ADVANCES IN
QUANTUM CHEMISTRY VOLUME 51
EDITORIAL BOARD
David M. Bishop (Ottawa, Canada) Guillermina Estiú (University Park, PA, USA) Frank Jensen (Odense, Denmark) Mel Levy (Greensboro, NC, USA) Jan Linderberg (Aarhus, Denmark) William H. Miller (Berkeley, CA, USA) John Mintmire (Stillwater, OK, USA) Manoj Mishra (Mumbai, India) Jens Oddershede (Odense, Denmark) Josef Paldus (Waterloo, Canada) Pekka Pyykkö (Helsinki, Finland) Mark Ratner (Evanston, IL, USA) Adrian Roitberg (Gainesville, FL, USA) Dennis Salahub (Calgary, Canada) Henry F. Schaefer III (Athens, GA, USA) Per Siegbahn (Stockholm, Sweden) John Stanton (Austin, TX, USA) Harel Weinstein (New York, NY, USA)
ADVANCES IN
QUANTUM CHEMISTRY EDITORS JOHN R. SABIN
ERKKI BRÄNDAS
QUANTUM THEORY PROJECT UNIVERSITY OF FLORIDA GAINESVILLE, FLORIDA
DEPARTMENT OF QUANTUM CHEMISTRY UPPSALA UNIVERSITY UPPSALA, SWEDEN
FOUNDING EDITOR
PER-OLOV LÖWDIN 1916–2000
VOLUME 51
AMSTERDAM • BOSTON • HEIDELBERG • LONDON • NEW YORK • OXFORD PARIS • SAN DIEGO • SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Academic Press is an imprint of Elsevier
Academic Press is an imprint of Elsevier 84 Theobald’s Road, London WC1X 8RR, UK Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands 525 B Street, Suite 1900, San Diego, CA 92101-4495, USA 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 2006 Copyright © 2006, Elsevier Inc. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email:
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ISSN: 0065-3276 For information on all Academic Press publications visit our web site at http://books.elsevier.com Printed and bound in USA 06 07 08 09 10
10 9 8 7 6 5 4 3 2 1
Contents
Contributors Preface
vii ix
The Usefulness of Exponential Wave Function Expansions Employing One- and Two-Body Cluster Operators in Electronic Structure Theory: The Extended and Generalized Coupled-Cluster Methods Peng-Dong Fan and Piotr Piecuch 1. Introduction 2. Practical ways of improving coupled-cluster methods employing singly and doubly excited clusters via extended coupled-cluster theory 3. Non-iterative corrections to extended coupled-cluster energies: Generalized method of moments of coupled-cluster equations 4. Virtual exactness of exponential wave function expansions employing generalized oneand two-body cluster operators in electronic structure theory Acknowledgements References
Angular Momentum Diagrams Paul E.S. Wormer and Josef Paldus 1. 2. 3. 4. 5. 6.
2 4 26 43 54 54 59 60 62 74 89 101 113 115 115 121
Introduction The essentials of SU(2) Diagrams Basic rules for angular momentum diagrams Irreducible closed diagrams Concluding remarks Acknowledgement Appendix: Summary of the graphical rules References
Chemical Graph Theory—The Mathematical Connection Ivan Gutman 1. 2. 3. 4. 5.
1
Prologue Introduction The first case study: Graph energy The second case study: Connectivity (Randi´c) index More examples v
125 125 126 127 130 132
vi
Contents
6. Concluding remarks Acknowledgement References
133 134 134
Atomic Charges via Electronegativity Equalization: Generalizations and Perspectives 139 Alexander A. Oliferenko, Sergei A. Pisarev, Vladimir A. Palyulin and Nikolai S. Zefirov 1. Introduction 140 2. Two approaches to electronegativity redistribution 141 3. Principle of electronegativity relaxation 149 4. Numerical examples 152 5. Chemical applications of atomic charges 153 6. Conclusions 153 References 154 Fast Padé Transform for Exact Quantification of Time Signals in Magnetic Resonance Spectroscopy Dževad Belki´c 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Introduction Challenges with quantification of time signals from MRS The quantum-mechanical concept of resonances in scattering and spectroscopy Resonance profiles The role of quantum mechanics in signal processing Suitability of the fast Padé transform for signal processing Fast Padé transforms inside and outside the unit circle Results Discussion Conclusion Acknowledgements References
Probing the Interplay between Electronic and Geometric Degrees-of-Freedom in Molecules and Reactive Systems Roman F. Nalewajski 1. 2. 3. 4. 5. 6. 7. 8. 9.
Introduction Summary of basic relations Electronic and nuclear sensitivities in geometric representations Minimum-energy coordinates in compliance formalism Compliant indices of atoms-in-molecules Atomic resolution—A reappraisal Collective charge displacements and mapping relations Concepts for reacting molecules Conclusion References
Subject Index
157 158 161 168 171 172 174 175 184 222 228 232 232
235 236 241 256 263 265 271 276 282 299 301 307
Contributors
Numbers in parentheses indicate the pages where the authors’ contributions can be found. Dževad Belki´c, Karolinska Institute, PO Box 260, S-171 76 Stockholm, Sweden, dzevad.belkic@ ki.se Peng-Dong Fan, Department of Chemistry, Michigan State University, East Lansing, MI 48824, USA and Quantum Theory Project, Department of Chemistry, University of Florida, PO Box 118435 Gainesville, FL 32611, USA Ivan Gutman, Faculty of Science, University of Kragujevac, 34000 Kragujevac, Serbia,
[email protected] Roman F. Nalewajski, Department of Theoretical Chemistry, Jagiellonian University, R. Ingardena 3, 30-060 Cracow, Poland Alexander A. Oliferenko, Pacific Northwest National Laboratory, Richland, WA 99352, USA and Department of Chemistry, Moscow State University, Moscow 119992, Russia,
[email protected]. Josef Paldus, Department of Applied Mathematics, Department of Chemistry, and Guelph-Waterloo Center for Graduate Work in Chemistry—Waterloo Campus, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1,
[email protected] Vladimir A. Palyulin, Department of Chemistry, Moscow State University, Moscow 119992, Russia Piotr Piecuch, Department of Chemistry, Michigan State University, East Lansing, MI 48824, USA and Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA,
[email protected] Sergei A. Pisarev, Department of Chemistry, Moscow State University, Moscow 119992, Russia Paul E.S. Wormer, Theoretical Chemistry, Institute for Molecules and Materials, Radboud University Nijmegen, Toernooiveld, 6525 ED Nijmegen, The Netherlands, pwormer@ teochem.ru.nl Nikolai S. Zefirov, Department of Chemistry, Moscow State University, Moscow 119992, Russia
vii
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Preface
Volume 51 of Advances in Quantum Chemistry deals with various aspects of mathematical versus chemical applications. Some parts belong to established scientific domains, where technical progress has been crucial for the development of modern quantum chemistry as well as the quantification problem in spectral resonance analysis. Other advances, though remarkably useful, have attracted controversial comments as, e.g., being too simple in execution to be correct. All told, should not the possible interconnections between mathematics and natural sciences in general and in chemistry in particular lead to more interesting and useful interpretations? As is usual, the editors leave the decision to the readers. The first article in the volume, by Peng-Dong Fan and Piotr Piecuch, concerns the calculation of molecular electronic structure to high accuracy, using a variety of one- and two-body schemes in the coupled cluster (CC) family of methods. Questions involving accurate calculation of potential surfaces, including regions representing bond breaking are addressed, and the use of a posteriori corrections leading to increased accuracy are taken up. Perhaps the most controversial question addressed in this chapter deals with the possibility of obtaining a “virtually exact” many-electron wavefunction using no more than two-body operators in a cluster expansion. Chapter 2 is devoted to “Angular Momentum Diagrams”, written by Paul Wormer and Josef Paldus. It represents a shorter version of what could have been a book on the Unitary Group Approach, UGA. It takes the reader from the purely mathematical work on invariant theory by Clebsch and Gordan, via Jucys and Bandzaitis to current versions of modern many-body theories applied to realistic quantum chemical systems, including recent developments by the authors and Piotr Piecuch, an author of the previous chapter. In Chapters 3 and 4, Ivan Gutman and Alexander Oliferenko respectively, portray Chemical Graph Theory (CGT). Chemical graphs deal with mathematical aspects of classical chemical structure theory building from, but not limited to, the classical concepts. It is argued that a clear symbiosis between CGT and several other modern areas of chemical theory including for the most part quantum chemistry should be expected. In Chapter 3 Gutman examines the impact that research done in CGT may have on mathematics advocating with two explicit case studies that the influence is not insignificant. Oliferenko et al., in the following chapter, rationalizes electrostatics and molecular descriptors in terms of CGT displaying interesting formulations where CGT meets mathematical physics. Dževad Belki´c advances quantum mechanical signal processing through the fast Padé transform (FPT) in Chapter 5. Here, it is shown that the FPT can analytically continue general functions outside their domain, and is an effective solver of generalized eigenproblems. The analogy between the spectral characteristics of synthesized time signals and Magnetic Resonance Spectroscopy (MRS) is employed to carry out reliable quantifications ix
x
Preface
of one of the most promising non-evasive diagnostic tools in medicine. Belki´c supports the theory with numerous tables and figures, displaying the mathematical problems (illconditioned) of initial MRS data and the remarkable reliability of the fast Padé transform to quantify spectral structures from isolated resonances to tightly overlapped and nearly confluent ones. The concluding chapter, by Roman Nalewajski, gives a mathematical view of molecular equilibria using a Density-Functional Theory (DFT) description. Both the externally closed (N -controlled) and open (μ-controlled) systems are explored within the Born– Oppenheimer approximation. The complementary Electron Following (EF) and Electron Preceding (EP) perspectives on molecular processes are examined, in which the external potential due to nuclei and the system electron density, respectively, provide the local statevariable of the molecular ground-state. We offer this volume of Advances in Quantum Chemistry to you with the confidence that it will be an interesting and informative read. Erkki J. Brändas and John R. Sabin Editors
The Usefulness of Exponential Wave Function Expansions Employing One- and Two-Body Cluster Operators in Electronic Structure Theory: The Extended and Generalized Coupled-Cluster Methods Peng-Dong Fan1,* and Piotr Piecuch1,2 1 Department of Chemistry, Michigan State University, East Lansing, MI 48824, USA 2 Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA
E-mail:
[email protected] Abstract In this paper, the applicability of exponential cluster expansions involving one- and two-body operators in high accuracy ab initio electronic structure calculations is examined. First, the extended coupled-cluster method with singles and doubles (ECCSD) is tested in the demanding studies of systems with strong quasi-degeneracies, including potential energy surfaces involving multiple bond breaking. The numerical results show that the singlereference ECCSD method is capable of providing a qualitatively correct description of quasi-degenerate electronic states and potential energy surfaces involving bond breaking, eliminating, in particular, the failures and the unphysical behavior of standard coupled-cluster methods in similar cases. It is also demonstrated that one can obtain entire potential energy surfaces with millihartree accuracies by combining the ECCSD theory with the non-iterative a posteriori corrections obtained by using the generalized variant of the method of moments of coupled-cluster equations. This is one of the first instances where the relatively simple single-reference formalism, employing only one- and two-body clusters in the design of the relevant energy expressions, provides a highly accurate description of the dynamic and significant non-dynamic correlation effects characterizing quasi-degenerate and multiply bonded systems. Second, an evidence is presented that one may be able to represent the virtually exact ground- and excited-state wave functions of many-electron systems by exponential cluster expansions employing general two-body or one- and two-body operators. Calculations for small many-electron model systems indicate the existence of finite two-body parameters that produce the numerically exact wave functions for ground and excited states. This finding may have a significant impact on future quantum calculations for many-electron systems, since normally one needs triply excited, quadruply excited, and other higher-than-doubly excited Slater determinants, in addition to all singly and doubly excited determinants, to obtain the exact or virtually exact wave functions. Contents 1. Introduction 2. Practical ways of improving coupled-cluster methods employing singly and doubly excited clusters via extended coupled-cluster theory 2.1. Extended coupled-cluster theory: A brief overview of the general formalism 2.2. Extended coupled-cluster methods with singles and doubles 2.2.1. The Piecuch–Bartlett ECCSD approach 2.2.2. The Arponen–Bishop ECCSD approach
2 4 4 7 8 9
* Present address: Quantum Theory Project, Department of Chemistry, University of Florida, PO Box 118435, Gainesville, FL 32611, USA.
ADVANCES IN QUANTUM CHEMISTRY, VOLUME 51 ISSN: 0065-3276 DOI: 10.1016/S0065-3276(06)51001-9
© 2006 Elsevier Inc. All rights reserved
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P.-D. Fan and P. Piecuch
2.3. Numerical results for multiple bond breaking 2.3.1. The Piecuch–Bartlett ECCSD approach 2.3.2. The Arponen–Bishop ECCSD approach 2.4. Conclusion 3. Non-iterative corrections to extended coupled-cluster energies: Generalized method of moments of coupled-cluster equations 3.1. The method of moments of coupled-cluster equations 3.2. The generalized MMCC formalism 3.3. The ECCSD(T), ECCSD(TQ), QECCSD(T), and QECCSD(TQ) methods and their performance in calculations for triple bond breaking in N2 3.4. Summary 4. Virtual exactness of exponential wave function expansions employing generalized one- and two-body cluster operators in electronic structure theory 4.1. Theory 4.1.1. The exp(X) conjecture 4.1.2. Formal arguments in favor of the exp(X) conjecture (ground states) 4.1.3. Extension of the exp(X) conjecture to excited states 4.2. Numerical results 4.3. Summary Acknowledgements References
13 13 22 24 26 26 31 34 42 43 43 44 46 49 50 54 54 54
1. INTRODUCTION Great advances have been made in ab initio quantum chemistry. Highly accurate calculations for closed-shell and simple open-shell molecular systems involving a few atoms are nowadays routine. This, in particular, applies to coupled-cluster (CC) theory [1–5], which has become the de facto standard for high accuracy calculations for atomic and molecular systems [6–13]. The basic single-reference CC methods, such as CCSD (CC approach with singles and doubles) [14] and the non-iterative CCSD + T(CCSD) = CCSD[T] [15] and CCSD(T) [16] approaches that account for the effect of tri-excited clusters by using arguments based on the many-body perturbation theory (MBPT), in either the spin-orbital [14–16] and spin-free [17–19] or orthogonally spin-adapted [20–22] forms, are nowadays routinely used in accurate electronic structure calculations. The idea of adding the a posteriori corrections due to higher-than-doubly excited clusters to CCSD energies, on which the CCSD[T] and CCSD(T) approaches and their higher-order analogs, such as CCSD(TQf ) (CC method with singles, doubles, and non-iterative triples and quadruples) [23], are based, is particularly attractive, since it leads to methods that offer an excellent compromise between high accuracy and relatively low computer cost, as has been demonstrated over and over in numerous atomic and molecular applications [7–10,12,13]. There are, however, open problems in CC theory. First and foremost is the pervasive failing of the standard single-reference CC methods, such as CCSD, CCSD[T], CCSD(T), and CCSD(TQf ), at larger internuclear separations, when the spin-adapted restricted Hartree– Fock (RHF) configuration is used as a reference, which limits the applicability of the standard CC approaches to molecules near their equilibrium geometries. Second is the large computer effort associated with the need for using higher-than-doubly excited clusters in calculations involving quasi-degenerate electronic states and bond breaking, particularly when larger many-electron systems are examined. Undoubtedly, it would be very useful to extend the applicability of the standard single-reference CC methods to entire molecular
Exponential Wave Function Expansions Employing One- and Two-Body Cluster Operators
3
potential energy surfaces (PESs) involving bond breaking, and quasi-degenerate electronic states in general, without invoking complicated and often time-consuming steps associated with the more traditional multi-reference approaches, in which one has to choose active orbitals and multi-dimensional reference spaces on an ad hoc molecule-by-molecule basis. Ideally, one would like to develop a straightforward theory which could provide a virtually exact description of many-electron wave functions with the exponential, CC-like, expansions involving one- and two-body operators only, since the molecular electronic Hamiltonian does not include higher-than-two-body interactions. There are several specific challenges in all those areas. First of all, the RHF-based CCSD method, on which the non-iterative CCSD[T], CCSD(T), and CCSD(TQf ) approaches are based, is inadequate for the description of bond breaking and quasi-degenerate states, since it neglects all higher-than-doubly excited clusters, including, for example, the important triply and quadruply excited T3 and T4 components. Second, the non-iterative triples and quadruples corrections defining the CCSD[T], CCSD(T), and CCSD(TQf ) methods aggravate the situation even further, since the usual arguments originating from MBPT, on which these non-iterative CC approximations are based, fail due to the divergent behavior of the MBPT series at larger internuclear separations (or when the strong configurational quasi-degeneracy and large non-dynamic correlation effects set in). In consequence, the ground-state PESs obtained with the CCSD[T], CCSD(T), CCSD(TQf ), and other standard non-iterative CC approaches are completely pathological when the RHF configuration is used as a reference (cf., e.g., Refs. [9,11–13,24–37] and references therein). The iterative extensions of the CCSD[T], CCSD(T), and CCSD(TQf ) methods, including, among many others, the CCSDT-n [21,38–41] and CCSDTQ-1 [42] approaches, and the non-iterative CCSDT + Q(CCSDT) = CCSDT[Q] [42] and CCSDT(Qf ) [23] methods, in which the a posteriori corrections due to T4 cluster components are added to the full CCSDT (CC singles, doubles, and triples) [43,44] energies, improve the description of PES in the bond breaking region, but ultimately all of these approaches break down at larger internuclear distances (see, e.g., Refs. [28,29,34]). One might try to resolve the failures of the standard single-reference CC approaches in the bond breaking region and for quasi-degenerate electronic states in more of a brute-force manner by including the triply excited, quadruply excited, pentuply excited, etc. clusters fully and in a completely iterative fashion, but, unfortunately, the resulting CCSDTQ (CC singles, doubles, triples, and quadruples) [45–48], CCSDTQP (CC singles, doubles, triples, quadruples, and pentuples) [49], etc. approaches are far too expensive for routine molecular applications. For example, the full CCSDTQ method requires iterative steps that scale as n4o n6u (no (nu ) is the number of occupied (unoccupied) orbitals in the molecular orbital basis). This N 10 scaling of the CPU operation count with the system size (N ) characterizing the CCSDTQ approach, combined with the enormous n4o n4u storage requirements for quadruply excited cluster amplitudes, restricts the applicability of the CCSDTQ method to very small systems, consisting of ∼2–3 light atoms described by small basis sets. For comparison, CCSD(T) is an n2o n4u (or N 6 ) procedure in the iterative CCSD steps and an n3o n4u (or N 7 ) procedure in the non-iterative part related to the calculation of the triples (T) correction, and the storage requirements for cluster amplitudes characterizing the CCSD(T) calculations are very small (no nu for singles and n2o n2u for doubles). In consequence, it is nowadays possible to perform the fairly routine CCSD(T) calculations for systems with up to 20–30 light atoms and a few heavier (transition metal) atoms. This indicates that in searching for new methods that would help to overcome the failures of the standard CC approaches in the bond breaking region and
4
P.-D. Fan and P. Piecuch
other cases of electronic quasi-degeneracies, one should focus on the idea of improving the results of the low-order CC calculations, such as CCSD, with the non-iterative corrections of the CCSD(T) type, since such methods have a chance to be applied to larger molecular systems in the not-too-distant future. In view of the above discussion, the question: Can one improve the quality of standard CC wave functions in the bond breaking region at the basic CCSD level of the singlereference CC theory? seems to be particularly important. In this paper, we review the results of our recent studies which clearly show that this can be accomplished by exploring the extended coupled-cluster (ECC) theory. The basic ECCSD results, particularly when multiple bonds are stretched or broken, are qualitatively much better than the corresponding standard CCSD results. However, they are not yet fully quantitative. This prompts another question: Can one improve the quality of the ECCSD results by adding a simple a posteriori correction to the ECCSD energy which is obtained by using the singly and doubly excited cluster amplitudes obtained with the ECCSD approach? In this paper, we demonstrate that the answer to this question may also be affirmative if we use, for example, the generalized method of moments of coupled-cluster equations (GMMCC). Eventually, of course, one would prefer to use only one- and two-body clusters to obtain an exact or virtually exact description of many-electron systems, since, as we have already mentioned, the Hamiltonians used in quantum chemistry do not contain higher-than-pairwise interactions. This prompts the third and the final question examined in this work: Can one obtain the exact or virtually exact many-electron wave functions by using exponential expansions involving at most two-body cluster operators?
2. PRACTICAL WAYS OF IMPROVING COUPLED-CLUSTER METHODS EMPLOYING SINGLY AND DOUBLY EXCITED CLUSTERS VIA EXTENDED COUPLED-CLUSTER THEORY 2.1. Extended coupled-cluster theory: A brief overview of the general formalism The extended coupled-cluster (ECC) theory is based on the asymmetric, doubly connected energy functional [50–60], (ECC)
E0
= Φ|H¯¯ |Φ,
(1)
where |Φ is the independent-particle-model reference configuration (e.g., the Hartree– Fock determinant) and † † † † † † (2) H¯¯ = eΣ e−T H eT e−Σ = eΣ H¯ e−Σ = eΣ H¯ C = eΣ H eT C C is the doubly transformed Hamiltonian, obtained by transforming the similarity-transformed Hamiltonian H¯ used in the standard CC theory, H¯ = e−T H eT = H eT C , (3) where H is the Hamiltonian and C stands for the connected part of the corresponding † operator expression, with the exponential operator e−Σ . T is the usual cluster operator, which is a particle–hole excitation operator generating the connected components of the
Exponential Wave Function Expansions Employing One- and Two-Body Cluster Operators
5
many-electron ground-state wave function |Ψ0 = eT |Φ,
(4)
Σ†
and is the auxiliary hole–particle deexcitation operator. In the exact theory, T is a sum of all many-body components Tn with n = 1, . . . , N , T =
N
(5)
Tn ,
n=1
where N is the number of electrons and Tn is defined as a1 ···an n Tn = tai11···i ···an Ei1 ···in ,
(6)
i1
