Molecular approach to solids

VIBRATIONALSPECTRA AND STRUCTURE Volume 23 MOLECULAR APPROACH TO SOLIDS EDITORIALBOARD Dr. Lester Andrews University...

Author: A.N. Lazarev | J.R. Durig

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VIBRATIONALSPECTRA AND STRUCTURE Volume 23

MOLECULAR APPROACH TO SOLIDS

EDITORIALBOARD Dr. Lester Andrews University of Virginia Charlottesville, Virginia USA

Dr. J.A. Koningstein Carleton University Ottawa, Ontario CANADA

Dr. John E. Bertie University of Alberta Edmonton, Alberta CANADA

Dr. George E. Leroi Michigan State University East Lansing, Michigan USA

Dr. A.R.H. Cole University of Western Australia Nedlands WESTERN AUSTRALIA

Dr. S.S. Mitra University of Rhode Island Kingston, Rhode Island USA

Dr. William G. Fateley Kansas State University Manhattan, Kansas USA

Dr. A. M{Jller Universit~it Bielefeld Bielefeld WEST GERMANY

Dr. H. Hs. GiJnthard Eidg. Technische Hochschule Zurich SWITZERLAND

Dr. Mitsuo Tasumi University of Tokyo Tokyo JAPAN

Dr. P.J.Hendra University of Southampton Southampton ENGLAND

Dr. Herbert L. Strauss University of California Berkeley, California USA

JAMES R. DUNG (SeriesEditor) CollegeOfArtsandsciences Universityof Missouri-KansasCity Kansas City, Missouri

A SERIES OF ADVANCESVOLUME23

MOLECULAR APPROACH TO SOLIDS

A.N. Lazarev St. Petersburg, Russia

1998

ELSEVIER Amsterdam

- Lausanne - New York - Oxford - Shannon - Singapore - Tokyo

ELSEVIER SCIENCE B.V. Sara Burgerhartstraat 25 PO. Box 211,1000AE Amsterdam The Netherlands

Library of Congress Cataloging in Publication Data Acatalog record from the Library of Congress has been applied for.

ISBN 0-444-50039-1 O 1998 Elsevier Science B.V. All rights resewed.

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, Elsevier Science B.V. , Copyright & Permissions Department, PO. Box 521,1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A,. should be referred to the copywright owner, Elsevier Science B.V., unless otherwise specified. @The paper used in this publication meetsthe requirements of ANSI/NISOZ39.48-1992 (Permanenceof Paper).

Printed in The Netherlands

PREFACE TO THE SERIES

It appears that one of the greatest needs of science today is for competent people to critically review the recent literature in conveniently small areas and to evaluate the real progress that has been made, as well as to suggest fruitful avenues for future work. It is even more important that such reviewers clearly indicate the areas where little progress is being made and where the chances of a significant contribution are minuscule either because of faulty theory, inadequate experimentation, or just because the area is steeped in unprovable yet irrefutable hypotheses. Thus, it is hoped that these volumes will contain critical summaries of recent work, as well as review the fields of current interest. Vibrational spectroscopy has been used to make significant contributions in many areas of chemistry and physics as well as in other areas of science. However, the main applications can be characterized as the study of intramolecular forces acting between the atoms of a molecule; the intermolecular forces or degree of association in condensed phases; the determination of molecular symmetries; molecular dynamics; the identification of functional groups, or compound identification; the nature of the chemical bond; and the calculation of thermodynamic properties. Current plans are for the reviews to vary, from the application of vibrational spectroscopy to a specific set of compounds, to more general topics, such as force-constant calculations. It is hoped that many of the articles will be sufficiently general to be of interest to other scientists as well as to the vibrational spectroscopist. As the series has progressed, we have provided more volumes on topical issues and, in some cases, single author(s) volumes. This flexibility has made it possible for us to diversify the series. Therefore, the course of the series has been dictated by the workers in the field. The editor not only welcomes suggestions from the readers, but eagerly solicits your advice and contributions.

James R. Durig Kansas City, Missouri

This Page Intentionally Left Blank

P R E F A C E T O V O L U M E 23

The current volume in the series Vibrational Spectra and Structure is a single topic volume on the vibrational spectra of molecules containing silicon in the solid state. The title of the volume Molecular Approaches to Solids has been treated by the workers in the Institute for Silicate Chemistry of the Russian Academy of Science in St. Petersburg for the past two decades. For the past 15 years, a number of publications have originated from the laboratory where quantum mechanical computations for suitably selected molecules have been utilized to explain the origins of some structure bonding interrelations and silicates and to evaluate their force constants. Most of the developments in this area have been published in the Russian literature and, therefore, remain relatively inaccessible to the Western scientists. Therefore, the current volume is a compilation of many of these studies with the ptttpose of reviewing the content of many of these publications and to summarize the essential conclusions of these studies. Unfortunately, after Professor Lazarev submitted the volume for publication in the series Vibrational Spectra and Structure he passed away. Therefore, some of the normal proof reading by the author has not been possible and it is hoped that the editor and one of his graduate students, Dr. James B. Robb II, have been able to provide adequate proofmg of the manuscript. Therefore, the editor would like to thank Dr. Robb for his assistance with this volume. The Editor would also like to thank his Administrative Assistant, Mrs. Linda Smitka for providing the articles in camera ready copy form and quietly enduring some of the onerous tasks associated with the completion of the volume. He also thanks his wife, Marlene, for copy-editing and preparing the subject index.

James R. Durig Kansas City, Missouri

~ 1 7 6

Vll


P R E F A C E BY T H E A U T H O R

The title problem of this book has been treated for many years as a problem of properties inherent of the condensed system from a free molecule and was restricted conventionally to the particular molecular crystal. Later, significant success in ab initio quantum mechanical computations of the equilibrium geometry and force constants of molecules promoted an interest for transferability of geometrical and dynamical parameters to chemically related crystals where similar first principle approaches were still in a state of development and their numerical applications remained rather cumbersome. For instance, for the last ten or fifteen years, a number of publications appeared where the quantum mechanical computations for suitably selected molecules were utilized in an attempt to explain the origin of some structure/bonding interrelations in silicates and to evaluate their force constants. In the author's opinion, insufficient attention was paid, however, to some subtle problems relating to a possible change in the physical meaning of formally similar quantities when being transferred from the description of a microscopically finite molecular system to a macroscopic body like crystal. This originates in the first place from the ambiguity of discerning between local and non-local interactions in a condensed system. These problems were the focus of several investigations in the author's laboratory of vibrational spectroscopy in the Institute for Silicate Chemistry at the Russian Academy of Sciences. Most of the developments in this area were discussed in a series of monographs and collections of papers issued in Russian and therefore remained hardly accessible to the Western audiences. This book is compiled as an attempt to review the content of those publications and to summarize their most essential conclusions. Treating the problem of molecular approaches to solids is, in general, an attempt to extend the notions developed originally in the study of molecules to the theory of crystals which deserves more attention. It relates first of all to some peculiarities of the intemal coordinate space adopted in the theory of molecular vibrations and applicable to the investigation of the lattice dynamics problems where the use of Cartesian atomic displacements dominated these studies since Bore's classic works. The application of internal coordinates to the description of crystals seems promising for a deeper understanding of the interrelation between the properties of phonon modes and ones related to the homogeneous macroscopic deformation of a lattice. It will be shown that by extending the notion of the microscopic shape of deformation widely used in spectroscopy to a description of the microscopic pattern of the uniform strain in crystals, it is possible to formulate uniformly the theory of all deformational properties of a crystal. This will be ix

adopted in a generalized approach to the so-called inverse vibrational problem (i.e., evaluation of the dynamical parameters from the experimental spectrum), and its applications being exemplified. The important consequences of the curvilinear (non-linearized) nature of internal coordinates, which are paid insufficient attention in the literature, will be discussed in some detail with particular interest to their relation to the formulation of the stability conditions. A description of the latter as a balance with external action on a crystal and its relation to the problem of phase transitions will be outlined as well. Another object pursued by this monogram is to highlight the most important problems related to the numerical implementation of proposed approaches to the crystals of more or less complicated structures and the various types of bonding. Thus, the structm'e of M. Smimov's crystal mechanics program developed in our laboratory is briefly outlined and some of its most typical applications discussed. This program ensures a practical realization of several modem approaches proposed in the book. The content of this book represents the application of molecular quantum mechanics to the investigation of both the equilibrium structure and deformational properties of the corresponding crystals. Chapter 1 attempts to rationalize some empirical regularities of the crystal chemistry of silicates and to propose a new one issuing from the quantum mechanical treatment of rather simple molecular systems. Chapter 2 outlines the problems of lattice dynamics with particular attention to the adoption of approaches originating from the theory of molecular vibrations and to the development of versatile computational routines. Chapter 3 is devoted to the application of the quantum mechanical treatment of some more complex molecular systems aimed at the evaluation of less localized interactions in the crystals which cover the second-, third- and fourth-order coordination around a given atom. Chapter 4 exemplifies the proposed approaches by their application to various silicates and oxides. A crucial problem of complementation of molecular force constants by accounting for the effects which relate to microscopic electrostatic field of a lattice is paid the most attention. The approaches save, at least in principle, the applicability of molecular force constants to lattice dynamics and they are illuminated in Chapters 2 and 4. Although the selection of the material reflects the personal views and preferences of the author, a treatment of particular problems follows in numerous cases one proposed originally by diverse members of the permanent staff of the laboratory. The most significant contributions originate from the specialists in solid state physics" Drs. A. P.

Mirgorodsky, M. B. Smimov, O. E. Kvyatkovskii and from Drs. I. S. Ignatyev, T. F. Tenisheva, B. F. Shchegolev who belong to the molecular branch of the laboratory. Their valuable discussions and proposals are thankfully acknowledged. Some important developments in experimental approaches originate mainly from Mr. V. F. Pavinich.

A. N. Lazarev St. Petersburg, Russia

xi


TABLE OF CONTENTS PREFACE TO THE SERIES ...................................................................................................

v

PREFACE TO VOLUME 23 ................................................................................................

vii

PREFACE BY THE AUTHOR...............................................................................................

ix

CONTENTS OF OTHER VOLUMES ...............................................................................

xvii

CHAPTER 1 QUANTUM CHEMISTRY OF MOLECULAR SYSTEMS RELATING TO THE CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

I . Computational Methods and Wave Functions ..................................................................

2

A. A Gradient Approach to Geometry Optimization ...................................................... 2 B . A Force Constant Determination................................................................................ 5 C . Atomic Wave Functions ........................................................................................... 10

I1. A Single Si-0 Bond at the Silicon Atom .........................................................................

12

Si-0 Bond in the Molecular Species of H, SiOX Type ........................................... The Oxygen Bridges in H,XOXH, Systems with X = C. Si .................................. The Effects of Additional Coordination of Bridging Oxygen Atom ....................... The Dynamical Properties of Oxygen Bridges ........................................................

12 18 28 32

I11. Systems with Tetrahedral Oxygen Coordination of Silicon ...........................................

35

A. B. C. D.

A. B. C. D. E.

SiOl- Oxyanion and Si(OH), Molecule ................................................................ Dynamical Properties ofthe SiO, Tetrahedron in Simple Systems ........................ Charge Redistribution in the Strained Silicon-Oxygen Tetrahedron....................... A Covalent Model..................................................................................................... An Ionic Model .........................................................................................................

35 42 48 53 54

IV. Quantum Mechanical Computations for Some Ionic Clusters and their Relation to the Crystal Chemistry of Silicates ...............................................................................

55

A . Partially Protonated Silicate Ions .............................................................................

64

References ...............................................................................................................................

...

xlll

78

XiV

CONTENTS

CHAPTER 2 INTRODUCTION TO THE DYNAMICAL THEORY OF CRYSTALS AND APPLICATION OF APPROACHES ORIGINATING FROM THE THEORY OF MOLECULAR VIBRATIONS I. The Elements ofDynamical Theory of Crystal Lattice ..................................................

84

A . Macroscopic Treatment of the Elastic and Dielectric Properties of Perfect 84 Crystals...................................................................................................................... B . Microscopic Treatment: Potential Energy Function and Charge Density Distribution Function and Their Model Representation.......................................... 90 C. A Comparison of Various Descriptions ofthe Electric Response Function .........103 I1. A Compatibility of Molecular Force Constants with the Explicit Treatment of Coulomb Interaction in a Lattice ................................................................................... A. Potential Energy Decomposition and Interrelation Between the Potential Energy Function and the Electric Response Function ........................................... B. Conditions of Compatibility of Molecular Force Constants with Explicit Separation of Coulomb Contribution to the Force Field........................................ C. Applications to Silicon Dioxide and Silicon Carbide .......................................... I11. Internal Coordinates in the Description of Dynamic Properties and Lattice Stability .......................................................................................................................

111 111 116 121

128

A. Vibrational Problem in Internal Coordinates and Indeterminacy of the Inverse Problem ...................................................................................................... 129 B. A Generalization of the Inverse Vibrational Problem and a Notion of the Shape of Uniform Strain of a Lattice ..................................................................... 141 C. The Microscopic Structure of Hydrostatic Compression and its Employment in the Generalized Formulation of the Inverse Vibrational Problem ................................................................................................................... 157 D. A Curvilinear Nature of the Internal Coordinates and its Certain Consequences.......................................................................................................... 161 E. A Relation of Internal Tension to Description of the Lattice Instability ............... 165

IV. Several Computational Problems ..................................................................................

171

A. Geometry Optimization and Potential Function Refinement ................................ B . Crystal Mechanics Program.................................................................................... C. The Operation of the Program ................................................................................

171 174 175

References .............................................................................................................................

183

CONTENTS

xv

CHAPTER 3 MOLECULAR QUANTUM MECHANICS IN THE EVALUATION OF INTERACTIONS OF LESS LOCALIZED ORIGIN I . The Ionic Charge of Oxygen in Silicon Dioxide and the Non-Bonding OxygenOxygen Interactions in Crystals.....................................................................................

192

A . The Point Ion Concept ........................................................................................... 192 B. The Dynamic Oxygen Charge in Disiloxane and the Applicability of the Point Ion Approximation ........................................................................................ 198 C. The Force Constants of Non-Bonded Oxygen-Oxygen Interaction ...................... 202 I1 . Tetramethoxysilane as a Model of the Silicon-Oxygen Tetrahedron in a Network of Partially Covalent Bonds............................................................................

213

A . Experimental Data and Spectral Assignments ....................................................... B. Quantum Mechanical Computation........................................................................ C. Frequency Fitting and the Force Constant Evaluation ...........................................

213 218 224

I11. The Disilicic Acid Molecule as a Model of the Fragment of a Silica Network ...........229 A. Electronic Structure and Equilibrium Geometry.................................................... B. Ab Initio Force Field Investigation and Intertetrahedral Interactions....................

229 234

References .............................................................................................................................

244

THE AB INITIO MOLECULAR FORCE CONSTANTS IN CHAPTER 4 LATTICE DYNAMICS COMPUTATIONS I . Molecular Force Constants in Dynamical Model of a-Quartz ..................................... A. B. C. D.

Force Constant Sets and Other Dynamical Parameters ......................................... Calculated Properties and their Comparison with Experiment.............................. Phonon Frequency Dispersion ................................................................................ A Representation of the Long-Range Coulomb Interaction in the Force Field Model Specified in Internal Coordinates ......................................................

248 248 255 269 271

I1. Ab Znitio Force Constants of Molecular Species in Lattice Dynamics of The Quartz-Like Aluminum Phosphate................................................................................

279

A. Experimental Phonon Spectra and Band Assignment ........................................... B . Related Molecular Systems and their Force Fields................................................ C. A Design of the Initial Approximation of the Force Field of Aluminum Phosphate ................................................................................................................

279 284 293

xvi

CONTENTS D . An Extension and Modification of the Initial Force Field .....................................

I11. Electrostatic Contribution to the Mechanical Modes of a More Polarizable Lattice: Pyroxene- Like Monoclinic Sodium Vanadate ............................................... A. B. C. D. E. F.

A Formulation of the Problem and Description of the Crystal .............................. Experimental Data and Spectral Assignments ....................................................... Normal Coordinate Calculation.............................................................................. The Origin of the Transversal v, VOV Modes Softening..................................... Some Further Perspectives ..................................................................................... Concluding Remarks ..............................................................................................

301 307 307 310 318 323 329 332

References ............................................................................................................................. 334 AUTHOR INDEX ..............................................................................................................

337

SUBJECT INDEX ..............................................................................................................

347

CONTENTS OF OTHER VOLUMES

VOLUME 10 VIBRATIONAL SPECTROSCOPY USING TUNABLE LASERS, Robin S. McDowell INFRARED AND RAMAN VIBRATIONAL OPTICAL ACTIVITY, L. A. Nafie RAMAN MICROPROBE SPECTROSCOPIC ANALYSIS, John J. Blaha THE LOCAL MODE MODEL, Bryan R. Henry VIBRONIC SPECTRA AND STRUCTURE ASSOCIATED WITH JAHN-TELLER INTERACTIONS IN THE SOLID STATE, M.C.M. O'Brien SUM RULES FOR VIBRATION-ROTATION INTERACTION COEFFICIENTS, L. Nemes

VOLUME 11 INELASTIC ELECTRON TUNNELING SPECTROSCOPY OF HOMOGENEOUS CLUSTER COMPOUNDS, W. Henry Weinberg

SUPPORTED

VIBRATIONAL SPECTRA OF GASEOUS HYDROGEN-BONDED COMPOUNDS, J. C. Lassegues and J. Lascombe VIBRATIONAL SPECTRA OF SANDWICH COMPLEXES, V. T. Aleksanyan APPLICATION OF VIBRATIONAL SPECTRA TO ENVIRONMENTAL PROBLEMS, Patricia F. Lynch and Chris W. Brown TIME RESOLVED INFRARED INTERFEROMETRY, Part 1, D. E. Honigs, R. M. Hammaker, W. G. Fateley, and J. L. Koenig VIBRATIONAL SPECTROSCOPY OF MOLECULAR SOLIDS-CURRENT TRENDS AND FUTURE DIRECTIONS, Elliot R. Bemstein

~

XVII

xviii


V O L U M E 12 HIGH RESOLUTION INFRARED STUDIES OF SITE STRUCTURE AND DYNAMICS FOR MATRIX ISOLATED MOLECULES, B. I. Swanson and L. H. Jones FORCE FIELDS FOR LARGE MOLECULES, Hiroatsu Matsuura and Mitsuo Tasumi SOME PROBLEMS ON THE STRUCTURE OF MOLECULES IN THE ELECTRONIC EXCITED STATES AS STUDIED BY RESONANCE RAMAN SPECTROSCOPY, Akiko Y. Hirakawa and Masamichi Tsuboi VIBRATIONAL SPECTRA AND CONFORMATIONAL ANALYSIS OF SUBSTITUTED THREE MEMBERED RING COMPOUNDS, Charles J. Wurrey, Jiu E. DeWitt, and Victor F. Kalasinsky VIBRATIONAL SPECTRA OF SMALL MATRIX ISOLATED MOLECULES, Richard L. Redington RAMAN DIFFERENCE SPECTROSCOPY, J. Laane

V O L U M E 13 VIBRATIONAL SPECTRA OF ELECTRONICALLY EXCITED STATES, Mark B. Mitchell and William A. Guillory OPTICAL CONSTANTS, INTERNAL FIELDS, AND MOLECULAR PARAMETERS IN CRYSTALS, Roger Frech RECENT ADVANCES IN MODEL CALCULATIONS OF VIBRATIONAL OPTICAL ACTIVITY, P. L. Polavarapu VIBRATIONAL EFFECTS IN SPECTROSCOPIC GEOMETRIES, L. Nemes APPLICATIONS OF DAVYDOV SPLITTING FOR PROPERTIES, G. N. Zhizhin and A. F. Goncharov

STUDIES

OF

RAMAN SPECTROSCOPY ON MATRIX ISOLATED SPECIES, H. J. Jodl

CRYSTAL


xix

VOLUME 14 HIGH RESOLUTION LASER SPECTROSCOPY OF SMALL MOLECULES, Eizi Hirota ELECTRONIC SPECTRA OF POLYATOMIC FREE RADICALS, D. A. Ramsay AB INITIO CALCULATION OF FORCE FIELDS AND VIBRATIONAL SPECTRA, G~za Fogarasi and Peter Pulay

FOURIER TRANSFORM INFRARED SPECTROSCOPY, John E. Bertie NEW TRENDS IN THE THEORY OF INTENSITIES IN INFRARED SPECTRA, V. T. Aleksanyan and S. Kh. Samvelyan VIBRATIONAL SPECTROSCOPY OF LAYERED MATERIALS, S. Nakashima, M. Hangyo, and A. Mitsuishi

VOLUME 15 ELECTRONIC SPECTRA IN A SUPERSONIC JET AS A MEANS OF SOLVING VIBRATIONAL PROBLEMS, Mitsuo Ito BAND SHAPES AND DYNAMICS IN LIQUIDS, Walter G. Rothschild RAMAN SPECTROSCOPY IN ENERGY CHEMISTRY, Ralph P. Cooney DYNAMICS OF LAYER CRYSTALS, Pradip N. Ghosh THIOMETALLATO COMPLEXES: VIBRATIONAL SPECTRA AND STRUCTURAL CHEMISTRY, Achim MUller ASYMMETRIC TOP INFRARED VAPOR PHASE CONTOURS AND CONFORMATIONAL ANALYSIS, B. J. van der Veken WHAT IS HADAMARD TRANSFORM SPECTROSCOPY?, R. M. Hammaker, J. A. Graham, D. C. Tilotta, and W. G. Fateley

xx


V O L U M E 16 SPECTRA AND STRUCTURE OF POLYPEPTIDES, Samuel Krimm STRUCTURES OF ION-PAIR SOLVATES FROM MATRIX-ISOLATION/SOLVATION SPECTROSCOPY, J. Paul Devlin LOW FREQUENCY VIBRATIONAL SPECTROSCOPY OF MOLECULAR COMPLEXES, Erich Knozinger and Otto Schrems TRANSIENT AND TIME-RESOLVED RAMAN SPECTROSCOPY OF SHORT-LIVED INTERMEDIATE SPECIES, Hiro-o Hamaguchi INFRARED SPECTRA OF CYCLIC DIMERS OF CARBOXYLIC ACIDS: THE MECHANICS OF H-BONDS AND RELATED PROBLEMS, Yves Marechal VIBRATIONAL SPECTROSCOPY UNDER HIGH PRESSURE, P. T. T. Wong

V O L U M E 17A SOLID STATE APPLICATIONS, R. A. Cowley; M. L. Bansal; Y. S. Jain and P. K. Baipai; M. Couzi; A. L. Verma; A. Jayaraman; V. Chandrasekharan; T. S. Misra; H. D. Bist, B. Darshan and P. K. Khulbe; P. V. Huong, P. Bezdicka and J. C. Grenier SEMICONDUCTOR SUPERLATTICES, M. V. Klein; A. Pinczuk and J. P. Valladares; A. P. Roy; K. P. Jain and R. K. Soni; S. C. Abbi, A. Compaan, H. D. Yao and A. Bhat; A. K. Sood TIME-RESOLVED RAMAN STUDIES, A. Deffontaine; S. S. Jha; R. E. Hester RESONANCE RAMAN AND SURFACE ENHANCED RAMAN SCATTERING, B. Hudson and R. J. Sension; H. Yamada; R. J. H. Clark; K. Machida BIOLOGICAL APPLICATIONS, P. Hildebrandt and M. Stockburger; W. L. Peticolas; A. T. Tu and S. Zheng; P. V. Huong and S. R. Plouvier; B. D. Bhattacharyya; E. Taillandier, J. Liquier, J.-P. Ridoux and M. Ghomi


xxi

V O L U M E 17B STIMULATED AND COHERENT ANTI-STOKES RAMAN SCATTERING, H. W. SchrStter and J. P. Boquillon; G. S. Agarwal; L. A. Rahn and R. L. Farrow; D. Robert; K. A. Nelson; C. M. Bowden and J. C. Englund; J. C. Wright, R. J. Carlson, M. T. Riebe, J. K. Steehler, D. C. Nguyen, S. H. Lee, B. B. Price and G. B. Hurst; M. M. Sushchinsky; V. F. Kalasinsky, E. J. Beiting, W. S. Shepard and R. L. Cook RAMAN SOURCES AND RAMAN LASERS, S. Leach; G. C. Baldwin; N. G. Basov, A. Z. Grasiuk and I. G. Zubarev; A. I. Sokolovskaya, G. L. Brekhovskikh and A. D. Kudryavtseva OTHER APPLICATIONS, P. L. Polavarapu; L. D. Barron; M. Kobayashi and T. Ishioka; S. R. Ahmad; S. Singh and M. 1. S. Sastry; K. Kamogawa and T. Kitagawa; V. S. Gorelik; T. Kushida and S. Kinoshita; S. K. Sharma; J. R. Durig, J. F. Sullivan and T. S. Little

VOLUME 18 ENVIRONMENTAL APPLICATIONS OF GAS CHROMATOGRAPHY/FOURIER TRANSFORM INFRARED SPECTROSCOPY (GC/FT-IR), Charles J. Wurrey and Donald F. Gurka DATA TREATMENT IN PHOTOACOUSTIC FT-IR SPECTROSCOPY, K. H. Michaelian RECENT DEVELOPMENTS IN DEPTH PROFILING FROM SURFACES USING FTIR SPECTROSCOPY, Marek W. Urban and Jack L. Koenig FOURIER TRANSFORM INFRARED SPECTROSCOPY OF MATRIX ISOLATED SPECIES, Lester Andrews VIBRATION AND ROTATION IN SILANE, GERMANE AND STANNANE AND THEIR MONOHALOGEN DERIVATIVES, Hans Btirger and Annette Rahner FAR INFRARED SPECTRA OF GASES, T. S. Little and J. R. Durig

xxii


V O L U M E 19 ADVANCES IN INSTRUMENTATION FOR VIBRATIONAL OPTICAL ACTIVITY, M. Diem

THE

OBSERVATION

OF

SURFACE ENHANCED RAMAN SPECTROSCOPY, Ricardo Aroca and Gregory J. Kovacs DETERMINATION OF METAL IONS AS COMPLEXES I MICELLAR MEDIA BY UV-VIS SPECTROPHOTOMETRY AND FLUORIMETRY, F. Fernandez Lucena, M. L. Marina Alegre and A. R. Rodriguez Fernandez AB INITIO CALCULATIONS OF VIBRATIONAL BAND ORIGINS, Debra J. Searles and Ellak I. von Nagy-Felsobuki

APPLICATION OF INFRARED AND RAMAN SPECTROSCOPY TO THE STUDY OF SURFACE CHEMISTRY, Tohru Takenaka and Junzo Umemura INFRARED SPECTROSCOPY OF SOLUTIONS IN LIQUIFIED SIMPLE GASES, Ya, M. Kimerfel'd VIBRATIONAL SPECTRA AND STRUCTURE OF CONJUGATED CONDUCTING POLYMERS, Issei Harada and Yukio Furukawa

AND

VOLUME 20 APPLICATIONS OF MATRIX INFRARED SPECTROSCOPY TO MAPPING OF BIMOLECULAR REACTION PATHS, Heinz Frei VIBRATIONAL LINE PROFILE AND FREQUENCY SHIFT STUDIES BY RAMAN SPECTROSCOPY, B. P. Asthana and W. Kiefer MICROWAVE FOURIER TRANSFORM SPECTROSCOPY, Alfred Bauder AB INITIO QUALITY OF SCMEH-MO CALCULATIONS OF COMPLEX INORGANIC SYSTEMS, Edx~ard A. Boudreaux

CALCULATED AND EXPERIMENTAL VIBRATIONAL SPECTRA AND FORCE FIELDS OF ISOLATED PYRIMIDINE BASES, Willis B. Person and Krystyna Szczepaniak


xxiii

V O L U M E 21 OPTICAL SPECTRA AND LATTICE DYNAMICS OF MOLECULAR CRYSTALS, G. N. Zhizhin and E. I. Mukhtarov

V O L U M E 22 VIBRATIONAL INTENSITIES, B. Galabov and T. Dudev


CHAPTER 1 QUANTUM CHEMISTRY OF MOLECULAR SYSTEMS RELATING TO T H E C R Y S T A L C H E M I S T R Y AND L A T T I C E D Y N A M I C S O F S I L I C A T E S

I.

Computational Methods and Wave Functions ............................................................ 2 A. A Gradient Approach to Geometry Optimization ....................................................2 B. A Force Constant Determination ..............................................................................5 C. Atomic Wave Functions ..........................................................................................10

II.

A Single Si-O Bond at the Silicon Atom ..................................................................... 12 A. Si-O Bond in the Molecular Species of H3SiOX Type .......................................... 12 B. The Oxygen Bridges in H3XOXH 3 Systems with X = C, Si ................................. 18 C. The Effects of Additional Coordination of Bridging Oxygen Atom ......................28 D. The Dynamical Properties of Oxygen Bridges .......................................................32

III. Systems with Tetrahedral Oxygen Coordination of Silicon .....................................35 A.

SiO 4- Oxyanion and Si(OH)4 Molecule ...............................................................35

B. Dynamical Properties of the SiO 4 Tetrahedron in Simple Systems .......................42 C. Charge Redistribution in the Strained Silicon-Oxygen Tetrahedron .....................48 D. A Covalent Model ...................................................................................................53 E.

An Ionic Model .......................................................................................................54

IV. Quantum Mechanical Computations for Some Ionic Clusters and their Relation to the Crystal Chemistry of Silicates ..........................................................55 A. Partially Protonated Silicate Ions ............................................................................64

References ..............................................................................................................................78

2 I.

LAZAREV C O M P U T A T I O N A L M E T H O D S AND WAVE FUNCTIONS

A. A Gradient Approach to Geometry Optimization In an adiabatic approximation, any deformation of a polyatomic molecule is expressed by the nuclei's motion along the electronic energy hypersurface. In each point of that surface specified by the internal coordinates of a system q the electronic energy is determined as: E(q) = (~F IHIq~ )

(1.1)

where W is a wave function of the ground electronic state at fixed nuclei and H is an adiabatic electronic Hamiltonian which includes the nuclei's Coulomb interaction. The stable geometries of a system correspond to the minima of E(q) dependence. Their searching by direct point-to-point energy calculation is a rather laborious problem and its solution may be considerably simplified if the analytic first derivative dE/dq were available. The electronic problem (E(q) determination) is investigated conventionally by variational approach. The exact expressions for the first derivatives (not for the higher-order ones) in this approach can be significantly simplified as originally proposed by Pulay [1 ]. The exact wave function W is substituted in this approach by some approximation q)(p), which depends on the variational parameters p. An approximate energy magnitude is determined by the functional: E~ e(p)= ( ~(p)[ H [~(p) )

(1.2)

where p values are deduced from the stationary conditions for e(p). The Hamiltonian H in eq. (1.2) is an explicit function of q and a variational function may depend on them explicitly as well. Then, the variational parameters p may depend on q implicitly. The expression (1.2) can be rewritten as" E(q) ~e(q,p(q)) = ( q~(q,p(q))[ H(q)Icp(q,p(q))) and its differentiation with respect to q produces:

(1.3)

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

~q

~q = tp -~q tp +2

\Oq

3

+2[0,~IHI~ ~ dp \Oq dq

(1.4)

The second and third terms in this expansion vanish if ~ is an exact solution of the Schrodinger equation (q~ = q0. The same is valid if neither a variational function nor the variational parameters are dependent on q, e.g., in a case of rigorous solution in the HartreeFock approximation. The remaining first term represents the Hellmann-Feinman theorem. This Hellmann-Feinman force is calculated relatively simply. Unfortunately, very high requirements to the quality of wave functions, which obey the Hellmann-Feinman theorem, make them practically inapplicable to calculations for polyatomic molecules.

The MO

LCAO approach, which dominates in quantum chemical computations, operates with a restricted basis set of atomic wave functions. Their centers are tightly bonded to nuclei and follow them at their shifts. In this case, all three terms of expansion (1.4) contribute to the gradient dE/dq. A wide application of a gradient approach to geometry optimization began since the effective methods to calculate the second and third terms in eq. (1.4) have been developed. The second term corresponds to the force originating from the condition of "rigid" following of AO to the nuclei's shifts. Its calculation includes the computation of multicenter integrals in the basis of atomic functions and of all their derivatives. In Pulay's program [2] this contribution to all forces is determined in one pass over all multicenter integrals which saves time and excludes the necessity of storage of a huge array of their derivatives. The variational parameters p are in the MO LCAO approach to the coefficients of MO expansion in AO. A determination of the third term in (1.4) implies the calculation of their changes at the geometry variation. This problem appears to be most complicated. It has been shown, however, that the use of stationary condition for e(p) allows the calculation of this term by means of dp/dq magnitudes determined from the orthonormality condition for MO [3 ]. Then, a calculation of that term is reduced to the determination of derivatives of AO's overlap integrals with respect to q.

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Pulay's approach saves computer time in the calculation of forces corresponding to any set of q values and thus considerably promotes the determination of equilibrium geometry of a polyatomic system. An iterative procedure known as the force relaxation method is adopted to the minimization of forces. At each step fq forces which correspond to the initial q set are calculated and the nuclei are then shifted to a new geometry q' in order to reduce the forces. That geometry is determined from the expression q' = q + Afq

(1.5)

where A is a matrix defined in a certain way. This procedure is known to be the more effective the nearer the A matrix is to the inverse matrix of derivatives of forces relative to q coordinates [4], i.e. the inverse force constant matrix Fo. A reliable approximation of an F o matrix can be designed practically for any molecule proceeding from the accumulated experience in normal coordinate calculation. In difference of the empirical normal coordinate calculations where for some reasons the redundant q sets are often adopted, the gradient approach requires that the internal coordinate set should be independent (non-redundan0 and complete. It is necessary to enable a determination of the inverse force constant matrix adopted in the force relaxation method and an unambiguous interrelation between internal and Cartesian coordinates. The Cartesian forces deduced from MO LCAO computation are interrelated linearly with the forces in q space through the B matrix (q = Bx) which is often met in theory of molecular vibration [5]. In a case of a favorably selected F o approximation, the forces are considerably reduced after a few iterations even for a molecule containing 10-15 atoms. The process is however retarded when approaching the equilibrium geometry. A refinement of the ab initio geometry is usually ceased if fq forces found from eq. (1.5) determine q'-q differences

lesser than 0.001A for bond lengths and 0.1 ~ for valence angles. It corresponds to residual forces ca. 0.005 mdyn. Of course, much smaller residual forces should be attained if a molecule possesses non-rigid degrees of freedom (slightly hindered intemal rotation etc.).


5

B. A Force Constant Determination

There exist, in principle, three possible approaches to the ab initio force constant calculation. A force constant defined as a second energy derivative relative to internal coordinates near the equilibrium position, F ij = d2E/dqidqj, is obtained most simply by the double differentiation in various points of the two-dimensional grid of qi and qj values. This method is very laborious because of the multiply repeated electronic energy calculation and suffers from the low accuracy of double numerical differentiation. The analytic expressions for the second energy derivatives have been already proposed and corresponding computational algorithms developed (see, e.g., [6,7]). Such an approach enables a good accuracy of the force constant calculation. It needs, however, considerable expense of computer time, which cannot be divided into several portions. As a compromise between the two above approaches, the gradient (force) method operates with analytic determination of forces and their further numerical differentiation. The force constants are found from the expression: Fij = d 2 E / d q i d q j = [fqi ( q ~

(q~

"

(1.6)

It implies the calculation of fqi forces acting in qi coordinate in two points along qj, qo+Aqj and qo-Aqj. The shift Aqj is selected as a trade-off of low accuracy in the subtraction of one small value from the close one, and enhanced error in numerical differentiation according eq. (1.6) at larger Aqj values, on the other. The empirically selected a Aqj magnitude constitutes usually 0.01A for the bond length and 2 ~ for the valence angle variation. Sometimes the reference (qo) geometry which deviates slightly from the equilibrium one is adopted for numerical differentiation by means of eq. (1.6) thus hoping to avoid the errors in the determination of quadratic force constants originating from the anharmonicity of theoretical potential energy hypersurface [8-10]. This approach proceeds from the approximate constancy of deviations in theoretical equilibrium geometry from the experimental one for a series of chemically similar molecules treated with the same basis set. The

6

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universality of this approach is, however, doubtful and it will be avoided further mainly for aesthetic reasons. Even in a case of large atomic wave function sets approaching the Hartree-Fock limit, the theoretical force constants deviate from the experimental ones. These deviations may be in principle of various physical origins and a neglect of electron correlation (configurational interaction) is only one of them. The overestimated values of ab initio force constants, which are obtained in most cases, are usually explained exactly by the adoption of the Hartree-Fock approximation.

However, a restriction to adiabatic approximation,

which is also another source of errors usually leads to overestimated force constant values. The non-adiabatic effects originating from the existence of low-lying excited electronic states have been found most important for some particular types of deformational modes in ethylenes and acetylenes [11 ]. A neglect of anharmonicity which is practically unavoidable in calculations for the middle size molecules may lead to a discrepancy between theoretical and experimental frequencies as well. Irrespective of the physical origin of those deviations, they are usually corrected by the introduction of empirical coefficients to theoretical force constants known as "scaling factors." Their magnitudes are found by frequency fitting, thus introducing some empirical element into the ab initio force constant determination. This approach is the more effective the nearer the scaling factors are to one. This depends mainly on a proper selection of an atomic basis set in quantum mechanical computation. The scaling factors ranging from 0.70.9 for most force constants and up to 1.2 in some particular cases, are believed to be suitable for any practical purpose. Secondly, the number of independent values of scaling factors should be reduced as much as possible in order to make their determination from frequency fitting more definite. The scaling factors are therefore oRen assumed to coincide for several coordinates of similar type. Then, a certain interrelation between scaling factors for diagonal force constants F ii, Fjj and off-diagonal ones, Fij, may be imposed [12,13], instead of def'ming them independ-


7

ently [9] (in some approaches, the theoretical off diagonal force constants have not been varied at all, and only diagonal ones were scaled). A most self-consistent and successful approach in practical application proposed in ref. [13] will be used throughout the text. It defines a scaling factor for Fij as a geometrical mean average of scaling factors for Fii and Fij. This may be expressed in a more general matrix form as the following transformation of the force constant matrix, 1

1

F scaled = c-2Fc~ ,

(1.7)

where c is a diagonal matrix of scaling factors. This approach is easily rationalized as an extension of the potential energy function in a space of internal coordinate q and it is invariant to their linear transformations. The scaling factors defmed in this way proved to be transferable in a series of molecules with some repeating units to a greater extent than the force constants themselves. In several cases the scaling factors for some more or less complex molecules have been suecessfully determined without any refinement simply as a combination of scaling factor sets each corresponding to a particular fragment of that molecule, their values being transferred from simpler molecules with no ambiguities in spectral assignment [ 11 ]. The IR intensities can be obtained in the gradient method without any additional computation simply by the numerical differentiation of the molecular dipole moment M. The dipole moment derivatives relative to internal coordinates, c3M/tgq, are transformed into derivatives relative to normal coordinates aM/aQ, by means of the matrix L of the "shapes" of normal modes: aM ~-~ aM L OVk~ = Z"a-~---- ik i qi

9

(1.8)

The integral band intensities in absorption A k are interrelated with aM/aQ values as" Ak

= n N (aM / aQk)2 3c 2

(1.9)

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where N is Avogadro's number. Taking into account the interrelation between the dipole 1

moment derivative in the CGSE system and the practically adopted D. ~-1. g - g (g is the atomic mass unit) dimensionality representation and substituting into (1.9) the values of constants, it is possible to express the interrelation between A k in cm. mo1-1 and aM/0Q in the above dimension as A k = 4.2273"10-6 (0M / 0Qk)2

.

(1.10)

It is known, however, that the sets of atomic wave functions adopted conventionally in the ab initio force constant computation are insufficient to reproduce the IR intensities with accuracy comparable to the precision of their experimental determination. Only the more complete and flexible atomic basis sets with polarizing functions and being complemented by diffuse s and p functions may be adequate in description of the charge redistribution excited by molecular deformation [14]. Nevertheless, even a rough ab initio IR intensity estimation may be of interest in certain cases. It can be adopted either in discussion of the origin of some polarization phenomena or to empirical spectral assignments. No approach to empirical correction of quantum mechanical IR intensities has been proposed yet. The forces acting upon the atoms in a space of their Cartesian coordinates are deduced from quantum mechanical MO LCAO computations as has been mentioned above. On the other hand, a space of internal coordinates is preferable both in the equilibrium geometry optimization and in description of the force constant matrix, which is more near to diagonal form just in this space. Correspondingly, a set of internal coordinates should be complete and independent one in order to ensure the unique interrelation between internal coordinates and Cartesian atomic displacements. The use of internal symmetry coordinates described in numerous books on molecular spectroscopy (see, e.g., [5]) solves the problem of exclusion of redundant coordinates. In a case of a more complex molecule, the independent internal coordinates are designed usually in a pictorial way as the local symmetry coordinates. Their detailed de-


9

scription for various typical molecular fragments has been given in refs. [8,15]. Only the set of internal symmetry coordinates for a particular case of a tetrahedral (Td) molecule XY 4 is given below: sl-l(

rl + r2 + r3 + r4 )

r~ (2tx 12 +2~34 - t~13 - ~23 - tx24 - ~14) s2' = ~--~ r~ (tx 13-ct23 + tx24 -tXl4 ) s2" =-~s ,

- 89 rl + r2 - r3 - r4 / 1

/,

s3,, = ~l~rl - r2 + r3 - r4)

(1.1 1)

1

s 3.... ~(rl - r2 - r3 +r4) ro s4' = ~ ( t X l 2 - ~ 3 4 ) s4" = ~r~ (a23 - Ctl4 ) ro s 4 .... ~ ( a 2 4

-a13)

The combinations of angle bending coordinates are multiplied here by the equilibrium bond length r o which excludes it from the kinematics matrix and reduces the dimensionality of corresponding force constants to one adopted in description of the bond stretching force constants (mdyn/A). It simplifies a comparison of force constants obtained by various authors and given in that dimensionality with no reference to the adopted r o value, which makes their recalculation to the angle units (mdyn/A) impossible. In non-cyclic molecules, the bond stretching coordinates are not involved in redundancy conditions and corresponding force constants can be determined separately in a case of physically meaningful difference between the bonds constituting the same symmetry set in the approximate local symmetry group. If the adopted local symmetry is lowered in a real complex molecule due to angular distortions, the expressions for independent angle bending coordinates are to be changed, rigorously speaking. E.g., in a case of non-equality

10

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of six tetrahedral angles, the coefficients at their values in the above expressions slightly differ from unity. This is neglected, however, in most numerical calculations. Of course, any sub-group of the approximate local symmetry group may be adopted in a treatment of the ab initio force constants of a particular system possessing lower total symmetry.

C. Atomic Wave Functions A complexity of molecular species to be studied (relatively large number of electrons) and limitations of available computational facilities restricted most calculations to a one determinant approximation of the Hartree-Fock method with the description of atomic orbitals by Gaussian basis functions of a split valence type. These functions describe the valence shell at a level approaching one obtained by means of well approved double-zeta Husinaga-Dunning (DZHD) basis set [ 16-18] being much more economic in computer time expenses. In a case of systems containing the atoms with lone pairs and at certain other circumstances the results obtained with the split valence basis functions are significantly improved when the basis set is complemented by so-called polarizing functions. The latter are introduced as the derivatives of exp(-rlr2), i.e., as the p-functions complementing s-AO, d-functions for p-AO etc. Among various split valence type basis sets for the first row elements the 4-21G set proved to be very economic and sufficiently precise to successfully reproduce the molecular geometry and force field [8]. This set at O and C was combined with the 3-3-21G basis set at Si in the first attempt to calculate the equilibrium geometry of some silicon containing molecules by the gradient method [19]. This set led, however, to considerably overestimated SiOX (X=C,Si) valence angles (the bond lengths were slightly overestimated as well) and was therefore complemented by 3d polarizing functions at Si and O with the same orbital exponent 113d = 0.8. The systematic errors in molecular geometry intrinsic to these sets could be estimated by comparison with corresponding experimental data. There exist, however, no experimental data on the molecular geometry of most systems treated before. It was thus reasonable to employ in our earlier computations the basis


11

sets adopted by Oberhammer and Boggs [20] hoping for the transferability of their estimations of geometrical errors. The basis set composed by 4-21G functions at O and C atoms and 3-3-21G functions at the Si atom (the latter was taken from ref. [19] where it was designed in a slightly different way than the one adopted by Oberhammer and Boggs) will be referred to as the set I. The same set complemented by polarizing 3d functions at Si and O atoms which were designed according to Pulay's recommendations [2] and with the same orbital exponents as in ref. [20] will be referred to as set II. The set II has been partially changed in the course of further computations. In particular, the orbital exponent of polarizing d functions was modified by inspecting its influence on the total energy, dipole moment, equilibrium geometry, and charge distribution in the methylsilane molecule, H3CSiH 3, since the results of similar computations with a much larger basis set were available [21 ]. The changes of corresponding properties in methyl silyl ether, H3COSiH 3, depending on that orbital exponent have been investigated as well. The set II deduced from these computations was characterized by the following orbital exponents of polarizing 3d

functions" r13d(0)=0.8, r13d(Si)=0.45.

The latter value coincides with

one recommended in the paper [22] specially devoted to this problem relating to the secondrow elements. The above basis sets have been additionally modified in certain cases either for the sake of economy which was most important for complex systems or in an attempt to represent the peculiarities of systems with excessive electronic charge. The most flexible true two-exponent DZHD basis set (12s9p/6s4p at Si and 9s5p/4s2p at O atoms) was used only in a few cases as the more time-consuming one and will be further referred to as set III. Most quantum mechanical computations presented below were accomplished using Pulay's TEXAS program [2].

The Mulliken's population analysis and the localization

procedure by Bois were adopted in attempts to rationalize the results of the electronic structure computation in terms of the charge distribution and bonding pattern. The localized molecular orbitals (LMO) of bonds and lone pairs (LP) were characterized by their conventional average radii ~ and removals d A of the center of electron density distribution

12

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(CEDD) of corresponding LMO from the atom A. The angular characterization of CEDD position in a LMO of the bond A-B relative to its (core-core) axis proved to be instructive in particular cases. The spatial orientation of LP LMO was described by the bondlLP and LPILP angles which were determined as the angles between the bond axis and the direction from the corresponding atom towards CEDD at LP LMO or between two such directions respectively. The AO's contributions to LMO were often applied to discussion of the nature of bonding.

II.

A S I N G L E Si-O BOND AT T H E S I L I C O N ATOM The systems containing only one Si-O bond at each silicon atom do not relate di-

rectly to silicates with a tetrahedral oxygen environment of Si. Their inspection may be, however, instructive for several reasons. It will be shown in the next chapters that the properties of the SiO4 tetrahedron can hardly be decomposed into the contributions from isolated bonds because of strong interactions along the edges. On the other hand, an evaluation of features which can be assumed intrinsic to a single Si-O bond is of interest for comparison of various approaches to that decomposition. Two types of systems having a single Si-O bond at the silicon atom are treated below.

A. Si-O Bond in the Molecular Species of H3SiOX Type The Si-H bond lengths are only 10% shorter than the Si-O bond while the importance of the direct H...H or H...O interactions can be supposed to diminish considerably relative to interactions in the silicon oxygen tetrahedron. A variation of the nature of the X atom (or of a group of atoms) permits one to investigate its influence on the properties of a single Si-O bond. A series of H3SiOX systems with X = H, Sill 3, Na has been selected and the electronic structure and equilibrium geometry calculated. To complete the row with steadily decreased electronegativity of X, the ionic system H3SiO" has been calculated as well. The atomic basis set I was adopted in these computations for simplicity since a comparison of their results with experimental data is impossible in most cases. It is known,


13

however, that this set considerably distorts the valence angle at oxygen (see, e.g., [11,20]) and it was therefore augmented by the inclusion of 3d polarizing functions only at this atom. A comparison of this set denoted I+d(O) with the more complete set II will be given for some particular systems in the next section. The results of computations are presented in Table 1.1. In difference of systems with partially covalent O-X bonds (X = Sill 3, H) and bent geometry of the Si-O-X bridge, the linear arrangement in the Si-O-Na fragment was found to be stable in a case of H3SiONa. The data characterizing this system are given in Table 1.1 in two versions. One corresponds to the total relaxation of internal forces like in all other systems while in another version the O-Na distance is fixed at 2.3A which is near to corresponding distances in sodium silicates (it is implied that a shorter O-Na equilibrium distance in the model system originates mainly from the absence of coordination polyhedron around sodium). The changes of the equilibrium Si-O bond length in this series coincide, at least qualitatively, with main trends met in silicate lattices [23 ]. Just as in silicates, the Si-O(H) bond is the longest one and a transition from the "bridging" type Si-O(Si) to the "terminal" Si-O(...cation) bond is accompanied by its shortening more than by 0.05A. This shortening may be rationalized in terms of the electronegativity diminishing of X. The overlap population is, however, the smallest in a case of the symmetrical Si-O-Si bridge in disiloxane. Its probable relation to peculiarities of the valence charge distribution will be discussed later. Judging by the CEDD position of LMO SiO in a bond, in all systems the Si-O bonds are polar. Their polarity may be characterized numerically by the relation of the CEDD LMO SiO removal from the oxygen atom to the equilibrium bond length:

doLMO SiO/reSiO, %

H3SiONa

H3SiOH

H3SiOSiH

28.7

27.7

26.0

It is seen that the bridging Si-O(Si) bond is most polar while the longest Si-O(H) bond occupies an intermediate position between it and the least polar terminal Si (cation) bond.

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TABLE 1.1 Electronic structure and equilibrium geometry of H3SiOX systems as calculated with I+d(O) basis set.

Computed values -Etotal, eV

H3SiO

H3SiONa*

H3SiONa

H3SiOSiH3

H3SiO

C3v

C3v

C3v

C2v

Cs

9900.076

reSiO, A

14278.548

14279.033

17770.149

9916.977

1.557

1.588

1.596

1.653

1.664

reOX

-

2.300"

1.941

1.653

0.959

reSiH

1.542

1.514

1.512

1.487t

1.488 t

-

180.0

180.0

142.1

114.7

ZOSiH

118.2

115.1

114.7

110.5 t

110.6 t

ZHSiH

99.4

103.3

103.8

108.2t

107.8 t

1x0.403

0.446

0.458

0.430

0.462

0.827

0.805

0.784

0.767

0.755

1x0.265

3x0.321

3x0.323

2x0.294

2x0.299

0.726

0.841

0.829

0.768

0.723

0.430

0.521

0.767

0.706

ZSiOX, degrees

Localization LMO SiO: do

LMO LP:

LMO OX:

LMO Sill:

do

do

do

ZLPIOILP, deg.

0.254

0.279

0.282

0.307 t

0.306 t

0.992

0.991

0.995

0.981 t

0.982t

3x115.8

3x115.6

1•

lx121.5

0.709

0.665

0.454

0.512

0.291

0.301

0.454

0.520

0.564

0.583

0.680 t

0.684 t

-$

Overlap Population Si-O

0.937

O-X Si-H

0.463


15

TABLE 1.1 (continued).

H3SiO C3v

H3SiONa* C3v

H3SiONa C3v

H3SiOSiH 3 C2v

H3SiO Cs

Si

1.122

1.195

1.202

1.228

1.183

O

-1.055

-1.064

-1.096

-1.128

-0.963

X

-

0.759

0.756

1.228

0.453

H

-0.356

-0.297

-0.287

-0.221 ~

-0.224

0.005

0.003

0.005

0.005

0.008

Computed values Net charge

Residual force, mdyn

*O-Nadistance is fixed. The residual force on Na (towards O) 0.322 mdyn. ~'Averagedover symmetricallynon-equivalentbonds, atoms etc. ~ALSO SiOIOILMOSiO = 85~ ALSO SiOIOILP= 129a.

The more polar the Si-O bond, the lesser the contribution in its LMO from 3s,3p-AO of silicon (Table 1.2). On the contrary, the contribution of 2s,2p-AO of oxygen increases in this direction and the contribution of silicon's AO into LMO of Si-H bonds increases as well. A shortening of Si-H bonds with an increase of the equilibrium Si-O bond length may be rationalized both in terms of enhanced bonding overlap and decreased H...O repulsion where a repulsion from the LP (which depends on the SiOX angle) is probably most important. Otherwise, the Si-H bonds in H3SiO moiety are shorter. Even more significant is a partial removal of the electronic charge at the oxygen atom on the outside of that system. Similar phenomenon will be met and discussed in more detail in the next chapters when investigating the behavior of the SiO4 tetrahedron depending on its excessive negative charge. The changes in the equilibrium of the HSiH and HSiO angles correlate with the above considerations. The former are greater and the latter are smaller at longer Si-O distances, thus corresponding to approximately tetrahedral bonding of silicon atom.

And

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TABLE 1.2 The coefficients of AD contributions to LMO of H3SiOX systems.

Atom

H3SiONa

and AOSi

Si-O

Si

3s'

0.18

3p'

0

H

H3SiOH

Si-H

H3SiOSiH 3

Si-O

Si-H*

Si-O

Si-H*

0.17

0.15

0.19

0.13

0.20

0.17

0.25

0.17

0.28

0.15

0.28

3s"

-

0.16

-

0.15

-

0.16

3s"

-

0.07

-

0.08

-

0.09

2s'

-0.10

-0.10

-0.11

2p'

0.36

0.39

0.38

2s"

0.32

0.33

0.38

2p"

0.46

0.48

0.48

Is'

0.26

0.26

0.26

Is"

0.50

0.46

0.47

*Averagedover EMO of symmetricallynon-equivalentbonds.

reversely, a strongly elongated Sill 3 group along the local three-fold axis pyramidal shape is found in systems with a less distant oxygen atom. This pattern is most pronounced in the ionic system H3SiO" included into the above series as a limiting case of the oxygen charge entirely distributed inside it (Table 1.1). The Sill 3 group with the most elongated Si-H bonds is strongly stretched in this system along its axis while the Si-O bond length is the smallest in the series. This bond length deduced from the computation is smaller than in any existing system (silicates, silicoorganic compounds) accessible to experimental investigation. The population analysis reveals the largest effective charges at hydrogen atoms and the largest Si-O overlap population in the series. On the other hand, the localization procedure explores a very surprising distribution of valence electrons among of valence electrons among the LMO. Three electron pairs symmetrical to the axis can be treated as bonding while the sole LP is directed along the axis, its CEDD


17

being not far from the oxygen's nucleus (cf. with ? LMO LP in other members of the seties). It means that the Si-O bond in H3SiO is a triple bond according to the results of the computation. There exist no indications of the real existence of this type bonding in any condensed system containing Si-O bonds. An arising of such a triple Si-O bond in a case of a tetrahedral oxygen environment of silicon seems improbable because of the repulsion between this type of bonding orbital and the LP of other oxygen atoms. This type of bonding would be easily recognized from the local geometry, which need a linear arrangement of the Si-O...cation fragment accompanying an unusually short Si-O bond, but it has been never met empirically. Even in the case of disilicates Li2Si205, Na2Si205 with layer complex anions possessing extremely short "terminal" Si-O" bonds, their lengths are still larger and none of the Si-O-cation angles equal 180 ~ Among our model systems treated in computations, in the case of H3SiONa, three LP are directed outward relative to the Si-O bond at any O...Na distance, probably because of the attraction to the positively charged sodium. The elongation ofLMO LP represents, in this case, a certain amount of bonding overlap with the cation. Thus, a false result of the computation originating probably from a casual selection of the basis set can be suspected. In reality, poor results are obtained when a deficiency of restricted basis sets in the description of systems with excessive negative charge is used and special types of polarizing functions have been proposed to avoid it. This problem will be discussed later. No inversion of bonding and lone pairs as described above occurs with the application of the wider basis set III complemented by 3d functions at oxygen. Even in this case, however, three LMO LP deviate considerably from the tetrahedral layout with a trend to the trigonal planar arrangement around the oxygen (Si[OILP angle equals 97~

A contri-

bution of silicon 3p AO is found in LMO of these LP, thus indicating their mixed nature. The removal of CEDD in the bonding (axial) pair from O towards Si is larger than in any other system and the equilibrium bond length remains extremely short (1.567A) in this basis

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set. No effects of similar kind were obtained with basis sets without polarizing d-functions at the oxygen. Nearly the opposite valence charge distribution is found in the calculation of the electronic structure of silanol, H3SiOH, with the longest Si-O bond in the series. The calculation with the I+d(O) basis set whose results are reported in Tables 1.1 and 1.2 has been extended to the calculation with the wider set III +d(O). The latter leads to a similar equilibrium geometry (reSiO=l.672A, reOH=O.954A, SiOH=ll8.3 ~ and a very low overlap population in the Si-O bond, 0.089 (cf.; 0.635 for the H3SiO" system in the same basis set). Fig. 1.1 reproduces the atomic positions and valence pair arrangement in the Si-O-H fragment obtained with that basis set. It is seen that the LPIOILP angle is larger than the tetrahedral one and the CEDD of the Si-O and O-H bonding pairs are shifted from the bond axes to inside the SiOH angle. Since the Si-O bond pair and two LP are positioned in vicinity of oxygen atom, the valence charge distribution around it resembles one in the free OH- anion. It means that the silanol molecule can be described as a system with considerable ionic contribution, H3Si+...OH ", where the hydroxide ion is slightly disturbed by the polarizing action of the positive charge at the silicon atom. The valence charge distribution in symmetrical Si-O-Si bridges will be discussed later. In general, it can be concluded that in SiOX systems the enhancement of electronegativity of the second ligand of oxygen leads to a concentration of valence charge around the oxygen, and increases the polarity of Si-O bond. In contrast to the Si-O- or Si-O--..M+ bonds, the bridging Si-O(Si) or Si-O(H) bonds are much more polar. This is why a discussion of the crystal chemistry of silicon dioxide and silicates in terms of predominantly ionic bonding, as it was adopted for a long time in the literature, proved to be rather realistic.

B. The Oxygen Bridges in H3XOXH 3 Systems with X = C, Si An advantage of the systems of this type is that they combine the relative simplicity of structure with their accessibility to experimental investigation as free molecules in the


,,

19

118 ~

Si

O6 H

Fig. 1.1

The arrangement of the valence pairs CEDD positions around the oxygen

atom in H3SiOH. gaseous phase. Most of them have been repeatedly studied by various structural and spectroscopic methods. The electronic structure and equilibrium geometry of dimethyl and disilyl ethers were computed by several authors employing various levels of quantum mechanical approaches [ 11,20,24-26] which simplifies an estimation of accuracy of their joint treatment with a middle quality basis set discussed below. Ignatyev's computation of a whole series [25] with the basis set II aimed both a comparison of theoretical molecular geometry with experimental data and a ref'mement of spectral assignments by means of "scaling" of ab initio quantum chemical force constants. It should be emphasized that this semi-empirical procedure seems to produce a better description of a real force field than its estimation by purely theoretical approaches employing laborious computations with the more extended basis sets, at least, if these are restricted to the Hartree-Fock method and with the adoption of the adiabatic approximation. Another reason for a particular interest to the force field of disiloxane, H3SiOSiH 3, originates as it has been mentioned above from the expected minimal influence of terminal groups on the intrinsic properties of the Si-O-Si bridge. This problem will be additionally investigated when discussing the best selection of molecular force constants for the description of local elastic properties of corresponding structural fragments in solids. The results of computations for the silicon containing molecules have been complemented by ones obtained with the simpler basis set I+d(O) in order to make them compara-

20

LAZAREV

ble with corresponding data of the previous section and to clarify its applicability to particular problems which need a quantum chemical computation of larger molecular systems. Table 1.3 compares the equilibrium geometry obtained with the basis set II with the experimental data, which was determined for dimethyl ether from the microwave spectrum [27] and for disilyl and methyl silyl ethers from the gas electron diffraction data [28,29]. All molecules possess the symmetry plane coinciding with the XOX plane. The C-H or Si-H bonds in the terminal XH 3 groups positioned symmetrically to that plane are labeled hereatter as C-H' or Si-H' in order to distinguish them from corresponding bonds lying in the plane which belong to another symmetry set and can differ in their properties. The LP are naturally symmetric to that plane and the C-H' or Si-H' bonds will be sometimes referred to as the tram-bonds implying their orientation relative to the LP. One more symmetry plane arises in dimethyl and disilyl ethers thus determining their C2v point group while methyl silyl ether belongs to the C s group. As is seen in Table 1.3, a computation with the basis set II reproduces the experimental geometry within 0.01A in bond lengths. Judging by the dimethyl ether where the problem has been investigated both experimentally and theoretically (see [ 11 ] for details), a computation reproduces correctly the deviations in the structure of terminal XH 3 groups from the local C3v symmetry. These are characterized by the differences between X-H and X-H' bonds, the latter being elongated due to the so called tram effect, or between OXH and OXH' angles with corresponding changes in HXH' and H'XH' angles. As it has been originally explored by Ignatyev [30,31 ], all these variations correlate with the LP orientation which can be characterized in a present case numerically by the angle between the bisectors of LPIOILP and XOX angles. A valence pair distribution around the bridging oxygen atom is shown in Fig. 1.2 by plotting their CEDD positions in a true scale (heavy dots). According to the results of computations, the local asymmetry of the methyl group is the largest in dimethyl ether and diminishes when passing to methyl silyl ether while the asymmetry of the silyl group in that molecule is less significant than in disiloxane (Table 1.3). It can be readily explained by the


21

TABLE 1.3 The equilibrium geometry ofH3XOXH 3 molecules calculated with the basis set II in comparison with experimental data.* Geometrical Parameters

H 3COCH 3

H 3COSiH 3

Theor.

Exp.

Theor.

Exp.

1.415

1.410

1.428

1.418

-

-

1.640

ZXOX, deg.

111.2

111.7

r3CH, A

1.081

r3CH'

H3SiOSiH 3 Theor.

Exp.

1.640

1.633

1.634

121.3

120.6

149.6

144.1

1.091

1.082

~ 1.080

1.087

1.100

1.085

J

r3SiH

-

-

1.470

1.471

1.486

r3SiH'

-

-

1.478

ZOCH, deg.

107.2

107.2

107.8

ZOCH'

111.6

110.8

111.1

ZHCH'

109.0

109.5

109.0

/H'CH'

108.4

108.7

108.8

/OSiH

-

-

107.4

/OSiH'

-

-

111.7

J

/HSiH'

-

-

109.2

~

107.4

J

reCO, A reSiO

1.485

1.475 111

108 l

109t

109.2

109.9

110.8 110

108.9

109.1 108.2 *No attention is paid to difference between so called rs- and rg-slructures [ 11] deduced from various experimental methods. tFrom the sum oftetrahedral angles. /H'SiH'

-

-

charged silicon atom (Fig. 1.2). The plane of the LP thus approaches the normal orientation to the Si-O bond and to parallel orientation relative to the C-O bond. This explanation treats the asymmetry of CH 3 and Sill 3 groups as a manifestation of trans effect in the bond pair/lone pair interaction, which can be rationalized in terms of their Coulomb interaction. The asymmetry of the methyl and silyl group structure manifests itself in dynamical properties (force constants, dipole moment derivatives) and in the peculiarities of valence

22

LAZAREV

I

18!1--~/-"x

C

Fig. 1.2 The arrangement of the valence pairs CEDD positions around the bridging oxygen atom in H3COCH3, H3COSiH3, H3SiOSiH 3 series.

(

~'l 5

Si

149.6

Si

Si

charge distribution (net charges of hydrogen atoms, overlap populations etc.). These details have been treated more or less exhaustively when discussing molecular spectra and structure [ 11] but they do not relate directly to silicates. A comprehensive list of computational data characterizing both the geometry and electronic structure of all three molecules is given in Table 1.4. The data obtained with a simplified basis set I+d(O) are included for the silicon containing molecules. It can be concluded that the results do not suffer very much from this simplification and the basis set I+d(O) can be applied (for the sake of economy) to calculations for more complicated systems of similar chemical nature.

TABLE 1.4 The electronic structure and spatial configuration of H3XOXH3molecules calculated with various basis sets.

Parameters of electronic structure and geometry

-Ebb,, eV r,CO,

A

reSiO LXOX, degrees reCH, A reCH' ReSiH r,SiH1 LOCH, degrees LOCH' LHCH'

LH'CH' LOSiH LOSifI'

H3COCH3

nf

H3COSiH3 11

I+d(O)

11'

H3SiOSiH3 I+d(O)

TABLE 1.4 (continued). Parameters o f electronic structure and geometry LHSH' LHfSiH' Localization LMO CO:

LMO SiO:

LMO LP:

LMO CH:

LMO CHf:

LMO SiH:

H3COCH, 11'

H3COSiH3 11'

H3SiOSiH3 I+d(O)

11'

I+d(O)

TABLE 1.4 (continued). Parameters of electronic structure and geometry LMO SiH:

do

i

LLPIOILP, degrees LCEDDLMo1 101CEEDLM02 Dipole moments, D hota~ (pexP)

pLMO CO &MO SiO pLM0 LP pLM0 CH** pLMO SiHt* Overlap Population C-0 Si-0 C-H

H3COCH3 11'

H,SiOSiH,

H,COSiH, 11'

I+d(O)

11'

I+d(O)

TABLE 1.4 (continued).

Parameters of electronic structure and geometry

H3COCH3 11'

H3COSiH3

II'

C-H' Si-H Si-H' Net charge

C Si

0 H(C)

H'(C) H(Si) H'(Si) Residual force, mdyn *The angIe between COSi bisector and dipole moment direction equals 45" (Fig. 1.3). Averaged over symmetrically non-equivalent LMO.

a*

J+d(O)

n

H,SiOSiH, I+d(O)


27

Let us now return to valence density distribution in XoO-X bridges (Fig. 1.2, Table 1.4). The non-uniformity in the charge distribution rises from X=C to X=Si: the negative charge at the oxygen increases and the CEDD of the bonding LMO are shifted towards this atom. The more polar the bonds become, the more their s-character increases as is seen from the AO contributions to the corresponding LMO. Consequently, the s-character of the LP decreases in that direction. A trend to a stable tetrahedral arrangement of four LMO around the oxygen, which can be deduced from Fig. 1.2, determines an important peculiarity of the bonding charge distribution in the bridges with significantly opened XOX angles. The CEDD of the bonding LMO do not follow the bond axes rotation at this opening, which is probably determined by an increased Coulomb repulsion of positively charged X atoms and thus a considerable portion of bonding density turns out to be concentrated inside the angle. The effect of off-axial location of bonding charge in the Si-O-Si bridges was first recognized by Newton and Gibbs [32-35] in their earlier quantum chemical computations of systems containing these bridges and discussed in its relation to peculiarities of valence charge distribution in silica and silicates deduced from the experimental X-ray diffraction data. The existence of the negatively charged area between the silicon atoms forming the Si-O-Si bridge has been adopted by Mirgorodsky and Smimov [36] to justify their dynamical lattice model of a-quartz where the Coulomb repulsion between these atoms was complememed by their non-Coulomb attraction, which played an important role in satisfying the conditions of equilibrium of the lattice. The population analysis results are complemented in Table 1.4 by the dipole moments. The total dipole moments are compared with corresponding experimental values and rather considerable discrepancies can be noted. It probably originates from the use of too narrow basis sets. Figure 1.3 explains the dipole moment orientation in methyl silyl ether where it is not restricted unequivocally by symmetry requirements. It might be instructive to inspect how the total dipole moment is composed by the moments of various

28

LAZAREV

] '

~

-

450

\ H

Fig. 1.3

The molecular di-

pole orientation in methyl silyl

kt I

Si

ether.

H' bonds and LP. A decomposition of the total dipole moment obtained from quantum mechanical computation adopts the procedure proposed by Malrieu [37]. In a neutral molecule where the number of electrons equal the nuclei's charge, a charge density distribution can be described as an assemblage of localized electrically neutral systems. If the charges at the bonding LMO are complemented by +1 e charges at the atoms constituting any bond and the charges at LMO LP and the cores are complemented by +2e charges at corresponding nucleus, then the total dipole moment is expressed as a vector sum of dipole moments of bonds, LP and cores. Such components are presented in Table 1.4, the largest of them being the moments of Si-H and Si-O bonds and of LP.

C. The Effects of Additional Coordination of Bridging Oxygen Atom Various coordination numbers of bridging oxygen atoms were met in crystallographic investigations of silicates. In most cases they are ranging from 2 to 4. The interaction with adjoining extra cations necessarily changes the valence charge distribution around the bridging oxygen atom in a complex silicate anion and thus influences its geometry and other properties. Some of these effects may be investigated numerically by quantum mechanical computations of an artificially designed system composed by the disiloxane molecule with two additional Li+ cations coordinated to the bridging oxygen atom. In order to retain C2v symmetry of the system, the Li+ cations were fixed in the symmetry plane normal to the SiOSi plane, each being removed from oxygen to a 2.0A distance. The LiOLi angle


29

equal to 120 ~ was kept fixed as well (this value was selected attempting to put the cations near the LP directions). It has been shown above that the use of the simplified basis set I+d(O) does not change the results of the computation for the disiloxane molecule. Therefore, this set was applied in a study of the disiloxane geometry relaxation in the system containing two lithium cations in fixed positions relative to the bridging oxygen atom. Table 1.5 shows how the charge distribution is changed in that system and how the internal coordinates of disiloxane come to new equilibrium values. In other words, the restarts of calculation should characterize the spatial distribution of tensions in this molecule excited by the additional cations approaching the oxygen atom. It is hoped that these processes resemble ones arising in silicate lattices and their model treatment helps to distinguish them from other effects of less localized origin which are particular for the crystals. As it follows from the inspection of Table 1.5, the additional coordination of the bridging oxygen in disiloxane considerably influences the Si-O bonds, which lengthen by 0.15A. This effect agrees qualitatively with a trend to longer Si-O(Si) bonds in silicates with the greater coordination number of bridging oxygen atoms, but it is evidently overestimated. It may originate from some simplifications introduced into our model: the total charge of the cluster is not compensated and the initial + 1e cation charge induces a strong charge flow from the oxygen which would be less significant in a larger cluster system with a completed oxygen polyhedron around Li + etc. A diminishing of the equilibrium SiOSi angle which is found in our cluster also correlates with well known trends in the crystal chemistry of silicates where these angles are usually larger, as the coordination number of the bridging oxygen atom decreases [23 ]. The data of Table 1.5 show that an approaching of cations to the oxygen atom polarizes the whole molecule. The CEDD of the LMO LP are shitted towards the cations and these LMO begin to partially play the role of bonding Li-O orbitals. On the other hand, the LMO SiO in the cluster is relatively nearer to the oxygen than in a free disiloxane molecule,

30

LAZAREV

TABLE 1.5 The electronic and geometric structure variation of disiloxane at the enhanced oxygen coordination.

Computed magnitudes rLiO, A (fixed)

FH3SiOSiH3] 2+ / \

L LiLiJ

H3SiOSiH3

2.0

ZLiOLi, deg. (fixed) reSiO, A

120.0 1.812

1.653

reSiH

1.557

1.483

reSiH'

1.475

1.489

ZSiOSi, degrees

134.4

142.1

/OSiH /OSiH'

102.5

109.0

105.4

111.2

ZHSiH'

114.3

108.8

ZH'SiH'

113.3

107.9

d0, A

0.432

0.430

r do

0.790

0.767

LMO LP:

0.341

0.294

r dH

0.770

0.768

LMO Sill:

0.379

0.310

1.022

0.980

0.357

0.306

0.991

0.982

ZLPIOILP, degrees

110.5

117.2

Overlap population: Si-O

0.199

0.454

Li-O

0.262

Si-H

0.732

0.691

Si-H'

0.739

0.674

0

-1.087

-1.128

Si

1.032

1.238

Li

0.842

H

-0.110

-0.217

H'

-0.110

-0.223

Localization LMO SiO:

LMO Sill"

Net charges:

dH,


31

thus indicating the increase of polar character of bonding (cf. do/reSiO values). These changes in valence dens@ distribution and a decrease of the SiOSi angle are accompanied by a variation of the AO-composition of bonds and LP. As it has been mentioned above, the cluster is unstable as a whole: at fixed Li + positions, the total energy is elevated by 1.5 eV even after the relaxation of internal forces arising in the disiloxane molecule.

This is why the deformation and polarization of that

molecule in the cluster should be treated as representing a trend to minimize the electrostatic repulsion between the cations and silicon atoms possessing the largest net charge inside the molecule. Both the elongation of the Si-O bond and the decrease of the SiOSi angle act towards the increase of Li+...Si distances at fixed Li+ positions. Another channel to relax that interaction is a lowering of the net charge at Si by the partial electron redistribution from H' and H atoms into their bonds with silicon whose overlap populations are increased in the cluster and the CEDD of their LMO is shifted towards Si. For all this, the AO-composition of Si-H and Si-H' bonds vary differently: the 3s-AO Si contribution increases in Si-H' bonds with some diminishing of Is H contribution while in Si-H bonds the 3p-AO Si increases contribution. The equilibrium bond lengths change in opposite directions. A shortening of Si-H' bonds can be rationalized in terms of the inverse "tram effect", i.e., as a consequence of the CEDD removal in LP from the oxygen which decreases the LP repulsion from the bonding pairs in trans bonds. The origin of considerable elongation of Si-H bonds is less clear. It may relate to the changes of AO-composition and to a cooperative nature of bond-bond interaction in a tetrahedral assemblage around silicon, which will be discussed in more detail when treating the properties of the SiO4 tetrahedron. Those variations in the structure of the terminal Sill 3 groups in the cluster probably indicate that the influence of the enhanced bridging oxygen coordination in silicates is not restricted to that bridge and may affect the adjacent bonds. In particular, the above data show that different action of the additional coordination of the bridging oxygen may be expected, depending on the orientation of the more removed Si-O bonds relative to the

32

LAZAREV

O.-.cation contacts. Even the opposite effects can be predicted: a shortening of "tram" bonds and a lengthening of bonds lying in the SiOSi plane. However, these effects can hardly be discerned in the analysis of various silicate structures being obscured by other changes in coordination, which usually accompany the changed coordination around bridging oxygen atom.

D. The Dynamical Properties of Oxygen Bridges The terminal Sill 3 groups attached to the Si-O-Si bridge in disiloxane are not very space consuming, and their mutual interaction may affect the equilibrium structure and dynamical properties of the bridge in a lesser extent than any other substituents at silicon. It means that a contribution to the force constants which belongs intrinsically to the bridge as itself is separated more easily than in larger molecular systems adopted in the estimation of force constants suitable for transfer to dynamical models of silicates. This problem will be discussed later. Another advantage of disiloxane for the ab initio determination of the force field of the Si-O-Si moiety originates from the exhaustive experimental data on its vibrational spectrum including the spectrum of the D-substituted species (see [38] for references). It restricts the ambiguity in the correction of errors in the quantum mechanical force constant evaluation by the scaling procedure. The applicability of the basis set II' to the calculation of the force field of various organic and silicoorganic ethers has been demonstrated by numerous examples treated in the book by Ignatyev and Tenisheva [11 ]. In order to make the force constant determination more definite, the scale factors for a series, H3COCH 3, H3COSiH3 and H3SiOSiH3 have been found jointly supposing the transferability of scale factors relating to a particular atomic group from one molecule to another [25]. The scale factors of the OSiH 3 group adopted in the evaluation of corresponding force constants in methyl silyl ether were determined at fixed scale factors of the OCH 3 group transferred from dimethyl ether. Those scale factors of the OSiH 3 group were found applicable to disiloxane without any additional refinement. It is thus hoped that the


33

ab initio force constants of disiloxane are sufficiently dependable. The more recent com-

putation [26] took into consideration the electron correlation and analysis of the anharmonicity of the SiOSi bending potential and confirmed Ignatyev's estimations of the harmonic force constants. A fraction of the disiloxane force constant matrix, which relates directly to in plane internal coordinates of the Si-O-Si bridge (and its orientation relative to the terminal Sill 3 groups) is presented below:

rSiO

6.141

~SSiOSi

0.186

0.068

rSiO

0.341

0.186

xSiH 3

-

-

rSiO

~SSiOSi

rSiO

5.345

0.160

0.297

0.058

0.160

-

5.345

-

6.141

-

-

0.002

-

xSiH 3 -

0.002

The upper right triangle contains the scaled force constants while the lower left one gives their unsealed values obtained from the quantum mechanical computation. A dimensionality of force constants mdyn/A, mdyn, mdyn.A is adopted for the stretch-stretch, stretch-bend and bend-bend interactions respectively. A very small diagonal SiOSi bending force constant is remark able. It can be interpreted jointly with a negligibly hindered Sill 3 internal rotation as a low elasticity of the oxygen shift in the bridge in both directions normal to the Si... Si direction. Such flexibility of Si-O-Si bridges (at least in absence of additional oxygen coordination) is well known in the crystal chemistry of silicates. A positive sign of the stretch-bend interaction force constant which is three times as large as the diagonal bending one also agrees with empirical correlation known in the crystal chemistry of silicates: this sign implies the arising of forces which shorten the Si-O bonds when the SiOSi angle increases. An analysis of crystallographic data shows statistically that the longer Si-O bonds correspond to the smaller SiOSi angles.

It can be concluded that there exist some intrinsic properties of those bridges

weakly dependent on the less local properties of a whole system containing them. Let us now investigate the polarization of the bridge arising under the bond deformation as it follows from the restricted computation (for simplicity) to the basis set I+d(O).

34

LAZAREV

That process can be analyzed in two different ways. The first one proceeds from the valence optical scheme which is often adopted in model IR intensity computation of molecular spectra. It decomposes the total dipole moment augmentation into a vector sum of increments which correspond, in a given case, to dipole moment variations along the bond axes at the stretching of one of them. Such decomposition of the total dipole moment augmentation leads to the following partial derivatives of bond dipoles relative to the bond stretch: aM1/c3r1 = 4.45, aM2/c3r1= -1.54 D/A. A relatively large off diagonal term indicates that the simplest diagonal approximation of valence optical scheme which assumes the total dipole variation to be located entirely inside the stretched bond is inapplicable to Si-O-Si bridge. Another decomposition of the total dipole moment augmentation may be deduced from the localization procedure by representing the total dipole as a sum of increments corresponding to dipoles of various LMO. In the present case, it means its decomposition into contributions of two LP, two Sill 3 groups with dipoles along the axis of the corresponding Si-O bond, and two Si-O bond LMO with dipoles deviating 15~ from the bond axes.

When stretching the first Si-O bond, the corresponding increments are (D/A):

0rnsio(1)/& 1 = 1.48, cOmsio(2)/c3rI = 0.97, igrnLp//gr1 = 0.08, /)rnsiH3(1)//grI = - 0 . 7 4 , O~nsiH3(2)/tgrI - 0.15. The dipole derivatives of the bonding LMO differ significantly from the bond dipole derivatives in the valence optical scheme; even the sign of the derivative of the bond adjacent to the stretched bond being opposite. Considerable polarization of Sill 3 groups including the one removed from the stretched bond indicates that the process involves the whole molecule. An insignificant contribution from the LP dipoles is explained by the approximate orthogonality of their sum to the Si-O bond dipole. It follows from the above discussion that a simple scheme of deformational polarization of the oxygen bridge hardly can be deduced from the quantum mechanical computation of disiloxane. It should be added in the conclusion of this section that most of the above considerations deduced from our earlier computations with moderate quality basis sets remain unaltered when more powerful modem methods are applied to the investigation of the same


35

systems. It has already been mentioned that a higher level of computation of the potential energy hypersurface of disiloxane [26] leads to essentially the same conclusions on its harmonic force constants as were obtained with a simpler basis set by means of empirical scaling. The electronic structure and equilibrium geometry of several members of the H3SiOX series have been calculated recently [39] using a complete basis set (with no frozen core orbitals) of the 6-31 G* type and electron correlation effects taken into consideration at the second order perturbation theory level by Moller-Plesset method (MP2). The following equilibrium geometry parameters were obtained: H3SiO-(C3v),

reSiO=10579A, reSiH=l.477A,

H3SiOSiH 3 (C2v), reSiO=l.656A,

reSiH=l.480A,

OSiH=ll8.0~ OSiH=108.3 ~

SiOSi=145.2 ~

OSiH=105.8 ~

reOH=O.969A,

reSiH'=l.485A , OSiH'=110.6~ H3SiOH (Cs),

reSiO=l.670A,

reSiH=l.477A,

SiOH=116.5 ~

reSiH'=l.487A, OSiH'=112.2 ~

A comparison of these values with the corresponding data of Tables 1.1 and 1.3 show a similarity of variation in the equilibrium geometry along the series although the absolute magnitudes differ. A trend to longer Si-O bonds and slightly shorter Si-H bonds obtained by Curtiss et al. [39] is remarkable. III. SYSTEMS W I T H T E T R A H E D R A L O X Y G E N C O O R D I N A T I O N OF SILICON

A. SiO 4- Oxyanion and Si(OH)4 Molecule Both of these systems are practically inaccessible to direct experimental investigation in a free state although the latter is present in a certain concentration in the alkaline silicate solutions (see [40] for more detail). These solutions have been investigated spectroscopically with the results previously being discussed [41 ]. Unfortunately, it is difficult to separate the bands of that molecule from the overlap with the bands of partially protonated silicate ions and of the products of their hydrolitic condensation. No experimental data on

36

LAZAREV

the equilibrium geometry of both systems are available. Nevertheless, they seem to be attractive for the theoretical investigation of interaction between the silicon-oxygen tetrahedron and surrounding atoms or ions in a condensed system as representing the limiting states of the tetrahedron: from a free oxyanion up to the tetrahedron incorporated into a three-dimensional network of partially covalent bonds. 4The electronic structure and equilibrium geometry of the SiO4 ion and Si(OH)4 molecules were originally calculated [42] using basis sets I and II with the orbital exponent for 3d-Si polarizing functions rl3d(Si)=0.8 transferred from ref. [20]. The results presented in Table 1.6 show an unexpectedly low positive net charge at silicon in the free SiO4- ion. When passing to the Si(OH)4 molecule, the net charge at silicon increases while the negative charges at the oxygen atoms decrease, the total negative charge of the tetrahedron being thus reduced more than twice. These findings were analyzed [43] and it has been concluded that the relatively rigid split valence type basis sets suffer from their inability to dispose the large excessive electronic charge in the peripheral area of the oxyanion which leads to a very overestimated effect of displacement of superfluous valence density into the internal part of the tetrahedron which artificially stabilizes it. Besides the underestimation of positive charge at Si, it leads to enhanced bonding overlap which favors the shortening of reSiO. A contraction of LMO LP and decrease of the oxygen-oxygen repulsion act in the same direction. The physical meaning of those computations was discussed [43] by comparison of description of one-electron and total energies of Si and O (3p) atoms in restricted basis sets with the results reported by Clementi and Roetti [44]. These calculations approached the Hartree-Fock limit. An application of restricted Hartree-Fock method in one-determinant approximation to computations for the tetrahedral oxyanions of second-row elements adds several electrons to the last filled MO up to the formation of a closed shell. That MO with t 1 symmetry is composed of the 2p-AO of the oxygen atoms. In LMO terms, it means the adding of electrons to the oxygen LP which "pushes out" a whole system of levels of the

TABLE 1.6 A silicon-oxygen tetrahedron in the ~i0:- ion and Si(OH)4molecule at various basis sets.

9 n

Electronic structure and geometry

LOSiO (x2), deg. LOSiO (x4) LMO SiO: LMO LP:

do, A

0.443

0.457

0.444

0.442

0.421

0.403

0.396

0.414

-

0.736 0.319

0.778 0.322

0.875 0.351

0.812 0.325

0.710 0.295*

0.744 0.307~

0.752 0.306*

0.755 0.309~

0.777 98

0.778 99

0.894 107

0.802 104

0.722* 104; 91

0.720* 95; 92

0.757~ 101; 95

0.749~ 98; 89

118(x3)

118(x3)

lll(x3)

121

112

110

110

1.268

1.240

2.126

1.987

2.015

2.066

2.489

2.335

0.637

0.57 1

0.378

0.526

0.540

0.466

0.243

0.420

-0.030

-0.028

-0.679

-0.105

-0.023*

-0.024~

-0.183~

-0.102~

r do, A -

r

LSiOIOlLP, deg. LLPlOlLP Net charge

Si

Overlap Si-0 Population 0 . . -0

115(~3)

'Averaged over symmetrically non-equivalent in the Sq point group sets of LP or of 0.. .O distances.

8 3

a

2 Lo

38

LAZAREV

oxyanion from the potential hole. The last filled MO obtained are positive one electron energies, i.e., the electrons cease to be bound, and a calculation of these oxyanions in frames of restricted Hartree-Fock approach is looking to be artificial. It was proposed to complement the atomic basis sets by the flat polarizing functions of the valence shell in order to get a better description of negatively charged systems [45]. Those slowly vanishing with distance functions are characterized by very small orbital exponents (rls,p=0.068 was proposed in a particular case of oxygen) and are specially adapted to description of the "mils" of MO with energies near the ionization limit. A basis of this type which will be referred to as I+s,p(O) has been adopted in the recalculation of both considered systems, the SiO 4- ion and the Si(OH)4 molecule [43 ], although its application should influence the results for the latter in a lesser extent. These results are also included into Table 1.6. Moreover, it follows from the above considerations that any other extension of the basis set would lead to the trends resembling one obtained by the addition of fiat polarizing functions of a valence shell. Therefore, a calculation of those systems has been repeated using the most flexible of our sets (DZHD set III). Table 1.6 represents these results. It should be mentioned that from the point of view set forth above, a discrepancy between the results obtained with basis sets I and III, must be less significant in a case of smaller excessive negative charge of the oxyanion to be considered. The difference between the results of their application to geometry optimization of the isoelectronic PO 34 ion is about half of that in the case of the silicate ion (cf. Table 1.6): rePO Basis set

(A)

I

1.617

III

1.640

Net charge P

Overlap P-O

O

population

+1.779

-1.193

0.560

+2.083

-1.271

0.587

It is interesting to note that in the case of PO 34- ion, the overlap population in the bond is slightly larger in a wider basis set while the opposite result was obtained for the silicate ion.


39

Let us compare in more detail the results of the complementation of the basis set I by flat polarizing functions with ones obtained by the transition to the more extended set III in the case of the SiO 4- ion. As seen from Table 1.6, there is a certain resemblance in the changes of valence electron distribution caused by both versions of the basis set extension although the total energy gain obtained by the transition to the set III is naturally much more significant than the one obtained with the set I+s,p(O). In both cases, a redistribution of excessive electron density towards the peripheral area of the oxyanion is most remarkable. The CEDD LP removal from the oxygen cores increase, and the average LP radii increase as well. An enhanced electrostatic repulsion between oxygen atoms (cf. their net charges in Table 1.6) constitutes the main origin of increase of equilibrium in Si-O bond lengths. The same conclusion can be deduced in both sets from a drastically enhanced nonbonding O...O overlap population whose tremendous value in the basis set I+s,p(O) originates evidently from the peculiarities of spatial configuration of polarizing functions of valence shell. The enhanced AO population of oxygen and decreased Si-O overlap population together with a shift of CEDD at its LMO towards the oxygen atom represent the same effect of the electron charge flow to the peripheral area of the tetrahedron in more flexible basis sets. It is interesting to note that a complementation of the basis set III by the flat polarizing functions of the oxygen valence shell leads to an increase of reSiO up to 1.758A which exactly coincides with that distance calculated using the set I+s,p(O). The results of the population analysis and the parameters of the LP and bond LMO render to be similar as well. It will be shown below that the results of computations are insignificantly affected by neglect of the electron correlation. Independently of the basis set adopted in calculation, a transition from the fi'ee tetrahedron with 4e excessive negative charge to the neutral Si(OH)4 molecule is accompanied by a considerable reduction of equilibrium Si-O bond length which ranges from 0.05-0.06A in the basis sets I and II up to 0.1A in the sets I+s,p(O) and III. Proceeding from the above considerations, the last value may be taken as a most realistic estimation of the decrease in tetrahedron size induced by the diminishing of its negative charge from -4 to 1.8 (the latter

40

LAZAREV

value being obtained by the atomic net charge summation over the tetrahedron in Si(OH)4 molecule). Despite the decreased Si-O bond length in the Si(OH)4 molecule, its polarity characterized by d0LMO SiO/reSiO relation is increased, which evidently relates to the outward transfer of electron density. The s-character of the Si-O bonds increases when passing from the SiO 4- ion to the Si(OH)4 molecule unless the structure of the bonds remains essentially the same. A contribution of 3s,3p-AO of silicon to the LMO of bonds reduces. A contribution of 2p-AO of oxygen reduces as well while a contribution of its 2s-AO rises. The enhanced s character of bonds may relate to their shortening together with weakened oxygen-oxygen repulsion. The electronic configuration around oxygen remains to t)e near sp3 in both systems where this atom forms one bond and three LP in the case of the SiO 4- ion or two bonds and two the LP in the Si(OH)4 molecule where the negative charge of that atom is considerably reduced. The mutual orientation of the LMO LP of the oxygen atoms in the SiO 44 ion fulfills a condition of maximal distance between any two LP of different atoms lying in the OSiO plane with the other four LP being positioned symmetrically to that plane. Two LP of any oxygen atom in the Si(OH)4 molecules belong to the same symmetrically equivalent set only in the case of the D2d point group which does not correspond to the deepest energy minimum. At the equilibrium geometry described by the S4 point group, these LP are only approximately symmetric to the SiOH planes. Some similarity may be found in the changes of electron distribution in the Si(OH)4 molecule obtained when passing from the basis set I to I+s,p(O), on one hand, and from the set I to III, on the other. Though these changes are much less significant than in a case of the SiO 4- ion, a difference in the equilibrium Si-O bond length is nearly an order of value smaller. The most economic basis set I renders to be a rather satisfactory approximation in the calculation of the structure and properties of electrically neutral systems, but its deficiency rapidly increases when being extended to negatively charged species, which should be taken in mind when analyzing the results presented in following sections.


41

A broadening of the basis set I by 3d polarizing functions at the Si and O atoms with equal orbital exponents (0.8) adopted in the basis set II, leads to similar changes in description of the electronic structure and equilibrium geometry of the SiO 4- ion and the Si(OH)4 molecule. It is accompanied by a shortening of Si-O bonds and an enhancement of their overlap population at the retained appearance of corresponding LMO. A transition to the set II reveals itself in the geometry calculation by the reduced (approximately to 115 ~ equilibrium SiOH angles. This value is much closer to the values found experimentally in the crystallographic investigations of acid silicates than the angles (ca. 130-135 ~) deduced from computation without any basis set possessing 3d functions at the oxygen atoms. This problem, and the problems of the correct reproduction of the potential of the internal rotation around the Si-O bonds (variation of dihedral OSiOISiOH angles) will be not discussed further. Since the basis set II has been applied for the calculation of the equilibrium geometry of existing molecules with Si-O bond lengths ranging from 1.634 to 1.681A, an estimation of systematic error in the bond length determination with this basis set is possible. The empirical correction for ab initio bond lengths in those molecules constitutes 'hrCalc' --~e IIsio=+0.015A. The calculated Si-O bond length in Si(OH)4 nearly coincides after this correction with an reSiO=l.635A obtained for that molecule by the "energetic" geometry optimization with a wider 6-31G basis set complemented by the polarizing d-functions at the Si and O atoms and p-functions at the H atoms [46]. After this correction of the bond lengths in SiO 4- and Si(OH)4 determined with the basis set II, an error in their determination using set I can be estimated. It varies from +0.9% for re _talc, ISIO=1.647A to +1.7% for rcalc, ISiO=I.708A in the above species. Such corrections would be applicable to the ree suits of computations for the cluster type systems treated below in a case of their insensitivity to the total charge of a system. Unfortunately, the latter is scarcely valid. It can be concluded from the above discussion that a restricted Hartree-Fock treatment of a free SiO 4- ion artificially over-stabilizes this system which cannot exist as itself. On the other hand, its protonation l~roduces the existing stable neutral molecule Si(OH)4

42

LAZAREV

where the SiO4 tetrahedron is stabilized by a partial transfer of its excessive negative charge to surrounding ions. This may relate to the mechanism of the SiO 4- ion's stabilization in condensed systems such as silicates. Therefore, the influence of that charge transfer will be additionally investigated in the following sections by the computation of clusters containing this oxyanion surrounded by a certain number of cations and by the study of partially protonated silicate ions. B. Dynamical Properties of the SiO 4 Tetrahedron in Simple Systems Various rough assumptions were adopted in the literature when attempting to estimate the eigenfrequencies of a "virtual" free SiO 4- ion from the spectra of condensed systems containing it in a bound state. One of approaches proceeded from the experimental spectrum of the Ba2SiO 4 crystal whose vibrational frequencies were more or less arbitrarily supposed to be weakly affected by the interactions in a lattice, and thus differ insignificantly from the frequencies of the free SiO 4- ion [47]. In Siebert's earlier estimations [48,49] the averaged frequencies of that ion in the spectra of various crystalline orthosilicates were adopted as an approximation to the frequencies of a free silicate ion. All these estimations were made in an assumption of the stability of the force constants of that ion irrespective of the nature of the condensed system containing it, and completely neglected the vibrational coupling of its internal modes with other vibrational movements in a crystal. The inadequacy of the former of these assumptions follows directly from the content of the above subsection while the latter one has been criticized in books [41,50] specially devoted to the normal coordinate analysis of the vibrational spectra of silicates. Another approach adopts the band frequencies in the spectra of alkaline aqueous silicate solutions as an estimation of the frequencies of a free silicate ion [51,52]. It is obvious that this approach also assumes the independence of the internal force field of that ion on the properties of the medium. Moreover, a stability of the internal mode frequencies of the SiO4 tetrahedron relative to degree of its protonation is implied implicitly in this approach, since the broad bands in the spectra of solutions were not resolved into the compo-


43

nents belonging to the SiO 4- ion and its partially protonated derivatives which necessarily coexist with it. It can be thus concluded that the "experimental" frequencies of the SiO 4ion presented in Table 1.7 are no more than a very rough estimations. On the other hand, no experimental frequencies of the Si(OH)4 molecule are well known. Correspondingly, the scaling procedure is hardly applicable to the theoretical force constants of both systems. A comparison of quantum mechanical force constants calculated by the gradient method in various basis sets with empirically estimated ones is given in Table 1.8 in terms of the tetrahedral symmetry co ordinates neglecting a slight distortion of the SiO4 group in the Si(OH)4 molecule (in a more rigorous approach the non vanishing interaction terms between the coordinates belonging to different irreducible representations of the T d point group would arise). As has been mentioned above, all force constants are reduced to the same dimensional representation (mdyn/A) in order to simplify a comparison of the data obtained by different authors. It should be taken into account that some authors [47,48] assumed the zero value of the off-diagonal term in the F 2 symmetry species in an attempt to reduce the ambiguity of the estimation of other force constants from the experimental frequencies. Among the empirical force field models of the SiO 4- ion, the one proposed by Handke [52] differs more significantly. It probably originates from the erroneous assignment of the band at 605 cm -1 in a water solution spectrum to the co4 mode of that ion though its origin may be explained more naturally by the probable hydrolitic condensation with arising of more complicated ions containing Si-O-Si bridges which possess vibrational modes just in this frequency area. The same trends in the theoretical force field variation under transition from the free SiO 4- ion to the Si(OH)4 molecule are met irrespective of the basis set adopted in quantum mechanical computation.

This transition is accompanied by the increase of the Si-O

stretching force constant, which correlates with the shortening of the bond. The bending force constants are, however, reduced at this transition. It can be tentatively explained by the reduced net charges at the oxygen atoms which diminishes a contribution of Coulomb

44

LAZAREV

TABLE 1.7 The experimental frequencies of vibrational modes of SiO4 group deduced from various sources and their theoretical magnitudes (basis II) for the SiO 44 ion and Si(OH)4 molecule. Vibrational modes

Frequencies Experimental [49]

[47]

Theoretical

[51 ]

[52]

Unsealed

Scaled*

SiO 4-

Si(OH)4

Si(OH) 4

eo3(F2)

935

910

906

906

923

1054

944

COl(A1)

775

826

819

780

764

876

783

co4(F2)

460

500

527

605

544

452

428

co2(E)

275

260

340

450

385

324

307

Scaling factor 0.8 for stretchingmodesand 0.9 for bending.

TABLE 1.8 The force constants of SiO4 tetrahedron. Force constant

Theoretical of the

Empirical (from the experimental

Theoretical of

SiO4-

frequencies)

the Si(OH)4

matrix elements

II

I

fl I(A1)

5.51

4.98

f22(E)

0.47

f33(F2)

[49]

[47]

[51]

[52]

I

II

4.23

5.67

6.30

6.32

5.74

7.19

7.24

0.43

0.30

0.24

0.20

0.36

0.57

0.26

0.34

4.29

4.48

3.04

4.03

3.50

4.20

5.30

6.53

6.24

f34

0.25

0.25

0.25

0.31

0.78

0.29

0.36

f44

0.82

0.79

0.58

0.62

0.80

0.74

0.91

0.47

0.51

rcalcsio

1.664

1.708

1.758

1.647

1.617

e

I+s,p(O)

"

"

-

"

repulsion along the edges of the tetrahedron into those force constants even at their slightly shorter lengths in the Si(OH)4 molecule. These considerations have been confirmed by corresponding numerical model estimations [43]. It was shown, in particular, that about


45

50% of the f22 and f44 force constant magnitudes deduced from quantum mechanical computation can be assigned to the electrostatic oxygen-oxygen repulsion. It means that an overestimation of the bending force constants of the SiO 4- ion in theoretical computation (Table 1.7) only partially originates from the intrinsic to the Hartree-Fock approximation trend to overestimate the diagonal force constants. Another source of that overestimation may be searched in the unrealistically large oxygen charges obtained by computation in this approximation. The experimental frequencies assigned to the SiO4- ion relate to condensed systems (crystals, solutions) and a partial transfer of the oxygen charge to the surrounding ions is very probable with a corresponding weakening of the oxygen-oxygen repulsion. It follows from the above considerations that some correlation may exist between the oxygen charges in silicate crystals and the OSiO bending force constants. However, it will be hardly revealed in the experimental spectroscopic data because of difficulties in the direct oxygen charge determination in complicated lattices and of vibrational coupling between the OSiO bending and lattice modes which complicates a sufficiently precise determination of the unperturbed frequencies of the former modes. The bond-stretching frequencies of the silicon oxygen tetrahedron calculated with ab initio quantum mechanical force constants agree unexpectedly well with the results of their

experimental determination (Table 1.7). It might be rather strange taking into account the above critical comments to the applicability of the restricted Hartree-Fock approach to the computation of the electronic structure and related properties of the silicate ion. The coincidence seems to be, however, a result of, more or less, accidental mutual compensation of errors of dual origin. One of them is the overestimation of the diagonal force constants in the Hartree-Fock computation with restricted basis sets. Another is that a destabilization of a free SiO 4- ion by its excessive negative charge may lower its bond stretching force constant. Furthermore, while the theoretical force constant evaluation relates to the free silicate ion, the experimental estimations of its vibrational frequencies are deduced from the spectra of systems where the real charge of that ion is smaller than -4e. These considerations agree

46

LAZAREV

with a larger equilibrium bond length in this ion deduced from theoretical computation than is obtained by the averaging of experimental bond lengths in crystals of orthosilicates. The above considerations are supported indirectly by the possibility to reproduce the "experimental" frequencies of silicate ion using the theoretical force constants of Si(OH)4 molecule with reasonable scaling factors as it is seen from the right-hand column of Table 1.7. The tentatively introduced scale factors compensate the errors of the Hartree-Fock computation while the charge of the tetrahedron in that molecule may not differ too much from the real charges of silicate ions in crystals. A further discussion of the theoretical values of stretching force constants can be given in terms of the Si-O bond force constant obtained by the recalculation of the symmetry force constants of Table 1.8 to the parameters of the general valence force field (GVFF) model [5] which is made unambiguously using the interrelations between the internal and symmetry coordinates presented above. A GVFF model has been numerously applied to empirical normal coordinate analysis for various silicates and molecules of silico-organic compounds, and a correlation between the Si-O force constant and the bond length was proposed [50]. This graph was later refined by means of additional data from [38,53] and other computations of the same authors and is plotted in Fig. 1.4. It should be emphasized that a common force constant/bond length dependence is proposed for the systems with tetrahedral oxygen coordination of silicon (silicates) and for the R3Si-OR' or R3Si-O-SiR 3 molecules with a single Si-O bond at any silicon atom. The non-linear character of the correlation is most clearly expressed in the area of rather long bonds (re > 1.65A) with the force constants almost twice as small as the ones corresponding to the shortest bonds (it should be taken in mind that, generally speaking, in empirical force constant determination by frequency fitting, the larger the magnitude is, the more precisely it is determined). The results of the above quantum mechanical computation of the force field of the SiO 4- ion and Si(OH)4 molecule are reduced to the Si-O bond stretch force constants in the GVFF model and plotted versus the corresponding equilibrium (theoretical) bond


fsio'

47

mdyn/]t .

Fig. 1.4 The empirical Si-O force constant~ond length correlation deduced from the normal coordinate computations

.

[11 ] for the silicoorganic molecules (+) or silicate crystals (x) in comparison with the results of ab initio computation fo

\Xx~"

++

|

i

I

|

1.60

!

!

SiO 4- ion and Si(OH)4 molecule ( D -

|

I

'

1.65

!

|

basis set I, O - basis set II).

I

I

|

1.70

qsio, A

|

!

I

I

!

1.75

lengths in Fig. 1.4. Two linear graphs are thus obtained for the computations in basis sets I and II. They can be corrected by reducing the theoretical bond lengths to the experimental ones by means of empirical corrections estimated for both basis sets. This procedure, which is shown in Fig. 1.4 by wavy arrows, leads to a practically coinciding graph for both sets whose slope closely resembles one obtained empirically for the bonds of moderate length. Still, the theoretical graphs need to be shifted along the bond length axis by about 0.05A to longer bonds. We shall return to the force constant/bond length correlation later when discussing the results of computations with a more flexible basis set. A single attempt was made to estimate the influence of electron correlation on the statements relating to the structure and the force field of the silicate ion deduced by means of the split valence type basis sets. The simplest set I was adopted in calculation by means of the Gaussian-80 program where the electron correlation was taken into consideration on the MP2 level with "frozen" cores. That computation led to reSiO=l.738A in the SiO 44ion instead of 1.708A obtained with the same set in the SCF approximation. The force constant of the Si-O bond turned out to be reduced to 4.16 mdyn/A from 4.60 mdyn/A in the

48

LAZAREV

SCF computation. The interrelation of SiO/reSiO still remained to correspond to the correlation graph deduced from the SCF computations (Fig. 1.3). It is thus hoped that the conclusions concerning the force field of the silicate ion and similar systems obtained from SCF computations will not change qualitatively with the transition to more elaborate computations taking into account the electron correlation. Although the total energy of SiO4- ion is reduced from-15916.999 to -15930.749 eV in our calculation, the force constant is insignificantly reduced and corresponds to a slightly larger bond length. It is well known that in numerous cases, the account of electron correlation considerably shifts the potential energy hypersurface along the energy axis weakly affecting its shape near the minima.

C. Charge Redistribution in the Strained Silicon-Oxygen Tetrahedron Besides the IR intensity calculation (and/or the piezoelectricity if a crystal is treated), a problem of electric polarization accompanying the strain of a system arises in any dynamical model which explicitly separates the contribution of Coulomb forces into mechanical properties. The direct ab initio determination of the charge redistribution in a deformed silicon-oxygen tetrahedron is thus instructive for the elucidation of the physical reliability of suppositions adopted in various model approaches to the lattice dynamics of silicates. The dipole moment derivatives relative to the internal coordinates were determined for both systems containing the SiO4 tetrahedron. Hereafter, the signs of those derivatives imply the positive increment of the bond elongation in the casc. of the stretching type coordinate and the angle enhancement in the case of the bending type coordinate. At normalized values of those adopted above, both types of dipole moment derivatives have the same dimensional representation (Debye/Angstr6m). The dipole moment derivatives relative to the normal coordinates, a ~ Q , are obtained by means of their shapes Lij as c3~)Qi = ~ J (a~/asj) Lij. The squares of these values determine the IR intensity of the corresponding normal mode. Unfortunately, neither for a free SiO 4- ion, nor for the Si(OH)4 molecule, the experimental IR intensities are known. It is possible to estimate the

la~/aQ31value for a high-


49

frequency co3 mode of the SiO 4 tetrahedron from the IR reflection spectrum of some orthosilicates, for instance, zircon ZrSiO 4 [53], where this mode is weakly coupled with lattice vibrations. These values are 3.36 and 3.84 D/A for two components of internal mode co3 with dipoles parallel and normal to the four-fold axis in that tetragonal crystal. However, their values represent, rigorously speaking, not only the internal polarization of the tetrahedron caused by its deformation but the total effect including the possible polarization of an adjacent area of the crystal. The theoretical values calculated with the basis set II are significantly smaller for both the silicate ion and the Si(OH)4 molecule: SiO~-

Si(OH)4

lap/aQ31

2.8

1.8

lap/aQ41

0.7

0.2

Comparison with the experimental co4 mode is not possible because of its coupling with the lattice modes. The dipole derivatives are larger if the basis set I+s,p(O) is employed with results that are believed to imitate the charge distribution obtained with the more flexible basis sets. For the silicate ion Itgla/O~31=3.52 and Ic31ahgQ41=l.48D/A values are obtained with the basis set I+s,p(O). The subsequent discussion of the mechanism of deformational polarization of the silicon oxygen tetrahedron does not operate with any experimental data and is based entirely on the theoretical c3p/tgsi magnitudes which represent explicitly the interrelation between the polarization and geometry variation of a system. The following values were obtained for the SiO 4- ion and the Si(OH)4 molecule by computation with various basis sets: SiO 4I

II

Si(OH)4 I+s,p(O)

I

II

a p/tgs3

-4.50

-5.60

-8.28

-3.90(-5.5)

-4.30(-5.9)

a i.t/as4

3.25

2.65

3.87

1.69(6.5)

1.11(5.9)

50

LAZAREV

The magnitudes relating to the SiO4 tetrahedron in Si(OH)4 molecule which can be compared with corresponding data for the silicate ion are given in brackets. These were deduced from the total dipole derivatives for that molecule by the exclusion of contributions from the hydrogen nuclei (their contribution is especially large for the bending coordinate s4), the change of the Ekkart condition formulation at the transition from one system to another being taken into ac count (see [43] for more detail). While the sign and absolute value of the dipole moment derivative of the tetrahedron with respect to the bond stretching coordinate remain practically the same in both systems, the dipole derivative of bending coordinate considerably increases when passing from the SiO 4- ion to the Si(OH)4 molecule. It can probably be interrelated with a similar character of bonding in the first case and with the changes in properties of the LP at the "transformation" of one of them into an Si-H bond in the second one. The results of the charge redistribution computation in the de formed tetrahedron can be applied to an estimation of the parameters of its valence optical model which is often employed in molecular spectroscopy for the IR intensity calculation. That model represents the total dipole moment and its deformational increments by expansion into the contributions of bonds. In quantum mechanical calculation, a decomposition of the total dipole moment into a sum of moments of various LMO [37] can be adopted as explained above. An insignificant contribution from the polarization of cores will be neglected and the total dipole moment will be treated as a vector sum of contributions from the bonds and LP. A similar decomposition of the total dipole moment of a charged system like the polyatomic anion is possible if the additional "ionic" charge is introduced at any atom whose number of valence electrons does not correspond to the number of LMO of its bonds and LP. In the particular case of the SiO 4- ion, this approach assumes adding -le excessive charges at each oxygen atom. The total dipole moment can then be represented by the sum of the moments of the four Si-O bonds, twelve moments of the LP and the "ionic" moment of four unit charges at the oxygen nuclei (a similar decomposition for the PO 3ion would require adding a +le "ionic" charge at the P atom).


51

Such decomposition of the dipole moment of the SiO 4- ion in the equilibrium geometry and at various strains corresponding to symmetry coordinates is presented in Table 1.9 using the results of the quantum mechanical computation in an sp-approximation (I+s,p(O) basis set). It is seen from Table 1.9 that the moments of the LP are comparable in their values with the moments of the bonding LMO, the same relates to the magnitudes of their variation in a strained tetrahedron. Therefore, any attempt to design some electro optic scheme, which would represent in its parameters the peculiarities of chemical structure and bonding, should operate both with the bonds and the LP. It can simply be done by treating the entities as the bond moments and the moments of atoms possessing a LP. Let us introduce a notion of effective bond dipole (EBD) in the SiO 4- ion, M i, in order to rationalize the data of Table 1.9 in terms of a standard valence optical scheme. This value will be defined as a vector sum of the dipole moment of corresponding bond LMO, m i, and of the moments of three LMO LP of the oxygen atom, Ilk. The total EBD is found thus as: 3 M i = m i + ~'~ lik. k=l

(1.12)

This definition is intrinsically restricted to the case of a free oxygen SiO44- ion since in more complicated systems where each oxygen atom forms more than one bond, the EBD direction may deviate from the Si-O bond axis. In particular, this definition is inapplicable to Si-O bonds in Si-O-Si bridges. It is seen from Table 1.9 that at the totally symmetrical A 1 stretch of the SiO44- ion, a very insignificant diminishing of the EBD occurs in each bond. In the F 2 type bond stretching motions, some bonds elongate while the others shorten. The EBD of the first mode are considerably increased and that of the second, decreased. The EBD are, however, changed at the angle bending motions of the tetrahedron as well, thus indicating the inapplicability of the diagonal approximation of the valence optical scheme. The same follows from the dependence of EBD of a given bond from the lengths of adjacent bonds.

52

LAZAREV

TABLE 1.9 A decomposition of the static dipole moment of the SiO4 ion and of its increments at the internal strains of various symmetry.* State of the

Internal coordinates

Dipole moments (Debye)

tetrahedron

rSiO,/k

ctOSiO, deg.

Equilibrium

1.7583 (4)

109.47(6)

A 1 (Ar)

1.7633 (4)

F 2 (Ar)

1.7633 (2)

Si-O (m)

LP (1)

EBD (M)

4.1803 (4)

3.3756 (12)

7.2275 (4)

[109.47]

4.1873 (4)

3.3731 (12)

7.2227 (4)

[109.47]

4.1976 (2)

3 3748 (2) l 7.2500 3.3777 (4) J

1.7533 (2)

[109.47]

4.1603 (2)

3.3764 (2) I 7.2050 3.3735 (4) J

F 2 (Aa)

[1.7583]

110.80(1)

4.1743 (2)

3.3480(2) ~ 7.2031 3.3836 (4) J

[1.7583]

108.08 (1)

4.1863 (2)

3.4032 (2) ] 3.3676 (4)

E (Ac~)

[1.7583]

110.60 (2) } 4.1804 (4) 108.90 (4)

(2)

(2)

(2)

7.2519 (2)

3.3583(4) } 7.2275 (4) 3.3842 (8)

*The multiplicityof a set (intemalcoordinates,LMO, dipole moments)is given in brackets; in square brackets are giventhe internalcoordinatesmagnitudesremainingconstantin a giventype of deformation. A further discussion of the data of Table 1.9 and of analogous data obtained with the more rigid basis set I will be devoted exclusively to a dependence of the bonds EBD on their lengths. The following values will be treated: t = Orni/Ori,

u = Omi/Orj,

T = OMi/Ori,

O = cOMi/o~j.

Their magnitudes in two basis sets are (D/A): t

I+s,p(O)

U

T

U

1.7

-0.4

1.6

-0.4

2.91

-0.50

3.08

-1.34


53

A transition from the moments of the bonds LMO to their EBD practically does not change these electrooptic parameters in a case of computation in the basis set I. In the more flexible basis set with flat polarizing functions of the oxygen shell (which considerably change the properties of LP). Such a transition leads to a significant increase of the off diagonal electro optic constant. It originates mainly from the rotation of lik vectors in a strained geometry of the tetrahedron. The approximate interrelation u ~ -0.2t and U ~ -0.33T follow from values obtained in quantum mechanical computation. Their origin can be explained in terms of two limiting type models of bonding. A comparison of these models may be helpful in the estimation of "ionic" and "covalent" contributions into the mechanism of polarization of the strained tetrahedron. D. A Covalent Model

A tetrahedral AB 4 molecular system is treated in which the A-B bonds are formed by the sp 3 hybrid AO of the central atom. The bond dipole variation at small deformations originates in this case from a rehybridization of sp 3 orbitals. A quantitative estimation of relative changes in the bond dipole moments can be deduced from the orthonormality condition. The dipole moment of an orbital Wi = ~ 1

(s+giPi) is found as:

2~'i Ixs(x)p(x)dv 1+~2. l

(1 13)

u ~ / tgmi and is approximately proportional to k i. Therefore, the relation T = - ~ i - ~ i is equal to 6q~j / tg~,i 63~j the relation - ~ i / 0 r i = 0~.i" From the orthonormality condition, a regular tetrahedron ()~i = ~/3) follows that c~)~j/c~;~i = _l. Thus, a covalent model predicts u = -It. Similar considerations allowed earlier deduce an interrelation between the diagonal and off-diagonal force constants which were adopted in the hybrid orbital force field (HOFF) model [54].

54

LAZAREV E. An Ionic Model

A system composed of the A Q+ cation tetrahedrally surrounded by four Bq- anions is treated. Neglecting the cation polarizability, the ith EBD is determined by the dipole moments of anions induced by the internal field E i of the system: M i = xE i where X is the polarizability of the anion. According to electrostatic laws: Qri ~ q irij Ei = r 7 + j=l'--" r3 .

(1.14)

Therefore, the following interrelations can be deduced: c3Ej = 1 / q 9 ~ _ ~ /gri r3

_ Q)

aE i ' t:3ri

1 5 r--~ q ~-~~]~ .

(1.15)

Now, the case of4q+Q = -4e in the tetrahedral SiO 4- ion is investigated. In this case: U/T = a E j / a E i tgri/~ii =-

0.382q 8e + 7.3 lq "

(1.16)

The expected U/T relations for several possible q and Q values are: q

Q

U/T

-1.2

0.8

-0.59

-1.3

1.2

-0.33

-1.4

1.6

-0.24

Thus, both simplified models allow the explanation of the origin of the signs and values of the interrelations between the diagonal and off-diagonal components of the tensor of electro-optic constants in frames of a generalized valence optical scheme as they were deduced from quantum mechanical computation.

Probably, both limiting type models

should be employed in the analysis of the deformational polarization of the SiO 4- ion. The covalent model seems to be more applicable in a study of the silicon AO contribution while the ionic model can be adopted in considerations relating to the oxygen polarizability contribution.


55

In conclusion, the parameters of the general valence optical scheme for a free SiO4ion derived from the data of Table 1.9 are presented: c3Mi/~i = 3.08 D/A, ~Mi/~j = -1.34 D/A, aMi/~czij = -1 D/radian, aMi/~Ctjk = 0. These values can be tested in the IR intensity calculation for some silicates.

IV. Q U A N T U M M E C H A N I C A L C O M P U T A T I O N S F O R S O M E I O N I C C L U S T E R S AND T H E I R R E L A T I O N T O T H E C R Y S T A L CHEMISTRY OF SILICATES Besides the simplest of systems treated above, a number of various ionic clusters built up by the SiO 4- ion surrounded by a certain number of cations or even simply of positive point charges in fixed positions were investigated by the same computational methods. A valence state of oxygen atoms in those clusters represents an intermediate case between the single-bonded oxygen in the silicate ion and the oxygen atom in the Si(OH)4 molecule where it participates in two partially covalent bonds. It will be shown below that the changes in state of the oxygen atom and the properties of a whole tetrahedron depend both on the number of surrounding cations (or point charges) and their positions relative to the tetrahedron. Nevertheless, some common trends can be found and elucidated when studying the relaxation of internal forces in the tetrahedron at fixed cation positions. It should be emphasized that in a general case, the equilibrium geometry of a cluster cannot be found, at least, at the interionic distances resembling the ones met in silicate crystals. In other words, it is impossible to design an equilibrium cluster with a closed electron shell which would reproduce a fraction of a crystal. It originates, first of all, from the specific "one-side" oxygen co ordination around cations in these clusters. The problems to be investigated by calculation" for such clusters are thus restricted to the changes in electronic structure and equilibrium geometry of the silicate ion under the influence of the nearest cations. Even these properties differ from ones characterizing these anions in a lattice since neither its electrostatic field nor the short-range interanionic interactions (oxygenoxygen repulsion) are considered in the numerical experiments treated below.

56

LAZAREV

Among various possible types of clusters, only the ones retaining T d symmetry of a whole system were selected. In this case, a structural relaxation of the tetrahedron under the influence of surrounding cations is reduced to a one-dimensional problem (reSiO being the unique structural parameter to be ref'med). It ensures a considerable economy in computer time expenses together with the use of the simplest basis set I For the same reasons Li § ions were selected as the cations, their AO being described by the basis set 3-21G [55]. Everywhere possible, the clusters possessing zero or minimal total charge were preferred. A designation of the type of a cluster adopted hereafter indicates its composition with the position of point charge q+ or cation Li § given in subscript. E.g.: 4q~]SiO 4- - 4 point charges on the three-fold axes positioned on the Si-O bond directions opposite to the tetrahedron apices, a "monodentate" positioning of q§ quasications. + 9 44q[3]$10 4

- 4 point charges on the three-fold axes positioned against the tetrahe-

dron faces, a "threedentate" position of q+. + 9 46q[2]S10 4

- 6 point charges on the two-fold axes positioned against the middles of

tetrahedron edges, a "bidentate" position of q+ (i.e., the two nearest oxygen atoms). This description of clusters is shown in Fig. 1.5 where the cation positions are indicated by various points of a cube in which the SiO 4- ion is inscribed with bonds lying on its spatial diagonals. The positions of types 1 and 3 are on the apices of that cube while the positions of the type 2 are in the centers of its faces. One more set of positions 1' on the middles of its edges (which also corresponds to the monodentate cation-to-anion coordination) is occupied in a more complicated cluster 12q~]SiO 4- with q+ charges situated around the three-fold axis of the tetrahedron, thus completing a tetrahedral surrounding of each oxygen atom. No calculations were performed for a similar cluster with 12 Li + cations. At the adopted cluster geometries, the parameters R(Si...q +) and R(O...q +) are interdependent and their interrelation is governed by the type of cluster. As a result, in


Av / , ,

2

1--/~-'--1 ' ,

I

r I I

,

2:o/\ L ..-"

57

t1' 3/,ol

A/

1, I

Fig. 1.5

The cationic sites in

clusters of various type.

i, I ....... I r/~

clusters of the first type described above, the R(Si...q+) distances are fixed near to the corresponding distances in silicate lattices, and the R(O...q+) distances are too short. At reasonable R(O...q+) distances in the second type clusters, the R(Si...q+) distances turn out to be implausibly short. These disadvantages are less exhibited in clusters with a larger number of cations, though the non-zero total charge is unavoidable in that case. It can be shown by simple electrostatic calculation in a point charge approximation that Coulomb forces acting on the oxygen atoms tend to compress the tetrahedron in most types of clusters. A single exclusion relates to the first type of clusters where the electrostatic forces stretch the Si-O bond (the force on the oxygen atom or on the cation will be denoted as the positive when it is directed outwards of silicon). At fixed coordinates of external charges, the total relaxation of forces on the oxygen atom may fail to be reached at some R(Si...q +) distances if a condition to maintain more or less reasonable interatomic distances is imposed. Numerous computations for various type cluster systems have been described by Shchegolev et al. [43]. Some of them are exemplified below before attempting to summarize their consequences which relate to the crystal chemistry of silicates. The few results of quantum mechanical computations for cluster type systems are collected in Table 1.10 with

TABLE 1.10 Some properties of silicon oxygen tetrahedron in ionic clusters (basis set I).

Properties of a system

si0f

4qhl~i01-

6 q b 1 ~ i ~ a - 4 ~ i h ~ i 0 : 6~ib$3i0:-

Si(OH)4 *

R(Si. - .q/Li), A R(0. .q/Li,H)

0.953

r,SiO LMO SiO: LMO LP:

1.708

1.647

do, A

0.457

0.403

f

0.778

0.744

do

0.322

0.307 (av.)

f

0.778

0.720 (av.)

LSiOlLP, deg. Net charge

99.1(x3)

Si

1.240

2.066

0

-1.310

-0.970

Li,H,q Overlap population Si-0 O*-.O 0-..LI,H

0.453 0.571

0.466

-0.028

-0.024

Force on Li/q, mdyn

*s4 symmetry of a whole system. **

95.0; 92.3

The total equilibrium of a whole system.

-

**

0.543

*


59

corresponding data for the free SiO 4- ion and Si(OH)4 molecule (which are treated as representing the possible limiting states of the silicon oxygen tetrahedron). All geometries in Table 1.10 correspond to the total relaxation of forces on the oxygen atoms. Unfortunately, for the reasons mentioned above, this condition for the clusters of the first type was fulfilled at non-coinciding R(Si...q +) and R(Si..-Li +) distances. For the same reasons, a comparison of properties of the clusters of first and third types hardly could be carded out at the same R(Si...q +) or R(Si-..Li +) distances. The data relating to the second type clusters are not included in Table 1.10, since the peculiarities of their arrangement (cation positions against the centers of the faces of the tetrahedron) lead to the arising of Si...Li overlap, which seem implausible in real silicate lattices. In accordance with predictions of purely electrostatic calculations, the relaxed size of + 9 4the SiO 4 tetrahedron is larger in the 4q[1]S10 4 cluster than in a free SiO 4- ion while in

the 6q(2]SiO 4- cluster, its size diminishes. The data relating to these clusters in Table 1.10 correspond to the case of practically coinciding magnitudes of forces on q+ quasi-cations directed towards Si in both cases (it was ensured by suitable selection of Si...q+ distances in these clusters). The electronic structure of the silicate ion is insignificantly changed in both clusters in comparison with that of a free ion. These changes are restricted to a certain redistribution of net charges at the unchanged degree of polarity in Si-O bonds with very slight changes of the LMO LP caused by their polarization from the q+ charges. When passing from these clusters to isostructural clusters with Li + cations possessing their own electron functions, 4Li~]SiO 4- and 6Li~2]SiO4- , only for the latter the condition could R(Si...Li +) = R(Si...q+) be satisfied. This transition is accompanied by much deeper changes in the electronic structure of the SiO 4- ion. In any type of cluster, a considerable part of excessive electronic charge is shitted outside the tetrahedron as can be seen from the net charges at the Li + ions. The total negative charge of the tetrahedron reduces nearly twice as a result of this transfer. The positive net charge at Si being increased while the negative charges at the oxygen atoms decrease. The LMO of the LP are rearranged and begin to partially play the role of Li-O bonding orbitals. Their overlap populations are

60

LAZAREV

probably over estimated in the clusters of a given type because of specific "one-side" coordination around the cations. The CEDD in LMO LP are slightly shifted outside the tetrahedron and these LMO tend to be directed towards cations. It is seen in the enhanced SiOIOILP angles in both types of Li containing clusters. Moreover, in a case of 6Li~2]SiO4

cluster, three "lobes" de-

scribing the LP of any oxygen are found to be turned around the Si-O bond in a way which corresponds to the orientation of the LP lying in a given OSiO plane inside that angle. The forces attracting the cations to the center of the cluster are considerably reduced, mainly be the cause of the decrease of the charges of cations. A decrease of the core-core repulsion between lithium and oxygen (it is probably most important in the clusters of first type due to a short R(O...Li +) distance) may play an additional role. It is important to emphasize that independently of the opposite influence of electrostatic forces from external charges in the 4Li~]SiO 4- and 6Li~2]SiO4- clusters on the positions of oxygen atoms, the equilibrium Si-O bond length reduces practically equally in both clusters (with respect to the free silicate ion). This sho~ening originates in the first instance from the reduced oxygen-oxygen repulsion. The Si-O bonding overlap, however, decreases despite the bond shortening which is evidently determined by a partial transfer of valence density outside the tetrahedron. The AO-composition of the Si-O bond LMO is changed insignificantly towards some increase of its s-character. The polarity of that bond slightly increases due to the shift of the CEDD in its LMO nearer to the oxygen atom. Generally speaking, the changes in the electronic structure of the silicate ion at the transition from q+- to Li+-clusters resemble ones described above when a free ion was compared with the Si(OH)4 molecule but are less drastic. Let us now discuss the consequences of the above computations which relate to the crystal chemistry of silicates, i.e., the interrelations between the composition of these crystals, their electronic structure, the and spatial geometry of their lattices. Special attention will be paid to considerations relating to the dynamical theory of these crystals.


61

It follows from the analysis of the above computations that an excessive negative charge destabilizes the silicon oxygen tetrahedron, its equilibrium size being larger, the larger the total charge, and reaching the maximum size in a free SiO 4- ion. It means that in condensed systems containing such ions, the influence of nearest environment, i.e., of outer ligands of oxygen atoms, consists mainly in their ability to form the efficient channels for the release of excessive charge. In the case of silicate crystals, there exist some other factors influencing the size of the tetrahedron whose action can not be reproduced in a treatment of quantum mechanical computations for any cluster or molecular system. One of them is the Madelung field of a lattice, which is understood here in a spirit of our approach to the lattice dynamics of ionic covalent crystals [56,57]. It treats that field as being constituted exclusively by the action of charges outside some nearest area of specific short range interactions where the Coulomb contribution hardly can be separated (this problem will be discussed in subsequent chapters). Irrespective of the intrinsic limitations to any attempt to deduce the properties of a macroscopic system-like crystal from a treatment of models of molecular type, the results of quantum mechanical computations of these systems help to explain the origin of some regularities in the structure of silicates. The main rules in the crystal chemistry of silicates have been formulated originally by Pauling [58] and remained practically unchanged (although being completed by Belov's rule [59] of ability of complex silicate ions to adapt their shape to the arrangement of the coordination polyhedra around cations). These rules should probably be completed by one more statement relating to the size of the silicon-oxygen tetrahedron in silicates: the average Si-O bond length in the tetrahedron is determined by its total charge in a given lattice, the geometry of the nearest environment playing a secondary role which affects the degree of scharacter in the Si-O bond and core-core repulsion between the oxygen and its outer ligands.

62

LAZAREV

The above rule allows the explanation of the empirical regularities of the mean Si-O bond length found in silicates. A statistical treatment of the mean bond lengths, ~ SiO, determined in 155 precise crystallographic structure investigations of more than 300 symmetrically non-equivalent tetrahedra has been carded out by Baur [60]. His analysis explored the following empirical correlation whose origin can be explained in terms of the following statements 1. A shortening of the ? SiO with an increase of the number of"shared" tetrahedron apices, i.e., the number of Si-O(T) bridges per one tetrahedron (T = Si, B, Al, P, Ga). Explanation: the increase in the number of most efficient channels for a discharge of the tetrahedron. 2. A shortening of the ? SiO with an increase in the average coordination number of the oxygen atom of a tetrahedron. Explanation: the increase in the number of channels for a charge flow from the tetrahedron. 3. A shortening of the f SiO with an increase in the angle of the silicon-oxygen outer ligand. Explanation: the increase in s-character in the Si-O bond and the compressive action of core-core repulsion between the oxygen and the outer ligand. A knowledge of the dynamic properties of systems investigated above by quantum mechanical methods can then be summarized in a form applicable to the lattice dynamics of silicates: 9 The theoretical force constants and equilibrium SiO bond lengths in a molecular species satisfactorily reproduce the slope of the linear part of their empirical correlation graph deduced from the normal coordinate calculation and frequency fitting for silicate crystals and silicoorganic molecules.


63

The theoretical OSiO bending force constants in the silicon-oxygen tetrahedron are very sensitive to the charges on the oxygen atoms and increase as the charge increases. The theoretical force constants of the stretch-stretch interaction at the common silicon atom constitute about 5% of the magnitudes of corresponding diagonal bond stretching force constants (the force constants reduced to conventional GYFF model are implied). The same is valid relative to the interaction of bonds with a common oxygen atom in Si-O-Si bridges. 9 A polarization of the SiO 4- ion at the bond stretching motions cannot be described in frames of the diagonal approximation of valence optical scheme: an increase in the dipole moment of the stretched bond is accompanied by the decrease of the dipole moments of the other bonds, both effects being of the same order of value. A charge density response to variation of the size (volume) of the silicon-oxygen tetrahedron can hardly be supposed to be located entirely inside the tetrahedron, and is probably spread on the adjacent cations. Only few comments to the statements whose applicability to the design of lattice dynamics models are given here and will be investigated in the next chapters. The first of these statements, relating to the force constant/bond length interrelation, will be treated in more de tail in the next sub-section employing additional computational data of a higher level. The second one can hardly be tested using the presently available results of lattice dynamic calculations for silicates because of the difficulties in the independent estimation of charges at various symmetrically non-equivalent oxygen atoms, and in a separate evaluation of the bending force constants for different OSiO angles. However, a special investigation of this problem by means of a suitably selected crystal with significantly different oxygen charges and possibly separated contributions of corresponding angles into various vibrational modes seem to be worthwhile.

64

LAZAREV

The third statement indicates that earlier magnitudes of the Si-O stretch/stretch interaction force constants deduced by frequency fitting in the normal coordinate calculation with the GVFF model (see, e.g., [50,53]) were strongly overestimated for interactions at a common silicon or oxygen atom. One can suspect that these force constants implicitly represented the empirical normal coordinate calculation influence on some effects which are not considered in either the GVFF model or in the ab initio treatment of dynamical properties of a molecular species. This problem will be paid attention when discussing a transferability of ab initio molecular force constants into the lattice dynamic models. The last two statements relate directly to one of the central problems in the model (phenomenological) approach to lattice dynamics which operates with various model representations of so called charge density response functions. These functions interrelate the charge density distribution with lattice strain and will be discussed later in some detail. It can be mentioned here that both of the above statements can be adopted in the variable charge model which treats the atomic charge in its simplest version as a function of the lengths of bonds issuing from a given atom [56]. This model was numerously applied to the IR intensity calculation in the spectra of silicates [50,53]. It is still less clear as to what extent the parameters of that model estimated from the IR intensity (and/or the piezoelectric constant) fitting are compatible with their values deduced from Coulomb contributions to the force constants in approaches separating those contributions explicitly.

A. Partially Protonated Silicate Ions It has been shown that important properties of silicon-oxygen tetrahedra can be treated in terms of ~ SiO averaged over a tetrahedron Si-O bond length which simplifies a comparison of variously distorted tetrahedra. Such distortions take place, in particular, in partially protonated silicate ions, which can be investigated by the same methods of quantum mechanics of molecular systems as were applied to more symmetrical ions and clusters. A series (HnSiO4)(4"n) with n = 0, 1, 2, 3, 4 whose highly symmetrical members were discussed above will be investigated step by step below.


65

It was emphasized earlier that a transition from the Si-O----cation or the Si-O- bonds to Si-O(H) bonds resemble change in the electronic structure, charge distribution, etc. and a transition to the bridging type Si-O(Si) bonds. Correspondingly, the series described below can be treated as a simple model for the investigation of differences between tetrahedra possessing various numbers of terminal bonds (Si-O) and bridging bonds (Si-O(Si)). The conclusions deduced from the computations of these relatively simple systems are probably applicable to such tetrahedra in complicated lattices of condensed silicates. Two types of changes in bonding can be studied in this approach. The first one relates to a difference between the Si-O and Si-O(H) bonds depending on the number of each type of bond in a complex anion. The other relates to the changes of the properties of a tetrahedron as a whole depending on the value of n. In particular, the applicability of the above regularities in the mean bond length variation can be studied. The number of parameters to be optimized when searching the equilibrium geometry of these systems is relatively large and the initial computation of a whole series [61 ] was restricted to the semi-empirical MNDO method for simplicity. A valence sp-basis and the standard set of parameters [62] were adopted. The results of the complete geometry optimization for all members of the series obtained by means of the gradient method are presented in Table 1.11. The main trends in variation of the mean bond length, ~ SiO, and the lengths of the Si-O- and Si-O(H) bonds are shown in Fig. 1.6 where these lengths are plotted versus the formal negative charge of the tetrahedron, 4-n. The mean Si-O bond length in the tetrahedron increases with the increase in 4-n at minor deviations from the linear relationship. This graph thus represents the earlier formulated dependence of the size of the tetrahedron on the possibility of its stabilization by removal of excessive negative charge. The same follows from the sequence of enthalpies of formation in Table 1.11. These are systematically increased with the rising of 4-n and become positive at 4-n > 3.

TABLE I . l l The structure and bonding variation in H.s~o$~-")series as deduced 6om MNDO calculation.

hoperties of a system mfomation, kcal/mol reSiO(H), A

-304.0 1.70 1

reSiO-

-295.4 1.737

-157.1 1.788

+112.5 1.859

1.642

1.666

1.706

1.763 1.763

f SiO

1.701

1.713

1.727

1.744

reOH

0.932

0.932

0.934

0.939

LO(H)SiO(H), degrees

109.5'

LO(H)SiO-

101.8

98.6

118.2

106.9*

102.6'

126.9

114.4'

113.8

108.9

LO-SiOLSiOH

Net charges

121.5 Si

Total charge on Si04

116.3

+510.5

109.5

1.348

1.128

0.980

0.877

0.803

-0.856

-1.472

-2.201

-3.053

-4.000

'Averaged over sqmmebically non-equivalent sets.


reSiO

1,8

67

X

Fig. 1.6 x

-

Si-O bond lengths in (HnSiO4) (4-n)series.

Semiempirical calculation:

a) Si-O(H) bonds, b) Si-O" bonds, c)

/o

1.7

The equilibrium

. -N"

fSiO over a tetrahedron, and d) the same bond lengths from the ab initio calculation.

I-I

0-r

1.6,

EI-d

6

i

3 4-n

The linear ~ SiO variation arises, however, as a result of two essentially non-linear dependencies of the Si-O(H) and Si-O bond lengths on the enhancement of 4-n. As is seen from Fig. 1.6, the AreSiO(H) and AreSiO- increments increase more and more at each step although the inequality of AreSiO(H) > AreSiO" remains the same. It can be rationalized in terms of a greater compliance of the tetrahedron at a higher negative charge, which destabilizes it, and a greater compliance of the Si-O(H) bonds which elongate more than the stiffer Si-O- bonds. Similarly, the mean length of the oxygen-oxygen edges of a tetrahedron is naturally larger at greater 4-n values while the lengths of the O'.--O-, O'-..O(H) and O(H).--O(H) type edges (taken separately) are decreased at each step of the negative charge enhancement. It originates from the decrease of the corresponding OSiO angles to the constant average angle in a tetrahedron and the changing of the number of various type edges. The number of crystallographically studied silicates with partially protonated orthosilicate ions is insufficient to empirically deduce the statistically significant trends in the bond length variation depending on the charge of the complex anion since the influence of

68

LAZAREV

differences in the type of lattice (Madelung field) and the nature and/or coordination of cations should be taken into consideration. Nevertheless, the relations rSiO(H) > rSiO" and ZO-SiO" > ZO'SiO(H) > Z(H)OSiO(H) do not contradict available experimental data. Moreover, the latter interrelation between the OSiO angles can be extended to the angles in condensed silicate ions by treating the Si-O(H) and Si-O(Si) bonds as belonging to the same type and denoting the corresponding oxygen atom as O br. The more general interrelation thus claims that larger OSiO angles are formed by shorter Si-O bonds and agrees with the regularities empirically deduced from the crystallographic data (see [63] and references therein). Other geometrical parameters of Si-O-H groups are changing systematically with an increase of the 4-n value as well: the SiOH angles diminish and the O-H bonds lengthen. The latter effect manifests itself most clearly at higher 4-n and can be probably treated as a consequence of the total rising instability of the complex ion. The equilibrium geometry variation along the series can be rationalized in terms of computed net charges (Table 1.11). The electronic charge of H in a system with a higher degree of protonation is, at any given step of deprotonation (isoelectronic substitution (OH)---~O-), only partially accepted by the arising O" type atom and its bond. Some portion of that charge is probably distributed over the whole system, thus enhancing its instability. Therefore, the increase in negative charge at O- on each step is accompanied by the increase of the negative charges at the remaining O(H) atoms while the positive charges at the Si and H decrease. The difference in the net charges of the O- and O(H) atoms, which increase with an increase of the 4-n value, may explain the above interrelations between different OSiO angles in the tetrahedron. Moreover, it may affect the interrelation between the bond lengths as well (acting through the repulsion in the edges). Similarly, the smaller SiOH angles at a larger 4-n can be interrelated with the reduced net charges of Si and H. Let us compare the increments of net charge moduli at various steps of 4-n enhancement:


A(4-n)

0---~1

2--,3

mlzlO-

+0.262

+0.121

Si

-0.220

-0.103

O(H)

+0.044

+0.037

H

-0.057

-0.047

69

It is seen that the increments of the O" and Si net charges tend to decay at higher 4-n values while much smaller increments of the O(H) and H net charges are more constant. It can be added that the nature of structural distortions in the SiO4 group in the course of transition from the SiO 4- ion to the Si(OH)4 molecule through a series of less symmetrical intermediate systems is possible to describe in terms of the so-called inductive effect. It implies the systematically increasing effective electronegativity (ability to attract the electrons from other atoms and bonds) of silicon in this series. It is seen from the increase of the net charge at Si on each step of the (O-)--,(OH) substitution which increases the number of more polar OH ligands around the silicon. These considerations agree with numerical estimations of the changes in the bond polarity by means of ab initio computations described below. In order to substantiate the results of semi empirical computations, some members of the series have been investigated by an ab initio approach: the middle Si(OH)202- system has been treated using the most flexible basis set III and the data obtained by its complete geometry optimization was compared with ones characterizing the SiO 4- ion and the Si(OH)4 molecule (Table 1.12). The interrelations between various types of OSiO angles in Si(OH)202- remain the same and their absolute values are near those deduced from the semi-empirical computation. It can relate to a similarity in the relationship between the oxygen net charges, although the difference between the O(H) and O- charges is significantly smaller in the ab initio computation and their absolute values differ considerably. The trends in the charge redistribution along the series are qualitatively similar in both approaches. It should be noted, however, that in both approaches, the sp-type atomic func-

70

LAZAREV

TABLE 1.12 Electronic structure and equilibrium geometry of some HnSiO~ra'-n) systems as deduced from ab initio computation with DZHD type basis set III.

Parameters of the system -Etota1, eV

Si(OH)4

Si(OH)202-

SIO4-

$4

C2

Td

16076.602

reSiO(H), A

16039.746

15981.707

1.642

1.781

reSiO"

-

1.618

1.738

SiO

1.642

1.700

1.738

reOH

0.951

0.959

ZO(H)SiO(H), deg.

109.5*

99.0

ZO(H)SiO-

-

106.9"

ZOSiO

-

126.8

ZSiOH

135.4

114.1

84.2

90.3

ZSiOHIOSiO Localization LMO SiO:

LMO LP:

do, A

do

/LPIOILP, deg. ZSiOILP

109.5

.

O(H)

O

0.414

0.404

0.460

0.442

0.755

0.769

0 804

0.812

0.309 t

0.319 t

0.316 t

0.325

0.749 t

0.735 t

0.813 t

0.802

110

114

117t

115

98, 89

106t

99 t

104

Overlap

Si-O

0.420

0.172

0.808

0.526

population

O...O

-0.102 t

-0.0865

-0.094~;

-0.105

O'H

0.632

0.547

Si

2.335

O

-0.970

-0.992

H

0.386

0.285

Net charge

1.967

*2x112.7, 4x107.9 in Si(OH)4;2x10808, 2x105.0 in Si(OH)202 ~'Averagedover nonequivalent sets. ~:O(H)...O(H)and O~ .O" edges respectively.

-1.277

2-

9

1.987 -1.497


71

tions were adopted and the addition of d-functions may lead to some changes in the numerical results. The results of the two approaches in the equilibrium bond length calculation can be compared using their graphical representation in Fig. 1.6. It is seen that the MNDO computation systematically overestimates the equilibrium Si-O bond length, the overestimation being more significant the shorter that bond is in the ab initio computation. As a result, the difference between reSiO(H) and reSiO- in Si(OH)202- is up to 30% larger than in semiempirical computation and the linear ~ SiO/4-n dependence becomes significantly sharper. The above trends in the reOH and the ZSiOH dependence upon 4-n are supported by ab initio computation (the absence of d-fimctions at O probably affects the calculated SiOH

angles in both cases). The data of Table 1.12 can be applied to a more detailed discussion of the electronic structure variation in the HnSiO(4-n)- series. A strong difference in the overlap population of the Si-O" and Si-O(H) bonds is substantiated in accordance with the difference in their equilibrium lengths. In all systems treated by the ab initio approach, the Si-O(H) bonds are more polar than Si-O" ones and can be estimated quantitatively by means of the results of a localization procedure: System

SiO 4-

Bond

Si-O-

Si-O

Si-O(H)

Si-O(H)

d0LMO SiO/re

25.3

28.4

22.5

25.2

Si(OH)20 22-

Si(OH) 4

These data show that the bonds of both types are more polar in the case of a larger bond length (cf. reSiO in Table 1.12), the CEDD in LMO of the Si-O(H) bonds being relatively nearer to the oxygen atom. The LMO LP seem to be more space extended in the case of the O(H) atom than in O'. The CEDD positions of the Si-O and O H bonds in all investigated systems are shifted from the bond axes inside the SiOH angle as it was noted for simpler systems above.

72

LAZAREV

The dipole moments of the LMO SiO determined using an approach that has been explained earlier are larger for the Si-O(H) bonds than for the Si-O- bonds. For both types of bonds, their dipole moments reduce with the shortening of the bond. The dipole moments of the LP are much more stable in their values. A relative constancy of non-bonding overlap in the edges of the tetrahedron deserves special comment since it is obtained at various net charges on the oxygen atoms, the corresponding bond lengths, and the OSiO angles. It may implicitly represent the conditions of equilibrium of the tetrahedron treated as a balance between the attraction along the bonds and the repulsion along the edges. The AO population in systems considered here gives some idea of the distribution of excessive negative charge accumulated in a system (Table 1.13). The most significant AO contributions into the LMO of bonds in those systems are presented in Table 1.14. It can be concluded that the charge flow from the SiO 4 tetrahedron in the SiO 4- ion arising to outer ligands at the oxygen originally involves 2p states of the latter while on the final step of the transition to the neutral Si(OH)4 molecule, this process involves 3p (and partially 3s) states of silicon as well. The 2s-AO oxygen contribution into the Si-O(H) bonds is larger than that into the O" bonds which agrees with the smaller polarity of the latter system. The 3s,3p-AO silicon contribution is, on the contrary, larger for the Si-O" bonds. There exist some differences in the composition of the LMO LP: a larger 2p-AO contribution is particular for the O" bonds. Several sufficiently precise X-ray or neutron diffraction structure determinations for silicates containing partially protonated ortho-silicate ions are presently known. Most of them relate to compounds in the system Na20-SiO2-H20 and include two series of hydrates with formulas Na2H2SiO4.XH20 (x = 4, 5, 7, 8) and Na3HSiOn.XH20 (x = 2, 5). The earlier data relating to the former series have been collected by Dent Glasser and Jamieson [64] whereas the subsequent investigations are described by Schmid et al. [65,66]. The crystal structures in the latter series were determined [67,68] and these data can be complemented by the structure of related Ca-hydrosilicate hydrate, Ca3(HSiO4)2.2H20 [69].


73

TABLE 1.13 AO populations in some HnSiO(4-n)- systems. Atom and AO Si

O

SiO44 -

Si(OH)20 22-

Si(OH)4

3s

0.650

0.739

0.741

3p

1.151

1.432

1.410

2s

1.819

1.853(O(H))

1.909(O-)

1.904

2p

5.155

5.143

5.372

5.597

TABLE 1 914 AO contributions into LMO of bonds in some of HnSiOh_~_n)_ra., systems. Atom and AO

Si(OH)4 O-H

Si

O

H

Si-O(H)

Si(OH)20 2O-H

Si-O(H)

SiO 4Si-O"

Si-O-

3s'

0.23

0.17

0.31

0.23

3p'

0.14

0.12

0.18

0.16

2s'

0.26

0.22

0.23

0.23

0.18

0.20

2p'

0.50

0.56

0.50

0.56

0.54

0.52

2s"

0.16

0.30

0.10

0.35

0.22

0.27

2p"

0.16

0.16

0.34

0.33

0.38

ls'

0

0

Is"

0.38

0.41

Although the structures of the hydrosilicate ions in these crystals are subject to effects which are not considered in their theoretical treatment (hydrogen bonding etc.), the experimental Si-O bond lengths obey the above dependence of the mean bond length in a tetrahedron.

The relation rSiO < rSiO(H) theoretically deduced is fulfilled as well.

Moreover, at least in the case of HSiO34- and H2SiO 42-, each type of bond is fulfilled and their theoretical sequence of lengths" rmeanSiO- in the former is larger than in the latter, and

74

LAZAREV

the same relates to the lengths of the Si-O(H) bonds. It should be noted here that one of the longest experimentally determined Si-O bond lengths has been found (1.703A) for the SiO(H) bond in Na3nSiO4.2n20 [68]. The results of the computations predict that the difference between rmeanSiO(H) and rmeanSiO- in the same ion should reduce with the lowering of its negative charge (4-n value). The appropriate averaging of the experimental bond lengths does not support this prediction. A lack of corresponding experimental data for the crystals with H3SiO 4- makes it difficult to decide if this discrepancy is consistent. As it relates to absolute magnitudes of the differences between the Si-O" and Si-O(H) bond lengths, their overestimation in theoretical computation is evident: the differences of averaged experimental bond lengths never exceed 0.1/1,. It is interesting to note that the ab initio computation overestimates it even more than a semi-empirical one.

It has already been mentioned that the experimental data confirm the interrelations between various types of OSiO angles in partially protonated silicate ions deduced by quantum mechanical computation.

The average OSiO- and O-SiO(H) angles in the

HSiO 3- ion constitute 112 and 106.5~ respectively, while the average O-SiO', O-SiO(H) and O(H)SiO(H) angles in H2SiO 2- ions are, according experimental data, 116.5, 108.5, and 105~ respectively. An extension of the O'SiO- angle over 125~ as predicted by the ab initio computation is thus not confirmed experimentally which may relate to the charge

redistribution between the oxygen atoms of a tetrahedron in a crystal. The theoretical Si-O bond lengths in three members of the HnSiO (44-n)- series that are treated with a more extended basis set than the one adopted in the above force constant calculation cover the interval over 0.15A. It therefore seems reasonable to calculate the corresponding bond stretching force constants in order to return to the discussion of the force constant/bond length correlation. The same experimental data as in Fig. 1.4 are employed when plotting the empirical correlation graph in Fig. 1.7 with an extended horizontal axis. It was attempted to smooth the experimental curve and to bring it closer to linearity as the anticipated spread of data permitted.


75

fsio' mdyn/A 7

x +

6, 5-l-l-+

43-

2

o

1.~iO

'

1.')0

'

1.80

reSiO, A Fig. 1.7 The Si-O force constant/bond length correlation deduced from the ab initio calculation in the (HnSiO4) (4-n)" series in comparison with the empirical correlation.

The empirical correlation graph of Fig. 1.7 is fairly well reproduced by the graph outlined by the means of four theoretical points corresponding to four different Si-O bonds in the systems investigated with the basis set III. This theoretical graph is still shifted from the empirical one, but in a lesser extent than in the case of calculations with a narrower basis set. It is reasonable to expect that appropriately scaled ab initio force constants of silicate ions will help to resolve the ambiguity of their empirical estimation from the vibrational spectra of silicates. To conclude this chapter, lets summarize some of the statements deduced from the quantum mechanical computation of molecular systems which may relate to the crystal chemistry and lattice dynamics of silicates. Some of them confirm and give the theoretical explanation of empirically deduced regularities. Others should be treated as the theoretical predictions which are still waiting for experimental validation. 1.

The stability of the tetrahedral coordination around the silicon atom can be treated in model description as a balance of attraction along the bonds, and a re-

76

LAZAREV

pulsion along the edges of the tetrahedron. The interactions between the lone pairs play the determinative role in the latter. 2.

A free SiO 4- ion is destabilized by the excessive electronic charge and its stability in condensed systems including crystals depends upon the number and efficiency of"channels" for discharge.

3.

The size of the SiO4 tetrahedron or its average bond length in a condensed system is a function of its negative charge and decreases with a decrease in the magnitude of that charge.

4.

In an unsymmetrically surrounded SiO 4 tetrahedron, the Si-O- bonds (Si-O 9.-M +) are shorter than the Si-O(Si) or Si-O(H) bonds and less polar, but the negative charges at the oxygen atoms are larger in the former.

5.

The equilibrium values of the OSiO angles and their force constants in a tetrahedron are larger as the charges at the corresponding oxygen atoms increase; the larger OSiO angles are formed by the shorter Si-O bonds.

6.

In partially protonated HnSiO(4-n)- ions, the average Si-O bond length decreases with a decrease of n, but at every value of n, the relations reSiO" < reSiO(H ) and ZO'SiO" > ZO-SiO(H) > Z(H)OSiO(H) are valid.

7.

The Si-O stretching force constant correlates approximately linearly with the equilibrium bond length, but at larger bond lengths the dependence becomes less steep.

8.

The deformational polarization of the silicon-oxygen tetrahedron does not obey the diagonal approximation of the valence optical scheme: as one bond is stretched, the total dipole moment variation is determined only partly by the change of dipole moment of that bond, and an equal contribution arises from changes of moments of the bonds at rest.

9.

In condensed systems, the charge density response to the internal deformation of the SiO 4 tetrahedron related to the change of its volume is not entirely localized inside the tetrahedron and extends at least to the nearest cations.

10.

The most polar character of the Si-O bonds in Si-O-Si bridges and interactions between the lone pairs and the bonding pairs around the oxygen atom cause the


unsymmetrical arrangement of valent density relative to the bond axes and its concentration inside the SiOSi angle. 11.

A transversal (angle bending) elasticity intrinsic to the Si-O-Si bridge as itself, is extremely low and may be determined in a more complicated system by interactions between distant atoms.

12.

The additional coordination of the oxygen atom in the Si-O-Si bridge to some other cations induces considerable elongation of the bonds in the bridge and may give rise to the shortening of the Si-O bonds adjacent to the bridge that are disposed in the trans orientation to the lone pairs of the bridging oxygen atom.

77

78

LAZAREV REFERENCES

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2.

P. Pulay, Theor. Chim. Acta, 50, 299 (1979). P. Pulay, Force Concept in Chemistry, (B. M. Deb, ed.), N. Y., Van Nostrand (1981) p. 449.

4.

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P. Pulay, J. Chem. Phys., 78, 5043 (1983). P. Pulay, G. Fogarasi, F. Pang and J. E. Boggs, J. Am. Chem. Soe., 101, 2550 (1979).

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I. S. Ignatyev and T. F. Tenisheva, Vibrational Spectra and Electronic Structure of Molecules with Carbon-Oxygen and Silicon-Oxygen Bonds (Russ.), Leningrad, Nauka Publ. (1991).

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13.

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G. Fogarasi and P. Pulay, Vibrational Spectra and Structure, Vol. 14, (J. R. Durig, ed.), Amsterdam, Elsevier (1985) p. 125.

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79

22.

M. M. Francl, W. J. Pietro, W. J. Hehre, J. S. Binkley, M. S. Gordon, D. J. Defrees and J. A. Pople, J. Chem. Phys., 77, 3654 (1982).

23.

F. Liebau, Structural Chemistry of Silicates. Berlin, Heidelberg, Springer (1985).

24.

C.E. Blom and C. Altona, Mol. Phys., 34, 557 (1977).

25.

I.S. Ignatyev, J. Mol. Struet., 172, 139 (1988).

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27.

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28.

A. Almenningen, B. Bastiansen, W. Ewing, K. Hedberg and M. Traetteberg, Aeta Chem. Seand., 17, 2455 (1963).

29.

C. Glidewell, D. W. H. Rankin, A. G. Robiette, G. M. Sheldrick, B. Beagley and J. M. Freeman, & Mol. Struet., 5, 417 (1970).

30.

I.S. Ignatyev, J. Mol. Struct., 197, 251 (1989).

31.

I.S. Ignatyev, J. Mol. Struet., 245, 139 (1991).

32.

M. D. Newton, in "Structm'e and Bonding in Crystals," Vol. 1, (M. O'Keeffe and A. Navrotsky, eds.), N. Y., Acad. Press (1981) p. 175.

33.

G.V. Gibbs, E. P. Meagher, M. D. Newton and D. K. Swanson, Ibid., p. 195.

34.

M.D. Newton and G. V. Gibbs, Phys. Chem. Miner., 6, 221 (1980).

35.

G.V. Gibbs, Am. Mineral., 67, 421 (1982).

36.

A. P. Mirgorodsky and M. B. Smimov in "Dynamical Properties of Molecules and Condensed Systems (Russ.)," (A. N. Lazarev, ed.), Leningrad, Nauka Publ. (1988) p. 63.

37.

J. P. Malrieu in "Localization and Delocalization in Quantum Chemistry," (R. Dandel, S. Diner, J. P. Malrieu and O. Chalvet, eds.), Boston, Reidel (1975) p. 263.

38.

A. N. Lazarev, I. S. Ignatyev and T. F. Tenisheva, "Vibrations of Simple Molecules with Si-O Bonds (Russ.)," Leningrad, Nauka Publ. (1980).

39.

L. A. Curtiss, H. Brand, J. B. Nicholas and L. E. Iton, Chem. Phys. Lett., 184, 215 (1991).

40.

R. K. Iler, "Chemistry of Silica: Solubility, Polimerization, Colloid and Surface Properties and Biochemistry of Silica," N.Y.-Chichester-Brisbane-Toronto, WileyIntersci. (1979).

41.

A. N. Lazarev in "Vibrational Spectra and Structure of Silicates," N. Y., Consultants Bureau (1972).

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42.

I. S. Ignatyev, B. F. Shchegolev and A. N. Lazarev, Dokl. Akad. Nauk SSSR (Russ.), 285, 1140 (1985).

43.

B. F. Shchegolev, M. B. Smimov and I. S. Ignatyev in "Dynamical Properties of Molecules and Condensed Systems (Russ.)," (A. N. Lazarev, ed.), Leningrad, Nauka Publ. (1988) p. 28.

44.

E. Clementi and C. Roetti, Atomic Data and Nuclear Data Tables, 14, 186 (1974).

45.

T. Clark, J. Chandrasekhar and G. W. Sputznagel, J. Comput. Chem., 4, 294 (1983).

46.

J. Sauer, Chem. Phys. Lett., 97, 275 (1983).

47.

P. Tarte, "Etude exp6rimentale et interpr6tation du spectre infra-rouge des silicates et des germanates. Application A des probl~mes structureaux relatifs fi l'tetat solides. M6m. Acad. Roy. Belgique, 35, N4a, 4b (1965).

48.

H. Siebert, Z. Anorg. Allg. Chem., 273, 170 (1953).

49.

H. Siebert, Z. Anorg. #Jig. Chem., 275, 225 (1954).

50.

A. N. Lazarev, A. P. Mirgorodsky and I. S. Ignatyev in "Vibrational Spectra of Complex Oxides (Russ.)," Leningrad, Nauka Publ. (1975).

51.

A. Muller and B. Krebs, J. Mol. Spectrosc., 24, 180 (1967).

52.

M. Handke, J. Moi. Struct., 114, 187 (1984).

53.

Vibrations of Oxide Lattices (Russ.), (A. N. Lazarev and M. O. Bulanin, ed.), Leningrad, Nauka Publ. (1980).

54.

I.M. Mills, Spectrochim. Acta, 19, 1585 (1963).

55.

J.S. Binkley, J. A. Pople and W. J. Hehre, J. Am. Chem. Soc., 102, 939 (1980).

56.

A. N. Lazarev, A. P. Mirgorodsky and M. B. Smimov, "Vibrational Spectra and Dynamics of Ionic-Covalent Crystals (Russ.), Leningrad, Nauka Publ, (1985).

57.

A. N. Lazarev, A. P. Mirgorodsky and M. B. Smimov, Solid State Commun., 58, 371 (1986).

58.

L. Pauling, "Nature of the Chemical Bond," 3rd edition, N. Y., Comell University Press, Ithaca, (1960).

59.

N. V. Belov, "Crystal Chemistry of Silicates with Large-Sized Cations (Russ.)," Moscow, Ac. Sci. Publ, (1961).

60.

W.H. Baur, Acta Crystallogr. B., 34, 1751 (1978).

61.

S. P. Dolin, B. F. Shchegolev and A. N. Lazarev, Dokl. Akad. Nauk SSSR, 298, 131 (1988).

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M.J.S. Dewar and W. Thiel, J. Am. Chem. Sot., 99, 4899 (1977).


81

63.

G.E. Brown and G. V. Gibbs, Am. Mineral., 55, 1587 (1970).

64.

L.S. Dent Glasser and P. B. Jamieson, Aeta Crystallogr. B., 32, 705, (1976).

65.

R.L. Schmid, J. Felsche and Mc Intyre, Acta Crystallogr. C, 40, 733, (1984).

66.

R.L. Schmid, J. Felsche and Mc Intyre, Acta Crystallogr. C, 41,638, (1985).

67.

Ju. I. Smolin, Ju. F. Shepelev and I. K. Butikova, Kristallografiya (Russ.), 18, 281

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R.L. Schmid, G. Huttner and J. Felsche, Acta Crystallogr. B., 35, 3024 (1979).

69.

K.M.A. Malik and J. W. Jeffery, Acta Crystallogr. B., 32, 475 (1976).


CHAPTER 2 INTRODUCTION

TO THE DYNAMICAL THEORY OF CRYSTALS

AND APPLICATION

OF APPROACHES

THEORY OF MOLECULAR

ORIGINATING

FROM THE

VIBRATIONS

The Elements of Dynamical Theory of Crystal Lattice ................................................ 84 A. Macroscopic Treatment of the Elastic and Dielectric Properties of Perfect Crystals ........................................................................................................................ 84 B. Microscopic Treatment: Potential Energy Function and Charge Density Distribution Function and Their Model Representation ............................................. 90 C. A Comparison of Various Descriptions of the Electric Response Function ............ 103 HI

A Compatibility of Molecular Force Constants with the Explicit Treatment of Coulomb Interaction in a Lattice .................................................................................. 111 A. Potential Energy Decomposition and Interrelation B~tween the Potential Energy Function and the Electric Response Function .............................................. 111 B.

Conditions of Compatibility of Molecular Force Constants with Explicit

C.

Separation of Coulomb Contribution to the Force Field .......................................... 116 Applications to Silicon Dioxide and Silicon Carbide ............................................... 121

Eft[. Internal Coordinates in the Description of Dynamic Properties and Lattice Stability ............................................................................................................................ A. Vibrational Problem in Internal Coordinates and Indeterminacy of the Inverse Problem ...................................................................................................................... B. A Generalization of the Inverse Vibrational Problem and a Notion of the Shape of Uniform Strain of a Lattice ........................................................................ C. The Microscopic Structure of Hydrostatic Compression and its Employment in the Generalized Formulation of the Inverse Vibrational Problem ....................... D. A Curvilinear Nature of the Internal Coordinates and its Certain Consequences ............................................................................................................. E. A Relation of Internal Tension to Description of the Lattice Instability ..................

128

IV. Several Computational Problems .................................................................................. A. Geometry Optimization and Potential Function Refinement ................................... B. Crystal Mechanics Program ...................................................................................... C. The Operation of the Program ...................................................................................

171

129

141 157 161 165

171 174 175

References ........................................................................................................................... 183

83

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

84

THE ELEMENTS OF DYNAMICAL THEORY OF CRYSTAL LATTICE Since the classic Born and Huang monograph [1 ], a number of excellent books devoted to the title problem appeared (see, e.g., [2-5]). Among them, a series [6-8] providing the most comprehensive treatment of a present state of affairs in this area of science should be specially advised. The main statements will be briefly restated in order to clarify the approach to the lattice dynamics of complex ionic-covalent crystals developed in subsequent chapters. This approach originates from ideas initially proposed in a Russian book [9] whose content is only available to the Western audience from review papers [ 10,11 ].

A. Macroscopic Treatment of the Elastic and Dielectric Properties of Perfect Crystals Any atom of a perfect crystal is specified by the positional vector R(Jj) where vector J determines the primitive cell and j enumerates the atoms in a cell. A crystal where the normal mode with the frequency co and wave vector q may be characterized by two variables, the electric field: Eexp {i[q. R(Jj)-o t]} and the atomic displacement:

u(j[q)exp {i[q- R(Jj)-o~t]}. The three dimensional vectors E and

u(jlq) determine the strength of the macroscopic elec-

trostatic field in a crystal and the displacements in jth Bravais lattice, respectively. The electrostatic, adiabatic and harmonic approximations are usually adopted in a theory of vibrations of relatively complex crystals if the numerical treatment is intended. In this approach, a basic equation determines the vibrational energy per unit cell as a quadratic function of E and u(jlq) [1,6,12]:

LAZAREV

85

1 V(q)= ~ Dal 3 (jklq)u= (jlq)ul3(klq)- Zal 3 (jlq)Ec~uB(j[q)1 - ~Zal3 (q)EaEl3

(2.1)

Hereafter, the Greek indices denote the Cartesian components while the Roman characters specify the atoms of a primitive cell. A summation over repeating indices lacking on the left side of the expression is usually implied. The coefficients on the right-hand side of eq. (2.1) are as follow. Dal 3 (jklq) are the elements of dynamic matrix D(q) with 3n x 3n dimension where n is the number of atoms in a primitive cell.* These coefficients describe the forces between atoms within the limited range of action arising at the lattice strain. They include the forces originating from the Lorentz field, but not by the macroscopic electric field. Zal 3 (j}q) are the elements of a 3 x 3n matrix of transverse effective charges Z(q) determining the polarization of a crystal at strains which do not lead to the arising of the macroscopic field E. And fmally, Za13 (q) are the elements of a 3 x 3 matrix of the electronic polarizability of a crystal. Correspondingly, the first term in eq. (2.1) specifies the energy of elastic forces acting between atoms at the distances not exceeding the radius of a Lorentz sphere. The second and third terms describe the energy of interaction of polarized medium with the macroscopic field E. The latter can formally be treated as an external term although in eq. (2.1) it represents the macroscopic field caused by the atomic displacements in a lattice. The second term represents the ionic contribution in that energy, and the third term corresponds to the electronic contribution. Assuming M(j) as the atomic mass in the jth sublattice, the equations of motion and the polarization of a crystal are obtained, respectively, by the differentiation of eq. (2.1) with respect to ua (Jlq), co2(q) M(j)ua (jlq) = Dafl (jklq)ufl (klq)- Zf~a (j[q)El3

(2.2)

*In a more conventional definition of D(q) matrix, the use of mass-weighted coordinates (introduction of M S1 M-jl factors) is implied.

86


and with respect to E a, Pa = ~[Zal3 (j[q)u[3 (jlq) + Zal3 (q)Ef~]

(2.3)

where f] is the volume of a primitive cell. According to Makswell theory, the E, P and q values are interrelated as: Ea=-

4nqaql3Pl3

[q[2 .

(2.4)

If the transversal (P_l_q) wave propagates along the a direction, it follows from eq. (2.4) that: Et=0

(2.5)

and the eqs. (2.2) and (2.3) are reduced to: o32t(q)M(j)u a (jlq) = D at 3 (jklq)ul3 (k]q)

(2.6)

Pat = ~ Zaf~ (jlq)uf~(j[q),

(2.7)

and

respectively. As is seen from eq. (2.6), only the forces of limited radius of action determine the frequencies of transversal modes. Eq. (2.7) explains the physical meaning of the macroscopic charge Zal 3 (jlq) as a change of the dipole moment of a cell along the a axis at the unit displacement of the jth sublattice along ~. In the case of longitudinal waves (Pllq), the P and E values are interrelated as: E l = -4nP 1 .

(2.8)

Making use of this expression, one obtains for polarization:

1 Zal3 (Jlq)uB (jlq)

p1 = ~ ~:el(q)

and for equations of motion

(2.9)

LAZAREV

87

co2 (q)M(j)u a(jlq)=

D~f~(jklq) +

4~ e-------i~ Z~ ~ (jlq)Za f~(klq)] u 13(klq)

nlzaa (q)

(2.10)

where I ~el0 t = l + 4 n ~ a ( q ) is an element of the tensor of the electronic dielectric permeability of a crystal. The indices t and l in expressions (2.5)-(2.10) refer corresponding values to the transversal and longitudinal vibrations, respectively. The coefficients before the atomic displacements u13(Jlq) in the right-hand side of eq. (2.9) constitute a matrix of longitudinal effective charges Z~I3(q). Its elements are 1 ( J l q=Za[3 ) (Jlq)/e~=(q) Z~13

(2.11)

An attenuation of the polarization vector on the transition from the transversal to longitudinal wave (which can be formally described by reduced elements of the

Z(q) matrix) origi-

nates from the screening of surface charges arising at the crystal vibrations by its electronic subsystem. The latter is disturbed by the macroscopic field E. Vibrations with wavelengths considerably longer than the lattice parameters, but much shorter than the linear size of a crystal are of most practical interest. These modes interact with electromagnetic radiation from the infrared to the ultra-violet interval. In order to analyze the properties of these vibrations, let us return to equations (2.2)-(2.4) supposing q--}0 therein. This condition will correspond to the independent expressions of q for macroscopic quantities. The equilibrium state of a crystal in a static external electric field E can be described by the equation of motion (2.2) where co is set to zero: Dal3 (jk)ul3 (k)= ZI3a (J)EI3 .

(2.12)

By making use ofeq. (2.12) for the exclusion oful3(J) values from eq. (2.3), and taking into account that ea~ = E + 4riP, it is possible to deduce an expression for the tensor of the static dielectric permeability of a crystal: 4n eal3 =8al3 +--~-)r

4n +--~- Za8 ( j ) D ~ ( j k ) Z ~ ( k ) .

(2.13)

88


The problem of the conversion of the D(q) matrix, which is singular at ~ 0 ,

is treated in

refs. [1,13]. Two contributions are usually discerned, the electronic: e el = ~5ctl3+ 4~:Xal3f)-1 a13 and the ionic: 47~ eion af3 =ear3_eelaf~=---~Za8 (j)D[3~ 1( j k ) Z N

(k).

(2.14)

The latter contribution is determined by the charge distribution response in a crystal on the nuclei's displacements into the new equilibrium positions under the influence of a static electric field. The longwave solutions of eqs. (2.6) and (2.10) are interrelated with the dielectric constants of a crystal by the Liddane-Sax-Teller relation [12]. It is written for orthorombic and higher symmetry crystals as:

H (r176 E~ i = 1,2,...m k,r

a

(2.15)

eaa

where the index a implies that the frequencies correspond to the modes polarized along that axis and m is the total number of these vibrations. A following expression can be deduced for _ion from eq. (2.15) by taking into account eq. (2.14):

el aa =ion eaa

ct .

(2.16)

This quantity can be expressed in the form [1 ]:

eion a~

i = ~ Aeaa , i

where:

(2.17)

LAZAREV

89 4riM-1 (j) ~~~i)5 [Zotl3(j)e~(j)] 2.

9

As~=

(2.18)

The e~ (j) term is a component of the ith eigenvector of the D matrix. The contributions Asia are the oscillator's strengths which determine the IR intensities of vibrational transitions with col frequencies. In the calculation for a particular crystal, the e~ot (q) quantity can be treated as a constant whose value is deduced from optical experiments. Then, a determination of the phonon spectrum of the crystal (which is a central problem of dynamical theory of a crystal lattice) is reduced to the design of the D(q) and Z(q) matrices. Besides the phonon spectrum and the dielectric properties of a crystal, knowledge of these matrices ensures the calculation of the macroscopic elastic and piezoelectric constants [1,14,15]. Both matrices are expanded in a series in terms ofa wavevector: Da[3 (jklq) = Doq3(jk) + iD~13(Jk)q 7 + 1 D~(jk)qTq 8 +...,

(2.19)

Zal3(jlq)= Za~ (j)+ iZ~l3(j)qv + . . . .

(2.20)

and

The elastic and piezoelectric constants are defined as 1 02v C{xl3,y8=-~- OU~13OUy8

(2.21)

and eo~,~

=

OP~ 8U~

(2.22)

where Uoq3 is the amplitude of the uniform strain of a lattice which is specified by the atomic displacements:

(4).

(2.23)

90

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS Using the above expansions, the following expressions for the elastic and piezoelec-

tric constants can be deduced:

c~,~ =[~,r~] + [r~,~5]- [~v,~s] + (~v,~5),

(2.24)

ea,~ = [ot,13y],

(2.25)

and

using the expressions:

[~,~] = ~ D ~ ,~5

(2.26)

(c~y,138)=1 D~a (j)D~ 1 (jk)D 8vl3 (k),

(2.27)

[ct,13T]= 1 {Z~I3+ Dr (j)D~I (jk)Zav(k)} ~13

(2.28)

~45 ~ ~8 D<xl3- . Dctl3(jk),

(2.29)

D ~ (j)= E D~'a~(jk) ,

(2.30)

and

J Z ~ =Z. Z ~ ( j ) . J

(2.31)

B. Microscopic Treatment: Potential Energy Function and Charge Density Distribution Function and Their Model Representation To begin with a microscopic model description of the dynamical properties of a latrice which would be adopted in numerical determination of the elements of D(q) and Z(q) matrices, we investigate their interrelation with the microscopic characteristics of a crystal, the V(R) and p(rlR) functions. These represent a dependence of the potential energy of a lattice and of the total charge density in a point r on the instantaneous nuclei's positions

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91

specified by the R vector, respectively. The absence or, more rigorously, the constancy of the macroscopic field E being implied. The series expansions of V(R) and p(rlR) in terms of small nuclei displacements from the equilibrium positions are to the second-order terms:

V(R)=Vo+Vcx(Jj)ua(Jj)+lvaf3(JjlKk)ua(Jj)uf3(Kk)

(2.32)

p(r[ R)= Po(r) + Pa (r[ J j)uct (J j)+ 1 pa[3(r [j j[ K k)ua (J j)ul3(K k).

(2.33)

and

V 0 is the energy of formation of a lattice, and has no relation to the treatment of its properties with respect to deformations, Va(Jj) is the force acting upon the (J j) atom taken with the opposite sign, and VaI3(JjlK k) are so-called force constants defined in a Cartesian space of atomic displacements. Also, the charge density distribution is represented by p0(r) which is the charge density in a point r in its equilibrium state, Pa (riJj) is the first-order charge density response function which describes the change of the charge density in a point r at the unit shitt of the (J j) atom along the ct axis, and pctfl(r~/jlK k) is the secondorder charge density response function. The pa(rlJ j) function will be referred to below simply as the charge density response function. The coefficients of expansions (2.32) and (2.33) should satisfy several interrelations which are deduced from the following conditions [1,16]: 1) A mutual compensation of forces acting upon any atom in the equilibrium state of a lattice, Vet (J j) = 0 .

(2.34)

2) The electroneutrality of a crystal, fPa (rlJ j) dr = 0 and

(2.35a)

92

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS Ipo (r)dr=O.

(2.35b)

3) The invariance of a lattice relative to translation, EVaf~ (JjlKk)=O, Jj

(2.36a)

0 O0 (r), E Pa (r [J j)=-~-Jj

(2.36b)

Ep~f~(rlJjlKk)= -~Pl3 0 (rlK k) ,

(2.36c)

JJ

and, relative to rotation,

V~f3(Jj[Kk)R~I(Kk ) - Vml (Jj]Kk)Rf~(Kk) =0,

(2.37a)

Pf3(r [J j)[ Ry (J J)- r~l] - P~l(r [S j)[ Rfs(J J)- rfs] = O,

(2.37b)

p~13(r IS Jl K k)Rv (K k) - Pctv(r IS Jl K k)RI3(K k) =

(2.37c)

8~13P~(rlSj)-8~oB(rlSj) . The V(R) function represents the energy of interparticle interactions in a lattice without taking into consideration the contributions originating from their interaction with the macroscopic electrostatic field E, and thus, this function determines the first term in eq. (2.1). Correspondingly, the elements of the D(q) matrix are expressed through the coefficients of expansion (2.32) as [1 ]: Daf3(jk [q)= ~ Val3(J j[ K k)exp {iq. [R (K k)- R (J j)]}. K

(2.38)

A polarization of a crystal can be described through the p(rlR) function as: Pet = N-lf/-1 IP(r [R)radr

(2.39)

where N is the number of primitive cells in a crystal. Making use of eq. (2.7), the following expression for the elements of the Z(q) matrix is obtained:

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93

Zal3(j[q)=N -1

I~jpf3(rlJj)r~ exp[iq. R(Jj)]dr.

(2.40)

The force constants Voq3(djlKk) and the response functions pcz(rldj) are shown to be the main microscopic features of a crystal which determine its dynamical properties in a harmonic approximation. It should be emphasized that the pa (rldj) function is essential in the determination of a variety of properties of a crystal relative to deformation. 9 The limiting (corresponding q--->0) values of the effective charge tensor which determine the IR intensities of the vibrational modes are found according to eq. (2.40) as the dipole moments of the response function: Z(~I3(j) = Ip[3 (r [J j)ra d r .

(2.41)

It has been shown by Martin [16] that from the definition of eq. (2.28) and eq. (2.37b), a possibility to express the contribution to piezoelectric constant originating from the external strain through the quadrupole moments of the response function are: Z~I3 = - 1(Qr

-Q~,al3 +Q~r

(2.42)

,

where Q a ~ = ~. Irapl3 (r [JJ)r~ d r . J

The interactions between spatially removed atoms are of electrostatic nature and therefore, for sufficiently distant atoms, the following expression of atomic force constant is valid: Val3(JjlKk) - I I

Pa(rlJj)Pf3(r'lKk)drdr' Ir- r'l

.

(2.43)

94


In a conventional approach to the lattice dynamics of ionic crystals, the applicability of the last of the above statements is extended to the shortest interionic distances; the explicit decomposition of the potential function of a lattice being implied in a form: V(R) = V C~176 (R) +V non-Coulomb (R) .

(2.44)

A non-vanishing contribution of the second term is supposed only at more- or less-restricted interionic distances. Since the Coulomb contribution trends to collapse a crystal, the second term representing the so-called short-range forces is often referred to as the repulsive one. Various approaches to the design of V(R) are briefly reviewed below; specifying them by the ideas underlying the adopted model representation of that function and of the electric response function (if the Coulomb contribution is separated explicitly as eq. (2.44) assumes). A wider review of numerous dynamical models may be found in refs. [ 17,18,19]. A very deep and physically consistent treatment of the problem in ref. [20] is recommended in particular. The analysis of dynamical models from a position of rigorous quantum mechanical theory of crystals has been done [21,22]. Before discussing various models which adopt the decomposition of eq. (2.44) of the potential function, one of the simplest and historically earliest atomic force constant approaches should be mentioned. If a treatment is restricted to deformations which do not lead to the arising of the macroscopic electric field, the force, s, acting upon the (J j) atom of a lattice at its deformation is expressed through the displacements of all atoms as a sum: sa(Jj) = -Votf~(Jj IK k)ul3(K k) .

(2.45)

A knowledge of Vaf~(Jj IK k) coefficients, which are treated as the adjustable parameters of the model, is sufficient to compose the equations of motion. These parameters should fit conditions (2.34), (2.36a), and (2.37a) and obey the symmetry requirements, but their magnitudes are not restricted by any other considerations. Their large number is usually restricted by the assumption of their vanishing at sufficiently large interatomic distances. In some computations, however, the interactions up to the fitth coordination sphere are taken into account [23].

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95

The atom-atomic potential approach, which is widely employed in a treatment of the equilibrium structure and lattice dynamics of complicated crystals including minerals, is the simplest way to interrelate various atomic force constants in terms of some analytic representation of the potential function (PF) V(R). This approach is essentially restricted to the central pair interactions. A PF is represented in this approach by the sum of a pair of potentials, each being dependent only on the distance between two atoms: V(R)= 1 ~..~. ~ij (Rij)

(2.46)

An analytic form of the ~ij pair potential is selected with, taking into consideration, the peculiarities of interacting atoms and the nature of bonding. Various type pair potentials are otten combined, more or less, intuitively in an attempt to represent the contributions to the PF of different physical origin. Although this approach originates historically from the investigation of the crystal state of noble gases and was later extended to the alkaline-earth halogenides and some other simple ionic crystals, it is sometimes attempted to adapt it even in a case of significantly covalent bonding [24] because of its obvious advantages in computations for rather complicated lattices. Some of the most important types of pair potentials adopted to represent the nonCoulomb contribution to decomposition (2.44) are listed below. The exchange repulsion of atoms or ions with closed electron shells was introduced in a form of the Born-Mayer potential, ~ij = Aexp(-Rij/Ro), or Bom-Lande potential, ~ij = A / R~ (n=9-12). The potential of the van-der-Waals attraction, 9ij = - C/R6+ D/R 8, which was originally introduced in order to describe the attracting forces in the crystals of noble gases, is supposed by some authors to be important for the description of interaction between anions in some ionic-covalent crystals. Since a covalent bond is assumed to be characterized intrinsically by its equilibrium bond length, R oij , and force constant, t~j, a potential representing the contribution of covalent bonding may be adopted in the form: 9

-

96


In numerous investigations, only the small deviations from the equilibrium geometry are considered and the explicit analytic approximation of the potential function is of secondary interest. All important properties of a system are determined by the first and second derivatives of tp(R) at reference geometry. This is why the Bom-Karman approach which operates with two parameters A= d2tp / dR2 and B = 1 ij dtp/dRij without specifying the dependence of tp(Rij) preferred in some works. The atom-atom potential computations based on the potential function presentation in the form (2.44) usually adopt the point ion approximation in description of the VCoulomb(R) contribution to dynamical properties. The area of applicability of this approximation, its shortcomings, and the possibility to estimate the charges in particular crystals will be discussed later. The rigid ion model (RIM) is the simplest one which adopts the point ion approximation. The ions are represented by the constant point charges, z i, and the contribution of their interaction into the PF of a crystal is calculated as: tpC~176

= ziz j / Rij .

(2.47)

A Coulomb contribution into the lattice energy may then be expressed as vC~176

=

12 ~. zi{~i, where t~i = .E . zjRij is a potential of the ith ion. 1

J~:l

Despite the apparent simplicity of this approach, its application to numerical calculation is labored by a slow convergence of the above lattice sums. For several years, considerable efforts were directed to a development of efficient methods to calculate the Coulomb sums. Ewald's method is presently approved as a most powerful and universal method. It should be noted that the long-range nature of Coulomb interaction, which determines a slow convergence of lattice sums, evokes some interesting physical problems. In particular, a sum determining the {~ipotential converges conditionally, and the result depends on the selection of a cell and sequence of summation. Moreover, in pyroelectric crystals with a non-vanishing dipole moment of a unit cell, this series diverges. A correct method to calculate the Coulomb field in pyroelectrics has proposed [25]. It has

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97

been proved [26] that a conditional convergence of the electrostatic lattice sums originates from the dependence of the internal field in dielectrics and on the shape and polarization state of a surface of a microsample. This is why a problem of selection of slowly (conditionally) convergent contributions in the Coulomb lattice sum is physically related to a separation in the electrostatic field of extemal (macroscopic) and intemal ones, the latter being the microscopic Lorentz field. A corresponding complementation of the standard Ewald procedure has been proposed [26]. The importance of the polarizability contribution to Coulomb interaction in ionic crystals has been realized, and the polarizable ion model (PIM) has been proposed as a direct extension of the RIM approximation. In this model, any atom is additionally specified to its point charge, z i, by a point dipole, Pi, induced by the electric field of a crystal and interrelated with the field strength at that ion, Ei, through the electronic polarizability tensor, ai: Pi = aiEi .

(2.48)

A Coulomb contribution into the interatomic potential is obtained in the PIM by complementing charge-charge interaction (2.47) by the charge-dipole and dipole-dipole terms: q~eoulomb _ zizj - Rij

+PiBijzj +pjBjiz i +piCijpj.

(2.49)

The most important supposition in the PIM is that ionic dipoles, Pi, are implied to follow the ionic positions adiabatically, and each atomic arrangement corresponds to a definite polarization state. A potential function in the PIM is no longer restricted to pair interactions only, and the term originating from the polarization energy of a crystal arises [17]: vP - 1~.

pia~lpi.

(2.50)

1

The p(R) dependence may be found from the adiabatic condition, dance with (2.48) one obtains:

t3v/Op- 0, and in accor-

98

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS Pi = "ai ~ (CijPj + Bijzj) 9 J

(2.51)

An effective PIM potential is thus obtained by the substitution of (2.51) into (2.49) and

(2.50). It can be concluded that an introduction of atomic polarizabilities in the PIM falls outside of the limits of a pair interaction approximation. Equation (2.49) bears a two-body character only formally since Pi and pj depend on the position and polarization of all other ions in a lattice. Another peculiarity of the PIM consists in the possibility of a treatment of the origin of high-frequency dielectric permeability. However, for the crystals with highly polarizable anions, the PIM predicts a so-called "polarization catastrophe" originating from the loss of the lattice stability due to the irresistible enhancement of mutually induced dipoles. An analysis given by Tolpygo [27] proved that polarization is restricted by the deformability of the electron shell with respect to forces of a short-range natta'e. Taking this effect into consideration, it is possible to treat the ionic dipole moment Pi as being dependent both on the field E i and the position of the ith ion relative to several other ions. In a first approximation, the dependence may be represented by adding one more term in eq. (2.48): Pi = aiEi + .~.,. bij (Ri- Rj) . j#l

(2.52)

This approach was utilized in the deformable dipole model (DDM) proposed by Hardy [28]. A main limitation of the DDM originates from the introduction of additional parameters (bij) in the description of deformational polarization with no simple approach to their estimation. A widely used shell model (SM) which is unique in being a model (in the true sense of the word) originates from similar considerations [29]. In this approach, the ionic charge is decomposed into the core charge, X i, and the electronic charge Yi, Xi+Yi = zi. The ionic point dipole moment, Pi, is replaced by a new variable, si, which is specified as a vector of displacement of the electronic density center relative to the core. These are interrelated as:

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99

Pi = Yisi

9

(2.53)

A polarization of an ion is described in this model by means of the elastic force connetting the shell and the core. Denoting it by the elasticity coefficient, k i, it is possible to write down a simple expression for the polarizability of a free ion in the SM: ai = ( y 2 / k i ) i

'

(2.54)

where I is a unit tensor. A description of the PF in the SM needs some comments. There exist no explicit V(R) expansion in this model. The energy of the system cores + shells is determined in the R and s variables. The expression for an energy includes the term representing the Coulomb interaction of ionic charges and dipoles which is similar to the corresponding contribution in the PIM (2.49). The excitation energy of shells at ionic polarization is expressed in the SM as vP = 1~. k is2 . The non-Coulomb interactions between cores to cores, shells 1

to shells, or cores to shells are introduced in the SM instead of a single ~non-Coulomb potential. The shells follow the core displacements adiabatically and the condition of minimal total energy w(R,s) = vp + 1 ~ij ~ / .i.~tPijCoulomb+gijshell-shell +gijc~176

determines the

+tpijcore-shell)

(2.55)

R(s) dependence. It resembles a searching of a polarization as a function of

deformation in the PIM. Two last terms in eq. (2.55) relate to the deformational polarization.

These are,

however, often omitted in practice of calculation with the so-called simple SM. It means that in the simple SM model, all non-Coulomb interactions are reduced to ones connecting the electron shells. Two important peculiarities of the SM should be noted. First, a PF which is determined in the SM through eq. (2.55) as V(R) =

w(R,s(R)) cannot be repre-

sented by the sum of the Coulomb and non-Coulomb contributions. In the SM, a deformational polarization is determined by the non-Coulomb potentials, and this polarization contributes to Coulomb interaction. Second, a range of action of non-Coulomb forces in the

100


SM is determined not by the speed of the tp~on-Coulomb decrease with distance, but by the interrelation of the core-shell and shell-shell elasticities. Even in a case when the tpshell-shell potential interconnects only nearest neighbor atoms, an effective non-Coulomb interaction 1---(kc~ ) exponentially decreases with distance (as e-~R where ~ = Ro\ fkshell_shel 1 [30]. An efficient and versatile SM was numerously applied to the calculation of phonon spectra of various crystals and the determination of the shapes of their vibrational branches. Unfortunately, its parameters are hardly interpreted in terms of chemical bonding. A further development of the SM was directed mostly to the inclusion of many-body interaction. It was aimed for models with more than one shell at the same atom [31 ] or with a deformable shell [32] including the so-called "breathing shell" [33]. This is most important in the investigation of crystals with anions of low symmetry that are unstable in a free state. Two important contributions published last year should be mentioned. Two oxygen shells were introduced [31 ] for oxides containing A-O-B bridges. The constant of interaction between the two shells was determined through the quadrupole polarizability and the PF structure represented as a directional character of bonding. Cohen et al. [34] proposed a potential induced breathing shell model (PIB SM) which assumed a dependence of the size and shape of the electron shell of the 0 2- ion on the crystal field. The properties of a shell were thus made dependent on the structure of a whole crystal. Generally speaking, the internal structure of the SM does not seem very suitable in the case of significant covalent bonding. A critical analysis of applicability of the SM to covalent crystals [35] led to another model specially adapted to this case. The bond charge model (BCM) operates with the positive charges of atomic cores and inertia-less negative point charges of covalent bonds and investigates the Coulomb interaction between all charges in a crystal. In a simple version of the BCM, the bond charge is fixed in the middle of a bond as has been done, e.g., in the calculation of vibrational branches for diamond [36]. A more elaborate version of the BCM [37] assumed the bound bond charges adiabatically: at any arrangement of the cores, the bond charges were supposed to find a posi-

LAZAREV

101

tion to minimize the total energy. Besides Coulomb interactions between all charges, the following non-Coulomb interactions were introduced: the elastic interaction of the bond charge with the nearest cores, and the three-body interaction of the type, bond charge-corebond charge. An excellent coincidence of theoretical vibrational branches with experimental ones using only four adjustable parameters, the bond charge and the three force constants of non-Coulomb interaction was obtained for diamond with this version of the BCM. This version of the BCM was also applied to crystals with heteropolar bonds assuming the different character of interaction of bond charges with cores of various electronegativity [38]. Unfortunately, the very attractive idea of the application of the adiabatic version of the BCM to more complex crystals will meet considerable difficulties. Dynamic models assuming a charge transfer on the distances comparable with the lengths of chemical bonds deserve special attention. It has been shown in a previous chapter that by the quantum mechanical treatment of systems resembling some fragments of silicate lattices, their deformation is accompanied by a significant valence charge redistribution which can be described in terms of a charge flow from one atom to another. However, the concepts of deformable and polarizable ionic shells do not represent some important peculiarities of the electron density disturbance in the internuclear space. The exchange repulsion between the shells of neighboring ions deforms them, and the local deficiency of negative charge arises in the area of the superposition of ionic wave functions. An idea to represent this effect by the introduction of"exchange" positive point charges which would shift at the displacements of corresponding nuclei, was proposed earlier in the first publication by Dick and Overhauser [29] where the SM was developed. The exchange charge idea was employed [39] to explain the origin of the pseudopolarizability of positive ions which was sometimes introduced in the SM computations. Even earlier, Lundqvist [40,41 ] proposed to localize the exchange charge at the positive ion and to represent its mobility by a variable magnitude of the total charge of that ion. His model was probably the earliest version of the variable charge model (VCM) or the charge transfer

102


model (CTM). This model was proven to be applicable to the estimation of the TO-LO splitting, and to explain the Cauchy relation violation. The concept of charge transfer was introduced [42] into the SM, which allowed the reproduction of the peculiarities of the shapes of the vibrational branches in the alkali halides. A detailed analysis of the CTM and other versions of the SM given in that paper, explored a resemblance of the variation in the pattern of interactions in a crystal originating from the introduction of a charge transfer, and from the radial deformability of a shell assumed by the breathing SM. The importance of the variable ionic charges in dynamical theory of crystals with partially covalent bonding was emphasized repeatedly by Tolpygo and co-workers [43,44,45]. From the analysis of the electronic wave function deduced in the valence bond approximation, a potential function for the crystals of zinc blonde type was obtained. The long-range part of that function was expressed through a Coulomb interaction of point charges and point dipoles on atoms, their magnitudes being the functions of the coordinates of neighboring nuclei. This complication represented, in the author's opinion, the peculiarities of the electronic response to the deformation in crystals with partially covalent bonds. In the calculation of the dynamic properties of GaAs, a complementation of the SM by variable charges was employed [46], and a considerable improvement in the reproduction of its elastic and particularly dielectric and piezoelectric constants was obtained. A coefficient of the charge transfer was deduced from firing the experimental data. Its sign was the opposite of the one deduced from the exchange charge approach and adopted in ref. [42] when applying the VOM to alkali halide crystals. This discrepancy was treated [46] as a manifestation of the difference in the electron density redistribution in crystals with various type of chemical bonding.

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C. A Comparison of Various Descriptions of the Electric Response Function A comparison of various descriptions of the electric response function to distinguish dynamic models of crystals which were classified above mainly by their approach to design of potential function. It is clear, however, that the electron density relaxation at the nuclei's displacements is a central problem of any dynamic theory of a lattice. Any condensed system may be treated as being constituted of rigid cores composed of nuclei and inner electrons and a gas of valence (outer) electrons whose density distribution is represented by the function xval(rlR). A total charge density distribution is then expressed as

o(rlR) =

~ zc~ (j) 5(R(Jj) - r) + xval(rlR) . J

(2.56)

The first term in this expression represents the core charge distribution by the point magnitudes, zcore(j), resided at the lattice nodes. Since these charges do not vary at the lattice strain, differentiation of (2.56) with respect to displacement (uct(dj)) produces the following decomposition of the electric response function:

pa(rlJj) = zcore (j) ~5~x(R(Jj) - r) + xval(r[Jj) where ~5~t(r) = aS(r)/Or~. The

(2.57)

0~(rlJj) quantity contains two contributions originating from

the shift of the constant zcore(j) charge, and from the response of valence electrons to that shift. The first contribution is very simply expressed by the point dipole localized at the (dj) nucleus, and the problem of the description of p~ (r[dj) is practically reduced to the problem of the description of the valence density response. In the alkali halide crystals, valence density is located around the anion positions, and a lattice is composed by anions with closed shells which possess in equilibrium a spherical symmetry similar to the atoms of noble gases. In this case, a presentation of the density distribution function of valence electrons by a superposition of the point (zval(j)) charges centered at the halogen nuclei is possible,

104

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS xval(rlR) = ~ zval (j) 8(R(Jj) - r ), J

(2.58)

and p(rlR) is expressed as: p(rlR) = ~ z ion (j) 8(R(Jj) - r) J where zi~

= zc~

(2.59)

+ zval(j), the last term being zero for the metal ions.

A supposition of the ions with undeformable shells tightly bonded to their nuclei corresponds to the RIM where the response function pot (rlJj) is localized in the R(Jj) point and is represented by the point dipole with zion(j) amplitude: p~ (rlJj) = zi~

8~ (R(Jj) - r) .

(2.60)

If the mobility and deformability of the shells are taken into consideration, the dependence of pa(rlJj) on the dipole moment of the (Jj) ion is complemented by its dependence, in a general case, on the moments of all other ions in a lattice. A response fimction bears a delocalized character and may be expressed as: p= (rlJj) = ~=13(KklJj) 8~ (R(Kk) - r)

(2.61)

where Z al3 (KklJj) is a total variation of the (Kk) ion dipole moment in the a direction at the unit displacement of the (Jj) ion in the 13direction. A substitution of (2.61) into eq. (2.40) leads to the following expression: Z~13(Jlq) =N-1 Z z~13(KklJj)exp[iq" R(Jj)] J,K,k

(2.62)

which interrelates the macroscopic Zal3(jlq) tensor with the microscopic characterization of ions composing a lattice. The 3nN-dimensional vector of the microscopic electric field strength E with the Ea (Jj) components is now introduced. Its interrelation with the electric moment vector P of the same dimension (whose components are the Cartesian components of dipole moments of ions P a(Jj)) can be expressed as: E =CP

(2.63)

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105

where C is a matrix of the Coulomb coefficients. By employing zi~ zi~

and m tensors determined as

= 0 pi~

(2.64a)

Pal3 (JjlKk) = cOppol (jj)/0EI 3 (Kk)

(2.64b)

rna[3 (JjlKk) = 0 pdef (,/j)/0u[3 (Kk),

(2.64c)

-L O [

and

a following expression for P can be written*: = (m + zion/) u + p E .

(2.65)

Here u is a 3nN-dimensional vector of atomic displacements in a crystal. A substitution of (2.65) into (2.63) produces the interrelation:

if, = (I- Cp)-I C(m + zi~

(2.66)

and polarization is expressed as: P =,~u

(2.67)

where = C -1 (1- Cp)-lC(m + zion/),

(2.68)

is a matrix with the ~, a13 (djlKk) elements (which determine the Z(jlq) tensor according eq. (2.62)) and 1is the unit matrix.

*A formal approach to the description of the lattice polarization adopted in ref. [4] is implied. It represents the dipole moment of a strained crystal by the combination of the microscopic dipole moments, P (3j), arising in the equilibrium ionic positions. Three contributions to P (dj) are discerned: the pi~ dipole originating from the displacement of the equilibrium ionic charge zi~ the pp~ dipole related to the electric polarizability of ions, and the/xlef(jj) dipole excited by the deformation of the ionic shell under the action of non-Coulomb forces. The clystal is thus described by three macroscopic tensors specified by eqs. (2.64 a-c). For further details see ref. [4].

106


The effective charge tensor is thus determined by a whole totality of polarization properties of ions, and may have non-vanishing elements in the case of zero static ionic charges zion(j). It occurs for some crystals with homeopolar bonds like graphite or tellurium. It should be mentioned that the concept of effective (dynamical) ionic charges was developed as a result of extensive discussion of the mechanism of lattice polarization and of ionic contributions in that process (see, e.g., [27,47,48]). Judging by the adopted type of the electric response function, a majority of dynamic models (RIM, PIM, DDM) and the simplest versions of the SM and BCM, can be all specified as the dipole type. The electric response is defined in these models as some combination of the point dipole functions. Such an approach seems reasonable in the case of ionic crystals with spatially localized wave functions of valence electrons in the vicinity of corresponding cores. In a more general case of ionic-covalent crystals, the applicability of the same approach is more questionable since the electric response may have a more spatially extended nature. A general approach to the description of the electric response function represents itself as an expansion in basis functions centered on the nuclei like it has been done in MO LCAO theory. This possibility was discussed by Sham [21] who presented a response function as: pct(r[Jj) = Cn(JjlKk)Xn(R(Kk ) - r)

(2.69)

where n is the number of the basis function. A conventional simplification adopted in dynamic models of lattices consists of a substitution of continuous functions by discrete quantities (which reduces the calculation of spatial integrals to the summation over a lattice). It is therefore reasonable to use ~n as the basis functions of multipole expansion, ~5-function and its derivatives. The above dipole type models can generally be characterized as models with a basis of expansion (2.69) restricted to

aS(r)/Ora functions,

i.e., p-functions. A complementation

of this basis by the c92~5(r)/ar2 function corresponds to the breathing shell approximation,

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while the introduction of the

107

025(r)/OreLOrf3terms corresponds to the models which take the

quadrupole deformation of ionic shells into consideration. As a qualitative illustration of the possibilities of various models to represent the perturbations of the charge density by the nuclei's displacements, Fig. 2.1 shows the response functions of the three linear atomic systems at two possible types of its symmetric and antisymmetric bond-stretch motions. The first model corresponds to the shifts of peripheral positively charged ions while the second only corresponds to the moving central negatively charged ion. The graphs a-e in Fig. 2.1 express the Pc~ function of that system in various models, each graph representing only the particular features of a given model which were lacking in previous graphs. E.g., Fig. 2.1b shows the contribution of Pet originating from the deformations of ions, but not of their displacements which is shown in Fig. 2.1a. Similarly, in Fig. 2.1d the contributions from the shit, s of peripheral ions are not shown since they have already been reproduced by the upper curves. The graphs (Fig. 2.1a, 2.1b, and 2.1c) express the response functions corresponding to the models of dipole type. In a simple version (RIM, Fig. 2.1a) the antisymmetric deformation produces the point dipole at the central atom while the symmetric deformation changes the quadrupole moment (two differently oriented dipoles arise at the terminal atoms). If a deformability of ions is introduced (DDM), the point "deformationar' dipoles are added to that pattern. As is seen from Fig. 2.1b, at the antisymmetric deformation, an additional dipole at the central atom reduces the total dipole of a system originating from the shit~ of that atom, while the additional dipoles at the peripheral atoms enhance the total dipole moment (a contribution from the atomic shit~ is implied to be the largest). The total dipole moment would be reduced more considerably in the case of an inverse sign of deformability of the peripheral atoms (which corresponds to anomalous polarizability) as is shown by the broken curves. In the case of the symmetric deformation, there is no contribution to the total dipole from any atom while the additional dipoles of the peripheral atoms may change the quadrupole moment of a system.

108


A z

S

-2z

9

z

0-~

z

9

0-~

9

~

-2z

z

0

+'0

a

z

9

-2z o

z

o-~

-2z o

z

z

-2z o

z

9

-2z o

z

d z+Az 9

o_A Fig. 2.1

-2z-Az+Az 0-~

z-Az 9

z-Az 0-~

-2z+2Az 0

z-Az

A

The charge density redistribution in a three atomic linear system as repre-

sented by various response functions. A - antisymmetric bond stretch, S - symmetric bond stretch.

A "deflection" of the anion's shell at its two-side compression, which is considered in the BCM and in the quadrupole SM (the QSM, a version of SM introduced [49] in order to describe the peculiarities of lattices containing the cations with filled d-shells, e.g., Ag +, Cu +) can be reproduced by including into the basis of expansion (2.69) the second derivatives of the [i-function. It corresponds to the arising of the point quadrupole at the central atom of our system in the case of its symmetrical deformation (Fig. 2.1c) and thus represents a quadrupole contribution in a pattern of charge redistribution which would need the introduction of anomalous deformability of cations in the DDM. Each atom is supposed to transfer some portion of its charge into the internuclear space in the BCM. Five charges should be treated correspondingly in our model: three atomic charges and two bond charges. A charge redistribution at the antisymmetric defor-

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109

marion is described in this model as a superposition of three point dipoles localized on me central atom and on the bonds while at the symmetric deformation, only two dipoles on the bonds arise (Fig. 2.1d). This model combines the features of the DDM and BSM which are displayed more clearly the nearer the equilibrium positions of bond charges are to the anion (central atom). If the s-type terms in expansion (2.69) are considered (i.e., the "monopole" term represented by 8-function is introduced), the local charges are assumed to be variable, and a description of the charge transfer is possible. The electroneutrality condition provides a balance between reducing some of the charges and the enhancement of others. A contribution of such terms into the response function of our system is clarified by Fig. 2.1e. In the case of the antisymmetric deformation, the enhancement of the charge at the central atom caused by the lengthening of one of the bonds is totally canceled by its reduction due to the shortening of another bond while at the terminal atoms, the additional charges of the opposite sign arise. It can be treated as a charge flow from one atom to another. The charges of the same sign arise on the terminal atoms at the symmetric deformation and twice the charge of the opposite sign arises at the central atom. This type of deformation corresponds to a charge flow from the peripheral atoms to the central atom and vice-versa. Formally, the introduction of monopole terms into the pa expansion leads to the CTM or VCM models discussed earlier. A comparison of contributions from various terms of expansion (2.69) shows that a basis composed of the derivatives of the 8-function (dipole and quadrupole terms) is suitable to describe the charge redistribution in ionic crystals as originating from the insignificant shiits of localized charges from their equilibrium positions. In the case of partially covalent bonding, the charge redistribution may have a less localized nature, and the basis of expansion (2.69) is to be complemented by the s-type (monopole) contributions in order to represent the charge transfer on the distances comparable with the bond length. This statement is supported, e.g., by the results of precise quantum mechanical calculations of the response function in the ionic-covalent GaAs crystal [50]. According to

110


these results, the pa (rlJj) function is described as an s-function whose center is shifted relative to the equilibrium position of (Jj) nucleus. Consequently, in a model approach, just this term should be treated in the response function while the p-term should be introduced as a correcting term to describe the shift of its maximum relative to the R(Jj) point. Our short review of dynamic models would be incomplete without the mention of modem trends to deduce the parameters of interatomic potentials from the "first principle" considerations. A Gordon and Kim [51 ] approach, based on the quantum mechanical computation of the PF in a simple cluster composed of a pair of ions with filled shells is presently most popular.

The electronic wave function in this approach is calculated in the

Hartree-Fock approximation for isolated ions, and the interaction energy is found using the local density functional method. The application of this approach to oxides evokes, however, some difficulties originating from the instability of the 0 2- ion with a closed shell in a free state. A special spherical potential referred to as a Watson sphere is adopted to stabilize it. The size of 0 2ion and the parameters of its interaction with other ions are very sensitive to the selection of the radius of that sphere. A physically consistent approach to this problem originating from the analysis of the Madelung potential of that ion in a given crystal has been proposed [34]. A possibility to take into account a variation of the Madelung potential and of the parameters of a repulsive potential in a strained crystal was treated as well in order to reproduce the many-body nature of the lattice potential. The last problem is of special importance in the case of oxides. It is known that a violation of the Cauchy relation occurs for elastic constants of simple crystals or analogous relations for more complex lattices and is not reproduced by any potential of pair central interaction. The same relates to the spatial arrangement of some crystals: the equilibrium geometry of a-quartz is not reproduced by means of any potential of pair interaction, which leads only to the 13-quartz structure. It is possible to reproduce the a-quartz structure only by introducing the non-central interactions through the breathing shell on oxygen [34] or

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111

two shells on this atom [31 ]. The same is obtained by the inclusion of a potential explicitly dependent on valence angles into the PF model [52]. II.

A C O M P A T I B I L I T Y OF M O L E C U L A R F O R C E CONSTANTS W I T H T H E E X P L I C I T T R E A T M E N T OF C O U L O M B I N T E R A C T I O N IN A LATTICE

A. Potential Energy Decomposition and Interrelation Between the Potential Energy Function and the Electric Response Function We now inspect the physical origin of the conventionally adopted decomposition (2.44) of the potential energy into Coulomb and non-Coulomb contributions and discuss some advantages of an alternative approach to the potential energy decomposition. A closer study of the interrelations between the V(R) function and the p(rlR ) function, the latter being treated in terms of decomposition (2.56), is appropriate. The V(R) function is implied to be constituted by two components, the nuclear and the electronic: V(R) = vnucl(R) + vel(R),

(2.70)

where the former is obtained as: vnucl(R) =

Z

J,K,j,k

z(j)z(k) IR(Jj)- R(Kk)I '

(2.71)

and the latter represents the ground-state energy of the electronic sub-system of a crystal.

veI(R) is expressed in the method of the local density function [53,54]

vel(R)

=

"r,

1~ ~ x(rlR)x(r'lR) +c[x], Ir-r'l

(rlR)w(rlR)dr + ~ dr dr'

as: (2.72)

where the electronic charge density distribution is described by the x(rlR) function and: w(rlR) = / ~ j,j

z(j) [R(Jj) - rl

is a potential of electron-nuclear interaction.

(2.73)

112


The first term in eq. (2.72) corresponds to the energy of interaction between electrons and nuclei while the second one represents the electron-electron Coulomb repulsion. The third term is some function of the

x(rlR)function which represents a summary of the

kinetic, correlation and exchange energy of electrons. The Hohenberg-Kohn theorem [53] states that the ground-state electronic energy has, at any fixed nuclei's arrangement, a minimum relative to the

x(rlR)variation and the

following expression is valid for the equilibrium geometry: 8vel(R)=

Here the w~

[Sx(rlR)__

(r)+

Ir-r' I

dr' + g(r) dr = O.

(2.74)

and g(r) magnitudes are determined by the expansions:

w(rlR) = w~

+ wet(rldj)ua(dj) + $1 wczl3(r~lKk)u= (Jj)uI3 (Kk) ,

(2.75)

and: G[x ~+Sx]=G[x ~]+

Ig(r)Sx(rlR)dr+

IIr (r'r'>Sx(rIR>Sx(r'lR>drdr'"

(2.76)

By making use of eqs. (2.70 and 2.72), the coefficients of expansion (2.32) can be now expressed as:

Vo >-VnUOl >+ (2.77)

7_-7 i and:

+

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113 nucl + x= (rl ~)wts(rl Kk)+ x=f3(rl,ljlKk)w~ (2.78)

+ IfXaf3(rl~lgk)x~

3(r'lgk) drdr'

Ir-r'l

+ Ig(r)xaf3(rldjJIKk)dr+IIF(r,r')xa(rl~)xf3(r'lKk)drdr'. The equality (2.74) helps to exclude all terms containing the

xa(r~) magnitudes

from eq. (2.77) and the terms with x~p(rlJjlgk) magnitudes from eq. (2.78). The above coefficients are thus reduced to the form: V~(,/j)=vnur

Ix~

,

(2.79)

and:

fr , v~t3(JJl~)= vnucl-ji ~ t ~l~)+ j[~~

+ x~(rl~)wfs(rlKk)+xf3(rlKk)w~(rl~)]dr

(2.80)

s( 9~(rl~)xp(r'lgk) / Ir-r'l1 + F(r,r')) ]drdr'. The latter expression for the force constant can be simplified by taking into consideration eqs. (2.73 and 2.71) and the decomposition (2.56). Its presentation in the form: Val3 (jjlKk)= I I pa (rl'/j)pl3(r'lKk) drdr' Ir-r'l (2.81)

114


explicitly separates a contribution to the force constant from Coulomb interactions between all charges in a lattice and thus substantiates eq. (2.43) for the force constants of interaction between removed ions. This term is determined, like the Zaf~(jlq) magnitude, by the response function of the total charge. The second term in (2.81) is defined by non-Coulomb interactions in a lattice. The above considerations may be treated as a proof of applicability of the energy decomposition (2.44) to the investigation of lattice dynamics problems. Unfortunately, some ambiguity arises in its application to the interactions between the nearest atoms which are partially covalently bonded in the most general case. If their electronic shells are overlapped, no simple and physically consistent approximation of the electric response function can be proposed. Also, the first term of eq. (2.81) is not applicable to the calculation of the Coulomb contribution to the force constant independently from the existence of suitable analytic approximation of the second term of eq. (2.81). It should be noted, however, that even in ionic-covalent crystals, the approximation (2.43) apparently remains valid for interactions between distant ions. Moreover, the simplest point ion concept may save its applicability at sufficiently large interionic distances which would considerably simplify the numerical calculations in the case of complicated lattices with numerous degrees of freedom. The area of applicability of the point ion concept for the crystals of oxides will be investigated more precisely in the next chapter by means of the methods of molecular quantum chemistry. An alternative approach to the potential energy decomposition may be proposed. Instead of the separation of contributions of different physical origin, it implies an explicit separation of contributions according to their spatial distribution. In order to justify that approach, the area of interactions described by the second term of expression (2.81) is investigated below. It has been shown [21 ] that the F(r,r') magnitude is interrelated with the electronic polarizability function of a crystal 9r

as:

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~(r,r')x(r', r")dr' = -5(r-r"),

115 (2.82)

and is, consequently, a kernel of the integral operator reciprocal to the x(r,r') operator. The latter determines the perturbation of the electron density 5x(r) in the point r excited by the variation of the total potential of the electric field in a crystal q)(r') in the point r': 5x(r) = ~x(r,r')Stp(r')dr'.

(2.83)

If &p(r') is localized in any point, fix(r) has a non-vanishing value in its vicinity whose extent is determined by the degree of locality of x(r,r') and is of the order of the length of the interatomic bond. In view of the interrelation (2.82), that vicinity is the area of localization of the F(r,r') function. It can be concluded that the area of interactions represented by the second term in eq. (2.81) does not exceed the total size of the areas of localization of the fimctions xa(r[Jj), xl3(rlKk), and F(r,r'). If the (Jj) and (Kk) nuclei are so spatially separated that the distance between the areas of localization of corresponding responses is larger than a characteristic size of the area of localization of F(r,r'), the second term in eq. (2.81) vanishes. In this case, the force constant is completely determined by the Coulomb interactions of responses of the total charge distribution. Let us suppose that the areas of spread of the xa(r~) responses are known and the size of the area of localization of the F(r,r') function (or of the function of electronic polarizability) is estimated as well. Then, for any (Jj) nucleus its neighborhood can be determined, satisfying a condition that for any other (Kk) nucleus positioned beyond this A(Jj) area, the force constants Va[3(djIKk) are dependent only on their Coulomb interactions. On the other hand, for (Kk) nuclei belonging to a A(Jj) area, the nature of interaction with that nucleus is more complicated. Additionally to electrostatic forces, the exchange and correlation interactions of electron shells and their overlap play an important role.

116


An introduction to the notion of the specific non-Coulomb interaction area A(dj) (intrinsic to any atom of a given lattice) implies a decomposition of the potential function, V(R), into two contributions according to their areas of action: V(R) = vsh~

+ vl~

(2.84)

The first term in this decomposition is implied to represent the interactions of any origin including Coulomb ones between (dj) atom and other atoms inside A(Jj) area, and the latter term being restricted to the interactions between distant atoms which are sufficiently determined in the Coulomb approximation. Proceeding from this idea, M. B. Smimov proposed an approach to the design of the dynamic model of an ionic-covalent crystal, which would be able to combine in an uncontradictory way, the introduction of molecular force constants with the explicit treatment of Coulomb interactions in a lattice. This approach has been originally outlined in ref. [55] and developed and complemented in refs. [9,10,56]. Its most characteristic features are discussed below and some applications are exemplified.

B. Conditions of Compatibility of Molecular Force Constants with Explicit Separation of Coulomb Contribution to the Force Field The problem of summation over a lattice will not be treated below, and the numbering of primitive cells is omitted for simplicity in subsequent expressions. It should be noted that the force constants in the above expression (2.81) was found linear to the electric response because of the adoption of the variational approach wnere a knowledge of exact

x(rlR) and w(rlR)

functions was implied. It is not the case if an approximate model repre-

sentation of those functions is adopted, and the second order electric response contribution to the force constants cannot be neglected. The Coulomb contribution to the second derivative of the potential energy function is expressed as:

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117

~ ~ [ Pa (rlJ)Pl31r'lk) Coulomb Pal3 (r[j[k)p~ = + '1 VaB (jlk) ir _ r [r - r

]

drdr'.

(2.85)

Assuming that the electron cloud perturbation caused by a shift of the jth ion is restricted to its vicinity, it is possible to arbitrarily suppose that the pa[3(r[j[k) magnitude has the area of localization whose size does not exceed the size of the area of actuality of forces represented by the first term of the PF decomposition (2.84). A condition of: Pal3(rLilk) - 0 ifk ~ A(j)

(2.86)

reduces the expression (2.85) to the form: v~oulomb(j[k)=

j'~pa(r)p~(r')drdr, Ir-r'l

ifk cA(j).

(2.87)

This expression represents the unique contribution to the force constant of interaction between distant atoms which is restricted to the first-order responses. It will be shown that the explicit separation of Coulomb contribution only for the distant atoms considerably moderates the requirements to the model representation of the response function adopted in the numerical computation. The second term of eq. (2.85) does not vanish, however, for the less distant (j) and (k) ions, satisfying the condition k e A(j). In this case, it represents the energy of perturbation of the charge density distribution inside the A(j) area subjected to the action of the electrostatic field of a whole lattice at the unit displacements of (j) and (k) ions, both belonging that area. In order to separate the contribution to the force constant related to the electrostatic field created by the charges of the rest lattice (beyond A(j) area), the second term of eq. (2.85) can be divided in the two terms (according to the portions of the space which determine the electrostatic field). The following expression is then obtained for the force constant in a case k e A(j):

118


Val3(jlk)=vn~176176

aB

+ Sp(zl3(rlJlk)

(jlk)+ f f P a (rlJ)Pl3(r'l k) drdr'

Ir-r'[

S

J" P~ (r')dr'dr F ~pal3(rljlk) over A (j)

(2.88)

P~ (r')dr'dr

beyond A(j)

The first three terms in this expression are combined in Smimov's approach as representing the contributions from the interactions inside A(j) area treated as a free molecular cluster. In other words, they represent the contribution to the force constant originating from the first term of the PF decomposition in a form (2.84). Denoting the totality of these three terms as 9,r~short (ilk), and the last term of (2.88) as vashort-long (jlk) in accordance with its "mixed" origin, the following structure of the force

constant of a crystal is deduced in a general form: IV sh~ r;~ k a + v s h ~ 1 7 6 Val3(jlk)= ~ a13 ~J' J a13

t";

~jlk), if k

eA(j) (2.89)

[ Via~ng (j[k), if k r A(j) where x/long . a13 (ilk) is determined by eq. (2.87). The physical meaning of the xrshort ,c~13 (ilk) force constant is thus identical to the force constant of a free A(j) molecule specified in a space of Cartesian atomic displacements. This force constant can be transferred to the dynamic model of a crystal from the force field of a suitably selected molecule whose force constants are determined either by direct quanturn mechanical calculation or by solution of the so-called inverse vibrational problem employing the experimental vibrational frequencies of that molecule. It should be emphasized that for a given crystal, the number of Afj) "molecules", each specified by its size, shape and dynamical parameters, is as large as the number of independent atomic sites in a lattice. Restrictions to the selection of these specific short-range interaction areas are not yet known unlike the ones determined by the symmetry of a lattice and invariance of the force constants to the ionic indices transposition.

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119

The vlong , a~ (jlk) contribution to the force constant represents the interactions between the sufficiently distant atoms whose interaction can be treated as being determined by Coulomb forces. It is calculated by conventional methods and the exclusion of some first terms from corresponding lattice sums (which is provided for by the discussed approach evokes no problems). The dual nature of the x/short-long "a13 (jlk) contribution originates from the existence of the energy of "submergence" of the A(j) cluster into the electrostatic field of a lattice and the variation of that energy at the charge redistribution inside the cluster. This contribution is calculated in the electrostatic approximation since it is determined by the interaction of moving charges inside the A(j) area with the distant ions of a lattice at rest. This contribution does not relate, however, to the long-range term of the PF presentation in equation (2.84) since it has been determined by the displacements of nearest ions. It does not vanish in any dynamic model of a lattice, assuming the existence of charges or dipoles on the atoms. It is natural to specify (in this approach) the short-range contribution to the crystal force constant in a space of internal coordinates which are conventionally adopted in the theory of molecular vibration. These molecular force constants are usually determined in a space of independent (non-redundant) internal coordinates if the quantum mechanical methods are employed in their theoretical evaluation (cf. Chpt. 1). No problems arise in this case when reducing these force constants to Cartesian space of atomic displacements adopted in calculation of other contributions to the crystal force constant (this and some related problems will be discussed in the next section). The subsequent speculations concerning the structure of the crystal force constant in terms of a given approach [56] substantiated a feasibility of complementing the short-range contribution by some molecular type interaction (off-diagonal) force constants. These correspond to internal coordinates including several nearest atoms outside the A(j) area. This complementation has been shown to be dependent upon the degree of locality of the adopted electric response function.

120


In order to calculate the long-range and "mixed" contributions to the force constant, which relate to Coulomb interaction, a summation over a lattice is necessary (with the exclusion of several terms as explained above). Since a treatment of crystals with considerable amount of covalent bonding was intended, and a rather delocalized nature of the electric response was assumed in a given case, an exhaustive analytic formulation of the VCM approximation was developed by Smimov [9] which adopted the Ewald approach to summation. Some other approximations beginning from the RIM and then to the PIM and DDM were employed in particular cases; and their preferences and disadvantages in the treatment of both the force constants and polarization phenomena (IR intensities and piezoelectric constants) were compared. The importance of the fulfillment of the static equilibrium conditions as a reference point of any physically consistent treatment of dynamic problems was discussed repeatedly. E.g., it has been emphasized [57,58] that a considerable amount of knowledge on the dynamic parameters of a complex lattice can be deduced from an investigation of its stability conditions (SC). In other words, their investigation formalizes the restrictions imposed on the PF of a crystal by its experimentally determined equilibrium geometry. This problem is not usually met in the empirical normal coordinate computations in molecular spectroscopy since the explicit analytic formulation of the PF is avoided by the adoption of the SC fulfillment as a basic assumption. This reduces the PP description to the problem of the force constants determination. In this approach, no ambiguity arises if the non-redundant internal coordinate set is employed, and the force constants found by li-equency firing are readily compared with ones obtained by quantum mechanical computation; where the real energy minimum is searched for the theoretical geometry before the force constant calculation (see Chpt. 1). If the PF of a system is defined with inclusion of at least one or several terms which specify a dependence of energy on the geometry by any analytic expression not ensuring the vanishing of relating forces in the equilibrium position, an investigation of the SC is unavoidable, and corresponding adjustment of dynamic parameters are needed. This circum-

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121

stance was apparently ignored in earlier attempts to combine the molecular type force constants with explicit treatment of Coulomb contribution to a dynamic matrix [59-61] and their physical consistency is therefore questionable. The investigation of the SC is an important item in the approach considered here. These are formulated in a present case as a balance of forces acting upon any atom,

c3Vshort o~ua(j)

=

c3vlong Oua (j) '

(2.90)

and a condition of stability of a lattice relative to the uniform strain ual 3 whose interrelation with atomic displacements, ua, has been specified by eq. (2.23)"

aV short 3Ua~

=

avlong OUa~

.

(2.91)

For any particular crystal, these SC lead to a system of equations linear to the first-order terms of the V short expansion in the internal coordinates (which represent the forces acting in a lattice at rest). A set of SC defines a system of equations linear to the first-order terms of the V sh~ expansion in internal coordinates. Its solution, which may be rather cumbersome in a case of complex lattices of low symmetry, leads to some interrelations between the first derivatives of the short-range interaction energy keeping jointly with other forces of the equilibrium of a lattice. A concept of coordinated internal tensions in a system with the number of internal coordinates larger than the number of independent SC will be discussed more extensively in the next section.

C. Applications to Silicon Dioxide and Silicon Carbide The applications to silicon dioxide and silicon carbide are still the unique examples of a consistent employment of the described approach. Since the earliest investigation by Elcombe [62], the lattice dynamics of a-quartz as one of a typical ionic-covalent crystal (whose properties were investigated exhaustively) was calculated with the explicit separa-

122


tion of Coulomb contribution to the PF by several authors in terms of the RIM and PIM approximations [63-65]. Although some common features can be seen in their results, the paper [63 ] will be mostly referred to below as the more consistent one because of the rigorous treatment of the SC problem. The previous computations were based on the presentation of the PF in the form of eq. (2.44) with the calculation of the Coulomb contribution in the point ion approximation. Irrespective of a certain difference in their initial supposition, their common disadvantage is apparently seen. Although the frequencies of transversal optical modes are reproduced reasonably well, the TO-LO splittings (or IR intensities) of the polar modes remain to be underestimated in these computations nearly by an order of value, thus indicating the inadequacy of adopted models for a treatment of dielectric behavior or piezoelectric properties. It should be noted that relatively small ionic charges were adopted in papers [63-65]: the charge at the oxygen was varied in a narrow interval from 0.35 to 0.5 e. Although the earlier computation by Elcombe with the oxygen charge approaching le seemed more preferable in reproduction of dielectric properties, discussion of the possibility of enhanced ionic charges was not given in those papers. It can be suspected, however, that larger ionic charges were not introduced because of their destabilizing action on the TO frequencies and on the conditions of static equilibrium when the latter was studied as well. In other words, it can be supposed that the difficulties met in the above computations represent the incompatibility of ionic charges adopted in the microscopic calculation of Coulomb contribution to the first and second derivatives of the PF with ones employed in the calculation of the macroscopic field which determine TO-LO splitting. That incompatibility may originate from the ambiguity of the separation of Coulomb contribution to the interaction between nearest ions (which is probably of special importance in the case of partially covalent bonding, and particularly from the inadequacy of the point ion approximation in the treatment of those interactions).

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123

In view of those inconsistencies, it seemed attractive to employ an approach proceeding from the presentation of the PF in the form (2.84) to the calculation of dynamic properties of a-quartz. This allowed the ambiguity of explicit separation of the Coulomb contribution to interaction between nearest ions entering the area of specific short-range interaction area to be avoided. That area was supposed to include the four nearest oxygen atoms for A(Si) while A(O) was extended to eight atoms (2 silicon and 6 oxygen) of two adjacent silicon-oxygen tetrahedra. The Coulomb interactions with atoms lying outside these short-range interaction areas and the macroscopic polarization of a crystal were calculated when applying the approach outlined above [9,10,56] by means of the simplest version of the VCM.

This

approach was used earlier in numerous IR intensity calculations for silicates along with the mechanical treatment of the TO frequency problem [66,67] (see the next section for a more detailed explanation). This version proposed the atomic charges in any bond to be dependent only on the length of that bond. The force field model employed in the description of the short-range contribution to the forces and force constants was not a molecular force field in a strict sense of word, i.e., it was not transferred as a whole from any chemically related molecule. It was specified in a space of internal coordinates (more exactly, their combinations symmetrical to translation) as an extended Urey-Bradley type force field whose most important contributions were estimated from corresponding values deduced from the normal coordinate calculations for the silicate lattices and molecules of silicoorganic compounds, empirical bond length/force constant correlations (see the previous chapter), etc. When investigating the SC with the adoption of the method of indefinite Lagrangian-multipliers to search for the unique solution of an excessive set of equations, a redundancy of the set of internal coordinates introduced for the description of the shortrange interactions was taken into consideration. The SC were defined as the conditions of a balance of electrostatic forces acting in a given approach between relatively distant atoms and of short-range forces of any origin represented by the tensions of internal coordinates.

124


These tensions were reduced for simplicity to a system of coordinated forces only in the two-body type coordinates (Si-O bond or O...O tetrahedron edge elongation). The initial solution of the SC and normal mode frequency problems [9,10] was slightly improved later [56] by some variation of the set of dynamic parameters providing a better description of macroscopic elastic properties of a-quartz. These results were employed in a detailed comparison of the reproduction of piezoelectric constants of that crystal which can be achieved by means of various model presentations of the electric response function. The static charge on the oxygen atom and the effective charge describing its increase at the bond elongation (l(az/al) where I is the bond length) were determined from the SC fulfillment and frequency fit, and both were found to be near 1e in the considered approach. These values are considerably larger than the ones adopted in computations discussed above and reasonably agree with quantum mechanical estimations for chemically related molecular systems discussed in Chpt. 1. It should be emphasized that the incompatibility of computed TO frequencies and TO-LO splittings does not arise in this approach. It is seen from Fig. 2.2 that the TO frequencies and IR intensities are reproduced with approximately the same degree of accuracy. The results of calculation of the compressibility and macroscopic elastic constants were less satisfactory: these quantities were systematically underestimated, their magnitudes not exceeding 70% of experimental ones. It was treated as a probable indication of the necessity of a further extension of the short-range interaction area in a given approach. This would lead, however, to a considerable complication in the SC formulation and enhancement of the number of adjustable parameters. On the other hand, a certain oversimplification of the electric response model can be noted. Although a less localized character of response than in earlier approaches was proposed, it did not reproduce the peculiarities of the deformational polarization of that of the SiO4 polarization deduced from the quantum mechanical computations of molecular systems in previous chapter.

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125

~10

(o)i I oIolF;I i

.

.

9

9

1 [

2 3

i

1

A 2 (z-dipoles)

,

I

,

I i

I

i

E (x,y-dipoles)

1

2 3

. I

1200

I ,

I

1000

,

I I

. I

,

I

800

,

600 co,

cm

I I

400

. ,

. I

200

,

0

"I

Fig. 2.2 Experimental and calculated spectra of ~t-quartz [11]. The positions of lines correspond to transverse frequencies of optical modes, the heights of lines being proportional to (co2 _ co2)/co 2 (solid points represent the frequencies in a non-polar symmetry species or the IR bands of minor intensity. 1) experimental data; 2) calculated by the conventional approach [63 ], 3) calculated by the approach proposed in refs. [9,10].

In difference to the above version of the VCM which supposes the uniform polarization of a bond at its elongation, the DDM approximation (expressing this process by the arising of two point dipoles at its ends) has been adopted in the application of the proposed approach to lattice dynamics of silicon carbide with a zinc blend structure [10,56]. This makes a direct comparison with the results of an earlier model calculation of lattice dynamics of [3-SIC based possible on the usual approach to the PF decomposition (eq. (2.44)) [68]. The three-dimensional network of Si-C bonds in this crystal is constituted by the regular alternation of SiC 4 and CSi4 lattice nodes which can be treated in the considered approach as the specific short-range interaction areas A(Si) and A(C), respectively.

126


The force constants were transferred from the experimentally determined force field of the tetramethylsilane molecule, Si(CH3)4. Corresponding data were not available for the tetrasilylmethane molecule, and more provisionally, the suitable force constants of disilylmethane, H2C(SiH3)2, were adopted as the short-range force constants of the CSi 4 node. The averaged values of these force constants were fixed, and no adjustment by fitting the SC or frequency was provided. The simplest version of the DDM assumed only two independent parameters specifying the charge distribution in a crystal and its dependence on the deformation. These are the static charge at the silicon atom (the equal charge of opposite sign at the carbon atom follows from the electroneutrality condition) and the deformable dipole parameter qr of the same dimensionality. The deformable dipole parameter is determined as the sum of charges characterizing the point dipoles which arise at Si and C at the Si-C bond elongation and are directed along the bond axis, y = 0Psi/01si C + 0pc/01si c. A satisfactory fit of TO and LO frequencies of the optical mode and of macroscopic elastic constants has been obtained by the adjustment of these two parameters. A correct prediction of nearly zero piezoelectric effect has been obtained as well. The properties of the silicon carbide crystal were reproduced in a standard approach to the PF decomposition [68] where the short-range contributions to the forces and force constants were determined by the frequency and elastic constants fitting. The values of some dynamic parameters seem, however, less convincing in that computation since they are in contradiction with the nature of bonding. Thus, the inverse sign of the y parameter can be shown to correspond (at the adopted zsi magnitude) to the diminishing of the bond dipole at the elongation of the Si-C bond, which seems doubtful. A quantum mechanical computation shows that in the H3Si-CH 3 molecule, the bond dipole increases with the increase of the bond length [56]. Also, the static atomic charges of the silicon carbide crystal were considerably larger in [68] which hardly agreed with the covalent nature of bonding in this crystal. It should be added, however, that the results of quantum mechanical computa-

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tions of molecules containing Si-C bonds indicate the oversimplified character of any electric response model restricted to the two-center character of this function. The nearly zero piezoelectric effect in [3-SIC was reproduced in ref. [68] as a result of mutual cancellation of two considerable contributions. It is thus more sensitive to the insignificant variation of parameters than the result obtained by the approach considered here, which deduces the small piezoelectricity from the infinitesimal magnitudes of all contributions. In general, it can be concluded that the proposed approach provides a physically consistent lattice dynamics model of [3-SIC with chemically meaningful values of the parameters. Since its application to more complex crystals is labored by the large number of equations determining the SC this approach will not be developed fimher in subsequent chapters. Also, some problems related to its application are not yet clarified. A less rigorous but more straightforward approach to the use of ab initio molecular force constants in the lattice dynamics will be attempted instead. It consists of the direct transfer of the force constants of suitable molecules into the dynamic model of a crystal with subsequent complementation of that model by the parameters representing some interactions of less localized origin (which are not met in a molecule or affect its structure and properties very negligibly). As will be shown in the next chapter, some of these interactions can be evaluated by the quantum mechanical calculation of artificially designed systems of molecular type, or of existing molecules of larger size. In terms of success, this approach allows the dynamic properties of complex crystals to be calculated rather easily, keeping all computational schemes interpretable in terms of chemical bonding space of internal coordinates. Moreover, it can be shown that some peculiarities of these coordinates determine their important advantages in a treatment of macroscopic elastic properties, lattice stability, and structural phase transitions. Since it is paid undeservedly low attention to the employment of internal coordinates in the treatment of the lattice dynamics, we shall review their application to crystals in some detail, and outline their corresponding computational routine in the next section.

128


III. I N T E R N A L C O O R D I N A T E S IN T H E D E S C R I P T I O N OF D Y N A M I C P R O P E R T I E S AND L A T T I C E S T A B I L I T Y The treatment of molecular vibrations by the so-called GF-matrix method has been realized more than 50 years ago. It originates from the investigation of the potential and kinetic energy of a vibrating molecule in a space of vibrational, (i.e., referenced to the equilibrium geometry) coordinates closely bound with the atomic arrangement and chemical bonding in a system. The idea of internal coordinates, which are often referred to in Russian literature as the "natural" ones, was practically proposed simultaneously by Eljashevich et al. [69] and Wilson et al. [70], and its application to the theory of molecular vibration was thoroughly developed in fundamental books. A more modern treatment of the molecular vibration given entirely in matrix language [71] will be referred to below. Subsequently, several more specialized books devoted to particular problems appeared. The books treating the vibrational amplitudes and their relation to the molecular structure determination [72] or the theory of the IR and Raman intensities [73] can be mentioned specifically. An extension of the internal coordinate approach to crystal vibrations has been originally outlined by Stepanov and Prima [74]. A detailed description of the GF-matrix formalism as applied to longwave optical vibrations of crystal lattices was given later [75] since it was rather often applied to the calculation of the vibrational spectra of various crystals. The advantages of this approach were most obvious in the treatment of complex crystal lattices with numerous degrees of freedom. E.g., the spectra of numerous complex oxides were analyzed by means of this method, and some correlations between the force constants and geometric parameters were deduced [76]. A computational scheme to calculate the macroscopic elastic properties of crystals in terms of internal coordinates has been proposed as well [77]. A deeper investigation of the interrelations between vibrational modes of a crystal and its macroscopic elastic properties treated in terms of internal coordinates proved to be useful in a study of various solid state problems [78-80]. Some new applications of the in-

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ternal coordinate approach in crystal physics related to the peculiarities of these coordinates intrinsically have been briefly reviewed [81 ] and will be discussed below in more detail.

A. Vibrational Problem in Internal Coordinates and Indeterminacy of the Inverse Problem A conventional approach to the vibrational problem in molecular spectroscopy (see, e.g., [71]) proceeds from the postulation of the SC fulfillment. The potential energy of a system is expressed in the harmonic approximation as a quadratic form in internal coordinates, g, which represent the changes of the bond lengths, valence angles, dihedral angles between planes determined by the pairs of bonds, etc., v(g) = ~1 (gFg)

(2.92)

where F is the force constant matrix. The kinetic energy t(g) is similarly expressed in a space ofrates g, t(g) = ~1 (gTg)

(2.93)

where T is the kinetic energy matrix whose elements are found from the spatial arrangement of a system and the atomic masses. A solution to the vibrational problem consists of finding the matrix L which brings the kinetic and potential energy matrices to diagonal form simultaneously: I = L+TL

(1is the unit matrix),

(2.94)

2 = L+FL

(L is the diagonal matrix).

(2.95)

In a mathematical sense, it is a linear transformation from the internal coordinates, g, to the normal coordinates, Q, which are interrelated through the L matrix as: gk = ~LklQl

(2.96)

gk = E Lkl(~l 9

(2.97)

1

and

1

130


The physical meaning of that transformation can be understood as a transition from coordinates g to coordinates Q, which reduces the vibrational Hamiltonian (H) to a simple form expressed as a sum of Hamiltonians representing the independent harmonic vibrators: H(Q)= 89

2 +~.IQ12) = ~-'H 1 .

(2.98) 1

The wavenumbers are determined by the eigenvalues ~1 as COl = ( 89n c ) ~ and the shapes are described in a space of internal coordinates by the eigenvectors L1. This is why they are oRen referred to as the shapes of the normal vibrational modes. In the harmonic approximation, only these vibrators can interact inelastically with the electromagnetic radiation at o 1 frequencies in the absorption process, and at Oexeitation + co1 in the scattering process. The existing computational routines make the sets of o 1 magnitudes and L 1 vectors easily accessible, even for rather complicated polyatomic molecular systems at any supposed values of the F matrix elements and known (or assumed) equilibrium geometry. This procedure is usually referred to as the direct vibrational problem. All of the above statements remain valid for a crystal if its surface vibrational states are excluded from consideration by the adoption of Bore's cyclic boundary conditions. Making use of the translation symmetry of a lattice, it is possible to design an infinite number of equations of the type (2.94, 2.95), each specified by a definite value of wavevector, q. The index, q, which should label any matrix in those equations, will be omitted in further expressions for simplicity. However, the internal coordinates of a crystal lattice, g, will be implied hereat~er as the translation symmetry coordinates consisting of a basis of some irreducible representation of the translation sub-group of a space group. These are deduced from the initial (local) internal coordinates of a lattice as is explained in several references [74-76]. The o(q) dependence (which determines the shape of so-called vibrational branches in reciprocal space) is investigated by the coherent inelastic scattering of slow neutrons on photons. The experimental procedure is rather cumbersome, and sufficiently complete sets

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of data have been obtained only for the crystals with a relatively simple structure and low or moderate vibrational frequencies. The complex crystal lattice can be treated as an assemblage of interpenetrating simple Bravais lattices, each representing a particular atom of a primitive cell. Among the normal modes of crystal lattice, only ones corresponding to the long-wavelength limit (q-->0), in which all those sublattices shift undistorted relative to another, possess a nonvanishing intensity in the first-order IR and Raman spectra. These spectra have been obtained for numerous crystals including the spectra of single-crystal specimens in polarized light, which ensures the assignment of experimental bands corresponding to various irreducible representations of the factor-group of a crystal. Correspondingly, in their theoretical treatment, the preferences of partial factorization of eqs. (2.94 and 2.95) by means of rotational symmetry operators of a space group can be utilized. A solution of the so-called inverse vibrational problem (IVP) is crucial in the appli-

cation of the data of vibrational spectroscopy to the investigation of bonding in a considered system. It provides a solution of eq. (2.95) relative to the F matrix elements using the experimental ~ values (co2). Unless the important advantage of the intemal coordinates originates from a closeness of the F matrix to the diagonal form in space, it is not exactly the diagonal matrix for any real system, and the off-diagonal elements represent the important peculiarities of interaction. Obviously, a unique solution of the IVP does not exist in this case since the number of the F matrix elements exceeds the number of experimentally accessible values. More rigorously, such a solution does not exist in any case from a formal point of view since eq. (2.95) includes the L vectors which are inaccessible to experimental determination. In the spectroscopy of organic molecules, these difficulties are partially passed over by searching for a joint solution of a large number of eqs. (2.95) relative to the same F matrix, each of those equations corresponding to a definite isotopically substituted species (H---~D substitution is mostly employed). In a more modem approach (discussed in the previous chapter) the initial approximation of the F matrix deduced from ab initio quantum

132


mechanical computation is very near a final one. The IVP solution is thus reduced to an insignificant refinement of the F matrix by the adjustment of several scale factors that does not alter the interrelations between its elements. Both approaches are hardly applicable to searching for a more definite solution of the IVP for inorganic crystals. A significant isotopic frequency shift can be obtained only for a few atoms like lithium and boron. A selective isotopic substitution of definite lattice sites (when a given atom occupies more than one set of equivalent sites) is practically never possible. On the other hand, the direct methods of ab initio quantum mechanical computation of the force constants of crystals are still in a state of development, and too laborious for a practical application to complicated lattices. This is why considerable attention is paid to the indirect information on the eigenvectors, L, which can be deduced from the IR intensity analysis. It should be restated that in the force constant approach, the variables of the energy (frequency) and polarization problems are completely separated since the explicit treatment of the Coulomb contribution to the PF is not attempted. However, these two problems remain to be interrelated through the shapes of the normal modes. The IR intensity of any mode is determined by the square of the polarization vector derivative (per unit cell) with respect to the normal coordinate, IdP/dQI2, and let the totality of all these derivatives constitute a matrix kt = dP/dQ. Then, its interrelation with the matrix of the shapes of normal modes can be expressed as = ZL

(2.99)

where Z is a tensor of effective charges defined in internal coordinates, dP/dQ. Its elements are evaluated from some particular model representation of the charge distribution in a system and its dependence on the deformation. (We will not develop similar considerations conceming the possible application of Raman intensities to the IVP solution because of the absence of a widely accepted and physically meaningful system of microscopic parameters interrelating them with the peculiarities of bonding).

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Just as in eq. (2.95), only the left-side values of eq. (2.99) are experimentally accessible. Their magnitudes are known to differ for various polar normal modes of a complex system by more than two orders of value. It was found empirically [67,76] that in a case of such a large difference in the IR intensities of modes belonging to the same irreducible representation (of a point group for a molecule or of a factor-group of a space group for a crystal), the peculiarities of their shapes were much more important in explanation of that difference than of any possible variations of effective charges. Both the adoption of any particular electro-optic scheme and the selection of magnitudes of its parameters (restricted by the consideration of their physical meaning) were of secondary importance in their effect on the results of the IR intensity calculation compared to the shapes of the corresponding modes. Consequently, it was decided to test the eigenvectors obtained in a trial solution of eq. (2.95) by their applicability to reproduce the IR intensities according to eq. (2.99), thus reducing the ambiguity of the IVP. This approach was sometimes successfully employed in the theoretical analysis of molecular spectra [82]. A typical example is met in the IR spectrum of a-quartz, whose dynamics will be treated in some detail later. Two high-frequency E modes of this crystal possess drastically different IR intensities, and their interrelation cannot be changed in the theoretical calculation of the spectrum by any assumption on the charge distribution. A correct reproduction of the sequence of weak and strong E modes may serve, however, for the interpretation of the force field parameters which influence the shapes of those vibrations. The difference in the polarity of those modes can be explained qualitatively by deducing their shapes from the shapes of the normal modes of a free SiO4 tetrahedron. Despite the difference caused by connection of the tetrahedra into a three-dimensional network, the polar E mode of a-quartz resembles (in its shape) the F 2 stretching mode of a free tetrahedron while the E mode with low polarity can be deduced from its dipole-less pulsation (A1) mode. Even the information on the direction of the polarization intrinsic to a particular mode of a complex lattice plays an important role in recognition of that mode; its importance being increased with the lowering of the symmetry of a crystal.

In the case of

134


high-symmetry isotropic crystals, all polar modes belong to the same irreducible representation, and their dipoles can be treated as being parallel in any particular direction. Two polar symmetry species exist in the uniaxial crystals, and their dipoles are either parallel or perpendicular to the optical axis. Three sets of polar modes exist in orthorhombic crystals with dipoles parallel to the corresponding crystal axis. The directions of dipoles in crystals of lower symmetry are, however, less symmetry-restricted. Thus, in monoclinic crystals with 2/m symmetry, all A u modes are polarized along the two-fold axis while the B u modes can be polarized in any direction in the plane normal to that axis. No symmetry restrictions exist for the direction of the dipole in the polar mode of the triclinic crystal. A quantitative determination of the IR intensities in crystals is conventionally connected with a measurement of the specular reflection from their suitably oriented faces. At nearly the normal incidence a reflectivity of a light polarized parallel to direction of dipoles of corresponding symmetry species is measured versus the frequency. The frequency dispersion of the dielectric permeability along the cx direction is expressed in the classic Lorentz oscillator approach, where the ionic contribution is represented by the sum of contributions from various phonon modes: oo 15Qt(~)=lso~ + 2

1

0) 2

4nPlO) 21 0) 2

-

- ico71

(2.100)

Here, 4xpl is the strength of 1th oscillator with co1TO frequency and damping factor ),1. The oscillator strength of a phonon mode is found in theoretical computation as: 4npl = IApl2/nf~MlO~ 2 , where M 1 has a dimensionality of mass and is determined from the eigenvector normalization (by means of eq. (2.94)), and f~ is the primitive cell volume. Thus, 4npr 2 in eq. (2.100) can be substituted for the intensity, S 1, if the effective charges are expressed in the units of the electron charge. A trial set of the oscillator parameters, S1, COl, and 71 is used to calculate e(co) and then the reflectance curve, R(o3), by means of the Fresnel's formula. The correct oscillator parameters are obtained by fitting the experimental R(o~) dependence. Sometimes, the fac-

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135

torized presentation of the e(03) dispersion is adopted. It characterizes any oscillator by four parameters: 031 (TO), Yl (TO), 031 (LO), and Yl (LO). The use of this approach, based on the Kramers-Kronig theorem, is preferred by some other authors (e.g., see [76] for details). All of the above approaches implied a reflectance of the system of parallel phonon oscillators, and were restricted in their application to the spectra of isotropic or uniaxial crystals. An extension of the Lorentz oscillator approach on the reflectance from a set of non-parallel coplanar oscillators has been proposed by Belousov and Pavinich [83 ] (see also [84]), and the computational routine for the treatment of reflection spectra of monoclinic crystals has been developed [85]. The expression for e(03) was adopted in the form:

Sleltxelfl EQt,~(03)=E~,~+)-~032 032 - i03y 1

--

(2.101) 1

where elcx and e113are the projections of the unit vector of the dipole moment of 1th oscillator. Each oscillator is specified in this approach by three of the above parameters and another one that determines the angle, 0, between the dipole direction and one of the crystallographic axes of the monoclinic plane. In order to evaluate all these parameters from the experimental data, a series of the IR reflection spectra from that plane is obtained, the polarization vector of the incident radiation rotating stepwise relative to the crystallographic axes. Several spectra corresponding to various polarization are treated jointly in the fitting procedure, which provides a set of refmed oscillator parameters of the B u modes. This approach was originally developed for a better understanding of the lattice dynamics of monoclinic pyroxene minerals, and it was initiated by the results of qualitative analysis of directional properties of B u vibrators in the IR reflection spectrum of diopside, CaMgSi206 [86].

The spectra of isostructural cx-spodumene, LiAISi206, were treated

quantitatively using the methods described above. The structure of this crystal, belonging to the C 2 / c - C6h space group with two formula units in the primitive cell, is shown in projections on the crystallographic planes in Fig. 2.3. The spectrum of a set of parallel di-

136


,7,_7 6

Y

9Si 2 /

o Li o

10

9

2~.

~

.-~1

~._

/

"N

"~

7-

AI

6

11 P9

,-,.7 17 .. - C I ~

16 11/6.. \

12

7

z l : 14\

s .X 3 13

117

10 ', ~'x." 19

19 ....':.'l ~

5 X

( )9

r

Fig. 2.3

~, "x~17

f.2.6

-,.

11 )

7

_. ~ ' ~

llq

The crystal structure of a-spodumene in projections on (001) and (010)

planes. The numbers enumerate the atoms non-equivalent relative to translation. A separation of the primitive cell is indicated.

poles in the A u species is shown in Fig. 2.4 and the results of its reproduction with the fired oscillator parameters of eq. (2.100) are presented as well. The spectra of B u vibrators obtained at various orientation of the polarization vector in ac plane are presented on Fig. 2.5 it together with the spectra repaired by means of eq. (2.101) [67]. The application of these data in searching for the physically consistent solution of the IVP is shown in Fig. 2.6. It is seen that only theoretical and experimental frequencies

LAZAREV R,%

137

100-

0 1300

12'00

11()0

10()0

960

860

700

600

54)0

4()0

360

2()0

14)0

CO,cm "1

Fig. 2.4 The IR reflectance of ot-spodumene for dipoles parallel to the two-fold axis (Au species) and its reproduction (points) by means of eq. (2.100). R, % 100

50 0 50 0

0

t

1200

11O0

1000

900

800

700

600

500

400

3,00

200

,

|

!

,

|

6;0

5;0

4;0

3;0

2;0

1O0

CO, cm -1 R, % 100 50

0

|

o

.

,

|

|

,

1200

1100

|

,

,

1000

900

|

|

8;0

7;0

|

100

CO, cm "l

Fig. 2.5 A series of IR reflection spectra of ot-spodumene obtained from the plane of monoclinic axis (Bu species) at various orientations of electric vector and their reproduction by means of eq. (2.101).

138


Ag exp

Ill!lill

calc Bg exp

ill ill I

Ill

calc A n

exp

I

calc

I

n u

I

I

I

I

exp

/--I Z

ll [I

9?

/

I

l

l l 0 0 1000 900

I

I

I

I

800

700

600

500

cm

Fig. 2.6

,

II I

II Z

I

i

I

,

I,l,,ll

I

?%

~-.. St

,,I

i

i

I

.e,

i

I

i

I

I

I

400

300

200

l

100

-1

A comparison of the experimental data on the spectrum of ot-spodumene

with the results of its calculation by means of adopted dynamical model [67]. Note that only frequencies can be compared in a case of Raman spectra (dotted lines correspond to weak bands excluded from the fundamentals).

are compared in the Ag and Bg species since the approach to the calculation of Raman intensities is not proposed. Both the calculated frequencies and IR intensities are fitted in the A u normal modes. These are complemented in the B u species by the dipole directions, which have been fitted as well. This data provides a more dependable solution of the IVP than one deduced solely from the frequency fit [67]. It should be emphasized that the data on the spectrum of diopside from [86] employed in the paper [87] with neglect of the direc-

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139

tional properties of B u dipoles led the author to a misassignment of the bands and, correspondingly, to an apparently wrong solution of the IVP. This approach has since been successfully applied to various other monoclinic crystals. Even in the case of very complicated potassium feldspar microcline, which is triclinic, and can only approximately be treated as monoclinic (C2/m), it was possible to identify most of the symmetry allowed (17 A u and 19 Bu) normal modes of the lattice containing two KAISi30 8 formula units in the primitive cell. The reflection spectra of a more or less perfect single crystal have been investigated in the polarized, light and the parameters of phonon oscillators evaluated by means of eqs. (2.100) and (2.101), respectively [88]. These data, complemented by the results of a single-crystal Raman spectra investigation [89], allowed a reliable dynamical model [90] to be proposed, and substantiated, apart from other considerations, by its ability to reproduce the experimental orientations of the B u dipoles. Conversely, it has been shown that the earlier set of force constants of that crystal [91 ] that were deduced only from the frequency fit, would lead to the shapes of normal modes in the B u species, and are incompatible with the experimental orientations of the dipoles irrespective of the adopted scheme of the charge distribution. More recently, the same approach was applied to the re-investigation of the spectrum of the simplest condensed silicate, thortveitite, SC2Si20 7 [92]. The original assignment and force field model [93] of this monoclinic (C2/m - C3h ) crystal (with one formula unit in the primitive cell) were deduced from the IR absorption study of the powder specimen. The investigation of the synthetic single crystal in the IR reflection and Raman scattering with the use of polarized radiation [92] provided much more experimental data which were employed in the normal coordinate treatment. Even the prediction of elastic constants and the compressibility of this crystal have been attempted using the refined set of force constants (cf. the next subsection) [94]. A relatively simple electro-optic model was adopted in all IR intensity calculations mentioned above, since these mostly pursued a visualization of implicit information on the shapes of the normal modes hidden in the corresponding experimental data. A deficiency

140


of the simplest concept of constant atomic (ionic) charges rigidly bound to corresponding nuclei was clarified in earlier computations [66], and these were complemented by the mechanism of the charge transfer related to the Si-O bond elongation in the complex anion or silicate network as it was originally proposed [95]. Among various model presentations of the electric response functions discussed in the beginning of this chapter, this approach corresponds to the simplest two-center version of the VCM. The dipole moment increment (per unit cell) which defines the IR intensity, S, of any mode is calculated in this approximation as: AP=

~ u(j)z~ overatoms

~ Alc over bonds

(2.102)

where the atomic displacements, u(j), and the bond elongations, Al, are determined by the eigenvectors, L, in the Cartesian or internal coordinate space, respectively. The parameters of the model are the equilibrium (constant) atomic charges z ~ and the effective charges

c = l~ z / a l (l ~

is the equilibrium bond vector). It is agreed to relate the

sign of the c charge to the oxygen charge increment; the negative sign thus corresponding to the increase of that charge on the bond elongation (there was some controversy in the earlier determinations of that sign for the Si-O bonds [67,76] originating from the indeterminacy of the AP sign when deduced from the experimental S magnitudes; see [9] for discussion). Similar parameters are not introduced for the bonds between the complex silicate (or any other) anion and monoatomic cations, which reduces the numbe- of independent parameters of the model at the cost of poor reproduction of the experimental IR intensities of the lowfrequency modes related to displacements of cations. It should be added that the expression (2.102) is decomposed in its practical application into the projections on the Cartesian axes. A very simple (and quite general) explanation of a particular success of the application of the IR intensity analysis to the IVP for the complex systems can be proposed. The more complex a system is, i.e., the larger the number of eigenfrequencies in the same interval, the less their exact magnitudes may be used as the intrinsic properties of particular

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141

mode (since their small differences are often determined by very faint differences in the force field). On the other hand, these modes can hardly be confused with any trial approximation of the force field in the solution of the vibrational problem if their shapes determine a sufficiently large difference in the direction and/or the magnitude of the polarization. Unfortunately, this additional information is usually insufficient to define the unique solution of the mechanical IVP. Some other restrictions, either originating from the first-principle considerations, or from the experimentally accessible quantities, would be of great importance to reduce the ambiguity.

B. A Generalization of the Inverse Vibrational Problem and a Notion of the Shape of Uniform Strain of a Lattice As a macroscopic body, any crystal is characterized by its size and shape, which can vary under external action. It is described in a case of homogeneous deformation changing all unit cells uniformly as a variation of lattice vectors: Aa i = Ua i

(2.103)

where U is the uniform strain matrix whose elements have been already interrelated with the atomic displacements u, (which represent the microscopic scale pattern of that process) through the positional vector, see eq. (2.23). It should be emphasized that, in a general case, a variation of structural parameters (atomic positions in various sublattices) does not occur similarly at the microscopically homogeneous deformation because of the inhomogeneity of the force field of a crystal. Since the earliest investigations in this area, it is adopted to separate (for convenience) two contributions, the external and internal, to the structure variation on the uniform strain. The structure variation is thus treated as being constituted by two sequential processes. The first of them changes all structural parameters as if the change of lattice vectors would lead to a microscopically homogeneous structure variation while the second one represents the rearrangement related to the relaxation of forces induced by the variation of the size or the shape of a cell.

142


The second of those processes relates to the F matrix of a considered crystal and therefore, it is worth reducing the description of the uniform strain to a form similar to one adopted in the treatment of the vibrational problem. It was originally proposed [78] and taken into consideration when developing corresponding computational routines [81,96,97] and applying them to particular problems [79,80,94,98]. The most important statements of this approach are outlined below. The uniform strain matrix is usually treated as a six-component vector, U, each component being specified by a corresponding number called the Voigt's index. Remembering the above interrelation between internal coordinates g and normal coordinates Q, g = LQQ (2.96), and specifying hereafter the matrix of the shapes of the normal modes by the Q index, it is possible to introduce a notion of the shape of uniform strain LU which interrelates a set of internal coordinates corresponding to zero wave vector [78] with U vector as:

g-LuU.

(2.104)

Here, LU is a matrix whose lines are the uniform strain vectors. The dimensionality of this matrix is evidently 6 x N where N is the total number of internal coordinates. The elements of the LU matrix can be calculated in the form of explicitly separating the external and internal contributions to the shape of homogeneous deformation of a lattice [56]: dgi ( 0gi ) ~1 ( t g Q l ) Lil tgUm U' dU---m = tgUm Q +

(2.105)

where I enumerates the normal modes. The subscript Q denotes that all Q1 are kept constant during differentiation, while the subscript U' implies all components of the U vector constant except the variable of differentiation. The first term in the right-hand side of eq. (2.105) is determined completely by the geometry of a lattice, and represents the external contribution to the LU matrix (the microscopic structure transforms similarly for any sublattice).

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143

The second term in (2.105) represents the internal contribution to LU which originates from the relative shifts of various sublattices, and is therefore dependent on the force field of a crystal. Rewriting eq. (2.104) as: LU = dg/dU, and taking into account the defmition of elastic constant matrix as: C = d2v/dUdU, one can interrelate it with the F matrix specified in the internal coordinates as [78]: C = L~jFL U .

(2.106)

It is clearly seen from this expression that the LU vectors are not the eigenvectors of the force constant matrix since the C matrix is not diagonal. If the left-side values in eqs. (2.95) and (2.106) are experimentally determined, their joint solution with respect to the F matrix may be searched, and thus the ambiguity of the IVP restricted. This opportunity has been exploited by Shiro [99] who calculated vibrational frequencies and elastic constants of a-quartz by the force constant method in an attempt to design a more physically consistent force field model. It should be emphasized, however, that this approach is tmable to completely remove the indeterminacy of the IVP since the LU vectors are generally inaccessible to direct experimental determination. The eq. (2.106) is expanded for the calculation of a particular matrix element Cmn in the form: Cmn = ~

dgi Fik dgk 9 dUm dU n

(2.107)

where m,n = 1, 2 . . . . 6 and are the Voigt indices. The Born and Huang's approach [1 ], which separates the external and internal contributions to the properties of a crystal relative to homogeneous deformation, has been adopted in eq. (2.105) to describe its shape in terms of the LU matrix elements. It should be reminded that in this approach, the Um components will denote below the external coordinates of a crystal treated as a homogeneous macroscopic body. Thus, the U space whose ordinates are specified by the Voigt indices and the orthogonal space of normal coordinates Q are employed in description of the homogeneous

144


deformation of a crystal representing its dual nature as a macroscopic body possessing microscopic internal structure. These two spaces are not orthogonal respective to each other, and non-vanishing C3Ql/c3Um values characterize the so-called optic-acoustic coupling [100]. That coupling is represented in the harmonic approximation of the potential energy description by mixed second derivatives Dim = c32v/tgQlc3Um. Their magnitudes are interrelated with the F matrix defined in a space of internal coordinates, g, by the following expression: g g i ) Q FikLkl Dim = . i,~k(k,tc3Um

(2.108)

where the meaning of the Q subscript is the same as in eq. (2.105). The interrelation of Dim values with the parameters of optic-acoustic coupling (C3Ql/C3Um)is determined as: aQl / aUm = - Dim / c~ 9

(2.109)

Using the above notation, the following expression for elastic constants that separates the external and intemal contributions can be proposed: ( Ogi )Q Q - ~ DlmDln Cmn = i,~k\0Um rik~0Un) 1 c~

(2.110)

Note that in this expression, both contributions depend on the force field of a crystal. The parameters Dim show, how the internal deformation induced by the external strain U m is projected on the orthogonal degrees of freedom of the microscopic structure represented by the normal modes QI. This expression is applicable to the analysis of the contributions of various phonon modes to elastic constants (see, e.g., [56]). The derivatives of elastic constants relative to the force constants are needed to search for a joint solution of eqs. (2.95) and (2.106). Instead of a relatively complicated and commonly used method of their computation [101 ], a simple expression has been proposed, using the notion of the shape of the uniform strain (which defines these derivatives as a direct product ofL U matrices):

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145

dC/dF = L U x LU .

(2.111)

This formula is similar to the corresponding expression for the derivatives of squares of vibrational frequencies, dA/dF = LQ x LQ [71]. It was rigorously proven in ref. [78]. It should be emphasized that a decomposition of elastic constants and other properties with respect to the uniform strain into external and internal contributions is a convenient, but rather formal approach, which can, in principle, be avoided [78]. In this case, the U m components are implied to include both the extemal homogeneous deformation of a crystal and its internal structural relaxation. The advantages of an approach to the IVP solution, which testified the shapes of normal modes deduced from the frequency fitting by the calculation of IR intensities, were discussed in the previous subsection. That approach, combining the treatment of mechanical deformation and polarization problems, can be extended on the homogeneous deformation of a lattice as well. The eq. (2.99) is expanded along any direction (Cartesian axis a) as: dPa dgi dP~t = E ~ g i ~1 dQ1 i

(2.112)

where l is the normal mode number and i enumerates the internal coordinates. The left-side values are the square roots of IR-intensities. A conception of the shape of the homogeneous deformation is useful in expressing the tensor of piezoelectric constants, e, in a form similar to eq. (2.99) for polarization induced by phonon modes: e=ZLu .

(2.113)

Any particular element of that tensor is determined by expression which is formally similar to eq. (2.112): dPa du

m

~

dPot i

dg i

(2.114)

146


where m are the Voigt indices and dPa/dgi are treated the same as the parameters of electrooptic model in eq. (2.112). As an application to particular crystal shows in numerous cases, any physically reasonable variation of the latter influences the calculated piezoelectric constants less than the peculiarities of the shape of the corresponding uniform strain. A dependence on the force field of a crystal determines the applicability of the piezoelectric effect to the IVP solution. If the above mentioned approach (which decomposes the homogeneous deformation into external and internal contributions) is employed, an expression for the piezoelectric constant with separated external and internal contributions can be deduced in terms of opticacoustic coupling:

etxm =

( c3Pct / - ~ dP~ Dlm aUm Q i ,__dQ 1 co21 "

(2.115)

Here, the first term representing the external contribution is independent of the force field. The internal contribution is obtained as a sum of magnitudes that determine the IR intensity of various normal modes, each being multiplied by the value proportional to the corresponding optic-acoustic coupling parameter. The eq. (2.115) is thus applicable to the investigation of the origin of the sign and magnitude of the piezoelectric effect, and its interrelations with IR intensities of various phonon modes (see, e.g., [56]). Moreover, it can be concluded that at favorable circumstances, the spectroscopic data can be employed to evaluate both the elastic and piezoelectric constants if they cannot be measured directly. This perspective also relates to the determination of the signs of those constants which are not deduced from some types of experiments. As it relates to determination of the force constants from spectroscopic data, the above considerations provide a generalized formulation of the IVP which will be referred to as the GIVP. If the properties of a crystal are specified in a space of internal coordinates, it implies a joint solution of a system:

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2 = LQ FLQ C = L U FL U

(2.116)

i.t = ZLQ e = ZL U

where the last two equations represent the polarization processes, and are employed to restrict the arbitrariness in the estimation of the shapes of the normal modes, and of uniform strains from the first two equations. This approach proved to be useful even in a relatively seldom met occasion when a complete set of spectroscopic data applicable to the IVP solution is available. An example treated below relates to lithium metagermanate, Li2GeO 3 [ 102]. This crystal attracted some attention because of its mechanical and electromechanical properties, and large perfect single crystals have been grown [ 103,104 ]. The structure of this orthorhombic (Cmc21-C 12 ) crystal, which has been recently refmed [105], is shown in projections on the crystallographic planes in Fig. 2.7. The numbering of non-equivalent atoms relative to translation being indicated. This wurtzite type lattice with tetrahedral coordination of any atom possesses two Li2GeO 3 traits in its primitive cell. The character of bonding allows the endless complex anionic chains along the c axis to be discerned, each being composed by a repeating one-dimensional primitive cell containing two linked tetrahedra, (Ge206)o o. The factor-group of a one-dimensional space group [ 106] of this chain possesses all symmetry operations of the factor-group of a threedimensional space group of the crystal (which was taken into consideration in the earlier attempt of spectral assignment). The longwave optical modes are distributed over the symmetry species as follows (the directions of dipole moments of polar modes are given in brackets): 9A l(Z) + 8A 2 + 7B 1 (x) + 9B2(Y ) .

148


1

, I

w

: I

,

I

I

I

I

1

~

I

1

I ~

1

i

,

I

'1

l I

I

,

,

3

I

I

2

I

~. 0 for all l and at any magnitude of wavevector q. Reversely, a condition ~,l < 0 defines

the energetic gain from the spontaneous deformation of a lattice along the LQI direction which leads to a displaced type structural phase transition. The Q1 mode is referred to, in this case, as a soft mode. The soft-mode concept is presently a basic one in the treatment of the origin of various second-order solid state transitions. Similar considerations on the nature of the ferroelastic phase transition can be proposed by the adoption of the notion of the shape of a uniform strain interrelated with the elastic constant matrix of a crystal by eq. (2.106). It should be restated, however, that the LU vectors are not the eigenvectors, and it is appropriate to begin with the problem of the diagonalization of the C matrix. An extemal strain N is introduced that it is interrelated with the uniform strain U as:

166


U=AN

(2.134)

,

where A is the matrix of the C-matrix eigenvectors. The latter brings the C matrix to a diagonal form: C' = A+CA

,

(2.135)

and the problem of stability relative to the external strain may be formulated as a condition of positive signs of the C matrix eigenvalues. If one of the C' matrix elements is negative, a spontaneous external deformation: Nn =

AnlmUm,

(2.136)

corresponds to the ferroelastic phase transition originating from the C n < 0 condition. The above expressions ensure the symmetry considerations relating to that transition: the symmetry of the N n deformation can be determined, and a linear combination of the C matrix elements vanishing in the transition point can be found [ 122,123 ]. The considerations on the interrelation between the internal tension in a lattice and the origin of ferroelastic instability caused by the hydrostatic compression are set forth following the original treatment [80]. A more detailed discussion of this and some related problems may be found in several papers of the book [124]. A conventional treatment of the uniform strain (2.103) is adopted. It treats that strain as being constituted by the external strain which represents the deformation of a homogeneous medium and the internal strain representing the mutual shit, s of various sublattices induced by the change of lattice vectors. The internal strain thus describes the relaxation of the microscopic structure of a crystal at variation of its size (or shape) as a macroscopic body and is described in a linear approximation by the following interrelation: u(U) = XuU.

(2.137)

In order to calculate the XU tensor, the U coordinates are to be introduced into the potential energy expansion (2.125):

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167

V(u, U) = ~o + Ouu + t/~tJU + 1 (~uuU2 + 2 ~uUUU + ~

U2) ,

(2.138)

where the higher-order terms are omitted. The SC (2.130), which needs zero forces on atoms, should now be complemented by the condition of the vanishing macroscopic stress, ~0u = 0

,

qxtj = 0

.

(2.139)

An adiabatic nature of the microscopic structure relaxation under the uniform external strain of a crystal which may be expressed in a form:

Ou U=const

= 0 ,

(2.140)

leads to the following expression for Xu: X U =r

-1

.

(2.141)

An expression for the PF of uniform strain is then obtained by substitution of (2.141) into eq. (2.138), the conditions (2.139) being taken into account: w(U) = l ( ~ . J U - (g~ju~ - uu 1 ~uU ) U2"

(2.142)

It is evident that the expression (in parenthesis) in this formula represents the elastic constant matrix, c=

~m-

r

-1

,

(2.143)

where the first and second terms correspond to the external and internal contributions, respectively. Now, proceeding to the description of elastic properties with the PF specified in a space of internal coordinates, it is necessary to complement eq. (2.123) by the introduction of the dependence ofg on U in explicit form: g = Bu u + BU U + 1 (Buuu2 + 2BuUUU + B u u U 2 ) .

(2.144)

A substitution of (2.144) into eq. (2.124) leads to formulas applicable to the investigation of the SC (2.139),

168

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS VgB u = 0

,

VgB U

= 0,

(2.145)

and to expressions for the ~uu, ~uU, and ~

tensors: (2.146)

9 uu = 8u+ vggSu + V g ~ u u ,

9 uu = Bu+ vggBu +

vg~.u

(2.147)

,

and = B ~ Vgg~u + V g B ~

.

(2.148)

These expressions make a calculation of elastic constants possible using eq. (2.143), and express the matrix of the shape of the uniform strain defined by eq. (2.104) through the Xu tensor: LU = BU + BuXU .

(2.149)

Among the above expressions, the eqs. (2.143) and (2.147-2.148) were originally deduced by Shiro [99] and represent a conventionally adopted scheme of the elastic constant calculation in internal coordinates. A notion of the LU tensor as the shape of the uniform strain specified by eq. (2.104) (see eq. (2.149) as well) permitted the proposal of a more simple and physically clear expression (2.106) deduced by Smimov and Mirgorodsky [78]. Now, it follows from the above considerations that in a presence of internal tensions in a lattice, the frequency of Qth normal mode is calculated as: d2g dg dg' t.O~ = ~ V g g , -d-~ + V g ~ dQ 2

(2.150)

where: d2g dQ 2

02g

du du'

0u0u' dQ dQ

and the UU' component of the elastic constant tensor is obtained as:

(2.151)

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169

dg dg' d2g CUU, = -d--~Vgg, dU' + Vg dUdU' '

(2.152)

dg tgg = ~+ dU c3U

(2.153)

where: Og du tgu d U '

and d2g

0 2g a 2g du 0 2g du a 2g du du' = ~ + ~ F t . dUdU' 0UOU' 0Uo~ dU' 0U'0u dU 0uc~' dU dU'

(2.154)

A curvilinear nature of the internal coordinates clearly reveals itself in these expressions. They are also applicable to the analysis of the Potential Energy Distribution (P.E.D.) of any normal mode or of a particular macroscopic strain of a crystal in terms of contributions of various origin. It was emphasized repeatedly by numerous authors [2,3,125,126] that the normal mode frequencies and elastic constants are volume-independent quantities in the harmonic approximation, and, correspondingly, no pressure- or temperature-dependence of their magnitudes exist. The above expressions show, however, that in a space of internal coordinates, no higher than the second order terms in the potential energy expansion are needed in order to deduce the volume dependence of those quantities. In the case of crystal under any external action, the SC (2.145) should be rewritten as [127]: VgB u = 0

,

,.next

V g B U = "" U

,

(2.155)

where ,.next ~ ' o unit is some static external stress. The trivial solution of the system (2.155) Vg = 0 does not exist in the case of crystal in a balance with the external field, and an investigation of solutions of that system relative to the Vg magnitudes provides their presentation as the functions of external stress. The contributions of these magnitudes to the normal mode frequencies (2.150-2.151) or to the elastic constants (2.152-2.154) specify their volume dependence.

170


In this way, a dependence of the phonon spectrum or of the elastic constants tensor upon any external action that changes the lattice parameters can be considered. In a particular case of the hydrostatic compression, the external pressure, p, is formally treated as a six-component vector and: gsext u = (p,p,p, O, O, O) .

(2.156)

A solution of the system of equations (2.155) provides Vg magnitudes as the functions of the hydrostatic pressure. At appropriate interrelations between Vgg values and pressuredependent static forces, Vg, the optic mode sottening or the ferroelastic type instability may arise, and a simple mechanical treatment of the origin of the phase transition is possible. This approach has been developed originally [80,124] in the investigation of the nature of the pressure-induced second order ferroelastic transition in paratellurite, TeO2. This crystal was known to undergo (at 9kbar)the tetragonal-to-orthorhombic (D 4 ~ D 4) transition with neither a discontinuous volume change nor a multiplication of the unit cell. It was deduced from the symmetry requirements that the elastic matrix eigenvalue C' = C 11 C12 should vanish at the transition point and a simple force field model reproducing this instability was proposed. This approach was successfully adopted later [128] to the pressure-induced phase transitions in ReO 3. It was proposed by Catti [ 129] to treat the thermal expansion formally as an extemal action changing the lattice parameters, and thus causing the temperature-dependent internal tensions. The uniform strain is expressed through the thermal expansion tensor A and temperature T as U=AT. It is possible to substitute ",.r,' uext =CAT into eq. (2.155) and to extend the above approach to a simple quasiharmonic treatment of the effects induced by the temperature variation.

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IV. S E V E R A L C O M P U T A T I O N A L P R O B L E M S

A. Geometry Optimization and Potential Function Refinement If the initial approximation of the PF of a crystal is specified in any way, two typical problems are usually investigated. Using the static SC in a form (2.139, 2.145) for a free cold crystal or (2.155) for a crystal in an external field, it is possible to calculate the equilibrium geometry. This formulation of a problem is adopted when a prediction of some unknown crystal phase is attempted proceeding from very general considerations on the nature of the interatomic potential. However, the equilibrium geometry is more otten determined experimentally. In this case, the deviation of the theoretically calculated geometry from the experimental one is utilized in a refinement of the PF parameters. The importance of the SC analysis in attempts to restrict the ambiguity of their determination was emphasized by

Boyer [58]. A general scheme of the structure variation at a given PF model up to fulfillment of the SC consists of an iterative repetition of the following steps: (i) a selection of an approximate geometry R(n); (ii) a calculation of the corresponding internal forces ~(un) and uniform stresses

(iii) an analysis of the non-vanishing forces and stresses and the design of a new trial geometry,

(2.157) The most frequently used version of this iterative formula is the one adopted in the Newton-Raphson method [130] which ensures the quadratic convergence. This approach requires the H matrix calculation (whose elements are the second derivatives relative to all geometric parameters) to be varied. In a given case, the H matrix is composed by the blocks:

172

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS 02 v

O2V

.......

n

cq2V

]

/ 02V

[

(2.158)

and the iterative formula in the Newton-Raphson method can be thus expressed as: ui~ / = H -1 ((*U)kx ]

(2.159)

If the investigation assumes a subsequent calculation of the phonon spectrum and elastic constants, the elements of the H matrix are to be determined in any case, and the application of the Newton-Raphson method to geometry optimization does not involve a considerable amount of additional computations. However, in a case of the geometry optimization for the crystals with complicated structm'e, the applicability of this method is restricted by the necessity to determine all elements of the H matrix. In this case, it seems more practical to divide the geometry optimization into two steps. At the first step, the optimization is performed along the internal degrees of freedom, uiot, keeping the external ones fixed (U~). The simplest diagonal approximation is applicable on this step: H = L/

(2.160)

where X is an empirical (averaged) coefficient ranging from 5 to 10 mdyn/A and I is the unit matrix. Then, the residual external stresses are relaxed using the relation:

H=C,

(2.161)

where C is the calculated elastic constant matrix. Both steps can be repeated when a more precise optimization is required. The approximation (2.160) is very economic computationally, but it only enables a convergence of the first order. This disadvantage is most important when non-rigid degrees of freedom are existing. In this case, the variable metric optimization method can be util-

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173

ized as some compromise between accuracy and simplicity. The Murtagh-Sargent algorithm [ 131 ] proved to be very suitable in the rapid geometry optimization for polyatomic molecules as has been shown by Mclver and Komomicki [132]. As follows from our own experience, this approach is successful in application to the crystal geometry optimization as well. Only few comments are given below to the problem of the refinement of the PF parameters by the SC investigation. In numerous SC investigations, the atom-atomic PF of the type (2.46) are adopted. The ionic charges are conventionally fixed as the integers, and the SC investigation in that approach is restricted to the determination of the parameters of the non-Coulomb part of the PF. In a cubic diatomic crystal, the single SC ((tI~tj)act = 0) is sufficient to determine one of the parameters of non-Coulomb potential. The number of independent SC rapidly increases with the complication of structure and loss of symmetry. In simple PF models, it can sometimes even be in excess of the number of the parameters to be determined. In this case, a problem of existence and consistency of the joint solution of all SC equations may arise. More importantly, however, the problem is how to find a physically meaningful potential in this way. A systematic study of a series of chemically related crystals containing similar structural elements often helps to avoid a casual solution applicable only to a particular structure. This approach has been implemented, e.g., by Parker et al. [ 133 ] who estimated the PF parameters for silicates using six reference lattices, and then applied them to a precise computational reproduction of the structure of fifteen complicated minerals with island, chain, or ring complex anions. Even in a case of coincidence of the number of SC and PF parameters, when a unique solution is possible in principle, the solution should be tested for its stability with respect to the uncertainty of experimental values and a small variation of the PP model. Moreover, it should be taken in mind that there is no physical reason to exactly reproduce the experimental equilibrium geometry by means of one calculated with a very simplified PF model. A requirement of the approximate coincidence of these two geometries would

174


be more reasonable. Such criterion constitutes a base of the well-known and widely applied Busing's method [ 134]. As a matter of fact, this approach adopts the least squares method to the PP parameter determination using the SC. It consists of the variation of the PF parameters conditioned by the minimization of the function: f= E (tDu)i2a + E (*U)~y is 13y

,

(2.162)

where the residual forces and external stresses represent the extent to which degree of accuracy the structure can be reproduced by a given PP model. The applicability of this approach is not restricted by the interrelation of the number of equations and the parameters to be determined. Another advantage relates to the determination of PF parameters jointly utilizing the data for various crystals. A more reliable PF evaluation is obtained, however, when the structural data are jointly treated with various physical properties. The Busing's approach is easily generalized, e.g., by adding some terms of the type (C calc - Cexp)2 to the minimized function where C is any physical value calculated with a given PF or determined experimentally.

B. Crystal Mechanics Program A versatile crystal mechanics program CRYME was compiled by M. B. Smimov [81,96,97,124] to ensure the numerical realization of most approaches described in the previous sections. The program is written in FORTRAN code and tailored to operate on an IBM PC. The program is applicable to the following: To the calculation of the equilibrium geometry, lattice dynamics, macroscopic elastic, piezoelectric, dielectric and thermodynamic properties of a crystal when its interatomic potential function and charge density distribution functions are specified.

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9 To the evaluation of the parameters of the above functions by fitting the available experimental data of the structure and deformational properties of a crystal. 9 To the prediction of some dynamic properties not easily accessible to direct experimental determination proceeding from the dynamic model based on the known properties. 9 To the explanation and theoretical treatment of the structural-response to an external force acting upon a crystal and changing its size and shape homogeneously. 9 To the investigation of the origin of instability of a lattice relative to the variation of external parameters which causes a change of lattice vectors. 9 To the examination and comparison of the stability conditions for various polymorph modifications of a certain compound. 9 To the proposal of the microscopic scale quantitative explanation of some structural features and peculiarities in physical properties relating to the nature of bonding. 9 To the deduction of the correlations between various structural and/or dynamic properties and to their application corresponding to the estimations for the crystals not yet studied experimentally.

C. The Operation of the Program CRYME provides for the calculation of bond lengths, valence angles, and dihedral angles from the input structural information, and further their use in the design of internal coordinates which are then utilized in the description of the potential and charge density distribution functions and in the computation of various dynamic properties. The following quantities are computed by CRYME:

176


9 Electric values. Electrostatic potential: the strength and gradient of the electrostatic field of a lattice. 9 Mechanic values. The forces acting upon the atom from any term of the PF and their derivatives with respect to atomic displacements (contributions to dynamic matrix). 9 Dynamic properties. The frequencies and shapes of the normal modes in the long-wavelength limit and at any given wave vector, phonon dispersion curves and vibrational density of states (DOS), partial derivatives of the vibrational frequencies with respect to the force constants and the potential energy distribution (P.E.D.), the dipole moment and polarizability derivatives of normal modes, the TO-LO split, tinge, and dielectric constants. 9 Elastic properties. Macroscopic elastic and piezoelectric constants and their derivatives relative to the parameters of the PF. The microscopic pattern of the structure relaxation in a hydrostatically compressed crystal or at any other external action changing the lattice vectors. 9 Thermodynamical properties. Heat capacity and Debye temperature. CRYME graphically represents the structm'e of a crystal, the shapes of normal modes, the phonon dispersion curves, the diagrams sketching the experimental and computed IR and Raman spectra, and the thermodynamic functions vs. temperature. The following approaches to dynamic problem are provided by the program: 9 A PF description, which does not imply the explicit separation of the Coulomb contribution, namely, the GVFF and UBFF, models. 9

Two-body interatomic potential functions (IPF) applicable to both calculations implied by the former item and to the description of the shortrange interaction in approaches providing for the explicit treatment of Coulomb contribution; their analytic expression in the form Aexp(-R/B),

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177

A/RB,A(R-B)2 or any possible combination of these terms is assured by the program. 9 An explicit calculation of the Coulomb contribution in terms of RIM, PIT, and SM approximations. CRYME provides employment of the databases on space symmetry groups, some atomic parameters and standard reference potentials. The user of the program does not need a detailed knowledge of how the input information should be formed. All input data are defined in a dialog "program-user". During the dialog, the new input files are opened, and all input information is written in those files in a formatted set. If the user wants to repeat some calculations, it is possible to use the created input files. When it is desirable to change some parameters, the input files can be edited. However, there are limitations of the version of CRYME tailored to operation on an IBM PC: 9 the number of atoms in a primitive cell - up to 30, 9 the number of various force constants - up to 50, 9 the number of various PF models - up to 10. A general outline of the program is given in Fig. 2.14 and only a few further comments will be given below. The computations can be carried out either without any symmetry restrictions originating from the point group operations or by making use of a whole space group of a crystal. The latter is entered in the program through its number in the International Tables [135] with the generation of that group by the program or, when necessary, by a manual introduction of all symmetry transformations. If no symmetry is used, the crystal system and the type of cell should be specified. Then, one should enter the cell parameters and interaxial angles. When the space group is specified, the program generates all equivalent positions. The fractional atomic coordinates in a cell are conventionally used. If no symmetry is in-

178

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS CRYSTAL STRUCTURE I

INIERNAL COORDINATES

' "l ","l

LATTICE SUMS

CHARGE DISTRIBUTION

'

i i

~

9

: tension adjustment and ' . P F refinement '.

i

}malysis of the lattice instability and structure refinement

.

"~ "~.

~9

adjustment

.,"of charges i~ ~,"I"

9

s,s"

STABILITY CONDITIONS

[

NORMALCOORDINATES

I

I

I I MICROSCOPIC STRUCTURE OF HYDROSTATIC COMPRESSION

I

k.__

I PIEZOELECTRICITY ~_cL._

Fig. 2.14 A general layout ofM. B. Smimov's Crystal Mechanics program.

troduced, it is possible to define the atomic positions in Cartesian axes. Denoted by the corresponding chemical symbols of all non-equivalent atoms in a cell, one can enter their standard parameters, the atomic masses, and ionic charges-and polarizabilities. These can be varied during the computations. In order to introduce the internal (vibrational) coordinates, the program forms (on the user's request) the list of bonds, i.e., of the interatomic distances up to R 1, the bond angles between bonds shorter than R 2 (R2< R1) and the dihedral angles which describe a torsion of

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179

the two bond angles having a common bond. The numbers of the introduced internal coordinates are then employed to specify various terms of the PF. The non-Coulomb interaction may be specified in the program by any arbitrary combination of the GVFF or UBFF force constants, and, if necessary, by some contribution from a suitable IPF. For convenience of the force constant description, a possibility to introduce the linear combinations of intemal coordinates is provided for. Any combination is specified by its type (composition) and coefficients. The IPF contribution to the force field is specified by the type of adopted analytic expression, and by setting the range of interatomic distances to be taken into consideration. The bond dipole and bond polarizability concepts are adopted in CRYME when calculating the spectral intensities in approaches, which do not provide for an explicit treatment of the Coulomb interaction in a lattice. Each bond is specified in this case by the bond dipole derivative and by the derivatives of the longitudinal and transversal bond polarizabilities. A calculation with explicit separation of the Coulomb contribution can be carried out in terms of RIM, PIM and SM approximations as has been mentioned above. The latter of them is reduced in the program to the so-caUed simple SM where all non-Coulomb interactions are implied to relate only to the shell-shell interactions. Since a treatment of the Coulomb interaction in~ olves the lattice sums computation, which is a rather time consuming routine, it is possible to perform a series of computations with the same lattice sums saving their initial values in a special file. The typical problems, which can be investigated by means of CRYME, are classified as follows: 9 The analysis of the SC. Formulation in the form (2.155) is adopted. Two different problems solved by the SC fulfillment are to be discemed. The first one consists of finding the PF parameters from the known crystal structure. The program provides for an adjustment of non-Coulomb PF parameters, which minimizes the mean square of forces or of forces and stresses. The program

180


calculates their partial derivatives with respect to the PF parameters, forms a Hessian matrix, and proposes the increments of the PF parameters thus diminishing the forces and stresses stepwise. The accuracy of fitting is selected by the user. If some additional restrictions should be taken into account, it is possible to set the limits of the variation of the PF parameters by noting the maximal and minimal values for each of them.

The introduction of weight factors helps to highlight some forces and

stresses. The program ensures a determination of the structure of internal tensions in a lattice. E.g., if the Bom-Karman approach, which operates with two types of the PF terms (A=d2V/dR 2 and B = = (dV/dR)l/R) is adopted, the number of introduced interactions may significantly exceed the number of independent SC, and a set of B parameters, which corresponds to zero forces and minimal tensions, should be found. Another type of problem investigated by means of the SC fulfillment relates to the determination of the equilibrium crystal structure corresponding to the fixed PF model. Only the condition of zero forces is treated in this routine. The user should choose between two geometry optimization methods: the force relaxation method (FRM) and the variable metric method (VMM). The former adopts the Newton-Raphson approach. On each step of the atomic coordinates optimization, both the forces and the metric matrix are calculated. The FRM procedure is more time consuming, but it ensures a quadratic convergence. The VMM approach does not need the second derivatives computation. At the first step, the metric matrix is reduced to a diagonal one. Then, it is redefined by the iterative MurtaghSargent method. This approach is faster, but not effective if the initial geometry is far from the equilibrium geometry. It is possible to avoid a redetermination of the metric matrix. In this case, the optimization method transforms into the steepest descent approach. The SC for macroscopic stresses is tested atter the elastic constant computation. The program calculates the residual external stresses and proposes new cell parameters, which are closer to the equilibrium ones.

LAZAREV

181

The program is arranged in a form suitable for the joint treatment of lattice vibrations and the properties of a crystal relative to the uniform strain as it was proposed by the GIVP formulation discussed above. The employment of phonon spectra, elastic and piezoelectric constants and the microscopic structure of hydrostatic compression to a ref'mement of the PF parameters or to deducing the unknown properties from the experimentally accessible ones is provided. CRYME enables the calculation of the normal modes (eigenfrequencies and eigenvectors) in the center of the Brillouin zone center either without any symmetry considerations or making use of the point symmetry factorization of the secular equation. The latter procedure is initiated by noting the character or the matrix element of symmetry transformation for each irreducible representation. The program then designs the symmetry coordinates, and reduces the secular equation to a set of independent equations of lower power. The same relates to the normal mode calculation at a fmite wave vector where its symmetry group can be taken into account. It is possible to calculate the whole set of vibrational branches in one pass if the initial and terminal values of the wave vector and the number of points per interval are indicated.

A graphical presentation of vibrational

branches is provided. For the longwave limit vibrations, the program determines their shapes in the amplitudes of atomic displacements or in internal coordinate increments and calculates the IR and Raman intensities if the corresponding electric response functions are specified. A graphical presentation of the shapes of normal modes is possible. The partial derivatives of the vibrational frequencies with respect to the force constants are calculated as well. A Coulomb contribution to the longwave frequencies may be determined if the adopted PF separates the corresponding interactions explicitly. The elastic properties are treated in conventionally accepted terms of the external strain/internal strain interrelation. It enables a decomposition of the microscopic internal strain into the normal mode contributions which may clarify the origin of some interrelations between various elastic or piezoelectric constants. If the force constant approach is

182


adopted, the program calculates the derivatives of the elastic constants relative to the force constants. 9 Thermodynamic properties.

CRYME accumulates information on the fre-

quency distribution over the zone and provides a histogram representation of the DOS if a proper grid in a reciprocal space is introduced. The heat capacity may then be the calculated and plotted versus temperature. A model approach, which combines a certain number of Debye and Einstein functions with continuum spectra [136], is provided as well.

LAZAREV

183

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.

~

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K. B. Tolpygo, Chemical Bonds in the Crystals of Semiconductors and Semimetals (Russ.), Minsk, Nauka i Technika (1973) p. 196.

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67.

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69.

M. V. Volkenstein, M. A. Eljashevich and B. I. Stepanov, Molecular Vibrations (Russ.), vols. I, II. Moscow, GITTL (1949).

70.

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Vibrational Intensities in Infrared and Raman Spectroscopy, (W. B. Person and G. Zerbi, eds.), Amsterdam-Oxford-N. Y., Elsevier, (1982).

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82.

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85.

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86.

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87.

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88.

V. P. Vodopjanova, V. F. Pavinich, A. E. Chisler and A. N. Lazarev, Izv. An. SSSR, Ser. Inorg. Materials, 19, 1129 (1983).

89.

V. P. Vodopjanova, A. P. Mirgorodsky and A. N. Lazarev, Izv. An. SSSR, Ser. Inorg. Materials, 19, 1891 (1983).

90.

V. P. Vodopjanova, Izv. Akad, Nauk SSSR, Ser. Inorg. Materials (Russ.), 22, 990 (1986).

91.

M.O. Stengel, Z. Kristallogr., 146, 1 (1977).

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93.

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94.

A. N. Lazarev, The Infra-Red Spectra of Minerals, (V. C. Farmer, ed.), London, Mineralog. Soc. (1974) p. 69.

95.

D.A. Kleinman and W. G. Spitzer, Phys. Rev., 125, 16 (1962).

96.

M.B. Smimov, Opt. SpektrosL, 65, 311 (1988).

97.

M. B. Smimov, Dynamical Properties of Molecules and Condensed Systems, (A. N. Lazarev, ed.), (Russ.), Leningrad, Nauka Publ., (1988) p. 95.

98.

A.N. Lazarev and A. P. Mirgorodsky, Phys. Chem. Miner., 18, 231 (1991).

99.

Y. Shiro, J. Sei. Hiroshima Univ., Ser. A-II, 32, 69 (1968).

100.

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101.

Y. Shiro and T. Miyazawa, B. Chem. Soe. Jpn., 44, 2371 (1971).

102.

V. A. Ryjikov and M. B. Smimov, Modern Vibrational Spectroscopy of Inorganic Compounds (Russ.), (E. N. Jurehenko, ed.), Novosibirsk, Nauka Publ. (1990) p. 155.

103.

T. Ikeda and I. Imazu, Jpn. J. Appl. Phys., 15, 1451 (1976).

104.

H. Hirano and S. Matsumura, Jpn. J. Appl. Phys., 13, 17 (1974).

105.

H. V611enkle, Z. Kristallogr., 154, 77 (1981).

106.

A. N. Lazarev, Vibrational Spectra and Structure of Silicates, N.Y.-London, Consultants Bureau, (1972).

107.

A. N. Lazarev and A. P. Mirgorodsky, Izv. An. SSSR, Ser. Inorg. Materials, 9, 2159(1973).

108.

A. N. Lazarev, V. A. Kolesova, L. S. Solntzeva and A. P. Mirgorodsky, Izv. An. SSSR, Ser. Inorg. Materials, 9, 1969 (1973).

188


109.

A. Lurio and G. Bums, Phys. Rev. B, 9, 3512 (1974).

110.

B. Orel, M. Klanjsek, V. Moiseenko, M. Volmyanskii, N. LeCalve and A. Novak, Phys. Status Solidi B, 128, 53 (1985).

111. V.A. Ryjikov and A. P. Mirgorodsky, Opt. Spektrosk, 64, 84 (1988). 112.

M. D. Volnjansky, O. A. Grzegorzevsky, A. Ju. Kudzin and S. A. Flerova, Fiz. Tverd. Tela, 17, 2487 (1975).

113.

M. D. Volnjansky, O. A. Grzegorzevsky, A. Ju. Kudzin and S. A. Flerova, Fiz. Tverd. Tela, 18, 863 (1976).

114.

R. M. Hazen in "Microscopic to Macroscopic," Revs. in Mineralogy, vol. 14. (S. W. Kieffer and A. Navrotsky, eds.), Washington, Mineralog. Soc., (1985) p. 317.

115.

S. R. Srinivasa, L. Cartz, J. D. Jorgensen, T. G. Worlton, R. A. Beyerlein and S. Billy, J. Appl. Crystallogr., 10, 167 (1977).

116. H.C. Urey and C. A. Bradley, Phys. Rev., 38, 1969 (1931). 117. M.B. Smimov, Opt. Spektrosk., 55, 1017 (1983). 118.

T. Shimanouchi, Pure Appl. Chem., 7, 131 (1963).

119.

B. Crawford and J. Overend, J. Mol. Spectrosc., 12, 307 (1964).

120.

M. Guissoni and G. Zerbi, Chem. Phys. Lett., 2, 145 (1968).

121. I.M. Mills, Chem. Phys. Lett., 3, 267 (1969). 122. R.A. Cowley, Phys. Rev. B, 13, 4877 (1976). 123.

"Light Scattering Near Phase Transitions," (H. Z. Cummins and A. P. Levanyuk eds.) in Modern Problems in Condensed Matter Sciences, (V. M. Agranovich and A. A. Maradudin, gen. eds.) vol. 5., Amsterdam, North-Holland, (1983).

124.

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125.

T. H. K. Barron, Lattice Dynamics, (R. F. Wallis, ed.), Oxford Pergamon Press, (1965) p. 247.

126.

R. Zallen, Phys. Rev. B, 9, 4485 (1974).

127. W.R. Busing and M. Matsui, Acta Crystallogr. A, 40, 532 (1984). 128. A.P. Mirgorodsky and M. B. Smimov, J. Phys." Cond. Matter, in press. 129.

M. Catti, Acta Crystallogr. A, 37, 72 (1981).

LAZAREV 130.

R. Pletcher, Optimization, N. Y., Plenum Press, (1969).

131.

A. Murtagh and W. H. Sargent, Comput. J., 13, 185 (1970).

189

132. J.W. Mclver and A. Komomicki, Chem. Phys. Lett., 10, 303 (1971). 133.

S. C. Parker, C. R. A. Catlow and A. N. Gormack, Acta Crystallogr. B, 40, 200 (1984).

134. W.R. Busing, Trans. Am. Crystallogr. Assoe., 6, 57 (1970). 135.

International Tables for X-ray Crystallography, Vol. I., (N. F. M. Henry and K. Lonsdale, eds.), Birmingham, Kynoch Press, (1952).

136.

S. W. Kieffer in "Microscopic to Macroscopic," Revs. in Mineralogy, vol. 14. (S. W. Kieffer and A. Navrotsky, eds.), Washington, Mineralog. Soc. (1985) p. 65.


CHAPTER 3 M O L E C U L A R Q U A N T U M M E C H A N I C S IN T H E E V A L U A T I O N O F INTERACTIONS OF LESS LOCALIZED ORIGIN

The Ionic Charge of Oxygen in Silicon Dioxide and the Non-Bonding Oxygen-Oxygen Interactions in Crystals ................................................................ 192

A. The Point Ion Concept .........................................................................................

192

B. The Dynamic Oxygen Charge in Disiloxane and the Applicability of the Point Ion Approximation ......................................................................................

198

C. The Force Constants of Non-Bonded Oxygen-Oxygen Interaction .................... 202 II.

Tetramethoxysilane as a Model of the Silicon-Oxygen Tetrahedron in a Network of Partially Covalent Bonds ..................................................................... 213

A. Experimental Data and Spectral Assignments ..................................................... 213 B. Quantum Mechanical Computation ..................................................................... 218 C. Frequency Fitting and the Force Constant Evaluation ........................................ 224 III. The Disilicic Acid Molecule as a Model of the Fragment of a Silica Network ...................................................................................................................... 229

A. Electronic Structure and Equilibrium Geometry ................................................. 229 B. Ab Initio Force Field Investigation and Intertetrahedral Interactions ................. 234 References ...........................................................................................................................

191

244

192

LAZAREV T H E I O N I C C H A R G E O F O X Y G E N IN S I L I C O N D I O X I D E AND T H E N O N - B O N D I N G O X Y G E N - O X Y G E N I N T E R A C T I O N S IN CRYSTALS

A.

The Point Ion Concept

Most of the model approaches to the microscopic dynamic theory of (more or less) complicated crystal lattices proceeds from the point ion approximation. These approaches (see, e.g., reviews in refs. [ 1,2]) describe a lattice as an assemblage of ionic charges which are treated as point sources of its electric field, and are adapted to the calculation of its electrostatic energy by the summation of their Coulomb interaction energies. This conception implies a localization of the electron density in the vicinity of a corresponding nucleus and a spherical symmetry of ions. Its applicability in this form is evidently restricted to systems such as alkali halides and some intrinsic shortcomings have been clarified. In other crystals, this approach was treated as a tentative one because of its relative simplicity further implying a more elaborate treatment, which would avoid its disadvantages. This approach was widely employed in lattice dynamics computations in a more general case of ionic-covalent crystals with Complex lattices (if their investigation assumed an explicit separation of the Coulomb contribution into the conditions of equilibrium and/or dynamic properties). In this case, however, the initial supposition of the point ion concept on the nature of the spatial distribution of electronic charge in a lattice is not valid. Even the idea of an ion as a spatially restricted charged system seems dubious for these crystals. Consequently, the physical meaning of ionic charge can hardly be defined, and a numerical value cannot be determined unambiguously. This problem was paid much attention for a long time either in general formulation [3-9] or in numerous applications to particular lattices. In our opinion, however, satisfactory general criterion of the applicability of the point ion concept to the calculation of various dynamic properties and estimation of errors originating from this approximation has not yet been proposed. Let us now begin from a more precise definition of the quantities adopted in the description of an ion as a point charge in the lattice. This is insistent because of a variety of

INTERACTIONS OF LESS LOCALIZED ORIGIN

193

terms introduced in the literature, some of them being insufficiently specified. When treating a crystal in an adiabatic approximation (that treatment is applicable to a molecule as well), it is represented by a system of nuclei (cores) immersed in some inhomogeneous distribution of electron density. As has been shown in the previous chapter, the dipole moment and electrostatic energy of a system are expressed respectively as:

P = ~p (rlR)rdr

(3.1)

Vel= ~ ~ P(rlR)p(r'lR)drdr'

(3.2)

and

where the p (fiR) function describes the total (nuclear and electronic) charge density in a point, r, at the nuclei's position specified in a generalized form by the atomic position vector, R. These formulas are applicable if the p (rlR) function is specified explicitly. The point ion concept corresponds to a simple version of this function def'med as a set of 5functions localized at corresponding nuclei. The following procedure is implied by t~at definition. The whole space where the p (rlR) function is defined should be partitioned into areas with each one including the ith nucleus and a certain portion of the electron density distribution around it. Then, the charge (zi) of the ith ion is defined as:

Zi = f p (fiR)dr

(3.3)

where f~i is the volume of the corresponding area. There exist, however, no rigorous approach to that partition of the internal space of a crystal since experimental or theoretical formulation of an ion frontier cannot be proposed in a unique way. It is possible to restrict a selection of ~i areas by imposing the requirement of a reasonable proximity of the experimental dipole moment to the magnitude obtained from the expression:

194

LAZAREV

P= E

Ri

f P(rlR)dr = E Rizi

i

(3.4)

i

with a sufficiently precise determination of the electrostatic contribution to the potential energy as a sum over a pair of interionic interactions by means of those charges"

vel = 1 i.~k 9

J" p(rl R)& ; p(rl R)dr ~i

~k IRi-Rkl

=1

2.

i~k

ZiZk

IRi_Rk I "

(3.5)

In these expressions, R i is a positional vector defming the center of the charge density distribution which is assigned to the ionic charge, z i. As follows from eq. (3.5), a potential of the electrostatic field, tp(r), created by the whole system in a point, r, can be defined through the ionic charges, zi; in particular, for the origin of coordinates: zi

~(o)= ~i I Ril

(3.6)

"

The charges (zi) have been introduced for the description of the static properties of a system, and therefore will be referred to hereafter as the static charges, z st . The static charge is a conditional quantity since in a general case all z st magnitudes can hardly be deduced from expressions (3.4-3.6). This would imply a separate measurement of the individual contribution of each type of ion to P and tp(r). It is possible, however, to determine the charge of any ion separately by the investigation of the interrelation in the changes of P and ~p(r) of a whole system induced by the shift of the ith nucleus with the magnitude of that shift, Aui. A quantity defined in this way relates to the deformational properties of a system and can be called the dynamic charge of the ion. The dynamic charge is a tensor whose elements [~zdyn ~tl3 (i) ) are specified by the expression which follows from eq. (3.4): zdY~ (i) = APJAul3(i ) . Here, Pa and ul3(i) are the Cartesian components of the corresponding vectors.

(3.7)

INTERACTIONS OF LESS LOCALIZED ORIGIN

195

The dynamic charges of ions can be defmed through the change of the electrostatic potential of a system, A~, caused by the shifts of the corresponding nuclei. It follows from eq. (3.6) that at IAu(i)l co_ relation being very indicative. The tetragonal I41/amd- D194hzircon crystal contains two ZrSiO4 formula units in the primitive cell. The atomic arrangement in this ABO 4 lattice, which is met in various phosphates, vanadates, and arsenates, is shown in Fig. 3.4. The right-hand side of that figure clarifies the mutual orientation of diminished tetrahedra. Each tetrahedron possesses D2d site symmetry being tetrahedrally surrounded by four oxy-anions. The spectra of zircon type crystals have been investigated repeatedly in both Raman and IR spectroscopy, and were reviewed in ref. [28]. A comparison of the ZrSiO 4 normal mode frequencies with ones corresponding to the free SiO~- ion (see Chapter 1 conceming the estimations of their magnitudes) is given in Fig. 3.5. The scheme represents the usual sequence of the factor-group analysis: a removal of degeneracy caused by the local (site) symmetry, and a splitting of intemal modes originating from their interactions. The internal modes of the bending type are coupled with lattice modes, and their separation adopted in Fig. 3.5 is more or less arbitrarily deduced from the normal coordinate calculation [21,28]. The enhancement of the frequencies of the internal modes in this area evidently originates from that coupling. The bond stretching internal modes are much less coupled with the lattice modes, and their splitting may be treated as Davydov's splitting. The low-frequency component of the pulsation (A1) mode is optica'.'y inactive, and only tentatively assigned to the weak absorption band at 730 or 770 cm "1. A tremendous splitting of this internal mode evidently follows from the high frequency of the unambiguously identified in-phase A lg component (the out-of-phase B2u component

IN'I~RACTIONS OF LESS LOCALIZED ORIGIN

A--B

207

B--A

A--B

B--A

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