popular. In its simplest form, QC ignores atomic vibrations and thus the entropic effects. This restricts QC to static problems at zero temperature. Dynamics of defects can be studied in a quasistatic setting. Finite temperature can be incorporated perturbatively [2, 4].

2.2.

MAAD – Macro Atomistic Ab initio Dynamics

MAAD (Macro Atomistic Ab initio Dynamics) was proposed in Refs. [5, 6] to simulate crack propagation in Silicon. The computational domain is decomposed into three parts: the continuum region away from the crack tip where the

* For recently development and source code, see http://www.qcmethod.com.

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linear elasticity model is solved using a finite element method, an atomistic region near the crack tip on which molecular dynamics m j x¨ j = − ∇x j V,

j = 1, 2, . . . , Natom ,

(7)

with the Stillinger–Weber potential is used, an a quantum mechanical region at the crack tip where the tight binding model (TB) is used to model bond breaking. This is done by writing the Hamiltonian in the form Htot = HFE + HMD + HFE/MD + HTB + HMD/TB ,

(8)

which represents the energy contribution from different regions and the interface between them. For brevity we will explain the calculation of the first three terms. In the (finite element FE) region, the variables are the displacement field u, and the expression for the Hamiltonian is standard: HFE =

Ne 1 uT Kuk + u˙ kT M u˙ k 2 k=1 k

(9)

Here K and M are the stiffness and mass matrices. The stiffness matrix can be obtained from the harmonic approximation of the interatomic potential. In the case of finite (but constant) temperature, these parameters are adjusted accordingly to be consistent with the atomistic system in the MD region. The Hamiltonian in the MD region is simply the total energy: HMD =

atom 1 N m i u˙ 2i + V 2 i=1

(10)

where ui is the displacement of the ith atom, V is the total potential energy in the MD region. A key ingredient in this procedure is a handshaking scheme at the continuum/MD (and MD/TB) interface. Specifically, near the continuum/MD interface the finite elements are refined all the way to the atomistic level so that their vertices coincide with the reference atomistic positions at the interface. The handshaking Hamiltonian HFE/MD accounts for the interaction across the interface. The energy is computed from the continuum side and the MD side using, respectively, the formulase (9) and (10) with half and half weight for each. The continuum region and the atomistic region are then evolved simultaneously in time. Energy transport across the interface has been ignored. By refining the finite element mesh to atomistic scale at the interface, MAAD also avoids the issue of phonon reflection that we will discuss at the end of this article.


2.3.

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CGMD – Coarse-Grained Molecular Dynamics

Coarse-grained molecular dynamics is a systematic procedure for deriving the effective Hamiltonian for a set of coarse-grained variables from the microscopic Hamiltonian [7]. Starting from a microscopic Hamiltonian HMD defined on the phase space and defining coarse-grained variable by uµ =

f j µ u j , pµ =

j

f j µp j ,

(11)

j

where f j µ are appropriate weights, the effective Hamiltonian for the coarsegrained variables are obtained from 1 E(uµ , pµ ) = Z

HMD e− HMD , kB T dx j dp j

(12)

where

= µ δ uµ −

fkµ uk δ pu −

k

fkµ pk ,

k

and Z is a normalization constant, T is the temperature. Consistency with the coarse-grained variables is ensured through the presence of the delta functions, similar to the imposition of the kinematic constraint in QC. Equation (12) plays the role of (3) at finite temperature, with Q defined via (11). The basic assumption in this formalism is that the small scale component is at equilibrium given the coarse-grained variables. Strictly speaking this is only true if the relaxation times associated with the small scales are much shorter than that or the coarse-grained variables. In general the coarse-grained energy in (12) is still difficult to compute. It has been computed for the case of harmonic potential in Ref. [7] and more generally in Ref. [8].

3.

Dynamics-based Method

So far we have discussed energy based methods. In these methods, the key is to obtain a multiscale representation of the total energy of the system. In QC, this is done via the representative atoms and the summation rule. In MAAD, this is done by handshaking the atomistic and continuum regions through a gradual matching of the grids. In CGMD, this is done via thermodynamically integrating out the contribution of the small scales. Hamilton’s equation is applied to the reduced Hamiltonian in order to model dynamics.

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An alternative approach is to model dynamics directly. Equilibrium states are obtained as steady states of the dynamics. This is essential if energy transport is coupled with the dynamics. At the present time, this approach is much less developed compared with energy-based approaches discussed earlier. So far the only general strategy seems to be that of Li and E[9], which is based on the framework of the heterogeneous multiscale method (HMM) developed by E and Engquist [10]. This will be discussed next. We will also discuss a related topic, namely how to impose matching conditions at the atomistic–continuum interface.

3.1.

Heterogeneous Multiscale Method

In order to develop a general multiscale methodology that can handle both dynamics and finite temperature effects, Li and E [9] relied on the framework of the heterogeneous multiscale method (HMM), which has been used for designing multiscale methods for several different applications including fluids.† there are two major components in HMM. The selection of a macroscale solver and the estimation of the needed macroscale data using the microscale solver. In general the macroscale solver should be chosen to maximize the efficiency in resolving the macroscale behavior of the system and minimize the complexity of coupling with the microscale model. In the context of solids, our starting point for both the macroscale and microscale models are the universal conservation laws of mass, momentum and energy in Lagrangian coordinates:   ∂t A − ∇x0 v = 0, 

ρ0 ∂t v + ∇x0 · σ = 0, ρ0 ∂t e + ∇x0 · j = 0,

(13)

Here A, v, e are the deformation gradient, velocity and total energy per particle respectively, ρ0 is the density. At the macroscale level, e.g., continuum mechanics, σ is the first Piola–Kirchhoff stress tensor and j is the energy flux. The first equation in (13) is merely a compatibility statement. The second and third equation express conservation of momentum and energy, respectively. After combining with proper constitutive relations these equations can be used to model nonlinear elasticity, thermoelasticity and even plasticity. At the microscopic level, i.e., molecular dynamics, these conservation laws

† For other applications of HMM, visit http://www.math.Princeton.edu/multiscale.


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continue to hold with the stress and energy given in terms of the atomistic variables by, 

 σ˜ (x0 , t) = 12 f xi (t) − x j (t) ⊗ x0i − x0j    i= /j    1    × δ(x0 − (x0j + λ(x0i − x0j )))dλ,  0  ˜j (x0 , t) = 1 v i (t) + v j (t) · f x j − xi x0i − x0j   4  i= /j     1    × δ(x0 − (x0j + λ(x0i − x0j )))dλ,

(14)

0

Here for simplicity we only provided these expressions for the case when the

atomistic potential is simply a pair potential: V =1/2 i =/ j φ xi (t) − x j (t) and f = −∇φ. It is well-known that pair potentials are quite inadequate for modeling solids, but one can find the formulas for more general potentials in Ref. [9]. These conservation laws suggest a new coupling strategy in the HMM framework at the level of fluxes: the macroscopic variables can be used as constraints for the atomistic system, the needed constitutive data – the fluxes, can be obtained from results from the atomistic model via ensemble time averaging after the microscale system equilibrates. This is the method proposed in Ref. [9]. Compared with QC or CGMD, HMM is more of a top-down approach in that it starts with an incomplete macroscale model, and uses the microscale model as a supplement to provide the missing data, the fluxes. In QC or CGMD, one starts with a full atomistic description with all the physical details. A coarse graining procedure is then applied to remove the unnecessary data in order to arrive at a coarse-grained model. We next describe the details of the HMM procedure.

3.1.1. Macroscale solver Since the macroscale model is a conservation law, the macroscale solver is a method for conservation laws. Although there are plenty of methods available for conservation laws, e.g., Ref. [11], many of them involve the computation of the Jacobian for the flux functions, and this dramatically increases the computational complexity in a coupled multiscale method when the continuum equation is not explicitly known. An exception is the central scheme of Lax–Friedrichs type, such as Ref. [12], which is formulated over a staggered-grid. As it turns out, this method can be easily coupled with molecular dynamic simulations.

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We first write the conservation laws in the generic form, ut + fx = 0,

(15)

We will confine our discussion to one dimensional continuum models since the extension to higher dimension is straightforward. A (macro) staggered grid is laid out as in Fig. 2. First order central scheme represents the solutions by piece-wise constants, which are the average values over each cell: unk

1 = x

x k+1/2

u(x, t n )dx.

x k −1/2

Time integration over xk , xk + 1 × t n , t n + 1 leads to the following scheme, +1 unk + 1/2 =

t n unk + unk + 1 − fk + 1 − fnk , 2 x

(16)

tn+2 fn+1 k 1/2

fn+1 k+1/2

[]

[] un+1 k 1/2

un+1 k+1/2

tn+1 fnk

fnk 1

[]

[] unk 1

tn

xk1

[] unk

xk

fnk+1

unk+1 xk+1

Figure 2. A schematic illustration of the numerical procedure for one macro time step: starting from piecewise constant solutions {unk }, one integrates (15) in time and in the cell [xk , xk+1 ]. The time step t is chosen in such a way that the waves coming from xk+1/2 will not reach xk , and thus for t ∈ [t n , t n+1 ), u(xk , t) = unk .To obtain the local flux, we perform a MD simulation using unk as constraints. The needed flux is then extracted from the MD fluxes via time averaging.


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where fnk

1 = t

tn +1

f(xk , t)dt

tn

This is then approximated by numerical quadrature such as the mid-point formula. A simple choice is f kn ∼ f (xk , t n ). The stability of such a scheme, which usually manifests itself in the form of a constraint on the size of t, can be appreciated from considering the adiabatic case f = f(u): if we choose the time step t small enough, the waves generated from the cell interface {x k + 1/2} will not arrive at the grid point {xk }, and, therefore, the solution as well as the fluxes at the grid points will not change until the next time step. With this specific choice of the macro-solver, we can illustrate the HMM procedure schematically in Fig. 2. At each macro time step, the scheme (16) requires as input the fluxes at grid point xk to complete the time integration, These flux values are obtained by performing local MD simulations that are consistent with the local macroscale state (A, v, e). The Eq. (13) is then integrated to next time step using (16).

3.1.2. Reconstruction Next we discuss how to set up the atomistic simulation to estimate the local fluxes. The first step is to reconstruct initial MD configurations that are consistent with the local macro state variables (A, v, e). The shape of the MD cell, and hence the new basis, is set up from the local deformation tensor. For example if the undeformed cell has basis E, then the ˜ new basis is E=AE. Assuming the deformation is uniform within the cell, the new basis then determines the displacement of each atom. From the atomic positions we can compute the potential energy. After subtracting the potential energy and the kinetic energy associated with the mean velocity from the total energy e, we obtain the temperature by assuming that the remaining energy is due to thermal fluctuation. Using the mean velocity and temperature we initialize the velocity of the atoms by Maxwell distribution.

3.1.3. Boundary conditions Of central importance is the boundary condition imposed on the microscopic system in order to guarantee consistency with the local macroscale variables. In the case when the system is homogeneous, the most convenient boundary condition is the periodic boundary condition. The MD cell is first

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deformed according to the deformation gradient A. Then the cell is periodically extended to the whole space.

3.1.4. Estimating the data The needed macroscale fluxes are estimated from the MD results by time averaging. To reduce the transient effects, we use a kernel that puts less weight on the transient period, e.g., 1 A K = lim t →+∞ t

t 0

s K (1 − )A(s)ds, t

K (θ) = 1 − cos (2π θ ).

(17)

Experience suggests that using this kernel substantially improves the quality of the data than straightforward averaging.

3.1.5. Dealing with defects In the presence of defects, QC and MAAD refine the grid to atomic level to account for defect energy. This procedure is seamless but can become rather complicated in simulating dynamics. HMM instead suggest keeping the macro-grid (which might be locally refined) in the entire computational domain but performing a model refinement locally near the defects. Away from the defects, the fluxes are computed using the procedure described before, or if an empirical model is accurate enough, one can simply compute the fluxes using the empirical model. Near the defects there are two cases to consider depending on whether there is scale separation between the local relaxation time around the defects and the time scale for the dynamics of the defects In the absence of such a time scale separation, the molecular dynamics simulation around the defects will be kept for all times. This imposes a limitation on the time scales that can be accessed using such a procedure. But if the atomistic relaxation times can be very long, there is really little one can do other than following the history of the atomistic features near the defects. Macro-scale fluxes can still be computed from the micro-scale fluxes via time averaging. In this case, since the atomistic region near the defect is necessarily macroscopically inhomogeneous, the atomistic boundary conditions need to the modified. Li and E [9] proposes using a biased Andersen thermostate at a border region that takes into account both the local mean velocity and local temperature. Finally, the overall deformation is controlled by fixing the outmost atoms. In the case when there is time scale separation, this procedure can be much simplified. In this case one can build the defect dynamics directly into the macro-solver and the atomistic simulations can be localized in space and time


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to predict the velocity of the defects and stress near the defects. Such a defect tracking procedure is implemented for twin boundary dynamics in Ref. [9].

3.1.6. Atomistic–continuum interface condition One issue that has received a great deal of attention is the matching condition at the atomistic–continuum interface. In a coupled MD-continuum calculation, the MD region is meant to be vary small but inevitably at finite temperature. The phonons generated in the MD region need to be propagated out in order to keep the fluctuations in the MD region under control. This is achieved through imposing appropriate boundary conditions at the atomistic– continuum interface that limits phonon reflection. The first attempt for deriving such boundary conditions is found in Ref. [12]. Cai et al. suggested obtaining the exact linear response functions at the interface by precomputing. This strategy is in principle exact under the harmonic approximation. But it is often too expensive since the linear response functions (which are simply Green’s functions) are quite nonlocal. When the MD region changes as a result of defect dynamics, these functions will have to be computed again. Further work along this line was done later by Wagner et al. Ref. [13]. To achieve an optimal balance between efficiency and accuracy, a local method was formulated in E and Huang [14, 15] with the idea of minimizing phonon reflection, giving a pre-determined stencil for the boundary condition. To explain the optimal local matching conditions, we consider the one dimensional case where the continuum model is the simple wave equation, ∂ 2u ∂ 2u = ∂t 2 ∂ x 2 and its discrete form, − 2u nj + u n−1 u n+1 j j = u nj +1 − 2u nj −1, j ≥ 1. (18) 2 t These equations can be obtained by linear zing (7). For simplicity we consider the case when the atomistic region is in the semi-infinite domain defined by x > 0 and j =0 is the boundary. To prescribe the boundary condition we express u n0 as u n0 =

ak, j u n−k j ,

a0,0 = 0.

k, j ≥ 0

We start with a pre-determined set S of {k, j }’s outside of which we set ak, j = 0. The set S is the stencil that we choose. Choosing the right S is a very crucial step in this procedure. Large S will lead to an increase in the complexity of

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the algorithm. But small S may not be enough for the purpose of suppressing phonon reflection. Once S is selected, {ak, j }are chosen to minimize the total reflection in appropriate norm. The reflection coefficient, or more generally the reflection matrix can be obtained by looking for solutions in the form of u nj = ei(nωt + j ξ ) + R(ξ )ei(nωt − j ξ ) . Using (18), we obtain

ak, j ei( j ξ −kωt ) − 1 , −i( j ξ −kωt ) − 1 k, j ak, j e

R(ξ ) = −

k, j

(19)

where ω = ω(ξ ) is the dispersion coefficient satisfying ωt ξ 1 sin = sin . t 2 2 Similar calculation can be done for general crystal structures in which case the phonon spectrum may consist of several branches. Having R(ξ ), ak, j can be obtained by minimizing the total phonon reflection, π

min

W (ξ )R(ξ )|2 dξ,

0

with appropriately chosen weight function W . In addition constraints are needed at ξ = 0 in the form of R(0) = 0, R (0) = 0, . . . , to ensure accuracy at large scale. As example, if one uses only the terms a1,0 and a1,1 , and W =1 with R(0)=0 at the boundary, one has, + tu n−1 u n0 = (1 − t)u n−1 0 1 .

(20)

If instead one keeps the terms {a j,k, j ≤ 3, k ≤ 2}, the minimization leads to the following coefficients:

(a j,k ) =

1.95264 −0.074207 −0.014903 −0.95406 0.074904 0.015621

.

In order to get better performance at high wave number, more coefficients (larger S) have to be included. The method has been applied to dislocation dynamics in the Frenkel– Kontorova model and friction between rough crystal surfaces. It has shown promise in suppressing phonon reflection.


4.

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Summary

We have based our presentation on dividing multiscale methods into energybased and dynamics-based methods. From the viewpoint of coarse-graining, there are also two different set of ideas. The first set of ideas, used in QC, CGMD and HMM, is to pre-define a set of coarse-grained variables. By expressing the microscopic model in terms of the coarse-grained variables, one finds a relationship that express the macroscale data in terms of the microscopic quantities. In QC, this relationship is (3). In CGMD, this relationship is (12). In HMM, this relationship is (14). This relationship is the starting point of the micro-macro coupling. The second set of ideas, used in MAAD and E and Huang [14], is to divide the computational domain into macro and micro regions. Separate models are used in different regions and an explicit matching is used to bridging the two regions. Most existing work on multiscale modeling of solids deals with single crystal with isolated defects. Going beyond single crystals requires substantial work. Dealing with polycrystals with grain boundaries and plasticity with many interacting dislocations seem to require new ideas in coupling.

References [1] E.B. Tadmor, M. Ortiz, and R. Phillips, “Quasicontinuum analysis of defects in crystals,” Phil. Mag. A, 73, 1529, 1996. [2] R.E. Miller and E.B. Tadmor, “The quasicontinuum method: overview, applications and current directions,” J. Comput.-Aided Mater. Des., in press, 2003. [3] J. Knap and M. Ortiz, “An analysis of the quasicontinuum method,” J. Mech. Phys. Solid, 49, 1899, 2001. [4] V. Shenoy and R. Phillips, “Finite temperature quasicontinuum methods,” Mat. Res. Soc. Symp. Proc., 538, 465, 1999. [5] F.F. Abraham, J.Q. Broughton, N. Bernstein, and E. Kaxiras, “Spanning the continuum to quantum length scales in a dynamic simulation of brittle fracture,” Europhys. Lett., 44(6), 783, 1998. [6] J.Q. Broughton, F.F. Abraham, N. Bernstein, and E. Kaxiras, “Concurrent coupling of length scales: methodology and application,” Phys. Rev. B, 60(4), 2391, 1999. [7] R.E. Rudd and J.Q. Broughton, “Coarse-grained molecular dynamics and the atomic limit of finite element,” Phys. Rev. B, 58(10), R5893, 1998. [8] R.E. Rudd and J.Q. Broughton, Unpublished, 2000. [9] X.T. Li and W.E, “Multiscal modeling of solids,” Preprint, 2003. [10] W.E and B. Engquist, “The heterogeneous multi-scale methods,” Comm. Math. Sci., 1(1), 87, 2002. [11] E. Godlewski, and P.A. Raviart, Numerical Approximation of Hyperbolic systems of Conservation Laws, Springer-Verlag, New York, 1996. [12] H. Nessyahu and E. Tadmor, “Nonoscillatory central differencing for hyperbolic conservation laws,” J. Comp. Phys., 87(2), 408, 1990.

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[13] G.J. Wagner, G.K. Eduard, and W.K. Liu, Molecular Dynamics Boundary Conditions for Regular Crystal Lattice, Preprint, 2003. [14] W.E and Z. Huang, “Matching conditions in atomistic-continuum modeling of material,” Phys. Rev. Lett., 87(13), 135501, 2001. [15] W.E and Z. Huang, “A dynamic atomistic-continuum method for the simulation of crystalline material,” J. Comp. Phys., 182, 234, 2002.

4.14 MULTISCALE COMPUTATION OF FLUID FLOW IN HETEROGENEOUS MEDIA Thomas Y. Hou California Institute of Technology, Pasadena, CA, USA

There are many interesting physical problems that have multiscale solutions. These problems range from composite materials to wave propagation in random media, flow and transport through heterogeneous porous media, and turbulent flow. Computing these multiple scale solutions accurately presents a major challenge due to the wide range of scales in the solution. It is very expensive to resolve all the small scale features on a fine grid by direct num-erical simulations. A natural question is if it is possible to develop a multiscale computational method that captures the effect of small scales on the large scales using a coarse grid, but does not require resolving all the small scale features. Such multiscale method can offer significant computational savings. We use the immiscible two-phase flow in heterogeneous porous media and incompressible flow as examples to illustrate some key issues in designing multiscale computational methods for fluid flows. Two-phase flows have many applications in oil reservoir simulations and environmental science problems. Through the use of sophisticated geological and geostatistical modeling tools, engineers and geologists can now generate highly detailed, three-dimensional representations of reservoir properties. Such models can be particularly important for reservoir management, as fine scale details in formation properties, such as thin, high permeability layers or thin shale barriers, can dominate reservoir behavior. The direct use of these highly resolved models for reservoir simulation is not generally feasible because their fine level of detail (tens of millions grid blocks) places prohibitive demands on computational resources. Therefore, the ability to coarsen these highly resolved geologic models to levels of detail appropriate for reservoir simulation (tens of thousands grid blocks), while maintaining the integrity of the model for purpose of flow simulation (i.e., avoiding the loss of important details), is clearly needed. 1507 S. Yip (ed.), Handbook of Materials Modeling, 1507–1528. c 2005 Springer. Printed in the Netherlands.

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In recent years, we have introduced a multiscale finite element method (MsFEM) for solving partial differential equations with multiscale solutions [1–4]. This method has been demonstrated to be effective in upscaling two-phase flows in heterogeneous porous media. The main idea of this approach is to construct local multiscale finite element base functions that capture the small scale information within each element. The small scale information is then brought to the large scales through the coupling of the global stiffness matrix. Thus, the effect of small scales on the large scales is captured correctly. In our method, the base functions are constructed by solving the governing equation locally within each coarse grid element. The local construction of the multiscale base functions offers several computational advantages such as parallel computing and local adaptivity in computing the base functions. These advantages can be explored in upscaling a fine grid model. One of the central issues in many multiscale methods is to localize the subgrid small scale problems. In the context of the multiscale finite element method, it is the question of how to design proper microscopic boundary conditions for the local base functions. Naive choice of microscopic boundary conditions can lead to large errors. The nature of the numerical errors due to improperly chosen local boundary conditions depends on the type of the governing equation for the underlying physical problem. For elliptic or diffusion dominated problems, the effect of the numerical boundary layers is strongly localized. For convection dominated transport, the errors caused by the improper microscopic boundary condition can propagate long distance and pollute the large scale physical solution. Below we will discuss multiscale methods for these two types of problems in some details.

1.

Formulation and Background

The flow and transport problems in porous media are considered in a hierarchical level of approximation. At the microscale, the solute transport is governed by the convection–diffusion equation in a homogeneous fluid. However, for porous media, it is very difficult to obtain full information about the pore structure. Certain averaging procedure has to be carried out, and the porous medium becomes a continuum with certain macroscopic properties, such as the porosity and permeability. With the modern geostatistical techniques, one can routinely generate a fine grid model as large as tens of millions of grid blocks. As a first step, one has to upscale the fine grid model to a coarse grid model consisting of tens of thousands of coarse grid blocks but still preserve the integrity of the original fine grid model. Once the coarse grid model is obt-ained, it can be used many times with different boundary conditions or source distributions for the purpose of model validation and oil field management. This could reduce the computational cost significantly.

Multiscale computation of fluid flow in heterogeneous media

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We consider a heterogeneous system which represents two-phase immiscible flow. Our interest is in the effect of permeability heterogeneity on twophase flow. Therefore, we neglect the effect of compressibility and capillary pressure, and consider porosity to be constant. This system can be described by writing Darcy’s law for each phase (all quantities are dimensionless) vj =

krj (S) K ∇ p, µj

(1)

where vj are Darcy’s velocity for the phase j (j = o, w; oil, water), p is pressure, S is water saturation, K is the permeability tensor, krj is the relative permeabilities of each phase and µj is the viscosity of the phase j. Darcy’s law for each phase coupled with mass conservation, can be manipulated to give the pressure and saturation equations ∇ · (λ(S)K ∇ p) = 0, ∂S + u · ∇ f (S) = 0, ∂t

(2) (3)

which can be solved subject to some appropriate initial and boundary conditions. The parameters in the above equations are given by krw (S) kro (S) + , µw µo krw (S)/µw , f (S) = krw (S)/µw + kro /µo u = vw + vo = −λ(S)K ∇ p. λ=

(4) (5) (6)

Typically, the permeability tensor K in an oil reservoir model contains many or continuous spectrum of scales that are not separable. The variation in the permeability tensor is also very large, with the ratio between the maximum and minimum permeability being as large as 106 . This means that flow velocity can be very large near certain fast flow channels. To avoid time-stepping restriction associated with an explicit method, a full implicit time discretization is usually employed for the saturation equation. Moreover, the geometry of the computational domain is quite complicated. All these complications make it difficult to apply standard fast iterative methods such as the multigrid method to solve the large scale elliptic equation for pressure. In fact, solving the elliptic problem seems to consume most of the computational time in practice. Thus developing an efficient multiscale adaptive method for solving the elliptic problem becomes essential in oil reservoir simulations.

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2.

T.Y. Hou

Multiscale Finite Element Method

We first focus on developing an effective multiscale finite element method for solving the elliptic (pressure) equation with highly oscillating coefficients. We consider the following elliptic problem L u : = − ∇ · (a (x)∇u) = f in ,

u = 0 on ∂,

(7)

where a (x) = (aij (x)) is a positive definite matrix, is the physical domain and ∂ denotes the boundary of domain . This model equation represents a common difficulty shared by several physical problems. For flow in porous media, it is the pressure equation through Darcy’s law. The coefficient a ε represents the permeability tensor. For composite materials, it is the steady heat conduction equation and the coefficient a ε represents the thermal conductivity. The variational problem of (7) is to seek u ∈ H01 () such that a(u, v) = f (v), ∀v ∈ H01 (),

(8)

where a(u, v) =

aij

∂v ∂u dx and f (v) = ∂x i ∂x j

f v dx.

We have used the Einstein summation notation in the above formula. The Sobolev space H01 () consists of all functions whose mth derivatives (m = 0, 1) are L 2 integrable over and which vanish at the boundary of . A finite element method is obtained by restricting the weak formulation (8) to a finite dimensional subspace of H01 (). For 0 < h ≤ 1, let Kh be a partition of by a collection of triangular element K with diameter ≤ h. In each element K ∈ Kh , we define a set of nodal basis {φ Ki , i =1, . . . , d} with d being the number of nodes of the element. The subscript K will be neglected when bases in one element are considered. In our multiscale finite element method, the base function φ i is constructed by solving the homogeneous equation over each coarse grid element: L φ i = 0 in K ∈ Kh .

(9)

Let x j ( j = 1, . . . , d) be the nodal points of K . As usual, we require φ i (x j ) = δi j , where δi j = 1 if i = j , and δi j = 0 for i =/ j . One needs to specify the boundary condition of φ i to make (9) a well-posed problem. The simplest choice of the boundary condition for φ i is the linear boundary condition. For now, we assume that the base functions are continuous across the boundaries of the elements, so that the finite element solution space V h , which is spanned by the multiscale bases φ Ki is a subspace of H01 (), i.e.,

V h = span φ Ki : i = 1, . . . , d; K ∈ Kh ⊂ H01 ().


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Except for special cases when the coefficient aij has periodic structure or is separable in space variables, we in general need to compute the multiscale bases numerically using a subgrid mesh. The multiscale finite element method is to find the approximate solution of (8) in V h , i.e., to find u h ∈ V h such that a(u h , v) = f (v), ∀v ∈ V h .

(10)

In the case when a (x) = a(x, x/) with a(x, y) being periodic in y, we have proved that the multiscale finite element method gives a convergence result uniform in as tends to zero [2]. Moreover, the rate of convergence in the energy norm is of the form O h + + (/ h)1/2 . We remark that the idea of using base functions governed by the differential equations has been used in the finite element community see e.g., [5]. The multiscale finite element method presented here is also similar in spirit to the residual-free bubble finite element method [6] and the variational multiscale method [7].

3.

The Over-sampling Technique

The choice of boundary conditions in defining the multiscale bases plays a crucial role in approximating the multiscale solution. Intuitively, the boundary condition for the multiscale base function should reflect the multiscale oscillation of the solution u across the boundary of the coarse grid element. To gain insight, we first consider the special case of periodic microstructures, i.e., a (x) = a(x, x/), with a(x, y) being periodic in y. Using standard homogenization theory [8], we can perform multiscale expansion for the base function, φ , as follows (y = x/) φ = φ0 (x) + φ1 (x, y) + θ (x) + O( 2 ), where φ0 is the effective solution, φ1 is the first order corrector. The boundary corrector θ is chosen so that the boundary condition of φ on ∂ K is exactly satisfied by the first three terms in the expansion. By solving a periodic cell problem for χ j y · a(x, y) y χ j =

∂ ai j (x, y) ∂ yi

(11)

with zero mean, we can express the first order corrector φ1 as follows: φ1 (x, y) = − χ j ∂φ0 /∂x j . The boundary corrector, θ , then satisfies x · a(x, x/) x θ = 0 in K with boundary condition

θ ∂ K = φ1 (x, x/)∂ K .

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The oscillatory boundary condition of θ induces a numerical boundary layer, which leads to the so-called resonance error [1]. To avoid this resonance error, we need to incorporate the multidimensional oscillatory information through the cell problem into our boundary condition for φ . If we set φ |∂ K = (φ0 + φ1 (x, x/))|∂ K , then the boundary condition for θ |∂ K becomes identically equal to zero. Therefore, we have θ ≡ 0. In this case, we have an analytic expression for the multiscale base functions φ as follows φ = φ0 (x) + φ1 (x, x/),

(12)

where φ1 (x, y) = −χ j (x, y)∂φ0 /∂x j , χ j is the solution of the cell problem (11), and φ0 can be chosen as the standard linear finite element base. This set of multiscale bases avoid the boundary layer effect completely. The analytic form of the multiscale base function also gives a more efficient way to construct the multiscale base functions. Numerical experiments by Andrew Westhead demonstrate a clear first order convergence of this method without suffering from resonance error. For more details, see www.ama.caltech.edu/∼ westhead/MSFEM. However, for problems that do not have scale separation and periodic microstructure, we cannot use this approach to compute the multiscale base functions in general. Motivated by our convergence analysis, we propose an over-sampling method to overcome the difficulty due to scale resonance [1]. The idea is quite simple and easy to implement. Since the boundary layer in the first order corrector is thin, O(), we can first construct intermediate sample bases in a domain with size larger than h + . Here, h is the coarse grid mesh size and is the small scale in the solution. From these intermediate sample bases, we can construct the multiscale bases over the computational element, using only the interior information of the sample bases restricted to the computational element. Specifically, let ψ j be the base functions satisfying the homogeneous elliptic equation in the larger sample domain S ⊃ K . We then form the actual base φ i by linear combination of ψ j φi =

d

ci j ψ j .

j =1

The coefficients ci j are determined by condition φ i (x j ) = δi j . The corresponding θ ε for φ i are now free of boundary layers. By doing this, we can reduce the influence of the boundary layer in the larger sample domain on the base functions significantly. As a consequence, we obtain an improved rate of convergence [1, 3].

4.

Convergence and Accuracy

To assess the accuracy of our multiscale method, we compare MsFEM with a traditional linear finite element method (LFEM for short) using a subgrid


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mesh, h s = h/M. The multiscale bases are computed using the same subgrid mesh. Note that MsFEM only captures the solution at the coarse grid h, while FEM tries to resolve the solution at the fine grid h s . Our extensive numerical experiments demonstrate that the accuracy of MsFEM on the coarse grid h is comparable to that of the corresponding well-resolved LFEM calculation at the same coarse grid. In some cases, MsFEM gives even more accurate results than LFEM. First, we demonstrate the convergence in the case when the coefficient has scale separation and periodic structure. In Table 1, we present the result for a(x/ε) =

2 + sin(2π x2 /ε) 2 + P sin(2πx1 ε) + (P = 1.8), 2 + P cos(2π x2 /ε) 2 + P sin(2π x1 /ε) f (x) = −1 and u|∂ = 0,

(13) (14)

where = [0, 1] × [0, 1]. We denote by N the number of coarse grid points along each dimension, i.e., N = 1/ h. The convergence of three different methods are compared for fixed ε/ h = 0.64, where “L” indicates that linear boundary condition is imposed on the multiscale base functions, “os” indicates the use of over-sampling, and LFEM stands for linear FEM. We see clearly the scale resonance in the results of MsFEM-L and the (almost) first-order convergence (i.e., no resonance) in MsFEM-os-L. Moreover, the errors of MsFEM-os-L are smaller than those of LFEM obtained on the fine grid. Next, we illustrate the convergence of the multiscale finite element method when the coefficient is random and has no scale separation nor periodic structure. In Fig. 1, we show the results for a log-normally distributed a ε . In this case, the effect of scale resonance shows clearly for MsFEM-L, i.e., the error increases as h approaches ε. Here ε ∼ 0.004 roughly equals the correlation length. Even the use of an oscillatory boundary conditions (MsFEMO), which is obtained by solving a reduced 1D problem along the edge of the element, does not help much in this case. On the other hand, MsFEM with over-sampling agrees very well with the well-resolved calculation. We have also applied the multiscale finite element method to study wave propagation in random media and singularly perturbed convection-dominated diffusion problems. For more details, see Refs. [9, 10]. Table 1. Convergence for periodic case N 16 32 64 128

ε 0.04 0.02 0.01 0.005

MsFEM-L ||E||l 2 Rate

MsFEM-os-L ||E||l 2 Rate

LFEM MN ||E||l 2

3.54e–4 3.90e–4 4.04e–4 4.10e–4

7.78e–5 3.83e–5 1.97e–5 1.03e–5

256 512 1024 2048

–0.14 –0.05 –0.02

1.02 0.96 0.94

1.34e–4 1.34e–4 1.34e–4 1.34e–4

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T.Y. Hou 1e⫺2

LFEM MFEM-O MFEM-L MFEM-os-L

l 2 -norm error

5e⫺3

1e⫺3

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64

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256

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Figure 1. The l 2 -norm error of the solutions using various schemes for a log-normally distributed permeability field.

5.

Recovery of Small Scale Solution from Coarse Grid Solution

To solve the transport equation in the two-phase flows, we need to compute the velocity field from the elliptic equation for pressure, i.e., u = − λ(S)K ∇ p. In some applications involving isotropic media, the cell-averaged velocity is sufficient, as shown by some computations using the local upscaling methods [11]. However, for anisotropic media, especially layered ones (Fig. 2), the velocity in some thin channels can be much higher than the cell average, and these channels often have dominant effects on the transport solutions. In this case, the information about fine scale velocity becomes vitally important. Therefore, an important question for all upscaling methods is how to take those fast-flow channels into account. For MsFEM, the fine scale velocity can be easily recovered from the multiscale base functions, which provide interpolations from the coarse h-grid to the fine h s -grid. To illustrate that we can recover the fine grid velocity field from the coarse grid pressure calculation, we use the layered medium which is plotted in Fig. 2. We compare the computations of the horizontal velocity fields obtained by two methods. In Fig. 3a, we plot the horizontal velocity field obtained by using a fine grid (N = 1024) calculation. In Fig. 3b, we plot the same horizontal velocity field obtained by using the coarse grid pressure calculation with a coarse grid (N = 64) and using the multiscale finite element bases to interpolate the fine grid velocity field. We can see that the recovered velocity field captures very well the layer structure in the fine grid velocity


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Figure 3. (a) Fine grid horizontal velocity field, N = 1024. (b) Recovered horizontal velocity field from the coarse grid calculation (N = 64) using multiscale bases.

field. Further, we use the recovered fine grid velocity field to compute the saturation in time. In Fig. 4a, we plot the saturation at t =0.06 obtained by the fine grid calculation. Figure 4b shows the corresponding saturation obtained using the recovered velocity field from the coarse grid calculation. Most of detailed fine scale fingering structures in the well-resolved saturation are captured very well by the corresponding calculation using the recovered velocity field from the coarse grid pressure calculation. The agreement is quite striking. We also check the fractional flow curves obtained by the two calculations. The fractional flow of the red fluid, defined as F = Sred u 1 dy/ u 1 dy (S being

1516 (a)

T.Y. Hou (b)

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Figure 4. (a) Fine grid saturation at t = 0.06, N = 1024. (b) Saturation computed using the recovered velocity field from the coarse grid calculation (N = 64) using multiscale bases.

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Figure 5. Variation of fractional flow with time. DNS: well-resolved direct numerical solution using LFEM (N = 512). MsFEM: over-sampling is used (N = 64, M = 8).

the saturation, u 1 being the horizontal velocity component), at the right boundary is shown in Fig. 5. The top pair of curves are the solutions of the transport problem using the cell-averaged velocity obtained from a well-resolved solution and from MsFEM; the bottom pair are solutions using well-resolved fine scale velocity and the recovered fine scale velocity from the MsFEM calculation. Two conclusions can be made from the comparisons. First, the


1517

cell-averaged velocity may lead to a large error in the solution of the transport equation. Second, both the recovered fine scale velocity and the cell-averaged velocity obtained from MsFEM give faithful reproductions of respective direct numerical solutions. We remark that a finite volume version of the multiscale finite element method has been developed by Jenny et al. [12]. They also found that by updating the multiscale bases adaptively in space and time, they can approximate the well-resolved solution accurately. The percentage of the multiscale bases that need to be updated is small, only a few percent of the total number of bases [13]. In some sense, the multiscale finite element method also offers an efficient approach to capture the fine scale details using only a small fraction of the computational time required for a direct numerical simulation using a fine grid.

6.

Scale-up of Two-phase Flows

The multiscale finite element method has been used in conjunction with some moment closure models to obtain an upscaled method for two-phase flows. In many oil reservoir applications, capillary pressure effect is so small that it is neglected in practice. Upscaling a convection dominated transport is difficult due to the nonlocal memory effect [14]. Here we use the upscaling method proposed in [15] to design an overall coarse grid model for the transport equation. In its simplest form, neglecting the effect of gravity, compressibility, capillary pressure, and considering constant porosity and unit mobility, the governing equations for the flow transport in highly heterogeneous porous media can be described by the following partial differential equations ∇ · (K (x)∇ p) = 0, ∂S + u · ∇ S = 0, ∂t

(15) (16)

where p is the pressure, S is the water saturation, K (x) = (Kij (x)) is the relative permeability tensor, and u = − K(x)∇ p is the Darcy velocity. The work of Efendiev et al. [15] for upscaling the saturation equation involves a moment closure argument. The velocity and the saturation are separated into a local mean quantity and a small scale perturbation with zero mean. For example, the Darcy velocity is expressed as u = u0 + u in (16), where u0 is the average of velocity u over each coarse element, u is the deviation of the fine scale velocity from its coarse scale average. If one ignores the third order terms containing the fluctuations of velocity and saturation, one can obtain an

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T.Y. Hou

average equation for the saturation S as follows

∂S ∂ + u0 · ∇ S = ∂t ∂x i

∂S Di j (x, t) , ∂x j

(17)

where the diffusion coefficients Di j (x, t) are defined by Dii (x, t) = |ui (x)| L 0i (x, t),

Di j (x, t) = 0, for i=/ j,

where |ui (x)| stands for the average of |ui (x)| over each coarse element. The function L 0i (x, t) is the length of the coarse grid streamline in the xi direction which starts at time t at point x, i.e., L 0i (x, t)

t

=

yi (s)ds,

0

where y(s) is the solution of the following system of ODEs dy(s) = u0 (y(s)), y(t) = x. ds Note that the hyperbolic equation (16) is now replaced by a convection– diffusion equation. One should note that the induced diffusion term is history dependent. In some sense, it captures the nonlocal history dependent memory effect described by Tartar in the simple shear flow problem [14]. The multiscale finite element method can be readily combined with the above upscaling model for the saturation equation. The local fine grid velocity u can be reconstructed from the multiscale finite element bases. We perform a coarse grid computation of the above algorithm on the coarse 64 × 64 mesh using a mixed multiscale finite element method [4]. The fractional flow curve using the above algorithm is depicted in Fig. 6. It gives excellent agreement with the “exact” fractional flow curve which is obtained using a fine 1024 × 1024 mesh. Upscaling the two-phase flow is more difficult due to the dynamic coupling between the pressure and the saturation. One important observation is that the fluctuation in saturation is relatively small away from the oil/water interface. In this region, the multiscale bases are essentially the same as those generated by the corresponding one-phase flow (i.e., λ = 1). These base functions are time independent. In practice, we can design an adaptive strategy to update the multiscale bases in space and time. The percentage of multiscale bases that need to be updated is relatively small (a few percent of the total number of the bases) [13]. The base functions that need to be updated are mostly near the interface separating the oil from the water. For those coarse grid cells far from the interface, there is little change in mobility dynamically. The upscaling of


1519

1

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t Figure 6. The accuracy of the coarse grid algorithm. Solid line is the well-resolved fractional flow curve. The slash-dotted line is the fractional flow curve using above coarse grid algorithm.

the saturation equation based on moment closure argument can be generalized to the two-phase flow with the enhanced diffusivity depending on the local small scale velocity field [15]. As we mentioned before, the fluctuation of the velocity field u can be accurately recovered from the coarse grid computation by using local multiscale bases.

7.

Multiscale Analysis for Incompressible Flow

The upscaling of the nonlinear transport equation in two-phase flows shares some of the common difficulties in deriving the effective equations for incompressible flow at high Reynolds number. The understanding of scale interactions for 3D incompressible flow has been a major challenge. For high Reynolds number flow, the degrees of freedom are so high that it is almost impossible to resolve all small scales by direct numerical simulations. Deriving an effective equation for the large scale solution is very useful in engineering applications, see e.g., [16, 17]. In deriving a large eddy simulation model, one usually needs to make certain closure assumptions. The accuracy of such closure models is hard to measure a priori. It varies from application to application. For many engineering applications, it is desirable to design a subgrid-based large scale model in a systematic way so that we can measure and control the modeling error. However, the strong nonlinear interaction of small scales and the lack of scale separation make it difficult to derive an effective equation.

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T.Y. Hou

We consider the incompressible Navier-Stokes equation ut + (u · ∇)u = −∇ p + ν u , ∇ · u = 0,

(18) (19)

with multiscale initial data u (x, 0) = u0 (x). Here u (t, x) and p (t, x) are velocity and pressure, respectively, ν is viscosity. We use boldface letters to denote vector variables. For the time being, we do not consider the effect of boundary and assume that the solution is periodic with period 2π in each dimension. For incompressible flow at high Reynolds number, small scales are generated dynamically through nonlinear interactions. In general, there is no scale separation in the solution. However, by decomposing the physical solution into a lower frequency component and a high frequency component, we can formally express the solution as the sum of a large scale solution and a small scale component. This decomposition can be carried out easily in Fourier space. Further, by rearranging the order of summation in the Fourier transformation, we can express the initial condition in the following form

u (x, 0) = U(x) + W x,

x ,

where W(x, y) is periodic in y and has mean zero. Here represents the cut-off wavelength in the solution above which the solution is resolvable and below which the solution is unresolvable. We call this a reparameterization technique. The question of interest is how to derive a homogenized equation for the averaged velocity field for small but finite . If the viscosity coefficient ν is of order one, then it can be shown that the high frequency oscillations will be damped out quickly in O() time. Even with ν = O(), the cell viscosity will be of order one and the oscillatory component of the velocity field is of order O(). In order for the oscillatory component of the velocity field persists in time, we need to have ν = O( 2 ). In this case, the cell viscosity is zero to the leading order. Since we are interested in the convection dominated transport, we set ν = 0 and consider only the incompressible Euler equation. The homogenization of the Euler equation with oscillating data was first studied by McLaughlin–Papanicolaou–Pironneau (MPP for short) [18]. In Ref. [18], MPP made an important assumption that the small scale oscillation is convected by the mean flow. Based on this assumption, they made the following multiscale expansion for velocity and pressure

t θ(t, x) t θ(t, x) + u1 t, x, , + ··· u (t, x) = u(t, x) + w t, x, , t θ(t, x) t θ(t, x) + p1 t, x, , + ··· p (t, x) = p(t, x) + q t, x, ,


1521

where w(t, x, τ, y), u1 (t, x, τ, y), q, and p1 are assumed to be periodic in both y and τ , and the phase θ is convected by the mean velocity field u ∂θ + u · ∇x θ = 0, θ(0, x) = x. (20) ∂t By substituting the above multiscale expansions into the Euler equation and equating coefficients of the same order, MPP obtained a homogenized equation for (u, p), and a periodic cell problem for (w(t, x, τ, y), q(t, x, τ, y)). On the other hand, it is not clear whether the resulting cell problem for w and q has a unique solution that is periodic in both y and τ . Additional assumptions were imposed on the solution of the cell problem in order to derive a variant of the k − model. The understanding of how small scale solution being propagated dynamically is clearly very important in deriving the homogenized equation. Motivated by the work of MPP, we have recently developed a multiscale analysis for the incompressible Euler equation with multiscale solutions [19, 20]. Our study shows that the small scale oscillations are convected by the full oscillatory velocity field, not just the mean velocity: ∂θ + u · ∇x θ = 0, θ (0, x) = x. (21) ∂t This is clear for the 2D Euler equation since vorticity, ω , is conserved along the characteristics θ (t, x) , ω (t, x) = ω0 θ (t, x), where ω0 (x, x/) is the initial vorticity, which is of order O(1/). Similar conclusion can be drawn for the 3D Euler equation. Now the multiscale structure of θ (x, t) is coupled to the multiscale structure of u . In some sense, we embed multiscale structure within multiscale expansions. It is quite a challenge to unfold the multiscale solution structure. Naive multiscale expansion for θ may lead to generation of infinite number of scales. Motivated by the above analysis, we look for multiscale expansions of the velocity field and the pressure of the following form u (t, x) = u(t, x) + w(t, θ(t, x), τ, y) + u(1) (t, θ(t, x), τ, y) + · · · , (22)

(1)

p (t, x) = p(t, x) + q(t, θ(t, x), τ, y) + p (t, θ(t, x), τ, y) + · · · , (23)

where τ = t/ and y = θ (t, x)/. We assume that w, and q have zero mean with respect to y. The phase function θ is defined in (21) and it has the following multiscale expansion: θ (1) θ = θ(t, x) + θ t, θ(t, x), τ, + ··· . (24)

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T.Y. Hou

This particular form of multiscale expansion was suggested by a corresponding Lagrangian multiscale analysis [19]. If one tried to expand θ naively as a function of x/ and t/, one would find that there is a generation of infinite number of scales at t > 0 and would not be able to obtain a well-posed cell problem. Expanding the Jacobian matrix, we get ∇x θ = B (0) +B (1) +· · · . Substituting the expansion into the Euler equation and matching the terms of the same order, we obtain the following homogenized equation

∂t u + u · ∇x u + ∇x · ww = −∇x p, u|t =0 = U(x), ∇x · u = 0,

(25) (26)

where ww stands for space-time average in (y, τ ), and ww stands for a matrix whose entry at the ith row and j th column is wi w j . The equation for w is given by

∂τ w + B (0) ∇y q = 0, τ > 0; (B

(0)

∇y ) · w = 0,

w|τ =0 = W(x, y),

(27) t = 0.

(28)

Moreover, we can derive the evolution equations for θ and θ(1) as follows

∂t θ + (u · ∇x )θ = 0, θ|t =0 = x,

(29)

∂τ θ(1) + (w · ∇x )θ = 0, θ(1)|τ =0 = 0.

(30)

From θ and θ(1) , we can compute the Jacobian matrix B (0) as follows: B (0) = (I − D y θ(1))−1 ∇x θ.

(31)

To check the convergence of our multiscale analysis, we compare the computational result obtained by solving the homogenized equation with that obtained by a well-resolved direct numerical simulation (DNS). Further, we use the first two terms in the multiscale expansion for the velocity field to reconstruct the fine grid velocity field. The initial velocity field is generated in Fourier space by imposing some power-law decay in the velocity field with a random phase perturbation in each Fourier mode. For this initial condition, we choose = 0.05. In Fig. 7a, we plot the initial horizontal velocity field in the fine mesh. The corresponding coarse grid velocity field is plotted in Fig. 7b. As we see in the spectrum plot in Fig. 9, there is no scale separation in the solution. We compare the computation obtained by the homogenized equation with that obtained by DNS at t = 0.5 in Fig. 8. We use the spectral interpolation to reconstruct the fine grid velocity field as a sum of the homogenized solution u and the cell velocity field w. We can see that the reconstructed velocity field (plotted only on the coarse grid) captures very well the fine grid velocity field obtained by DNS using a 512 × 512 grid. We also compare the accuracy

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Figure 8. Horizontal velocity fields at t = 0.5.

in the Fourier space, which is given in Fig. 9b. The agreement between the well-resolved solution and the reconstructed solution from the homogenized equation is excellent in both low frequencies and high frequencies. Further, we compare the mean velocity field obtained by the homogenized equation with that obtained by direct simulation using a low pass filter. The results are plotted in Figs. 10 and 11, respectively. We can see that the agreement between the two calculations is very good up to t = 1.0. Similar results are obtained for longer time calculations. The above multiscale analysis can be generalized to problems with general multiscale initial data without scale separation and periodic structure. This can be done by using the reparameterization technique in the Fourier space, which we described earlier for the initial velocity. This reparameterization technique

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T.Y. Hou

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Figure 10. Mean velocity fields at t = 1.0.

can be used repeatedly in time. The dynamic reparameterization also accounts for the dynamic interactions between the large and small scales. The difficulty associated with finding the local microscopic boundary condition can be overcome. Preliminary computational results show that the multiscale method can capture accurately the large scale solution and the spectral property of the small scale solution for a relatively long time computations. Our ultimate goal is to use the multiscale analysis to design an effective coarse grid model that can capture accurately the large scale behavior but with a computational cost comparable to the traditional large eddy simulation

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(LES) models [16, 17]. To achieve this, we need to take into account the special structures in the fully mixed flow, such as homogeneity and possible local self-similarity of the flow in the interior of the domain. When the flow is fully mixed, we expect that the Reynolds stress term, i.e., ww , reaches to a statistical equilibrium relatively fast. As a consequence, we may need to solve for the cell problem in τ for only a small number of time steps after updating the effective velocity in one coarse grid time step. Moreover, we need not solve the cell problem for every coarse grid for homogeneous flow. It should be sufficient to solve one or a few representative cell problems for fully mixed flow and use the solution of these representative cell solutions to compute the Reynolds stress term in the homogenized velocity equation. If this can be achieved, it would lead to a significant computational saving.

8.

Discussions

Multiscale methods offer several advantages over direct numerical simulations on a fine grid. First, the multiscale bases are very local. This makes it very easy to implement the method in parallel computing. Also the memory requirement is less stringent compared with direct numerical simulations since the base functions can be computed locally and independently. Secondly, we can use an effective adaptive strategy to update the multiscale bases only in the region that is needed. Thirdly, the multiscale methods offer an effective tool in deriving upscaled equations. In oil reservoir simulations, it is often the

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case that multiple simulations of the same reservoir model must be carried out in order to validate the fine grid reservoir model. After the upscaled model has been obtained, it can be used repeatedly with different boundary conditions and source distributions for management purpose. In this case, the cost of computing the multiscale base functions is just an over-head. If one can coarsen the fine grid by a factor of 10 in each dimension, the computational saving of the upscaled model over the original fine model could be as large as a factor 10 000 (three space dimensions plus time). It remains a great challenge to develop a systematic multiscale analysis to upscale the convection-dominated transport in heterogeneous media. While the upscaled saturation equation based on perturbation argument and moment closure approximation is simple and easy to implement, it is hard to estimate its modeling error as the fluctuations in velocity or saturation are not small in practice. New multiscale analysis need to be developed to account for the longrange interaction of small scales (the memory effect). Recently, we have developed a novel multiscale analysis for convection-dominated transport equation [2]. The analysis is based on a delicate multiscale analysis of the transport equation. The multiscale analysis for two-phase flows is not as complicated as that for the incompressible Euler equation. There is no need to introduce a multiscale phase function here, and the fast variable, y = x/, which characterizes the small scale solution, enters only as a parameter. This makes it easier for us to generalize the analysis to problems which do not have scale separation. We remark that there are other different approaches to multiscale problems, see e.g., [22–27]. Some of these methods assume that the media have periodic microstructures or scale separation, and explore these properties in their multiscale methods, while others use wavelet approximations, renormalization group techniques, and variational methods.

9.

Outlook

Looking forward, the main challenge in developing multiscale methods seems to be the lack of analytical tools in studying nonlinear dynamic problems that are convection-dominated and whose solutions do not have scale separation or periodic microstructures. For convection-dominated transport problems that do not have scale separation, it is very difficult to construct local multiscale base functions as we did for the elliptic-or diffusion-dominated problems. Incorrect local microscopic boundary conditions for the local multiscale base functions can lead to order one errors propagating down stream and create fluid dynamic instabilities. Systematic multiscale analysis needs to be carried out to account for the long-range interaction of small scales.


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To bridge the gap between the classical homogenization theory where scale separation is required and those practical applications where we do not have scale separation, we need to develop a new type of multiscale analysis. The new multiscale analysis should not require a large separation of scales. By using the dynamic reparameterization technique, we can always divide a multiscale solution into a large scale component and a small scale component. Interaction of the large scales and small scales can be effectively modeled by using a two-scale analysis for a short time increment. Then we use the reparameterization technique to decompose the solution again into a large scale component and a small scale component. Thus interaction of large and small scale solution occurs iteratively at every small time increment. Over a long time, we can account for interactions of all scales. We are currently pursuing this approach with the hope to develop a systematic multiscale analysis for incompressible flow at high Reynolds number.

References [1] T.Y. Hou and X. Wu, “A multiscale finite element method for elliptic problems in composite materials and porous media,” J. Comput. Phys., 134, 169–189, 1997. [2] T.Y. Hou, X. Wu, and Z. Cai, “Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients,” Math. Comput., 68, 913–943, 1999. [3] Y.R. Efendiev, T.Y. Hou, and X. Wu, “Convergence of a nonconforming multiscale finite element method,” SIAM J. Numer. Anal., 37, 888–910, 2000b. [4] Z. Chen and T. Hou, “A mixed finite element method for elliptic problems with rapidly oscillating coefficients,” Math. Comput., 72, 541–576, 2002. [5] I. Babuska, G. Caloz, and E. Osborn, “Special finite element methods for a class of second order elliptic problems with rough coefficients,” SIAM J. Numer. Anal., 31, 945–981, 1994. [6] F. Brezzi and A. Russo, “Choosing bubbles for advection-diffusion problems,” Math. Models Methods Appl. Sci., 4, 571–587, 1994. [7] T.J.R. Hughes, “Multiscale phenomena: Green’s functions, the Dirichlet-toNeumann formulation, subgrid scale models, bubbles and the origins of stabilized methods,” Comput. Methods Appl. Mech. Engrg., 127, 387–401, 1995. [8] A. Bensoussan, J.L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, 1st edn., North-Holland, Amsterdam, 1978. [9] T.Y. Hou, “Multiscale modeling and computation of incompressible flow,” In: J.M. Hill and R. Moore (eds.), Applied Mathematics Entering the 21st Century, Invited Talks from the ICIAM 2003 Congress, SIAM, Philadelphia, pp. 177–209, 2004. [10] P. Park and T.Y. Hou, “Multiscale numerical methods for singularly-perturbed convection–diffusion equations,” Int. J. Comput. Meth., 1(1), 17–65, 2004. [11] L.J. Durlofsky, “Numerical calculation of equivalent grid block permeability tensors for Heterogeneous porous media,” Water Resour. Res., 27, 699–708, 1991. [12] P. Jenny, S.H. Lee, and H. Tchelepi, “Multi-scale finite volume method for elliptic problems in subsurface flow simulation,” J. Comput. Phys., 187, 47–67, 2003.

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[13] P. Jenny, S.H. Lee, and H. Tchelepi, “Adaptive multi-scale finite volume method for multi-phase flow and transport in porous media,” Multiscale Model. Simul., 3, 50–64, 2005. [14] L. Tartar, “Nonlocal effects induced by homogenization,” In: F. Culumbini (ed.), PDE and Calculus of Variations , Birkhäuser, Boston, pp. 925–938, 1989. [15] Y.R. Efendiev, L.J. Durlofsky, and S.H. Lee, “Modeling of subgrid effects in coarsescale simulations of transport in heterogeneous porous media,” Water Resour. Res., 36, 2031–2041, 2000a. [16] J. Smogorinsky, “General circulation experiments with the primitive equations,” Mon. Weather Rev., 91, 99–164, 1963. [17] M. Germano, U. Pimomelli, P. Moin, and W. Cabot, “A dynamic subgrid-scale eddy viscosity model,” Phys. Fluids A, 3, 1760–1765, 1991. [18] D.W. McLaughlin, G.C. Papanicolaou, and O. Pironneau, “Convection of microstructure and related problems,” SIAM J. Appl. Math., 45, 780–797, 1985. [19] T.Y. Hou, D. Yang, and K. Wang, “Homogenization of incompressible Euler equations,” J. Comput. Math., 22(2), 220–229, 2004b. [20] T.Y. Hou, D. Yang, and H. Ran, “Multiscale analysis in the Lagrangian formulation for the 2-D incompressible Euler equation,” Discr. Continuous Dynam. Sys., 12, to appear, 2005. [21] T.Y. Hou, A. Westhead, and D. Yang, “Multiscale analysis and computation for two-phase flows in strongly heterogeneous porous media,” (in preparation), 2005a. [22] M. Dorobantu and B. Engquist, “Wavelet-based numerical homogenization,” SIAM J. Numer. Anal., 35, 540–559, 1998. [23] T. Wallstrom, S. Hou, M.A. Christie, L.J. Durlofsky, and D. Sharp, “Accurate scale up of two-phase flow using renormalization and nonuniform coarsening,” Comput. Geosci., 3, 69–87, 1999. [24] T. Arbogast, “Numerical subgrid upscaling of two-phase flow in porous media,” In: Z. Chen (ed.), Numerical Treatment of Multiphase Flows in Porous Media, Springer, Berlin, pp. 35–49, 2000. [25] A. Matache, I. Babuska, and C. Schwab, “Generalized p-FEM in homogenization,” Numer. Math., 86, 319–375, 2000. [26] L.Q. Cao, J.Z. Cui, and D.C. Zhu, “Multiscale asymptotic analysis and numerical simulation for the second order Helmholtz equations with rapidly oscillating coefficients over general convex domains,” SIAM J. Numer. Anal., 40, 543–577, 2002. [27] W.E. and B. Engquist, “The heterogeneous multi-scale methods,” Comm. Math. Sci., 1, 87–133, 2003.

4.15 CERTIFIED REAL-TIME SOLUTION OF PARAMETRIZED PARTIAL DIFFERENTIAL EQUATIONS Nguyen Ngoc Cuong, Karen Veroy, and Anthony T. Patera Massachusetts Institute of Technology, Cambridge, MA, USA

1.

Introduction

Engineering analysis requires the prediction of (say, a single) selected “output” s e relevant to ultimate component and system performance:∗ typical outputs include energies and forces, critical stresses or strains, flowrates or pressure drops, and various local and global measures of concentration, temperature, and flux. These outputs are functions of system parameters, or “inputs”, µ, that serve to identify a particular realization or configuration of the component or system: these inputs typically reflect geometry, properties, and boundary conditions and loads; we shall assume that µ is a P-vector (or P-tuple) of parameters in a prescribed closed input domain D ⊂ R P . The input–output relationship s e (µ) : D → R thus encapsulates the behavior relevant to the desired engineering context. In many important cases, the input–output function s e (µ) is best articulated as a (say) linear functional of a field variable u e (µ). The field variable, in turn, satisfies a µ-parametrized partial differential equation (PDE) that describes the underlying physics: for given µ ∈ D, u e (µ) ∈ X e is the solution of g(u e (µ), v; µ) = 0,

∀ v ∈ X e,

(1)

where g is the weak form of the relevant partial differential equation† and X e is an appropriate Hilbert space defined over the physical domain ⊂ Rd . Note * Here superscript “e” shall refer to “exact.” We shall later introduce a “truth approximation” which will

bear no superscript. † We shall restrict our attention in this paper to second-order elliptic partial differential equations; see Outlook for a brief discussion of parabolic problems. 1529 S. Yip (ed.), Handbook of Materials Modeling, 1529–1564. c 2005 Springer. Printed in the Netherlands.

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in the linear case, g(w, v; µ) ≡ a(w, v; µ) − f (v), where a(·, ·; µ) and f are continuous bilinear and linear forms, respectively; for any given µ ∈ D, u e (µ) ∈ X e now satisfies a(u e (µ), v; µ) = f (v),

∀ v ∈ X e (linear).

(2)

Relevant system behavior is thus described by an implicit “input–output” relationship s e (µ) = (u e (µ)),

(3)

evaluation of which necessitates solution of the partial differential equation (1) or (2). Many problems in materials and materials processing can be formulated as particular instantiations of the abstraction (1) and (3) or perhaps (2) and (3). Typical field variables and associated second-order elliptic partial differential equations include temperature and steady conduction–Poisson; displacement and equilibrium or Helmholtz elasticity; {velocity, temperature} and steady Boussinesq incompressible Navier–Stokes; wavefunction and stationary Schrödinger via (say) Hartree–Fock approximation. The latter two equations are nonlinear, while the former two equations are linear; in subsequent sections we shall provide detailed examples of both nonlinear and linear problems. Our particular interest – or certainly the best way to motivate our approach – is “deployed” systems: components or processes that are in service, in operation, or in the field. For example, in the materials and materials processing context, we may be interested in assessment, evolution, and accommodation of a crack in a critical component of an in-service jet engine; in real-time characterization and optimization of the heat treatment protocol for a turbine disk; or in online thermal “control” of Bridgman semiconductor crystal growth. Typical computational tasks include robust parameter estimation (inverse problems) and adaptive design (optimization problems): in the former – for example, assessment of current crack length or in-process heat transfer coefficient – we must deduce inputs µ representing system characteristics based on outputs s e (µ) reflecting measured observables; in the latter – for example, prescription of allowable load or best thermal environment – we must deduce inputs µ representing “control” variables based on outputs s e (µ) reflecting current process objectives. Both of these demanding activities must support an action in the presence of continually evolving environmental and mission parameters. The computational requirements on the forward problem are thus formidable: the evaluation must be real-time, since the action must be immediate; and the evaluation must be certified – endowed with a rigorous error bound – since the action must be safe and feasible. For example, in our aerospace crack example, we must predict in the field – without recourse to a lengthy computational investigation – the load that the potentially damaged structure

Real-time solution of parametrized partial differential equations

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can unambiguously safely carry. Similarly, in our materials processing examples, we must predict in operation – in response to deduced environmental variation – temperature boundary conditions that will preserve the desired material properties. Classical approaches such as the finite element method cannot typically satisfy these requirements. In the finite element method, we first introduce a piecewise-polynomial “truth” approximation subspace X (⊂ X e ) of dimension N . The “truth” finite element approximation is then found by (say) Galerkin projection: given µ ∈ D, s(µ) = (u(µ)),

(4)

where u(µ) ∈ X satisfies g(u(µ), v; µ) = 0,

∀ v ∈ X,

(5)

or, in the linear case g(w, v; µ) ≡ a(w, v; µ) − f (v), a(u(µ), v; µ) = f (v),

∀ v ∈ X (linear).

(6)

We assume that (5) and (6) are well-posed; we articulate the associated hypotheses more precisely in the context of a posteriori error estimation. We shall assume – hence the appellation “truth” – that X is sufficiently rich that u(µ) (respectively, s(µ)) is sufficiently close to u e (µ) (respectively, s e (µ)) for all µ in the parameter domain D. Unfortunately, for any reasonable error tolerance, the dimension N needed to satisfy this condition – even with the application of appropriate (parameter-dependent) adaptive mesh refinement strategies – is typically extremely large, and in particular much too large to provide real-time response in the deployed context. Deployed systems thus present no shortage of unique computational challenges; however, they also provide many unique computational opportunities – opportunities that must be exploited. We first consider the “approximation opportunity.” The critical observation is that, although the field variable u e (µ) generally belongs to the infinitedimensional space X e associated with the underlying partial differential equation, in fact u e (µ) resides on a very low-dimensional manifold Me ≡{u e (µ) | µ ∈ D} induced by the parametric dependence; for example, for a single parameter, µ ∈ D ⊂ R P=1 , u e (µ) will describe a one-dimensional filament that winds through X e . Furthermore, the field variable u e (µ) will typically be extremely regular in µ – the parametrically induced manifold Me is very smooth – even when the field variable enjoys only limited regularity with respect to the spatial coordinate x ∈ .∗ In the finite element method, the approximation space X is * The smoothness in µ may be deduced from the equations for the sensitivity derivatives; the stability and

continuity properties of the partial differential operator are crucial.

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much too general – X can approximate many functions that do not reside on Me – and hence much too expensive. This observation presents a clear opportunity: we can effect significant dimension reduction in state space if we restrict attention to Me ; the field variable can then be adequately approximated by a space of dimension N N . However, since manipulation of even one “point” on Me is expensive, we must identify further structure. We thus next consider the “computational opportunities”; here there are two critical observations. The first observation derives from the mathematical formulation: very often, the parameter dependence of the partial differential equation can be expressed as the sum of Q products of (known, easily evaluated) parameter-dependent functions and parameter-independent continuous forms; we shall denote this structure as “affine” parameter dependence. In our linear case, (2), affine parameter dependence reduces to a(w, v; µ) =

Q

q (µ) a q (w, v),

(7)

q=1

for q : D → R and a q : X × X → R, 1 ≤ q ≤ Q. The second observation derives from our context: rapid deployed response perforce places a predominant emphasis on very low marginal cost – we must minimize the additional effort associated with each new evaluation µ → s(µ) “in the field.” These two observations present a clear opportunity: we can exploit the underlying affine parametric structure (7) to design effective offline–online computational procedures which willingly accept greatly increased initial preprocessing – offline, pre-deployed – expense in exchange for greatly reduced marginal – online, deployed – “in service” cost.∗ The two essential components to our approach are (i) rapidly, uniformly (over D) convergent reduced-basis (RB) approximations, and (ii) associated rigorous and sharp a posteriori error bounds. Both components exploit affine parametric structure and offline–online computational decompositions to provide extremely rapid deployed response – real-time prediction and associated error estimation. We next describe these essential ingredients.

2. 2.1.

Reduced-Basis Method Approximation

The reduced-basis method was introduced in the late 1970s in the context of nonlinear structural analysis [1, 2] and subsequently abstracted, analyzed, * Clearly, low marginal cost implies low asymptotic average cost; our methods are thus also relevant to (non real-time) many-query multi-optimization studies – and, in fact, to any situation characterized by extensive exploration of parameter space.


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and extended to a much larger class of parametrized PDEs [3, 4] – including the incompressible Navier–Stokes equations [5–7] relevant to many materials processing applications. The RB method explicitly recognizes and exploits the dimension reduction afforded by the low-dimensional and smooth parametrically induced solution manifold. We note that the RB approximation is constructed not as an approximation to the exact solution, u e (µ), but rather as an approximation to the (finite element) truth approximation, u(µ). As already discussed, N , the dimension of X , will be very large; our RB formulation and associated error estimation procedures must be stable and (online) efficient as N → ∞. We shall consider in this section the linear case, g(w, v; µ) ≡ a(w, v; µ) − f (v), in which s(µ) and u(µ) are given by (4) and (6), respectively; recall that a is bilinear and f , , are linear. We shall consider a “primal–dual” formulation particularly well-suited to good approximation and error characterization of the output; towards this end, we introduce a dual, or adjoint, problem: given µ ∈ D, ψ(µ) ∈ X satisfies a(v, ψ(µ); µ) = −(v),

∀ v ∈ X.

(8)

Note that if a is symmetric and = f , which we shall denote “compliance,” ψ(µ) = −u(µ). In the “Lagrangian” [4] RB approach, the field variable u(µ) is approximated by (typically) Galerkin projection onto a space spanned by solutions of the governing PDE at N selected points in parameter space. For the primal probpr pr lem, (6), we introduce nested parameter samples S N ≡ {µ1 ∈ D, . . . , µ N ∈ D} pr and associated nested RB approximation subspaces W N ≡span{ζn ≡ u(µn ), 1 ≤ n ≤ N } for 1 ≤ N ≤ Nmax ; similarly, for the dual problem (8), we define corredu sponding samples S Ndudu ≡ {µdu 1 ∈ D, . . . , µ N du ∈ D} and RB approximation du du du du ∗ spaces W N du ≡span{ζndu ≡ ψ(µdu n ), 1 ≤ n ≤ N } for 1 ≤ N ≤ Nmax . (Procedu dures for selection of good samples SN , S N du and hence spaces W N , W Ndudu will be discussed in subsequent sections.) Our RB approximation is thus: given µ ∈ D, s N (µ) = (u N (µ)) + g(u N (µ), ψ N du (µ); µ),

(9)

where u N (µ) ∈ W N and ψ N du (µ) ∈ W Ndudu satisfy a(u N (µ), v; µ) = f (v),

∀ v ∈ WN ,

(10)

and a(v, ψ N du (µ); µ) = −(v),

∀ v ∈ W Ndudu ,

(11)

* In actual practice, the primal and dual bases should be orthogonalized with respect to the inner product associated with the Hilbert space X, (·, ·) X ; the algebraic systems then inherit the “conditioning” properties of the underlying partial differential equation.

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respectively. We emphasize that we are interested in global approximations that are uniformly valid over a finite parameter domain D. We note that, in the compliance case – a symmetric and = f such that ψ(µ) = −u(µ) – we may simply take N du = N , S Ndu = S N , W Ndu = W N , and hence ψ N (µ) = −u N (µ). In practice, in such a case we need never actually form the dual problem – we simply identify ψ N (µ) = −u N (µ) – with a corresponding 50% reduction in computational effort. Typically [8, 9], and in some very special cases provably [10], u N (µ), ψ N (µ), and s N (µ) converge to u(µ), ψ(µ), and s(µ) uniformly and extremely rapidly – thanks to the smoothness in µ – and thus we may achieve the desired accuracy for N, N du N . The critical ingredients of the a priori theory are (i) the optimality properties of Galerkin projection,∗ and (ii) the good approximation properties of W N (respectively, W Ndudu ) for the manifold M ≡ {u(µ) | µ ∈ D} (respectively, Mdu ≡ {ψ(µ) | µ ∈ D}).

2.2.

Offline–Online Computational Procedure

Even though N , N du may be small, the elements of (say) W N are in some sense “large”: ζn ≡ u(µpr n ) will be represented in terms of N N truth finite element basis functions. To eliminate the N -contamination of the deployed performance, we must consider offline–online computational procedures [7– 9, 11]. For our purposes here, we continue to assume that our PDE is linear, (6), and furthermore exactly affine, (7), for some modest Q. In future sections we shall consider a nonlinear example as well as the possibility of nonaffine operators. To begin, we expand our reduced-basis approximation as u N (µ) =

N

du

u N j (µ)ζ j ,

ψ N du (µ) =

j =1

N

ψ N du j (µ)ζ jdu .

(12)

j =1

It then follows from (9) and (12) that the reduced-basis output can be expressed as s N (µ) =

N

du

u N j (µ) (ζ j ) −

j =1

N

ψ N du j (µ) f (ζ jdu )

j =1 N du

+

Q N j =1 j =1 q=1

u N j (µ)ψ N du j (µ)q (µ)a q (ζ j , ζ jdu ),

(13)

* Galerkin optimality relies on stability of the discrete equations. The latter is only assured for coercive

problems; for noncoercive problems, Petrov–Galerkin methods may thus be preferred [12].


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where the coefficients u N j (µ), 1 ≤ j ≤ N , and ψ N du j , 1 ≤ j ≤ N du , satisfy the N × N and N du × N du linear algebraic systems N j =1 du

N j =1

Q

(µ)a (ζ j , ζi ) u N j (µ) = f (ζi ), q

q=1 Q

q

1 ≤ i ≤ N,

(14)

q (µ)a q (ζidu , ζ jdu ) ψ N du j (µ) = −(ζidu),

1 ≤ i ≤ N du .

q=1

(15) The offline–online decomposition is now clear. For simplicity below we assume that N du = N . In the offline stage – performed once – we first solve for the ζi , ζidu , 1 ≤ i ≤ N ; we then form and store (ζi ), f (ζi ), (ζidu), and f (ζidu ), 1 ≤ i ≤ N , and a q (ζ j , ζi ), a q (ζidu , ζ jdu ), 1 ≤ i, j ≤ N , 1 ≤ q ≤ Q, and a q (ζi , ζ jdu ), 1 ≤ i, j ≤ N , 1 ≤ q ≤ Q.∗ Note all quantities computed in the offline stage are independent of the parameter µ. In the online stage – performed many times, for each new value of µ “in the field” –we first assemble and subsequently invert the N × N “stiff ness matrices” qQ= 1 q (µ) a q (ζ j , ζi ) of (14) and qQ= 1 q (µ) a q (ζidu , ζ jdu ) of (15) – this yields the u N j (µ), ψ N du j (µ), 1 ≤ j ≤ N ; we next perform the summation (13) – this yields the s N (µ). The operation count for the online stage is, respectively, O(Q N 2 ) and O(N 3 ) to assemble (recall that the a q (ζ j , ζi ), 1 ≤ i, j ≤ N , 1 ≤ q ≤ Q, are pre-stored) and invert the stiffness matrices, and O(N ) + O(Q N 2 ) to evaluate the output (recall that the (ζ j ) are pre-stored); note that the RB stiffness matrix is, in general, full. The essential point is that the online complexity is independent of N , the dimension of the underlying truth finite element approximation space. Since N, N du N , we expect – and often realize – significant, orders-of-magnitude computational economies relative to classical discretization approaches.

3. 3.1.

A Posteriori Error Estimation Motivation

A posteriori error estimation procedures are very well developed for classical approximations of, and solution procedures for, (say) partial differential equations [13–15] and algebraic systems [16]. However, until quite recently, * In actual practice, in the offline stage we consider N = N du du max and N = Nmax ; then, in the online stage,

we extract the necessary subvectors and submatrices.

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there has been essentially no way to rigorously, quantitatively, sharply, and efficiently assess the accuracy of RB approximations. As a result, for any given new µ, the RB (say, primal) solution u N (µ) typically raises many more questions than it answers. Is there even a solution u(µ) near u N (µ)? This question is particularly crucial in the nonlinear context – for which in general we are guaranteed neither existence nor uniqueness. Is |s(µ)−s N (µ)| ≤ tol, where tol is the maximum acceptable error? Is a crucial feasibility condition s(µ) ≤ C (say, in a constrained optimization exercise) satisfied – not just for the RB approximation, s N (µ), but also for the “true” output, s(µ)? If these questions cannot be affirmatively answered, we may propose the wrong – and unsafe or infeasible – action in the deployed context. A fourth question is also important: Is N too large, |s(µ) − s N (µ)| tol, with an associated steep (N 3 ) penalty on computational efficiency? An overly conservative approximation may jeopardize the real-time response and associated action – with corresponding detriment to the deployed systems. We may also consider the approximation properties and efficiency of the (say, primal) parameter samples and associated RB approximation spaces, S N and W N , 1 ≤ N ≤ Nmax . Do we satisfy our global “acceptable error level” condition, |s(µ) − s N (µ)| ≤ tol , ∀µ ∈ D, for (close to) the smallest possible value of N ? And a related question: For our given tolerance tol , are the RB stiffness matrices (or, in the nonlinear case, Newton Jacobians) as well-conditioned as possible – given that by construction W N will be increasingly colinear with increasing N ? If the answers are not affirmative, then our RB approximations are more expensive (and unstable) than necessary – and perhaps too expensive to provide real-time response. In short, the pre-asymptotic and essentially ad hoc or empirical nature of reduced-basis discretizations, the strongly superlinear scaling (with N , N du ) of the reduced-basis online complexity, and the particular needs of deployed realtime systems virtually demand rigorous a posteriori error estimators. Absent such certification, we must either err on the side of computational pessimism – and compromise real-time response – or err on the side of computational optimism – and risk sub-optimal, infeasible, or potentially unsafe decisions. In Refs. [8, 9, 17, 18], we introduce a family of rigorous error estimators for reduced-basis approximation of a wide class of partial differential equations (see also Ref. [19] for an alternative approach). As in almost all error estimation contexts, the enabling (trivial) observation is that, whereas a 100% error in the field variable u(µ) or output s(µ) is clearly unacceptable, a 100% or even larger (conservative) error in the error is tolerable and not at all useless; we may thus pursue “relaxations” of the equation governing the error and residual that would be bootless for the original equation governing the field variable u(µ). We now present further details for the particular case of elliptic linear problems with exact affine parameter dependence (7): the truth solution satisfies


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(4), (6), and (8), and the corresponding reduced-basis approximation satisfies (9)–(11). (In subsequent sections we shall consider the extension to nonlinear problems through a detailed example; we shall also briefly discuss nonaffine problems.)

3.2.

Error Bounds

We shall need several preliminary definitions. To begin, we denote the inner product and norm associated with our Hilbert space X as (w, v) X and √

v X = (v, v) X , respectively; we further define the dual norm (of any bounded linear functional h) as h(v) .

v X

h X ≡ sup v∈X

(16)

We recall that we restrict our attention here to second-order elliptic partial differential equations: thus, for a scalar problem (such as heat conduction), H01 () ⊂ X e ⊂ H 1 (), where H 1 () (respectively, H01 ()) is the usual space of derivative-square-integrable functions (respectively, derivative–square– integrable functions that vanish on ∂, the boundary of ) [20]. A typical choice for (·, ·) X is (w, v) X =

∇w · ∇v + wv,

(17)

which is simply the standard H 1 () inner product. We next introduce [12, 18] the operator T µ : X → X such that, for any w in X , (T µ w, v) X = a(w, v; µ), ∀ v ∈ X . We then define σ (w; µ) ≡

T µ w X ,

w X

and note that β(µ) ≡ inf sup

a(w, v; µ) = inf σ (w; µ),

w X v X w∈X

(18)

γ (µ) ≡ sup sup

a(w, v; µ) = sup σ (w; µ);

w X v X w∈X

(19)

w∈X v∈X

w∈X v∈X

we also recall that β(µ) w X T µ w X ≤ a(w, T µ w; µ),

∀ w ∈ X.

(20)

Here β(µ) is the Babuˇska “inf–sup” stability constant – the minimum singular value associated with our differential operator (and transpose operator) – and

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γ (µ) is the standard continuity constant. We suppose that γ (µ) is bounded ∀ µ ∈ D, and that β(µ) ≥ β0 > 0, ∀ µ ∈ D. We note that for a symmetric, coercive bilinear form, β(µ) = αc (µ), where αc (µ) ≡ inf

w∈X

a(w, w; µ) ,

w 2X

is the standard coercivity constant. Given our reduced-basis primal solution u N (µ), it is readily derived that the error e(µ) ≡ u(µ) − u N (µ) ∈ X satisfies a(e(µ), v; µ) = −g(u N (µ), v; µ),

∀ v ∈ X,

(21)

where −g(u N (µ), v; µ) ≡ f (v) − a(u N (µ), v; µ) (in this linear case) is the familiar residual. It then follows from (16), (20), and (21) that

e(µ) X ≤

ε N (µ) , β(µ)

where ε N (µ) ≡ g(u N (µ), · ; µ) X ,

(22)

is the dual norm of the residual. We now assume that we are privy to a nonnegative lower bound for the ˜ ˜ inf–sup parameter, β(µ), such that β(µ) ≥ β(µ) ≥ β β(µ), ∀µ ∈ D, where β ∈]0, 1[. We then introduce our “energy” error bound

N (µ) ≡

ε N (µ) , ˜ β(µ)

(23)

the effectivity of which is defined as η N (µ) ≡

N (µ) .

e(µ) X

It is readily proven [9, 18] that, for any N , 1 ≤ N ≤ Nmax , 1 ≤ η N (µ) ≤

γ (µ) , ˜ β(µ)

∀ µ ∈ D.

(24)

From the left inequality, we deduce that e(µ) X ≤ N (µ), ∀µ ∈ D, and hence that N (µ) is a rigorous upper bound for the true error∗ measured in the

· X norm – this provides certification: feasibility and “safety” are guaranteed. From the right inequality, we deduce that N (µ) overestimates the true * Note, however, that these error bounds are relative to our underlying “truth” approximation, u(µ) ∈ X, not to the exact solution, u e (µ) ∈ X e .


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∗ ˜ error by at most γ (µ)/β(µ), independent of N – this relates to efficiency: an overly conservative error bound will be manifested in an unnecessarily large N and unduly expensive RB approximation, or (even worse) an overly conservative or expensive decision or action “in the field.” We now turn to error bounds for the output of interest. To begin, we note that the dual satisfies an “energy” error bound very similar to the primal result: du , for 1 ≤ N du ≤ Nmax

ψ(µ) − ψ N du (µ) X ≤ du N (µ),

∀ µ ∈ D;

du du ˜ here du N ≡ ε N (µ)/β(µ), and ε N (µ) = − (·) − a(·, ψ N du (µ); µ) X is the dual norm of the dual residual. It then follows† that

|s(µ) − s N (µ)| ≤ sN (µ),

∀µ ∈ D,

(25)

where

sN (µ) ≡ ε N (µ) du N (µ).

(26)

du ˜ It is critical to note that sN (µ) = β(µ) N (µ) N (µ): the output error (and output error bound) vanishes as the product of the primal and dual error (bounds), and hence much more rapidly than either the primal or dual error. From the perspective of computational efficiency, a good choice is ε N (µ) ≈ ε du N (µ); the latter also (roughly) ensures that the bound (25), (26) will be quite sharp. In the compliance case, a symmetric and = f , we immediately obtain

du N (µ) = N (µ), and hence (25) obtains for

sN (µ) ≡

ε 2N (µ) , ˜ β(µ)

∀ µ ∈ D (compliance);

(27)

here, we obtain the “square” effect even without (explicit) introduction of the dual problem. For a coercive further improvements are possible [9]. The real challenge in a posteriori error estimation is not the presentation of these rather classical results, but rather the development of efficient computational approaches for the evaluation of the necessary constituents. In our particular deployed context, “efficient” translates to “online complexity independent of N ,” and “necessary constituents” translates to “dual norm of the primal residual, ε N (µ) ≡ g(u N (µ), ·; µ) X , dual norm of the dual residual, ε du N (µ) ≡ − (·) − a(·, ψ N du (µ); µ) X , and lower bound for the inf–sup ˜ constant, β(µ).” We now turn to these issues. * The upper bound on the effectivity can be large. In many cases, this effectivity bound is in fact quite pessimistic; in many other cases, the effectivity (bound) may be improved by judicious choice of (multipoint) inner product (·, ·) X – in effect, a “bound conditioner” [21]. † The proof is simple: |s(µ) − s (µ)| = |(e) − g(u (µ), ψ (µ); µ)| = | − a(e(µ), ψ(µ); µ) − g(u (µ), N N N N ψ N (µ); µ)| = |g(u N (µ), ψ(µ) − ψ N (µ); µ)| ≤ ε N (µ) du N (µ).

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Offline–Online Computational Procedures

3.3.1. The dual norm of the residual We consider only the primal residual; the dual residual admits a similar treatment. To begin, we note from standard duality arguments that ε N (µ) ≡ g(u N (µ), ·; µ) X = e(µ) ˆ X,

(28)

where eˆ (µ) ∈ X satisfies (e(µ), ˆ v) X = −g(u N (µ), v; µ),

∀ v ∈ X.

(29)

We next observe from our reduced-basis representation (12) and affine assumption (7) that −g(u N (µ), v; µ) may be expressed as −g(u N (µ), v; µ) = f (v) −

Q N

q (µ)u N n (µ)a q (ζn , v),

∀v ∈ X.

q=1 n=1

(30) It thus follows from (29) and (30) that eˆ (µ) ∈ X satisfies (e(µ), ˆ v) X = f (v) −

Q N

q (µ) u N n (µ) a q (ζn , v),

∀ v ∈ X.

(31)

q=1 n=1

The critical observation [8, 9] is that the right-hand side of (31) is a sum of products of parameter-dependent functions and parameter-independent linear functionals. In particular, it follows from linear superposition that we may write e(µ) ˆ ∈ X as e(µ) ˆ =C+

Q N

q (µ) u N n (µ) Lqn ,

q=1 n=1

for C ∈ X satisfying (C, v) X = f (v), ∀ v ∈ X, and Lqn ∈ X satisfying (Lqn , v) X = − a q (ζn , v), ∀ v ∈ X , 1 ≤ n ≤ N , 1 ≤ q ≤ Q; note from (17) that the latter are simple parameter-independent (scalar or vector) Poisson, or Poisson-like, problems. It thus follows that 2

e(µ) ˆ X = (C, C) X +

Q N

q (µ) u N n (µ) 2(C, Lqn ) X

q=1 n=1

+

Q

N

q =1 n =1

q

(µ) u N n (µ)

q (Lqn , Ln ) X

.

(32)


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The expression (32) – which we relate to the requisite dual norm of the residual through (28) – is the sum of products of parameter-dependent (simple, known) functions and parameter-independent inner products. The offline– online decomposition is now clear. In the offline stage – performed once – we first solve for C and Lqn , 1 ≤ n ≤ N , 1 ≤ q ≤ Q; we then evaluate and save the relevant parameter-independent q inner products (C, C) X , (C, Lqn ) X , (Lqn , Ln ) X , 1 ≤ n, n ≤ N , 1 ≤ q, q ≤ Q. Note that all quantities computed in the offline stage are independent of the parameter µ. In the online stage – performed many times, for each new value of µ “in the field” – we simply evaluate the sum (32) in terms of the q (µ), u N n (µ) and the precalculated and stored (parameter-independent) (·, ·) X inner products. The operation count for the online stage is only O(Q 2 N 2 ) – again, the essential point is that the online complexity is independent of N , the dimension of the underlying truth finite element approximation space. We further note that, unless Q is quite large, the online cost associated with the calculation of the dual norm of the residual is commensurate with the online cost associated with the calculation of s N (µ).

3.3.2. Lower bound for the inf–sup parameter Obviously, from the definition (18), we may readily obtain by a variety of techniques effective upper bounds for β(µ); however, lower bounds are much more difficult to construct. We do note that in the case of symmetric coercive ˜ operators we can often determine β(µ) (≤ β(µ) = αc (µ), ∀µ ∈ D) “by inspection.” For example, if we verify q (µ) > 0, ∀ µ ∈ D, and a q (v, v) ≥ 0, ∀ v ∈ X , 1 ≤ q ≤ Q, then we may choose [8, 21] for our coercivity lower bound ˜ β(µ) =

q (µ) min αc (µ), ¯ q∈{1,...,Q} q (µ) ¯

(33)

for some µ¯ ∈ D. Unfortunately, these hypotheses are rather restrictive, and hence more complicated (and offline-expensive) recipes must often be pursued [17, 18]. We consider here a construction which is valid for general noncoercive operators (and thus also relevant in the nonlinear context [22]); for simplicity, we assume our problem remains well-posed over a convex parameter set that includes D. To begin, given µ¯ ∈ D and t = (t(1) , . . . , t( P) ) ∈ R P – note t( j ) is the value of the j th component of t – we introduce the bilinear form T (w, v; t; µ) ¯ = (T µ¯ w, T µ¯ v) X +

P p=1

t( p)

Q ∂q q=1

∂µ( p)

µ¯

µ¯

(µ) ¯ a (w, T v) + a (v, T w) q

q

(34)

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and associated Rayleigh quotient F(t; µ) ¯ = min v∈X

T (v, v; t; µ) ¯ ; 2

v X

(35)

it is readily demonstrated that F(t; µ) ¯ is concave in t [24], and hence D µ¯ ≡ P ¯ µ) ¯ ≥ 0} is perforce convex. We next introduce semi-norms {µ ∈ R |F(µ − µ; | · |q : X → R+,0 such that |a q (w, v)| ≤ q |w|q |v|q , Q

C X = supw∈X

q=1

∀w, v ∈ X, 1 ≤ q ≤ Q,

|w|2q

w 2X

(36) ,

for positive parameter-independent constants q , 1 ≤ q ≤ Q, and C X ; it is often the case that 1 (µ) = Constant, in which case the q = 1 contribution to the sum in (34) and (36) may be discarded. (Note that C X is typically independent of Q, since the a q are often associated with non-overlapping subdomains of .) Finally, we define

(µ; µ) ¯ ≡ CX

max

q∈{1,...,Q}

q (µ) − q (µ) ¯ q

 ∂ (µ − µ) ¯ ( p) (µ) ¯ , − ∂µ( p) p=1 P

q

(37)

for µ ≡ (µ(1) , . . . , µ( P) ) ∈ R P. We now introduce points µ¯ j and associated polytopes P µ¯ j ⊂ D µ¯ j , 1 ≤ j ≤ J, such that D⊂

J

P µ¯ j ,

(38)

j =1

min

ν∈V

µ ¯j

F(ν − µ¯ j ; µ¯ j ) − max (µ; µ¯ j ) ≥ β β(µ¯ j ), µ∈P

µ¯ j

1 ≤ j ≤ J. (39)

Here V µ¯ j is the set of vertices associated with the polytope P µ¯ j – for example, P µ¯ j may be a simplex with |V µ¯ j | = P + 1 vertices; and β ∈ ]0, 1[ is a prescribed accuracy constant. Our lower bound is then given by ˜ β(µ) =

max

j ∈{1,...,J }|µ∈P

µ ¯j

β β(µ¯ j ).

(40)

˜ ˜ In fact, β(µ) of (40) may not strictly honor our condition β(µ) > β β(µ); however, as the latter relates to accuracy, approximate satisfaction suffices.


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˜ (Recall that β(µ) appears in the denominator of our error bound; hence, even a relative inf–sup discrepancy of 80%, β ≈ 1/5, is acceptable.) It can be eas˜ ily demonstrated that β(µ) ≥ β(µ) ≥ β β0 > 0, ∀µ ∈ D, which thus ensures well-posed and rigorous error bounds. We now turn to the offline–online decomposition. The offline stage comprises two parts: the generation of a set of points and polytopes–vertices, µ¯ j and P µ¯ j , V µ¯ j , 1 ≤ j ≤ J ; and the verification that (38) (trivial) and (39) (nontrivial) are indeed satisfied. We focus on verification; generation – quite involved – is described in detail in [23]. To verify (39), the essential observation is that the expensive terms – “truth” eigenproblems associated with F, (35), and β, (18) – are limited to a finite set of vertices, J+

J

|V µ¯ j |,

j =1

in total; only for the extremely inexpensive – and typically algebraically very simple – (µ; µ¯ j ) terms must we consider minimization over the polytopes. The online stage (40) is very simple: a search/look-up table, with complexity logarithmic in J and polynomial in P. We close by remarking on the properties of F(µ − µ; ¯ µ) ¯ that play an important role. First, F(µ − µ; ¯ µ) ¯ ≤ β 2 (µ), ∀µ ∈ D µ¯ (say, for the case in which q (µ) = µ(q) , 1 ≤ q ≤ Q = P): this ensures the lower bound result. Second, F(t; µ) ¯ is concave in t (note that in general β(µ) is neither (quasi-) concave nor (quasi-) convex in µ [24]): this ensures a tractable offline computation. Third, F(µ − µ; ¯ µ) ¯ is “tangent”∗ to β(µ) at µ = µ¯ – the cruder estimate (µ; µ) ¯ is a second-order correction: this controls the growth of J (for example, relative to simpler continuity bounds [17]).

3.4.

Sample Construction: A Greedy Algorithm

Our error estimation procedures also allow us to pursue more rational constructions of our parameter samples S N , S Ndudu (and hence spaces W N , W Ndudu ) [18]. We consider here only the primal problem – in which our error criterion is

u(µ) − u N (µ) X ≡ e(µ) X ≤ tol ; similar approaches may be developed for du the dual – ψ(µ) − ψ N du (µ) X ≤ tol , and hence the output – |s(µ) − s N (µ)| ≤ s tol. We denote the smallest primal error tolerance anticipated as tol, min – this must be determined a priori offline; we then permit tol ∈ [tol, min, ∞[ to be specified online. We also introduce F ∈ D nF , a very fine random sample over the parameter domain D of size n F 1. * To make this third property rigorous we must in general consider non-smooth analysis and also possibly

a continuous spectrum as N → ∞.

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We first consider the offline stage. We assume that we are given a sample S N , and hence space W N and associated reduced-basis approximation (procedure to determine) u N (µ), ∀µ ∈ D. We then calculate µ∗N = arg maxµ ∈ F

N (µ) – N (µ) is our “online” error bound (23) that, in the limit of n F → ∞ queries, may be evaluated (on average) in O(N 2 Q 2 ) operations; we next append µ∗N to S N to form S N + 1 , and hence W N + 1 . We now continue this process until N = Nmax such that N∗ max = tol,min, where N∗ ≡ N (µ∗N ), 1 ≤ N ≤ Nmax . In the online stage, given any desired tol ∈ [tol, min, ∞[ and any new value of µ ∈ D “in the field”, we first choose N from a pre-tabulated array such that N∗ ≡ N (µ∗N ) = tol. We next calculate u N (µ) and N (µ), and then verify that – and if necessary, subsequently increase N such that – the condition

N (µ) ≤ tol is indeed satisfied. (We should not and do not rely on the finite sample F for either rigor or sharpness.) The crucial point is that N (µ) is an accurate and “online-inexpensive” – O(1) effectivity and N -independent asymptotic complexity – surrogate for the true (very-expensive-to-calculate) error u(µ) − u N (µ) X . This surrogate permits us to (i) offline – here we exploit low average cost – perform a much more exhaustive (n F 1) and, hence, meaningful search for the best samples S N and, hence, most rapidly uniformly convergent spaces W N ,∗ and (ii) online – here we exploit low marginal cost – determine the smallest N , and hence, the most efficient approximation, for which we rigorously achieve the desired accuracy.

4. 4.1.

A Linear Example: Helmholtz-Elasticity Problem Description

We consider a two-dimensional thin plate with a horizontal crack at the (say) interface of two lamina: the (original) domain o (z, L) ⊂ R2 , shown in Fig. 1, is defined as [0, 2] × [0, 1] \ Co , where Co ≡ {x1 ∈ [b − L/2, b + L/2], x2 = 1/2} defines the idealized crack. The left surface of the plate is secured; the top and bottom boundaries are stress-free; and the right boundary is subjected to a vertical oscillatory uniform traction at frequency ω. We model the plate as plane-stress linear isotropic elastic with (scaled) density unity, Young’s modulus unity, and Poisson ratio 0.25; the latter determine the (parameter-independent) constitutive tensor E i j k . Our P = 3 input is µ ≡ (µ(1) , µ(2) , µ(3) ) ≡ (ω2 , b, L); our output is the (oscillatory) amplitude of the average vertical displacement on the right edge of the plate.

* We may in fact view our offline sampling process as a (greedy, parameter space, “L ∞ (D)”) variant of the

POD economization procedure [25] in which – thanks to N (µ) – we need never construct the “rejected” snapshots.


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L b

Figure 1. (Original) domain for the Helmholtz elasticity example.

The governing equation for the displacement u o (x o ; µ) ∈ X o (µ) is therefore a o (u o (µ), v; µ) = f o (v), ∀ v ∈ X o (µ), where X o (µ) is a quadratic finite element truth approximation subspace (of dimension N = 14,662) of X e (µ) ≡ {v ∈ (H 1 (o (b, L)))2 | v|x1o = 0 = 0 }; here a (w, v; µ) ≡

o

wi, j E i j k v k, − ω2 wi v i ,

o (b,L)

(v i, j denotes ∂v i /∂ x j and repeated physical indices imply summation), and f o (v) ≡ x o = 2 v 2 . The crack surface is hence modeled extremely simplisti1 cally – as a stress-free boundary. The output s o (µ) is given by s o (µ) = o (u o (µ)), where o (v) = f o (v); we are thus “in compliance”. We now map o (b, L) via a continuous piecewise-affine transformation to a fixed domain . This new problem can now be cast precisely in the desired abstract form, in which , X , and (w, v) X are independent of the parameter µ: as required, all parameter dependence now enters through the bilinear and linear forms; in particular, our affine assumption (7) applies for Q = 10. In the Appendix we summarize the q (µ), a q (w, v), 1 ≤ q ≤ Q; the bound conditioner (·, ·) X ; and the resulting continuity constants q and semi-norms | · |q , 1 ≤ q ≤ Q, and norm equivalence parameter C X . The (undamped, nonradiating) Helmholtz equation exhibits resonances. Our techniques can treat near resonances, as well as large frequency ranges, quite well [18, 23]. For our illustrative purposes here, we choose the parameter domain D (⊂ R P = 3 ) ≡ (ω2 ∈ [3.2, 4.8])×(b ∈ [0.9, 1.1]) × (L ∈ [0.15, 0.25]); D contains no resonances – β(µ) ≥ β0 > 0, ∀µ ∈ D – however, ω2 = 3.2 and 4.8 are close to corresponding natural frequencies, and hence the problem is distinctly noncoercive.

4.2.

Numerical Results

We first consider the inf–sup lower bound construction. We show in Fig. 2 β (µ) and F(µ− µ; ¯ µ) ¯ for µ= ¯ µ¯ 1 =(4.0, 1.0, 0.2); for purposes of presentation 2 we keep µ(1) = (ω = 4.0) fixed and vary µ(2) (= b) and µ(3) (= L). We observe 2

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0.02 0.01 0

⫺0.01 ⫺0.02 0.25 0.225 0.2

L

0.175 0.15 0.9

0.95

1

1.05

1.1

b

Figure 2. β 2 (µ) and F(µ − µ; ¯ µ) ¯ for µ¯ = (4, 1, 0.2) as a function of (b, L); ω2 = 4.0.

that (in this particular case, even without (µ; µ)), ¯ F(µ − µ; ¯ µ) ¯ is a lower bound for β 2 (µ); that F(µ − µ; ¯ µ) is concave; and that F(µ − µ; ¯ µ) is tan2 ¯ Thanks to the latter, we can cover D (for ¯β = 0.2) such gent to β (µ) at µ = µ. that (38) and (39) are satisfied with only J = 84 polytopes; in this particular case the P µ¯ j , 1 ≤ j ≤ J, are hexahedrons such that |V µ j | = 8, 1 ≤ j ≤ J . Armed with the inf–sup lower bound, we can now pursue the adaptive sampling strategy described in the previous section. We recall that our problem is compliant, and hence we need only consider the primal variable (and (µ) = ε N (µ)). For tol, min subsequently set ψ N du = N (µ) = −u N (µ) and ε du N du = N pr = 10−3 and n F = 729 we obtain Nmax = 32 such that Nmax ≡ Nmax (µ Nmax ) = 9.03 × 10−4 . We present in Table 1 N,max,rel , η N,ave , sN,max , and ηsN,ave as a function of N . Here N,max,rel is the maximum over Test of N (µ)/ u Nmax max , η N,ave is the average over Test of N (µ)/ u(µ) − u N (µ) X , sN,max,rel is the maximum over Test of sN (µ)/|s Nmax |max , and ηsN,ave is the average over Test of sN (µ)/|s(µ) − s N (µ)|. Here Test ∈ (D I )343 is a random parameter sample of size 343; u Nmax max ≡ maxµ ∈ Test u Nmax (µ) X = 2.0775 and |s Nmax |max ≡ maxµ∈Test |s Nmax (µ)| = 0.089966; and N (µ) and sN (µ) are given by (23) and (27), respectively. We observe that the RB approximation – in particular, for the output – converges very rapidly, and that our rigorous error bounds are in fact quite sharp. The effectivities are not quite O(1) primarily due to the relatively crude inf–sup lower bound; but note that, thanks to the rapid convergence of the RB approximation, O(10) effectivities do not significantly affect efficiency – the induced increase in RB dimension N is quite modest. We turn now to computational effort. For (say) N = 24 and any given µ (say, (4.0, 1.0, 0.2)) – for which the error in the reduced-basis output s N (µ)


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Table 1. Numerical results for Helmholtz elasticity N

N,max,rel

η N,ave

sN,max,rel

ηsN,ave

12 16 20 24 28

1.54 × 10−1 3.40 × 10−2 1.58 × 10−2 5.91 × 10−3 2.42 × 10−3

13.41 12.24 13.22 12.56 12.44

3.31 × 10−2 2.13 × 10−3 4.50 × 10−4 4.81 × 10−5 9.98 × 10−6

15.93 14.86 15.44 14.45 14.53

relative to the truth approximation s(µ) is certifiably less than sN (µ) (= 4.94 × 10−7 ) – the Online Time (marginal cost) to compute both s N (µ) and sN (µ) is less than 0.0030 the Total Time to directly calculate the truth result s(µ) = (u(µ)). The savings will be even larger for problems with more complex geometry and solution structure, in particular in three space dimensions. As desired, we achieve efficiency due to (i) our choice of sample, (ii) our rigorous stopping criterion sN (µ), and (iii) our affine parameter dependence and associated offline–online computational procedures; and we achieve rigorous certainty – the reduced-basis predictions may serve in “deployed” decision processes with complete confidence (or at least with the same confidence as the underlying physical model and associated truth finite element approximation). The true merit of the approach is best illustrated in the deployed–real-time context of parameter identification (crack assessment) and adaptive mission optimization (load maximization); see Ref. [24] for an example.

5.

A Nonlinear Example: Natural Convection

Obviously nonlinear equations do not admit the same degree of generality as linear equations. We thus present our approach to nonlinear equations for a particular quadratically nonlinear elliptic problem: the steady Boussinesq incompressible Navier–Stokes equations. This example permits us to identify the key new computational and theoretical ingredients; then, in Outlook, we contemplate more general (higher-order) nonlinearities.

5.1.

Problem Description

We consider Prandtl number Pr = 0.7 Boussinesq natural convection in a square cavity (x1 , x2 ) ∈ ≡ [0, 1] × [0, 1]; the Pr = 0 limit is described in greater detail in [22, 26]. The governing equations for the velocity U = (U1 , U2 ), pressure p, and temperature θ are the (coupled) incompressible steady Navier– Stokes and thermal convection–diffusion equations. Our single parameter

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(P = 1) is the Grashof number, µ ≡ Gr, which is the ratio of the buoyancy forces (induced by the temperature field) to the momentum dissipation mechanisms; we consider Gr ∈ D ≡ [1.0, 1.0 × 104 ]. This flow is a model problem for Bridgman growth of semi-conductor crystals; future work shall address geometric (angle, aspect ratio) and Pr variation, and higher Gr – all of which are important in actual materials processing applications. In terms of the general mathematical formulation, (5), u(µ) ≡ (U1 , U2 , p, θ, λ)(µ), where λ is a Lagrange multiplier associated with the pressure zero-mean condition. Our solution u(µ) resides in the space X ≡ X U × X p × X θ × R, where X U ⊂ (H01 ())2 , X p ⊂ L 2 () (respectively, X θ ⊂ {v ∈ H 1 () |v|x1 = 0 = 0}) is a classical P2 −P1 Taylor–Hood Stokes (respectively, P2 scalar) finite element approximation subspace [5]; X is of dimension N = 2869. We associate to X the inner product and norm (w, v) X =

∂χ ∂φ ∂ Wi ∂ Vi + Wi Vi + rq + + χφ + κα ∂x j ∂x j ∂ xi ∂ xi

√

and w X = (w, w) X , respectively, where w = (W1 , W2 , r, χ, κ) and v = (V1 , V2 , q, φ, α). The strong (or distributional) form of the governing equations is then √

Gr u j

√ ∂p √ ∂u i ∂ 2ui = − Gr + Gr θδi2 + , ∂x j ∂ xi ∂x j∂x j

i = 1, 2,

∂u i = λ, ∂ xi √ ∂ 2θ ∂θ Gr Pr u j = , ∂x j ∂x j∂x j

with boundary–normalization conditions u|∂ = 0 on the velocity, p = 0 on the pressure, and ∂θ/∂n|1 = 1, θ|0 = 0, ∂θ/∂n|s = 0 on the temperature; the flow is thus driven by the flux imposed on 1 . Here δij is the Kroneckerdelta, ∂ is the boundary of , and 0 = {x1 = 0, x2 ∈ [0, 1]} (left side), 1 = {x1 = 1, x2 ∈ [0, 1]} (right side), and s = {x1 ∈ ]0, 1[ , x2 = 0} ∪ {x1 ∈ ]0, 1[ , x2 = 1} (top and bottom). It is readily derived that λ = 0; however, we retain this term as a computationally convenient and stable fashion by which to impose the zero-mean pressure condition on the truth finite element solution. Our output of interest is the average temperature over 1 : s(Gr) = (u(Gr)), where (v = (V1 , V2 , q, φ, α)) ≡

φ;

1

note that s −1 (Gr) is the traditional “Nusselt number”.

(41)


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The weak form of our partial differential equations is then given by (5), where g(w, v; Gr) ≡ a0 (w, v; Gr) + 12 a1 (w, w, v; Gr) − f (v), a0 (w 1 , v; Gr) ≡

+

∂ Wi1 ∂ Vi − ∂x j ∂x j

a1 (w 1 , w 2 , v; Gr) ≡

√

Gr −

∂ Wi1 q + κ1 ∂ xi √

Gr −

q +α

χ V2 −

r1

1

(42)

r

1 ∂ Vi

∂ xi

,

(43) 1 ∂ Vi

W j1 Wi2 + W j2 Wi

+ Pr f (v) ≡

∂χ ∂φ + ∂ xi ∂ xi 1

∂χ W j2

1

∂x j

+

φ;

∂x j

∂χ W j1

2

∂x j

φ ,

(44) (45)

1

here w 1 = (W11 , W21 , r 1 , χ 1 , κ 1 ), w 2 = (W12 , W22 , r 2 , χ 2 , κ 2 ) , and v = (V1 , V2 , q, φ, α). Note that, even though = f , we are not in “compliance” as g is not bilinear, symmetric; however, we are “close” to compliance, and thus might anticipate rapid output convergence. We next observe that a0 (w 1 , v; Gr) and a1 (w 1 , w 2 , v; Gr) satisfy (a nonlinear version of) our assumption of affine parameter dependence (7). In particular, we may write a0 (w 1 , v; Gr) =

Q0

q

q

0 (Gr)a0 (w 1 , v),

(46)

q=1

a1 (w 1 , w 2 , v; Gr) =

Q1

q

q

1 (Gr)a1 (w 1 , w 2 , v),

(47)

q=1

√ 1 2 = 2 and Q = 1. In particular, (Gr) = 1, (Gr) = Gr, and 11 (Gr) = for Q 0 1 0 0 √ Gr; the corresponding parameter-independent bilinear and trilinear forms should be clear from (43) and (44). We shall exploit (46) and (47) in our offline–online decomposition. We define the derivative (about z ∈ X ) bilinear form dg(·, ·; z; Gr) : X × X → R as dg(w, v; z; Gr) ≡ a0 (w, v; Gr) + a1 (w, z, v; Gr)

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which clearly inherits the affine structure (46) and (47) of g; we note that, for our simple quadratic nonlinearity, g(z + w, v; Gr) = g(z, v; Gr) + dg(w, v; z; Gr) + (1/2) a1 (w, w, v; Gr). We then associate to dg(·, ·; z; Gr) our Babuˇska inf–sup and continuity “constants” dg(w, v; z; Gr) ,

w X v X dg(w, v; z; Gr) , γ (z; Gr) ≡ sup sup

w X v X w∈X v∈X β(z; Gr) ≡ inf sup w∈X v∈X

respectively; these constants now depend on the state z about which we linearize. We shall confirm a posteriori that a solution to our problem does indeed exist for all Gr in the chosen D; we can further demonstrate [22] that the manifold {u(Gr)|Gr ∈ D} upon which we focus is a nonsingular (isolated) ∗ solution branch, √ and thus β(u(Gr)) ≥ β0 > 0, ∀ Gr ∈ D. We can also verify γ (z; Gr) ≤ 2 Gr (1 + ρU (ρU + Prρθ ) z X ), where

V L 4 () ,

V X U

ρU ≡ sup

V ∈X U

ρθ ≡ sup

φ∈X θ

φ L 4 ()

φ H 1 ()

(48)

are embedding constants [27, 28]; for V ∈ X U , V L n () ≡ Sobolev n/2 1/n ( (Vi Vi ) ) , 1 ≤ n < ∞, (W, V ) X U ≡ (∂ Wi /∂ x j )(∂ Vi /∂ x j ) + Wi Vi , 1/2 and V X U ≡ (V, V ) X U . We present in Fig. 3(a) a plot of s(Gr); as expected, for low Gr we obtain the conduction solution, s(Gr) = 1; at higher Gr, the larger buoyancy terms create more vigorous flows and hence more effective heat transfer. We show in Fig. 3(b) the velocity and temperature distribution at Gr = 104 ; we observe the familiar “S”-shaped natural convection profile.

5.2.

Reduced-Basis Approximation

For simplicity of exposition we shall not address here the adjoint in the nonlinear (approximation or error estimation) context [22], and we shall thus only consider RB treatment of the primal problem, (5) and (42). Our RB (Galerkin)

* We note that our truth approximation is div-stable in the sense that the “Brezzi” inf–sup parameter, β Br ,

is bounded from below (independent of N ):

β Br ≡

inf

{q∈X p |

sup

q(∂Vi /∂xi )

q=0} V ∈X U V X U q L 2 ()

> 0;

this is a necessary condition for “Babuˇska” inf–sup stability of the linearized operator dg(·, ·, z; Gr).


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Figure 3. (a) Inverse Nusselt number s(Gr) as a function of Gr; and (b) velocity and temperature field for Gr = 104 .

approximation is thus: for given Gr ∈ D, evaluate s N (Gr) = (u N (Gr)), where p u N (Gr) ≡ (U N , p N , θ N , λ N )(Gr) ∈ W N ≡ W NU × W N × W Nθ × W Nλ satisfies g(u N (Gr), v; Gr) = 0,

∀ v ∈ WN ,

for and g defined in (41) and (42)–(45). There are two new ingredients: correct choice of W N to ensure div-stability; and efficient offline–online treatment of the nonlinearity. We first address W N . To begin, we assume that N = 4m for m a positive intpr eger, and we introduce a sequence of nested parameter samples S N ≡ {µ1 ∈ pr D, . . . , µ N/4 ∈ D} in terms of which we may then define the components of ¯ W N . It is simplest to start with W p ≡ span{p(µn ), 1 ≤ n ≤ N/4, and p}, where p¯ = 1 is the constant function; we then choose W NU ≡ span{U (µpr n ), 2 U ), 1 ≤ n ≤N/4}, where for q ∈ L (), Sq ∈ X satisfies S p(µpr n (Sq, V ) X U =

∂ Vi q, ∂ xi

∀ V ∈ XU ;

W Nθ

λ ≡ span{θ(µpr we next define n ), 1 ≤ n ≤ N/4}; and, finally, W N ≡ R. Note that W NU must be chosen such that the RB approximation satisfies the Brezzi div-stability condition; for our problem, the domain and hence, the span of the supremizers do not depend on the parameter, and therefore the choice of W NU is simple – the more general case is addressed in [29]. We obp serve that dim(W NU ) = (N/2), dim(W N ) = (N/4) + 1, dim(W Nθ ) = (N/4), and dim(W Nλ ) = 1, and hence dim(W N ) = N + 2.∗

* In fact, we can explicitly eliminate (the zero coefficient of) p¯ and λ (= 0) from our RB discrete equations, N pr p and thus the effective dimension of W N is N . In the RB context, for which each member p(µn ) of W N is explicitly zero-mean, the services of the Lagrange multiplier are no longer required.

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For our nonlinear problem, the essential computational kernel is the inner Newton update: given a kth iterate u kN (Gr), the Newton increment δu kN (Gr) v; u kN (Gr); Gr)=−g(u kN (Gr), v; Gr), ∀v ∈ X . If we now satisfies dg(δu kN (Gr), N = n=1 u kN n (Gr) ζn – where W N = span{ζn , 1 ≤ n ≤ N } – and expand u kN (Gr) N k δu N (Gr) = j =1 δu kN j (Gr) ζ j , we obtain [17] the linear set of equations N j =1

Q0

q

q

0 (Gr) a0 (ζ j , ζi )

q=1

+

Q1 N n=1 q =1

q q 1 (Gr)u kNn (Gr)a1 (ζ j , ζn , ζi )

= − g(u kN (Gr), ζi ; Gr),

δ kN j (Gr)

1 ≤ i ≤ N,

where (from (42))

−g(u kN (Gr), ζi ;

Gr) = f (ζi ) −

N j =1

1 + 2

Q1 N

Q0

q

q

0 (Gr) a0 (ζ j , ζi )

q=1

q q u kN n (Gr)1 (Gr)a1 (ζ j , ζn , ζi )

u kN j (Gr)

n=1 q=1

is the residual for v = ζi . We can now directly apply the offline–online procedure [7–9] described earlier for linear problems, except now we must perform summations both “over the affine parameter dependence” and “over the reduced-basis coefficients” (of the current Newton iterate about which we linearize).∗ The operation count for the predominant Newton update component of the online stage is then – per Newton iteration – O(N 3 ) to assemble the residual, −g(u kN (Gr), ζi ; Gr), 1 ≤ i ≤ N , and O(N 3 ) to assemble and invert the N × N Jacobian. The essential point is that the online complexity is independent of N , thanks to offline generation and storage of the requisite parameter independent quantities q (for example, a1 (ζ j , ζn , ζi )). For this particular nonlinear problem, there is relatively little additional cost associated with the nonlinearity. However, our success depends crucially on the low-order polynomial nature of our nonlinearity: in general, standard Galerkin procedures will yield N n + 1 complexity for an nth order (n ≥ 2) polynomial nonlinearity. Although symmetries can be invoked to modestly improve the scaling with N and n [18], in any event new approaches will be * In essence – we shall see this again in the error estimation context – our quadratic nonlinearity effectively introduces N additional “parameter-dependent functions” and “parameter-independent forms” associated with the coefficients of our field-variable expansion and our trilinear form, respectively; however, these new parameter contributions are correlated in ways that we can gainfully exploit.


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required for nonpolynomial nonlinearities; we discuss these new procedures for efficient treatment of general nonaffine and nonlinear operators in Outlook.

5.3.

A Posteriori Error Estimation

The motivation for rigorous a posteriori error estimation is even more selfevident in the case of nonlinear problems. Fortunately, there is a rich mathematical foundation upon which to build the necessary computational structure. We first introduce the former; we then describe the latter. For simplicity, we develop here error bounds only for the primal energy norm, u(µ)−u N (µ) X ; we can also develop error bounds for the output – however, good effectivities will require consideration of the dual [22].

5.3.1. Error bounds We require some slight modifications to our earlier (linear) preliminaries. µ µ In particular, we introduce TN : X → X such that, for any w ∈ X , (TN w, v) X = dg(w, v; u N (µ); µ), ∀v ∈ X ; we then define σ N (w; µ) ≡ TNµ w X / w X . Our inf–sup and continuity constants – now linearized about the reduced-basis solution – can then be expressed as β N (µ) ≡ β(u N (µ); µ) = infw ∈ X σ N (w; µ), and γ N (µ) ≡ γ (u N (µ); µ) = supw ∈ X σ N (w; µ), respectively; as before, we shall need a nonnegative lower bound for the inf–sup parameter, β˜ N (µ), such that β N (µ) ≥ βÑ (µ) ≥ 0, ∀ µ ∈ D. As in the linear case, the dual norm of the residual, ε N (µ) of (22), shall play a central role; the (negative of the) residual for our current nonlinear problem is given by (42) for w = u N (µ). We also introduce a new √ combination of parameters τ N (µ) ≡ 2ρ(µ)ε N (µ)/βÑ2 (µ), where ρ(µ) = 2 GrρU (ρU + Prρθ ) depends on the Sobolev embedding constants ρU and ρθ of (48); in essense, τ N (µ) is an appropriately “nondimensionalized” measure of the residual. Finally, we define N ∗ (µ) such that τ N (µ) < 1 for N ≥ N ∗ (µ); we require N ∗ (µ) ≤ Nmax , ∀ µ ∈ D. (The latter is a condition on Nmax that reflects both the convergence rate of the RB approximation and the quality of our inf–sup lower bound.) We recall that µ ≡ Gr ∈ D ≡ [1.0, 1.0 × 104 ]. Our error bound is then expressed, for any µ ∈ D and N ≥ N ∗ (µ), as

N (µ) =

βÑ (µ) 1 − 1 − τ N (µ) . ρ(µ)

(49)

The main result can be very simply stated: if N ≥ N ∗ (µ), there exists a unique solution u(µ) to (5) in the open ball

βÑ (µ) B u N (µ), ρ(µ)

≡

Ñ (µ) β z ∈ X z − u N (µ) X < ; ρ(µ)

(50)

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N.N. Cuong et al.

furthermore,

u(µ) − u N (µ) X ≤ N (µ).

(51)

The proof, given in Ref. [22], is a slight specialization of a general abstract result [30, 31] that in turn derives from the Brezzi–Rappaz–Raviart (BRR) framework for the analysis of variational approximations of nonlinear partial differential equations [32]; the central ingredient is the construction of an appropriate contraction mapping which then forms the foundation for a standard fixed-point argument. On the basis of the main proposition (50) and (51) we can further prove several important corollaries related to the wellposedness of the truth approximation (5), and – similar to the linear result (24) – the effectivity of our error bound (49) [22]. We note that, as ε N (µ) → 0, we shall certainly satisfy N ≥ N ∗ (µ); furthermore the upper bound to the true error, N (µ) of (49), is asymptotic to ε N (µ)/βÑ (µ). We may derive these limits directly and rigorously from (49) and (51), or more heuristically from the equation for the error e(µ) ≡ u(µ) −u N (µ), dg(e(µ), v; u N (µ); µ) = −g(u N (µ), v; µ) − 12 a1 (e(µ), e(µ), v; µ). (52) We conclude that the nonlinear case shares much in common with the limiting linear case. However, there are also important differences: even for τ N (µ) < 1, we must (in general) admit the possibility of other solutions to (5) – solutions outside B(u N (µ), βÑ /ρ(µ)) – that are not near u N (µ); and for τ N (µ) ≥ 1, we cannot even be assured that there is indeed any solution u(µ) near u N (µ). This conclusion is not surprising: for “noncoercive” nonlinear problems the error equation (51) may in general admit no or several solutions; we can only be certain that a small (isolated) solution exists, (50) and (51), if the residual is sufficiently small. The theory informs us that the appropriate measure of the residual is τ N (µ), which reflects both the stability of the operator (βÑ (µ)) and the strength of the nonlinearity (ρ(µ)). As in the linear case, the real computational challenge is the development of efficient procedures for the calculation of the necessary a posteriori quantities:∗ the dual norm of the residual, ε N (µ); the inf–sup lower bound, βÑ (µ); and – new to our nonlinear problem – the Sobolev constants, ρU and ρθ . We now turn to these considerations.

* Typically, the BRR framework provides a nonquantitative a priori or a posteriori justification of asymp-

totic convergence. In our context, there is a unique opportunity to render the BRR theory completely predictive: actual a posteriori error estimators that are quantitative, rigorous, sharp, and (online) inexpensive.


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5.3.2. Offline-online computational procedures The dual norm of the residual. Fortunately, the duality relation of the linear case, (29), still applies – g(w, v; µ) of (42) is nonlinear in w, but of course linear in v. For our nonlinear problem, the negative of the residual, (42), for w = u N (µ), may be expressed in terms of the reduced-basis expansion (12) as −g(u N (µ), v; µ) = f (v) −

N

u N n (µ)

n=1

Q0

q

q

0 (µ)a0 (ζn , v)

q=1

Q1 N

1 q q 1 (µ) u N n (µ)a1 (ζn , ζn , v) , + 2 q =1 n =1

(53)

where we recall that µ ≡ Gr. If we insert (53) in (29) and apply linear superposition, we obtain e(µ) ˆ =C+

N

u N n (µ)

n=1

 Q0  

q

0 (µ)Lqn +

q=1

Q1 N q =1 n =1

q

q

 

1 (µ)u N n (µ)Qn n , 

where C ∈ X satisfies (C, v) X = f (v), ∀ v ∈ X , Lqn ∈ X satisfies (Lqn , v) X = q q q − a0 (ζn , v), ∀ v ∈ X , 1 ≤ n ≤ N , 1 ≤ q ≤ Q 0 , and Qn n ∈ X satisfies Qn n = q −a1 (ζn , ζn , v)/2, ∀ v ∈ X , 1 ≤ n, n ≤ N , 1 ≤ q ≤ Q 1 ; the latter are again simple (vector) Poisson problems. It thus follows that [22] 2

e(µ) ˆ X

= (C, C) X +

N

u N n (µ) 2

Q0

n=1

× 2

Q1

q=1

q

q

1 (µ)(C, Qn n ) X +

q=1

+

N

u N n (µ) 2

n =1

+

N n =1

q

0 (µ)(C, Lqn ) X +

Q0 Q1 q=1 q =1

u N n (µ)

Q1 Q1 q=1 q =1

q

Q0 Q0 q=1 q =1 q

q

N

u N n (µ)

n =1

q

q

0 (µ)0 (µ)(Lqn , Ln ) X q

0 (µ)1 (µ)(Lqn , Qn n ) X

q q q q 1 (µ)1 (µ)(Qn n , Qn n ) X

from which we can directly calculate the requisite dual norm of the residual through (28). We can now readily adapt the offline–online procedure developed in the linear case; however, our summation “over the affine dependence” now involves a double summation “over the reduced-basis coefficients”. The operation count for the online stage is thus (to leading order) O(Q 21 N 4 ); the essential point is that

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N.N. Cuong et al.

the online complexity is again independent of N – thanks to offline generation and storage of the requisite parameter-independent inner products (for examq q ple, (Qn n , Qn n ) X , 1 ≤ n, n , n , n ≤ N , 1 ≤ q, q ≤ Q 1 ). Although the N 4 online scaling is certainly less than pleasant, the error bound is calculated only once – at the termination of the Newton iteration – and hence in actual practice the additional online cost attributable to the residual dual norm computation is in fact not too large. However, the quartic scaling with N is again a memento mori that, for higher order (than quadratic) nonlinearities, standard Galerkin procedures are not viable; we discuss the alternatives further in Outlook. Lower bound for the inf–sup parameter. Our procedure for the linear case can be readily adopted: we need “only” incorporate the N additional parameterdependent “coefficient functions” – in fact, the RB coefficients – that appear in the linearized-about-u N (µ) derivative operator. Hence, for our nonlinear problem, the bilinear form T of (34) and Rayleigh quotient F of (35) now contain sensitivity derivatives of these additional “coefficient functions”; furthermore, the (µ, µ) ¯ function of (37) – our second-order remainder term – now includes the deviation of the RB coefficients from linear parameter dependence. Further details are provided in Ref. [22] (for Pr = 0) for the case in which W N ≡ W NU is divergence-free. Sobolev continuity constant. We present here the procedure for calculation of ρU ; the procedure for ρθ is similar. We first note [27, 28] that ρU = ˆ ξˆ ) ∈ (R+ , X U ) satisfies (1/δˆmin )1/2 , where (δ, (ξˆ , V ) X U = δˆ

ξˆ j ξˆ j ξî Vi ,

∀V ∈ X U ,

ξˆ 4L 4 () = 1,

and (δˆmin , ξˆmin ) denotes the ground state. To solve this eigenproblem, and in particular to ensure that we realize the ground state, we pursue a homotopy procedure. Towards that end, we introduce a parameter h ∈ [0, 1] (and associated small increment h) and look for (δ(h), ξ(h)) ∈ (R+ , X U ) that satisfies

(ξ(h), V ) X U = δ(h) h

ξ j (h)ξ j (h)ξi (h)Vi

+ (1 − h)

ξi (h)Vi , ∀V ∈ X U ,

h ξ 4L 4 () + (1 − h) ξ 2L 2 () = 1;

(54)

(δmin (h), ξmin (h)) denotes the ground state. We observe that (δmin (1), ξmin (1))= (δˆmin , ξˆmin ); and that (δmin (0), ξmin (0)) is the lowest eigenpair of the standard


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(vector) Laplacian “linear” eigenproblem. Our homotopy procedure is simple: we first set h old = 0 and find (δmin (0), ξmin (0)) by standard techniques; then, until h new = 1, we set h new ← h old + h, solve (54) for (δmin (h new ), ξmin (h new )) by Newton iteration initialized to (δmin (h old), ξmin (h old )), and update h old ← h new . For our domain, we find (offline) ρU = 0.6008, ρθ = 0.2788; since ρU and ρθ are parameter-independent, no online computation is required.

5.3.3. Sample construction The greedy algorithm developed in the linear case requires some modification in the nonlinear context. The first issue is that, to evaluate our error bound N (µ), we must appeal to our inf–sup lower bound; however, in the nonlinear case, this inf–sup lower bound, βÑ (µ), is defined with respect to the linearized state u Nmax (µ) [22]. In short, to determine the “next” sample point µ N+1 we must already know S Nmax – and hence µ N+1 . To avoid this circular reference during the offline sample generation process, we replace our inf–sup lower bound with a crude (for example, piecewise constant over D) approximation to β(u(µ)); once the samples are constructed, we revert to our rigorous (and now calculable) lower bound, βÑ (µ). The second issue is that, in the nonlinear context, our error bound is not operative until τ N (µ) < 1; hence, the greedy procedure must first select on arg maxµ∈F τ N (µ) – until τ N (µ) < 1 over D – and only subsequently select on arg maxµ ∈ F N (µ) [Prud’homme, private communication]. The resulting sample will ensure not only rapid convergence to the exact solution, but also rapid convergence to a certifiably accurate solution.

5.4.

Numerical Results

We present in Table 2 u(µ˜ N ) − u N (µ˜ N ) X / u(µ˜ N ) X , N,rel (µ˜ N ) ≡

N (µ˜ N )/ u N (µ˜ N ) X , and η N (µ˜ N ) ≡ N (µ˜ N )/ e(µ˜ N ) X for 8 ≤ N ≤ Nmax = 40; here µ˜ N ≡ arg max

µ∈Test

u(µ) − u N (µ) X

u(µ) X

and Test is a random parameter grid of size n Test = 500. We observe very rapid convergence of u N (µ) to u(µ) over D (more precisely, Test ) – our samples S N are optimally constructed to provide uniform convergence. The output error decreases even more rapidly: maxµ ∈ Test |s(µ) − s N (µ)|/s(µ) = 1.34 × 10−1 , 2.80 × 10−4 , and 9.79 × 10−7 for N = 8, 16, and 24, respectively; this “superconvergence” is a vestige of near compliance. As regards a posteriori error estimation, we observe that N ∗ (µ˜ N ) = 24

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N.N. Cuong et al. Table 2. Convergence and effectivity results for the natural convection problem; the “*” signifies that N ∗ (µ˜ N ) > N, which in turn indicate that τ N (µ˜ N ) ≥ 1 N

u(µ˜ N ) − u N (µ˜ N ) X

u(µ˜ N ) X

N,rel (µ˜ N )

η N (µ˜ N )

8 16 24 32 40

3.28 × 10−1 1.45 × 10−2 1.80 × 10−4 8.05 × 10−7 4.60 × 10−8

* * 7.47 × 10−4 7.60 × 10−6 8.69 × 10−7

* * 4.15 9.44 18.93

is relatively small – we can (respectively, can not) provide a definitive error bound for N ≥ 24 (respectively, N < 24); more generally, we find that N ∗ (µ) ≤ 24, ∀ µ ∈ D. We note that the effectivities are quite good∗ – in fact, considerably better than the worst-case predictions of our effectivity corollary. (The higher effectivity at N = 40 is undoubtedly due to round-off in the online summation.) The results of Table 2 are based on an inf–sup lower bound construction with J = 28 elements: points µ¯ j and polytopes (here segments) P µ¯ j , 1 ≤ j ≤ J . The accuracy of the resulting lower bound is reflected in the modest N ∗ (µ) and the good effectivities reported in Table 2. Most of the points µ¯ j are clustered at larger Gr, as might be expected. Finally, we note that the total online computational time on a Pentium M 1.6 GHz processor to predict u N (Gr), s N (Gr), and N (Gr) to a relative accuracy (in the energy norm) of 10−3 is – ∀ Gr ∈ D – 300 ms; this should be compared to 50 s for direct finite element calculation of the truth solution, u(Gr), s(Gr). We achieve computational savings of O(100): N is very small thanks to (i) the good convergence properties of S N and hence W N , and (ii) the rigorous and sharp stopping criterion provided by N (Gr); and the marginal computational complexity to evaluate s N (Gr) and N (Gr) depends only on N and not on N – thanks to the offline–online decomposition. The computational savings will be even more significant for more complex problems particularly in three spatial dimensions; it is critical to recall that we realize these savings without compromising rigorous certainty.† * It is perhaps surprising that the BRR theory – not really designed for quantitative service – yields such sharp results. However, it is important to note that, as ε N (µ) → 0, N (µ) ∼ ε N (µ)/β˜ N (µ), and thus the more pessimistic bounds (in particular ρ) are absent – except in τ N (µ). † We admit that the extension of our results to much larger Gr is not without difficulty. The more complex flow structures and the stronger nonlinearity will degrade the convergence rate and a posteriori error bounds – and increase N and J ; and (inevitable) limit points and bifurcations will require special precautions.


6.

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Outlook

We address here some of the more obvious questions that arise in reviewing the current state of affairs. As a first question: How many parameters P can we consider – for P how large are our techniques still viable? It is undeniably the case that ultimately we should anticipate exponential scaling (of both N and certainly J ) as P increases, with a concomitant unacceptable increase certainly in offline but also perhaps in online computational effort. Fortunately, for smaller P, the growth in N is rather modest, as (good) sampling procedures will automatically identify the more interesting regions of parameter space. Unfortunately, the growth in J is more problematic: we shall require more efficient construction and verification procedures for our inf–sup lower bound samples. In any event, treatment of hundreds (or even many tens) of truly independent parameters by the global methods described in this chapter is clearly not practicable; in such cases, more local approaches must be pursued.∗ A second question: How can we efficiently treat problems with non-affine parameter dependence and (more than quadratic) state-space nonlinearity? Both these issues are satisfactorily addressed by a new “empirical interpolation” approach [33]. In this approach, we replace a general nonaffine nonlinear function of the parameter µ, spatial coordinate x, and field variable u(x; µ), H(u; x; µ), by a collateral RB expansion: in particular, we approxµ); x; µ) – as required in our RB projection for u N (µ) – by imate H(u N (x; M H M (x; µ) = m=1 dm (µ)ξm (x). The critical ingredients of the approach are H = {µH , . . . , µH }, and approximation (i) a “good” collateral RB sample, S M 1 M H H space, span{ξm = H(u(µm ); x; µm ), 1 ≤ m ≤ M}, (ii) a stable and inexpensive interpolation procedure by which to determine (online) the dm (µ), 1 ≤ m ≤ M, and (iii) effective a posteriori error bounds with which to quantify the effect of the newly introduced truncation. It is perhaps only in the latter that the technique is somewhat disappointing: the error estimators – though quite sharp and very efficient – are completely (provably) rigorous upper bounds only in certain restricted situations. Finally, a third question, again related to generality: What class of PDEs can be treated? In addition to the elliptic equations discussed in this paper, parabolic equations can also be addressed satisfactorily from both the approximation and error estimation points of view [24, 34, 35]:† much of the elliptic technology directly applies, except that time now appears as an additional parameter; this parabolic framework can be viewed as an extension of * We do note that at least some problems with ostensibly many parameters in fact involve highly coupled

or correlated parameters: certain classes of shape optimization certainly fall into this category. In these situations, global progress can be made. † To date we have experience with only stable parabolic systems such as the heat equation; unstable systems present considerable difficulty, in particular if long-time solutions are desired.

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time-domain model reduction procedures [19, 25, 36]. Unfortunately, treatment of hyperbolic problems does not look promising: although RB methods can perform quite well anecdotally, in general the underlying smoothness (in parameter µ) and stability will no longer obtain; as a result, both the approximation properties and error estimators will suffer. We close by noting that the offline aspects of the approaches described are both complicated and computationally expensive. The former can be at least partially addressed by appropriate software and architectures [37]; however, the latter will in any event remain. It follows that these techniques will really only be viable in situations in which there is truly an imperative for real-time certified response: a real premium on (i) greatly reduced marginal cost (or asymptotic average cost), and (ii) rigorous characterization of certainty; or equivalently, a very high (opportunity) cost associated with (i) slow response – long latency times, and (ii) incorrect (or unsafe) decisions or actions. There are many classes of materials and materials processing problems and contexts for which the methods are appropriate; and certainly there are many classes of materials and materials processing problems and contexts for which more classical techniques remain distinctly preferred.

Appendix A Helmholtz Elasticity Example We first define a reference domain corresponding to the geometry b = br = 1 and L = L r = 0.2. We then map o (b, L) → ≡ o (br , L r ) by a continuous piecewise-affine (in fact, piecewise-dilation-in-x1 ) transformation. We define three subdomains, 1 ≡ ] 0, br − L r /2 [ × ] 0, 1 [ , 2 ≡ ] br − L r /2, br + L r / ¯ = ¯1∪ ¯2∪ ¯ 3. 2 [× ] 0, 1[, 3 ≡ ]br + L r /2, 2 [×] 0, 1 [, such that We may then express the resulting bilinear form a(w, v; µ) as an affine sum (7) for Q = 10; the particular q (µ), a q (w, v), 1 ≤ q ≤ 10, as shown in Table 3. (Recall that w = (w1 , w2 ) and v = (v 1 , v 2 ).) The constitutive constants in Table 3 are given by c11 =

1 , 1 − ν2

c22 = c11 ,

c12 =

ν , 1 − ν2

c66 =

1 , 2(1 + ν)

where ν = 0.25 is the Poisson ratio (and the normalized Young’s modulus is unity); recall that we consider plane stress and a linear isotropic solid. We now define our inner product-cum-bound conditioner as (w, v) X ≡

c11

∂v 1 ∂w1 ∂v 2 ∂w2 ∂v 2 ∂w2 ∂v 1 ∂w1 + c22 + c66 + c66 ∂ x1 ∂ x1 ∂ x2 ∂ x2 ∂ x1 ∂ x1 ∂ x2 ∂ x2

+ w1 v 1 + w2 v 2

=

Q q=2

a q (w, v) ;


1561

Table 3. Parametric functions q (µ) and parameter-independent bilinear forms a q (w, v) for the two-dimensional crack problem q (µ)

q 1

1

c12

a q (w, v) ∂v 1 ∂w2 ∂v ∂w1 + 2 ∂ x1 ∂ x2 ∂ x2 ∂ x1

+ c66 2

br − L r /2 b − L/2

3

Lr L

4

2 − br − L r /2 2 − b − L/2

5

b − L/2 br − L r /2

6

L Lr

7

2 − b − L/2 2 − br − L r /2

8

−ω2

9

L −ω2 Lr

10

c11 1

c11 2

c11 3

c22 1

c22

b − L/2 br − L r /2

2 − b − L/2 −ω2 2 − br − L r /2

2

c22

3

∂v 1 ∂w1 ∂ x1 ∂ x1 ∂v 1 ∂w1 ∂ x1 ∂ x1 ∂v 1 ∂w1 ∂ x1 ∂ x1 ∂v 2 ∂w2 ∂ x2 ∂ x2 ∂v 2 ∂w2 ∂ x2 ∂ x2 ∂v 2 ∂w2 ∂ x2 ∂ x2

∂v 1 ∂w2 ∂v ∂w1 + 2 ∂ x2 ∂ x1 ∂ x1 ∂ x2

+ c66

1

2

3

1

2

+ c66

+ c66

+ c66

+ c66

+ c66

3

∂v 2 ∂w2 ∂ x1 ∂ x1 ∂v 2 ∂w2 ∂ x1 ∂ x1 ∂v 2 ∂w2 ∂ x1 ∂ x1 ∂v 1 ∂w1 ∂ x2 ∂ x2 ∂v 1 ∂w1 ∂ x2 ∂ x2 ∂v 1 ∂w1 ∂ x2 ∂ x2

w1 v 1 + w2 v 2 1

w1 v 1 + w2 v 2 2

w1 v 1 + w2 v 2 3

thanks to the Dirichlet conditions at x1 = 0 (and also the wi v i term), (·, ·) X is appropriately coercive. We now observe that (µ) = 1 ( 1 = 0) and we can thus disregard the q = 1 term in our continuity bounds. We may then choose |v|2q = a q (v, v), 2 ≤ q ≤ Q, since the a q (·, ·) are positive semi-definite; it thus follows from the Cauchy–Schwarz inequality that q = 1, 2 ≤ q ≤ Q; furthermore, from (36), we directly obtain C X = 1.

Acknowledgments We would like to thank Professor Yvon Maday of University Paris VI for his many invaluable contributions to this work. We would also like to thank

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Dr Christophe Prud’homme of EPFL, Mr Martin Grepl of MIT, Mr Gianluigi Rozza of EPFL, and Professor Liu Gui-Rong of NUS for many helpful recommendations. This work was supported by DARPA and AFOSR under Grant F49620-03-1-0356, DARPA/GEAE and AFOSR under Grant F49620-03-10439, and the Singapore-MIT Alliance.

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