Molecular approach to solids

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

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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

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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

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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

CONTENTS OF OTHER VOLUMES

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

CONTENTS OF OTHER VOLUMES

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

CONTENTS OF OTHER VOLUMES

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

CONTENTS OF OTHER VOLUMES

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

CONTENTS OF OTHER VOLUMES

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

CONTENTS OF OTHER VOLUMES

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

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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.

4

LAZAREV

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.).

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

LAZAREV

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-

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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)

8

LAZAREV

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-

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

LAZAREV

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

LAZAREV

(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,

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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.

14

LAZAREV

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

16

LAZAREV

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

18

LAZAREV

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

,,

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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)

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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,

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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.

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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.

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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-

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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-

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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).

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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.

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

*

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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.

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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.

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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.

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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.

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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:

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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].

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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.

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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

CRYSTAL CHEMISTRY AND LATTICE DYNAMICS OF SILICATES

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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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.

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25.

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26.

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27.

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28.

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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.

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31.

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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.

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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.

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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.

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44.

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

45.

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46.

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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).

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

49.

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

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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.

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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).

62.

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

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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).

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

This Page Intentionally Left Blank

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

LAZAREV

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:

LAZAREV

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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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103

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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.

LAZAREV

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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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~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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

,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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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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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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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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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.

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INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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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147

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

1

, I

w

: I

,

I

I

I

I

1

~

I

1

I ~

1

i

,

I

'1

l I

I

,

,

3

I

I

2

I

~.<<..~....~-~ . . . . . . -%~..,b.----~9-

g i

I. 1

l

I

,

,

I

I

I

..6.6

~,

I

,

1

.,

-~-~--.~.-

Z

/ +

;x

Y

o.O ~

-,.~/

-u', 9

T

oGe Fig. 2.7

oO

T

9

;X

-Li

The structure of lithium metagermanate in crystallographic projections.

Solid lines correspond to Ge-O bonds while the discontinuous lines represent the shortest Li-O contacts. The numbers enumerate the atoms non-equivalent relative to translation.

All modes are Raman active, which favors the determination of their TO and LO frequencies independently from the results of oscillator analysis of IR reflection spectra for three polar species. The original attempts to calculate the normal modes of lithium metagermanate [76,107] were based only on the IR absorption spectra of polycrystalline specimens [76,108]. These have been later complemented by the results of a single-crystal investigaI

tion of the Raman spectra and the IR reflection spectra with a Kramers-Kronig analysis [109]. A more complete set of single-crystal data, including ones obtained for isotopically

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enriched 6Li2GeO 3 and 7Li2GeO 3, was obtained [110].

149

Still, the assignment of some

modes seemed doubtful and some fundamentals remained unidentified. The misassigrmmnts of previous investigations have been corrected, and the exhaustive identification of all symmetry allowed fundamentals have been attained both for 6Li2GeO3 and 7Li2GeO 3 [ 102,111 ] using suitably oriented large single crystals. In order to possibly obtain a complete separation of TO and LO modes from various irreducible representations, the following orientation and polarization conditions were adopted in the Raman investigation (the Porto's symbols being adopted to describe the scattering geometry with overscribed bars indicating the negative direction of the corresponding axis): A TO - x(zz)y, A LO - yz(xx)y2 A 2 - x ( y x ) y , z(yx)y, z(xy)x B TO - y(zx)z, B LO - xy(z, xy)~y, i. e., in the presence of B TO B TO - x(zy)z, B LO - xy(z, xy)xy, i. e., in the presence of B TO

Here, the double indices denote the bisectors of corresponding angles. Figs. 2.8 and 2.9 illustrates only a few of those spectra. The IR reflection spectra obtained for the incident beam polarized along one of the crystal axes are shown in Figs. 2.10-2.12 where the dotted curves correspond to the spectra repaired by means of fitted magnitudes of phonon oscillators. These were deduced using the factorized presentation of the e(co) dispersion. A complete set of TO phonon frequencies and squares of the dipole amplitudes for polar modes has been obtained for both isotopic species. Only the frequencies below 600 cm -1 were sufficiently sensitive to isotopic substitution. Their shills were, however, very informative on the localization of the corresponding modes since the internal modes of polyatomic chain anions, even in this frequency area, often remained unaffected by the

Fig. 2.8 Raman spectrum of 7Li2Ge03,A'? bands (upper curve) and forbidden at given geometry of scattering is indicated.

bands (lower curve). A tentative origin of weak bands

Fig. 2.9 Rarnan spectrum of 7Li2Ge03,B caption to Fig. 2.8.

TP bands (upper curve) and B LP bands mixed with B To (lower curve). See also the note in

152

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

R, %1oo 80

60 40 20 |

|

9

!

!

|

80

0

900

J

700

800

600

500

400

300

2;0

1;0

co,cm-I Fig. 2.10

IR reflection of 7Li2GeO3 (upper curve) and 6Li2GeO3 (lower curve) at

EIIc (A1 bands). Solid curves correspond to the experimental spectra while the dotted ones represent their reproduction by fitting eq. (2.100).

R , % 100806040200

I

I

I

I

I

I

I

I

I

80604020-

0

Fig. 2.11

I

900

I

800

I

700

I

600 500 (0, cm -1

I

400

I

300

I

200

I

100

IR reflection of 7Li2GeO3 (upper curve) and 6Li2GeO3 (lower curve) at

Ella (B1 bands). Other explanations see the caption to Fig. 2.10.

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153

R,% lOO

|

0 900

|

800

!

|

700

|

|

!

600

500

400

300

200

100

~, cm 1

Fig. 2.12

EIIb (B2 bands).

IR reflection of 7Li2GeO3 (upper curve) and 6Li2GeO3 (lower curve) at Other explanations see the caption to Fig. 2.10.

isotopic substitution. On the other hand, in the case of coupling between the internal and external modes, the isotopically induced variation of the IR intensities could be treated as a manifestation of the shapes of the normal modes. This set of experimental data was employed in an attempt to find a solution of the IVP. The initial approximation of the force field model was composed of GVFF type force constants of internal degrees of freedom of the complex anion complemented by a number of two-body (bond-stretching type) force constants for the rest of lattice.

This represented

the interactions at the shortest distances in lithium-oxygen tetrahedra and non-bonding O..-O, Li...Li and Ge-..Li interactions up to 3.5A. The edges of GeO 4 tetrahedra were excluded since their elasticity was already represented in the GVFF model containing threebody OGeO force constants. Even with a minimal number of off-diagonal force constants, the set includes more than 30 parameters. The bond-stretching type force constants were assumed to correlate with the equilibrium lengths of the corresponding interatomic dis-

154

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

tances. The simple electro-optic scheme described in the previous subsection was adopted in the IR intensity calculation. This set was refined by frequency fitting (including the reproduction of isotopic shills) with restrictions originating from the IR intensity computation, some considerations mentioned above, and the implicit requirement of similarity with the force fields of chemically related substances. The quality of reproduction of the experimental spectrum of one of the isotopic species is shown in Fig. 2.13. A poor reproduction of the IR intensities at lower frequencies may originate from the inadequate description of the shapes of the normal modes and from the deficiency of the adopted electrooptic model which does not provide for any mechanism of charge redistribution in a deformed lattice outside the complex anion. The elastic constants of the crystal have been calculated (Table 2.1). At the reliable estimation of interrelation between the diagonal and off-diagonal elastic constants, the first were significantly overestimated, and the anisotropy of the elastic properties failed to reproduce. The compressibility of the crystal was correspondingly underestimated. Before attempting to determine the set II of force constants by the joint fitting of frequencies and macroscopic elastic constants, the number of independently varying force constants was reduced by the supposition of equality of several force constants issuing from their physical meaning and peculiarities of bonding and structure. Thus, less than half of the number of independent parameters was involved in the adjustment of the set II force constants. The results (Fig. 2.13) show approximately the same quality of reproduction of the experimental spectrum, and in Table 2.1 where a certain improvement in the description of macroscopic elastic properties is shown. In addition, it leads to a remarkably better description of the piezoelectric properties (although they were not included into fitting procedure and calculated for both force constant sets with the same magnitudes of electro-optic parameters).

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155

a

I. ,

II

.l

, i

l

l

,I

l b

I,

l

,

,l

,I

,

I

I

,

,

i

C

,

I

I

I

I

800

,

I

I

I

700

I

I

600

I

.

I

500

,

,

I

I

l,

I

400

I

(

II

300

|

I

200

cm-I Fig. 2.13 A diagrammatic comparison of the: a) experimental optical spectrum of 7Li2GeO3; b) calculated with adoption of the set I force constants; and c) set II. The positions of lines correspond to TO frequencies and the heights of the lines are proportional to IR intensities.

156

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

TABLE 2.1 The elastic and piezoelectric constants of lithium metagermanate calculated with two versions of the force field and the reproducibility of its phonon frequencies Calculated Quantity

Calculated Magnitudes I

Experimental Data

II

Cll , (Newton/m2).10-10

23.1

16.9

13.4 [112,113]

C22

20.3

16.7

14.2

C33

17.5

17.2

15.7

C44

4.5

5.1

4.4

C55

4.8

4.9

5.6

C66

5.1

5.2

3.6

C12

7.9

6.8

3.3

C13

6.1

5.5

3.4

C23

6.8

4.9

4.6

g, (m2/Newton) 910-10

0.90

1.06

1.38

e31, Coulomb/m2

-0.35

-0.31

-0.38 [ 103 ]

0.671112,113]

e32

-0.37

-0.39

-0.42

0.82

e33

-0.79

0.41

1.36

0.67

el 5

-0.33

-0.30

-0.17

0.56

e24

-0.32

-0.29

-0.03

0.14

ACOaverage

2.85

3.08

-

AcomaximaI

13.54

15.32

-

Frequency deviation, %

Comments: The experimental piezoelectric constants deduced from the measured piezomoduli by their multiplication with the corresponding elastic constant. Only the absolute magnitudes of piezomoduli were determined [112,113].

It can be concluded from this example that even in a case when the maximal amount of spectroscopic information is available, the inclusion of the properties relative to the homogeneous deformations considerably reduces the ambiguity of the IVP solution. A closer inspection shows that the two sources of information on the force field of a crystal are of complementary nature. The phonon frequencies are determined mainly by the strongest interactions at shorter distances while the macroscopic elastic properties are more sensitive

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157

to weaker interactions at longer distances and their inclusion into the IVP leads to a less localized representation of the force field. This is of special importance for the complex lattices whose vibrational branches are usually unknown.

C. The Microscopic Structure of Hydrostatic Compression and its Employment in the Generalized Formulation of the Inverse Vibrational Problem The hydrostatic compression of a crystal is unique among other types of uniform strain relating to the possibility of direct experimental observation of its microscopic pattern. The methods of x-ray or neutron diffraction structure investigation under high pressure are presently well developed both in the single crystal and powder versions, and corresponding data on the structure variation in compressed crystals are available in numerous cases (see a review of these data in ref. [114]). The derivatives of various structural parameters, including the internal coordinates relative to external pressure, can be deduced from these data. The introduction of a notion of the shape of hydrostatic compression is thus worthwhile, and its description by means of the Lp vector can be proposed. This vector is interrelated with internal coordinates as: g = Lpp.

(2.117)

In this notation, the compressibility of a crystal can be expressed in a form analogous with (2.95) and (2.106) [78]: ~ = L;FLp .

(2.118)

The formulation of the GIVP as a solution of the system (2.116) is thus complemented by eq. (2.118). There exist, however, a basic difference between this and other similar expressions. Only the F matrix on the right-hand side is inaccessible to direct experimental determination while the Lp vector is found from diffraction data as Lp = dg/dp which correspond to the compressibilities of various di- and three-atomic fragments of a lattice.

158

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

In the macroscopic description of the compressibility of an anisotropic body the scalar X magnitude is complemented by the three-dimensional vector XU; whose components are determined as X i - dUi/dp, where i = 1, 2, 3 (the Voigt's index). The interrelation between Lp and L o is expressed by means of the X U vector in the form: Lp = LU, X U ,

(2.119)

where L U, is the L U matrix with the last three columns omitted. The experimental X and X i values can be obtained by the inspection of the dependence of the unit cell size upon the pressure. These values, either experimental or theoretical, can be deduced from their interrelation with the elements of the C-1 matrix [2]: Xi = Z C~1 k

(i,k = 1,2,3) ,

(2.120)

1

X = Z

xi"

(2.121)

In a theoretical treatment of the microscopic structure of hydrostatic compression, the

dg/dp

derivatives are obtained as: dg dp

=

dg dU . dU dp

(2.122)

A simple example may clarify the importance of a joint treatment of eqs. (2.116 and 2.118) in the investigation of the IVP.

Here, the endless diatomic linear chain

-A-B-A-B-A-B- with the unit cell specified by the lattice constant a and reduced mass m is treated. The internal coordinate set is comprised of the changes of the two non-equivalent bonds, A-B and B-A (in the external case, a molecular crystal with AB molecules can be implied): denoted as gl and g2, each being treated as a translation symmetry coordinate. A simple VFF model is constituted by three force constants, the diagonal force constants, fl

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159

and f2, and the interaction constant, f12. This set can not be determined solely from the solution of the equation for the longwave optical frequency: = m-l(fl + f2- 2f12). The same relates to the evaluation of the effective charges, z 1 and z 2, introduced as: z = dP/dg since a single experimental value determined by the IR intensity of that mode is accessible: 1

~t=m

2 (Z 1 _ Z2 ) .

Let us now take into consideration two other expressions for the elastic constant, C, and the piezoelectric constant, e: C = a 2 (fl f2- t"122)/(ti + f2- 2f12) , e = z l l 1+z212

,

where 11 = (f2 - f12)/(f1 + f2 - f12) and 12 = (t"1 - f12)/(f1 +f2 - t"12). If the microscopic structure of the compressibility of a chain is known, then one more equation containing an experimental value can be added to the left-hand side: 11/12 = (f2- fl2)/(fl - f12) 9 The 11 and 12 values in the above expressions are the dg/dU derivatives. Thus, 5 magnitudes introduced to specify the F and Z matrices of this particular system can be uniquely defined by the set of above equations. The use of the experimentally determined microscopic pattern of the hydrostatic compression is even more important in the case of lacking sufficiently complete spectroscopic data. The available spectroscopic information on the silicon oxynitride, Si2N20 , which attracted attention as a perspective ceramic material was restricted to the IR absorption spectrum of the powder specimen and no single crystals were accessible. However, its

160

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

crystal structure and the structure variation under hydrostatic compression have been determined [115] by neutron diffraction on the polycrystalline sample. The crystal is orthorhombic (Cmc21 - C12v) with two formula units in the primitive cell. The three-dimensional network of partially covalent bonds is composed of folded Si2N2 sheets along the yz plane, and interconnected by the Si-O-Si bridges with the Si...Si directions along the x plane. Only the oxygen atoms have a low coordination number (two) in this structure, the coordination numbers of silicon and nitrogen are four and three, respectively. The initial set of force constants adopted in the theoretical treatment of the lattice dynamics of this crystal [79] was restricted to a diagonal approximation of the F matrix composed of the elasticities of several two-body (bond stretching type) coordinates for the first-order neighbors, non-bonding atom-atomic interactions at the distances less than 3.5A, and the force constants of valence angles. The simple electro-optic model described above was adopted to calculate the polarization phenomena, the IR intensities, and the piezoelectric effect. The possibility to reproduce the experimental pattern of the structure variation in a hydrostatically compressed crystal was adopted as a guide in the refinement of the original set of the force constants and the introduction of a minor number of interaction constants. The reproduction of the frequency distribution in the experimental spectrum played a secondary role. A rather reliable dynamic model of the crystal dedgced in this way allowed, in particular, the origin of the anisotropy of compressibility to be visualized, to reveal a very strong influence of the weak non-bonding oxygen-oxygen interaction at the distance of 3.36A on the elastic properties, and the microscopic structure of hydrostatic compression. Moreover, the deduced dynamic model was employed in an attempt to theoretically predict the macroscopic elastic and piezoelectric constants of this material.

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161

D. A Curvilinear Nature of the Internal Coordinates and its Certain Consequences

In accordance with the adoption of the harmonic approximation, attention has not been paid to the curvilinear nature of the internal coordinates 9 In all previous considerations conventionally used in the force constant approach, a simple linearized interrelation between internal coordinates g and Cartesian coordinates u could be implied: (2.123)

g = Buu.

A calculation of the B matrix, which is completely determined by the geometry of a system, is described in numerous books on the theory of molecular vibration (see, e.g., [71 ]). There exist, however, certain problems where a more careful investigation of the interrelations between the curvilinear coordinates g and the Cartesian coordinates u is needed 9 It means that the expressions of the potential function (PF) defmed in both coordinate basis sets as ~(u) = v(g(u)) are to be analyzed more precisely 9 The above functions are expanded in a power series near the equilibrium position:

g(U) = Buu + } Buuu2 + ...

(2.123)

v g) = V o + r,gg + 1 vg g2 ...

(2.124)

V(u) = ~0 + ~u u + } ~uu u2"'"

(2.125)

By substituting (2.123) into (2.125) and setting (2,124) equal to (2.125), one obtains the following expressions for the coefficients of the V(u) expansion: ~u = VgBu ,

(2.126)

e'uu = rg~uu + Vgga 2

,

(2.127)

Ouuu = VgBuuu + VggBuBuu + VgggB3 , 9

,

,

,

,

,

,

,

,

,

,

,

(2.128)

162

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

Two very important conclusions immediately follow from the inspection of these expressions. First, a non-linear g(u) dependence produces a non-vanishing contribution to the nth order Cartesian force constant from all lower order force constants specified in the internal coordinate basis up to the nth order: Vg, Vgg.... Vg...gn. This conclusion is valid irrespective of the independent or dependent (redundant) nature of the set of internal coordinates {g}. In particular, a decomposition of ~(u) up to the second order term in u is expressed through the force constants in the internal coordinate space as: tl/(u) = V 0 + VgBuu+ ~1 (Vgg Bu2 + VgBuu)U2'

(2.129)

where a contribution from the linear term in g into the second order force constant in u is explicitly separated. Second, the defmition of the harmonic approximation (which is understood as a restriction of expansion of eq. (2.124) or (2.125) to the second order term) is physically not the same in the g and u coordinate basis set. A potential determined by the quadratic form in internal coordinates corresponds to an anharmonic potential in Cartesian coordinates. The stability conditions (SC) require zero forces on the atoms, #u = O. It is possible to define the SC using eq. (2.126) as:

VgBu = O .

(2.130)

A set of expressions (2.130) is usually treated as a system of equations linear to Vg. If the number of independent SC (2.130) exceeds or is equal to the number of nonequivalent Vg components, there exist a unique solution of that system, Vg = 0, which means the absence of internal tensions in the equilibrium geometry: there exist no forces along any internal coordinate. Even in this case, a presentation of the potential energy by the quadratic form in internal coordinates corresponds according to the existence of nonvanishing higher-order terms in its expansion in Cartesian coordinates (2.128). This effect may be thought of as a "kinematic" anharmonicity.

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If the number of independent SC (i.e., equations of the type (2.130)) is less than the number of non-equivalent Vg components, which is often met when a PF is represented by the VFF model, a collection of non-zero solutions of that system exists, and the idea of coordinated internal tensions in a system is introduced to specify them. It means that the absence of forces on any atom in the equilibrium state is ensured by the mutually dependent stresses of the internal coordinates. These considerations lead to the Urey-Bradley force field (UBFF) [116], in which a PF may be generally expressed as: VUBFF = VVFF + Vgg .

(2.131)

It was hoped that an introduction of internal tension would allow some of the numerous off-diagonal force constants, which appeared usually in the VFF approach to be avoided. In this way, the empirical normal coordinate analysis based on the frequency fitting would be considerably simplified and a more definite IVP solution obtained. On the other hand, an explicit separation of the contribution from the internal tension to the second order force constants from the rest of the internal coordinates may help to rationalize the origin of some elastic properties of a system whose force field is described in a space of intemal coordinates. It has been shown, for example, by application of the UBFF model to the planar BF 3 molecule (its total set of force constants was determined earlier from the vibration-rotation spectra), that the elasticity of its out-of-plane mode is constituted by 80% of the tension in planar coordinates. A redundancy of the internal coordinate sets is most appreciable in the case of crystals if their PF are represented by means of pair atom-atomic potentials. Such PF models defined by eq. (2.46) are described by means of q~ij potentials, which are approximated by some analytic expressions with non-vanishing first derivatives. Since numerous interatomic distances should be taken into consideration in this approach, the internal coordinate set {g} - {Rij) is considerably redundant. A harmonic force constant is thus expressed in this case according eq. (2.129) as:

164

INTRODUCTION TO DYNAMICAl. THEORY OF CRYSTAI,S d2~dRdR dR2dudu ,

+

d~dR 2 dRdudu'

.

(2.132)

Since Born and Huang [1 ], the dcp/dR quantities are referred to in lattice dynamic theory as the tangential force constants. It implies that the tensions created by the pair interactions in a lattice specify the origin of transversal elasticity of such fragments as linear chains or atomic planes even in the absence of any additionally introduced second-order force constants. The physical meaning of the UBFF and redundant coordinate sets was repeatedly discussed in the spectroscopic literature [117-121]. Several authors believe an inclusion of linear terms into the harmonic PF defined in internal coordinate to be superfluous. They usually appeal to the approximate character of the normal coordinate calculation, and to the restriction of harmonic approximation to small amplitudes which saves, in their opinion, the applicability of linear g(u) dependence. The relation of the vibrational problem to other physical properties of a system is not implied in these considerations. A problem of inclusion of linear terms in the harmonic PF was commented in Califano's book [71 ] as being "simply a matter of convenience and of personal preference". In our opinion, the use of the UBFF is the more essential. The less the investigated problem is restricted to the reproduction of experimental frequencies, a deeper understanding of the interrelation between the spectrum and other properties is intended. The UBFF is physically consistent in the treatment of internal coordinates (g) as exact curvilinear coordinates of a system, and allows a straightforward decomposition of the PF into the contributions of its fragments. It should be noted, however, that presently, the UBFF and related problems attract less attention in theory of molecular vibration because of the development of the ab initio force constant calculation which utilize the independent coordinate sets (cf. Chpt. 1).

The importance of the rigorous treatment of the curvilinear nature of internal coordinates should be particularly emphasized in the case of crystals. A condition of invariance of the potential energy to small angle rotation is conventionally formulated as a set of interrelations:

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E ~nRo = A (i0n_1 over atoms

165

(2.133)

where A is an antisymmetric matrix and R o are the equilibrium coordinates. In spite of these relations, various terms in eqs. (2.125-2.128) cannot be neglected arbitrarily. Otherwise, the rotational invariance can be lost. It is impossible, e.g., to investigate the internal tensions when analyzing the SC (2.130) with neglect to their contribution to the force constants according to eq. (2.129). Moreover, the internal tensions can be shown to relate to the properties of a crystal treated as a macroscopic body. Their investigation is of particular interest if the SC formulation is extended to a crystal being in a balance with some external action. These problems will be briefly outlined below and several applications reviewed.

E. A Relation of Internal Tension to Description of the Lattice Instability A stability of the lattice relative to its microscopic internal deformation requires an emergence of a restoring force along any direction of distortion. The total set of these distortions is covered by a set Of LQ vectors corresponding to a whole Brillouin zone. The stability to internal deformation can thus be formulated in terms of eq. (2.95) as a condition ~1 > 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:

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INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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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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.

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INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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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175

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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.

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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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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

REFERENCES

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.

~

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67.

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70.

E. B. Wilson, J. C. Decius and P. C. Cross, Molecular Vibrations, N. Y., McGrawHill, (1955).

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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.

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

83.

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86.

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88.

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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.

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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).

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

104.

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105.

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

106.

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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

INTRODUCTION TO DYNAMICAL THEORY OF CRYSTALS

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.

Dynamic Theory and Physical Properties of Crystals (Russ.), (A. N. Lazarev, ed.), St. Petersburg, Nauka Publ., (1992).

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).

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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.

This Page Intentionally Left Blank

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<
(3.8)

where A(p(r,ct) is a change of the potential in the point removed to the distance r from the ith nucleus in the ot direction, that change being induced by the shift of this nucleus in the 13 direction. n (i) can be deduced from eq. (3.1) in a form: A more rigorous definition of zdY (~13 zdyn a13 (i) = aPa//gul3(i) =

rap~ (rli)dr,

(3.9)

where Pa is the ath component of the P vector and: p~(rli) = ~p

(rlR)/iguf~(i)

(3.10)

is the charge density response function. It describes the change of the charge density in the point r at the unit shift of ith nucleus in the 13 direction. It is seen from eqs. (3.9-3.10) that the dynamic charge of the ith ion should be treated as a moment of the charge response of a whole system to a unit shift of a particular ion and cannot represent its intrinsic property. A deep difference between the notions of static and dynamic ionic charges thus follows directly from the above definitions. The static charge (z~t) is really a charge of localized electronic and nuclear assemblage restricted to the f~i area, while the dynamic charge (zdyn) characterizes the non-local properties of a whole system. These are determined bo,h by the shift of the ith nucleus and a self-consistent electron density redistribution over a system induced by the change of the potential of a given ion in all of the internal area of that system. As a result, new magnitudes of p(rlR) and q~(r) should arise. It is seen from eq. (3.9) that the dynamic charge of the ith ion is determined by the integration over the area where pl3 (rli), 0 and that area has no relation to the definition of the f)i area specifying the static charge. Therefore, an introduction of the notion of dynamic ionic charge zdyn(i) does not require any definition of an ion as the entity in a static

196

LAZAREV

sense. The idea of dynamic (effective) charge is widely employed in solid state physics [10,11 ] while in the theory of molecular structure and dynamic properties, a similar quantity is usually referred to as a tensor of dipole derivatives [12]. Let us now investigate the physical meaning of the dynamic charge in terms of the point ion concept. The following expression is obtained using the definition (3.9) and eq. (3.4): zdyn(i)=St~ fp(rlR)dr+ERt~(k)fP~(rli)dr ~13 ~i k ~k

(3.11)

A condition (aRt~(k)/c~u~(i) - 5ikSt~) which is implied by the adopted approach, was taken into consideration when deducing that expression. The first term in expression (3.11) represents a contribution to the dynamic charge of the ith ion originating from its displacement as a rigid (non-deformable) ion and thus corresponds to its static charge (cf. eq. (3.3)). The second term describes a contribution originating from the polarization of all ions induced by the shift of the ith nucleus. In a case of a quite rigid connection of electrons with their nuclei, their distribution is independent on the positions of other nuclei, and a tensor (zdyn(i)) coincides with a scalar z st. In other words, z st is a uniform (scalar) part of the zdyn(i) tensor. Eq. (3.11) thus represents a possibility to decompose the dynamic charge of any ion into a sum of its own static charge and the Fnlarization of surrounding ions caused by its shift. In accordance with the physical meaning of z st and zdyn(i), the former determines such properties of a system like the static dipole moment and the electrostatic contribution to the energy of interaction between the ions at rest. The latter describes its properties relative to the deformation. In the case of a crystal, these are the phonon frequencies, the ionic part of the dielectric permeability, the elastic and piezoelectric constants etc. Excluding some of the simplest systems, an approach to the unambiguous experimental estimation of both of the above quantities (separately) for various kinds of ions does not exist. It is difficult, therefore, to judge on the preferences and disadvantages of their theoretical evaluation

INTERACTIONS OF LESS LOCALIZED OR/GIN

197

in various model approaches adopted in the literature on the lattice dynamics. This is why it seems reasonable that an attempt to apply the ab initio quantum mechanical calculations to their computation for a suitably designed molecular system as has been done by Mirgorodsky and Shchegolev [13]. Their most important statements are reproduced and discussed below. Besides the numerical estimation of the static and dynamic charges, the quantum mechanical computations of molecular systems can be applied to a judgement on the degree of applicability of the point ion approximation as itself. This conception can hardly be justified in a general case; even to relative z st values which correspond to the charges localized in the area near to atomic size. It is still more doubtful concerning the dynamic charges which represent the effects of the essentially delocalized origin. Returning to expressions (3.7 and 3.8), which specify the dynamic charge, eq. (3.7) defines zdyn(i) as some property related to the ith ion, but not specifying the area of its localization. Equation (3.8) proceeds, in principle, from a treatment of an ion as the point charge since the Aq) magnitude in that formula is obtained as a potential of the dipole, which arises due to a shift of the zdyn(i) charge on the distance of the shift of the ith nucleus. In a real case, the Aq) increment !ncludes the contributions from the higher order multiples, and the result of the determination of the dynamic charge from eq. (3.8) should be dependent on the degree of remoteness of the point where Atp is measured from the point where the ith ion is positioned. Beginning with a certain distance, the Aq) increment is defined (with acceptable accuracy) by the dipole, AP, arising in the R(i) point. It means that the asymptotic limit of the zdyn(i) magnitude determined by eq. (3.8) coincides with its estimation by means of eq. (3.7). In other words, sufficiently far from the ith ion, a small shift (ui) of its nucleus induces the same variation of electric field as would originate from the u i shift of the point charge obtained from eqs. (3.7) and (3.8). These considerations give a rigorous estimation of a "critical distance," which is sufficient for a precise determination of the Coulomb contribution to the interionic interaction in the point ion approximation. It should be restated that in the previous chapter, an approach to lattice dynamics was outlined which avoided

198

LAZAREV

any explicit separation of the Coulomb contribution to interactions at short interionic distances by means of the introduction of the notion of specific short-range interaction area. The above considerations help to estimate its size compatibility with the treatment of longrange Coulomb interactions in the point ion approximation.

B.

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

A similarity in the equilibrium geometry and the electronic structure of the Si-O-Si bridge in disiloxane with analogous bridges in silicon dioxide was taken as a basis of an attempt to adopt this molecule as a system suitable for the estimation of the dynamic oxygen charge in crystals directly from the molecular quantum mechanical calculation. Pulay's TEXAS program, as has been widely used in Chapter 1, was employed in the calculations discussed below. Besides the analytic evaluation of forces, that program provides for the computation of derivatives of the molecular dipole moment relative to the deformation a,ld of the electrostatic potential in any point in the vicinity of the investigated molecule. These possibilities considerably simplify its application to the problems considered in this section. Another simplification originates from adoption of a rather restricted basis set (I + d(O)) whose applicability to disiloxane and the errors estimated by comparison has been shown earlier both with experimental data and with the results of computation employing a more extended basis set. The use of a set containing polarization functions only on the oxygen atom seemed reasonable since only the changes of the dipole moment and electrostatic potential induced by the oxygen displacement were investigated. It should be emphasized that the main source of errors in the estimations discussed below probably originates from the restricted size of the molecule adopted in the calculations of quantities which are believed to be transferable to a crystal. Therefore, the errors intrinsic to the adopted basis set are probably of secondary importance. In particular, the influence of the terminal Sill 3 groups in the disiloxane molecule should be taken into consideration.

INTERACTIONS OF LESS LOCALIZED ORIGIN

199

Fig. 3.1 shows the geometry of disiloxane and the Cartesian axes adopted in further considerations" the origin of coordinates coincides with the oxygen atom position, the x axis lying in the SiOSi plane parallel to the Si... Si direction and the y axis being normal to that plane. The z axis coincides with the bisector of the SiOSi angle. The parameters of the equilibrium molecular geometry obtained with the basis set I + d(O) have been presented earlier (Table 1.4). The increments of the total dipole moment corresponding to the shift of the oxygen atom along each Cartesian axis were determined numerically by using the results of quantum mechanical computation for the distorted molecular geometry. The magnitude of the oxygen shift was taken as equal to 0.04A. It ensured a dipole moment variation significantly larger than the error of its calculation. On the other hand, it does not exceed the amplitude of the thermal motion too much. Proceeding from the definition (eq. (3.7)) of the dynamic charge, it is possible to determine the elements of this tensor by employing the dipole moment increments found by quantum mechanical computation: zdynal3= APa/Auf I "

(3.12)

These elements are (in electron charge units)'Zxx-dyn = -2.35 e, Zyy-dyn=-0.85 e, and Zzz-dyn_-0.89 e. All off-diagonal elements are negligibly small. This result can be treated as independent, and deduced from the "first principles" validation of the dynamic charge tensor estimation obtained empirically for a-quartz in refs. [14,15]. A variable charge model (VCM) was adopted in those investigations to describe the charge density response function. Similar ideas on the character of the anisotropy of dynamic oxygen charge in the Si-O-Si bridge originates in the VCM approach from an assumption that its dipole moment variation is governed mostly by the changes of the bond lengths. Correspondingly, the oxygen shift along x, which changes the lengths of both bonds substantially, induces the greatest dipole moment variation. Also, the oxygen shift along y, i.e., normally to the bonds, should weakly affect the dipole moment. An empirical

200

LAZAREV

Z

T "X

Fig. 3.1

Molecular geometry of disiloxane (C2v) and adopted directions of Carte-

sian axes.

fitting of the dynamic properties of ct-quartz in refs. [14,15] produced VCM parameters corresponding to the diagonal tensor of the dynamic charge of oxygen with the following elements" Zxx-dyn = -2.4e, Zyy-dyn=-1.0e, and Zzz-dyn = -1.5e. These values resemble ones obtained by the ab initio approach to the dynamic charge of oxygen in disiloxane, excluding a more significant difference for zz component. It should be mentioned that similar estimations of the elements of the dynamic oxygen charge tensor have been deduced earlier by Harrison [6] from an elementary quantum mechanical treatment of the electronic structure of the Si-O-Si bridge. Moreover, similar estimations have been obtained analogously for the dynamic oxygen charge in the Te-O-Te bridge [7]. Thus, one can suppose that the above ab initio estimation of the dynamic oxygen charge is applicable to a variety of systems with two-fold coordination around this atom. The smallest of the above ab initio dynamic oxygen charge elements may be treated as a tentative magnitude of the static (point ion) charge of oxygen in disiloxane. It corresponds to the oxygen displacement normal to its bonds, and practically does not affect their mutual arrangement (only a certain excitation in the out-of-plane Si-H' bonds can be availed). If so, the above estimation of other dynamic charge elements shows that a very significant difference between static and dynamic charges is possible with a high degree of

INTERACTIONS OF LESS LOCALIZED ORIGIN

201

anisotropy. It means that a scalar description of the ionic ch,,rge adopted in various dynamic models (see the references in ref. [ 16]) and attempts to employ its value in the calculation of static and dynamic properties of ionic-covalent crystals can hardly be fruitful. Another approach to the ab initio determination of the dynamic oxygen charge in disiloxane originates from eq. (3.8) which interrelates the dynamic charge of an ion with a variation of the electrostatic potential around it induced by the shift of the nucleus of that ion. The elements of the dynamic charge tensor are found in this case as: dyn _ Aq~(a,r) 2 Zal3 - AUl3 r .

(3 13)

Here, Aq~ (a,r) is a potential variation in the point removed by the distance (r) along the a axis from the equilibrium oxygen position (the origin of coordinates). This is induced by the Au shift of the oxygen nucleus in the 13direction. Employment of this expression is ensured by the possibility of direct ab initio quantum mechanical computation of electrostatic potential in any point in the vicinity of the investigated molecular system. In order to use this expression, a series of quantum mechanical computations of the electrostatic potential has been performed. A variation of the electrostatic potential alo~g the x, y, and z axes at several distances from the oxygen atom of the disiloxane molecule was determined at the shifts of that atom by 0.04A along the corresponding axis. The interval of distances from 1.5 to 4.2A covered the range of radii of the three first coordination spheres around the oxygen ion, which are most often met in the lattices of silicon dioxide dyn and silicates. The Zal3 tensor designed in this way proved to be diagonal like in the case of the employment of eq. (3.12) for its evaluation. In the present case, its elements turned out to be the functions of r. Fig. 3.2 represents the zdyn(r) dependence in the investigated interval while the horizontal lines correspond to the estimation of these elements from eq. (3.12) for dipole derivatives. It is seen that these lines are in satisfactory agreement with values corresponding to the asymptotic limits of zdyn(r) curves obtained by means of eq. (3.13). Various diagonal terms of the dynamic charge tensor approach their limits in a different way, but at the dis-

202

LAZAREV

tances near 3.5A, all of them become practically constant. It means that at r > 3.5A, a small shit~ of the oxygen nucleus excites the same electrostatic field variation as an equal shiR of a point charge (which correspond to appropriate element of the zdyn(oo) tensor). This estimation may be transferred to the lattice dynamics of oxides. The above considerations appear to validate the applicability of the point ion approximation to the description of the contribution of electrostatic interaction in an ioniccovalent crystal at distances exceeding some critical magnitude (as it was proposed in an approach to lattice dynamics which combined an explicit treatment of those interactions with the adoption of molecular force constants to description of the short-range interactions [14,16]). It should be emphasized, however, that in both above approaches to the oxygen dynamic charge estimation, a response of a whole electronic sub-system to the shift of a particular atom was investigated. Thus, the above estimations should be dependent on the size and structure of the adopted molecular system. This problem will be studied later.

C.

The Force Constants of Non-Bonded Oxygen-Oxygen Interaction

The force constants of non-bonding interaction between oxygen ions play an important role in any model presentation of the PF of oxide lattices which assumes the potential energy decomposition with respect to the internal degrees of freedom of a crystal. Their importance in the extension of the applicability to dynamic properties of crystals of the GVFF or UBFF models (originating from the theory of molecular vibration) has been emphasized in the previous chapter when discussing the GIVP formulation and its application to the evaluation of dynamic parameters. As has been mentioned above, these force constants most significantly contribute to the macroscopic elastic properties of oxides because of the larger interatomic distances than in valence bonds, and to the microscopic pattern of the homogeneous deformation of a crystal. Their contribution to the potential energy of phonon modes relates mainly to the low frequency lattice modes, and exerts a more restricted influence on most of the internal

INTERACTIONS OF LESS LOCALIZED ORIGIN

203

z, -e 2.5-

/

X /

~ x

dyn

Zxx

•

2.0-

1.5-

•

1.0-

X

Z dyn

-0~./...~0-----0 + - 1 - / ~ ~ _-~----------~-0 - ~ - -~0+ / , X - -

0.5

Fig. 3.2

1

~

yy + dyn 0 Z zz

I

I

I

2

3

4

r,X

A dependence of the dynamic charge elements of oxygen in disiloxane

upon the distance, r, as determined from eq. (13).

modes of complex anions. The contributions of the O...O interactions to the P.E.D. of any mode are usually spread over a variety of these distances existing in a lattice, and the influence of the particular one on the frequency is not easily estimated. However, in some crystals with unusually short O...O distances, which are smaller than the length of the tetrabedron edge in a complex anion, their existence reveals itself in very prominent effects on the frequencies of the internal modes. These effects were discussed in several papers of the book [17] in their relation to the problem of the driving forces of the so-called Davydov splitting (or factor-group splitting) of internal modes of complex ions in crystals and the main conclusions were summarized later in ref. [ 16]. According to Davydov's earlier supposition, a splitting of the internal vibrational states of molecules or polyatomic ions in crystals originates from their elec-

204

LAZAREV

trostatic resonance interaction. In this approach, the sources of that correlation field are seen in the excitations of internal modes, and an adoption of the multiple decomposition leads to accounting for the dipole-dipole interaction as a simple approximation. Thus it restricts the problem to a splitting of polar intemal modes. This approach was adopted by Decius [18] who extended it to include induced dipoles in a series of subsequent publications devoted to the splitting of internal modes of polyatomic ions in crystals (see, e.g., [19]). This approach met criticism from several authors [20-26] reasoning that there was an insufficient distance between the neighboring complex ion in a lattice for the applicability of the dipole-dipole approximation and an underestimation of a direct (short-range) interaction between adjacent peripheral atoms of various complex ions. It was argued, e.g., that in the dipole-dipole approximation, the splitting of any internal mode is proportional to its IR intensity with some factor containing the lattice sum multiplied by the inverse reduced mass of the internal vibrator. It followed from simple estimations that the differences in the reduced masses for various internal modes are generally much smaller than the differences in the IR intensity. Since for most crystals the lattice sum is the same for the internal modes belonging to a given irreducible representation of the site-group, their splittings should correlate with the IR intensities which is not supported by the experimental data*. The same is valid for the internal modes in teepleite, Na2B(OH)4CI, where the polyatomic B(OH)~ anions are "diluted" by the chlorine ions possessing no internal degrees of freedom, and thus more spatially separated [ 16,27]. Moreover, the splittings of the internal modes of the complex anion were found to be dependent on the localization of those modes: ones localized in the central BO 4 "core" of the anion are split significantly less than ones localized in the outer O-H groupings. It was treated as an indication of the importance of localized interactions between adjacent anions.

*The lattice sums may differ in the same irreducible representation in the case of crystals with variously oriented dipoles of normal modes in a certain representation. The peculiarities of the IR spectra of such crystals have been discussed in some detail in the previous chapter.

INTERACTIONS OF LESS LOCALIZED ORIGIN

205

However, their importance revealed itself most clearly in the splitting of non-polar or weakly polar internal modes of the pulsation type. A tremendous splitting, Aco, which reaches sometimes 10% of the unperturbed co0 magnitude of these modes in some crystals simplifies the analysis of its origin. A treatment of the simple model of a one-dimensional crystal with two AB molecules in the unit cell gives a very pictorial explanation of the origin of such splitting of the internal mode, and emphasizes its difference from the splitting of the resonance nature implied by Davydov's approach.

Suppose for simplicity that

m A "~ m B. It is possible to estimate the frequencies of two crystal modes corresponding to the sole internal mode, co0, as follows. In the case of the resonance nature of vibrational interaction, a non-vanishing term in the potential energy expansion exists, /92V/tgrABtgrBA ~ 0. The in-phase co+ and out-ofphase co. modes are split approximately symmetric with respect to the unperturbed internal frequency COo;co+~--e00 + Aco and co. ~--o~0 - Aco. The sign of the splitting (o3+~ co_ interrelation) being determined by the sign of the mixed second potential energy derivative. Another mechanism of splitting originates from the elastic properties of a lattice, e.g. from the stiffness of intermolecular contacts (atom-atomic interactions/92Vhgr2AA,/92V/tgr2p. ). In this case, the interrelation co+ > co_ necessarily holds; the crystal mode frequencies being thus co+- ~0 + A~ and co.- co0. The two above possibilities can hardly be discerned in the case of an unknown co0 frequency of a free molecule. The problem is solved, however, if the lattice mode frequency f~ and/or the macroscopic elastic constant of a crystal are determined since they are governed predominantly by the intermolecular interactions. All preceding considerations can be deduced from the inspection of the shapes of the normal modes of our onedimensional crystal which are shown in Fig. 3.3. It is seen, in particular, that the out-ofplane mode co_ weakly affects the intermolecular atom-atomic distances, while these are changed significantly at the in-phase co+ vibration (the co0 >> ~ interrelation is implied). It can be concluded from the above considerations that in a real three-dimensional crystal with dense packing of complex oxy-anions, a considerable splitting of the internal

206

LAZAREV

modes affecting the interanionic oxygen-oxygen distances may occur. In contrast with the basic assumption of Davydov's approach, dependence of splitting on the polarity of internal modes is not assumed, and even the non-polar modes are significantly split if they are accompanied by the variation of external oxygen-oxygen distances. As follows from the above model treatment, the splittings of this origin are easily recognized by the unsymmetdeal arrangement of the components relative to the unperturbed ~0 frequency: the co+ > 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

<----

<----

~

~

<---

<----

~

--~

<----

<----

~

~

4---

<----

<----

-9

(I)_

Fig. 3.3 Normal modes of one-dimensional AB crystal.

Y i1

7

20/

't

o-4

7

oo

~t

~ 'o-4~

~.

.....

t'

&-~--o

$ 8~ -'101"12(]9-" la(]86 20- . . . . . . . . . . . .

Y

_____~

O~

~',,, ,-r ~....... ~ ...... ? / ~

zl :o

:,~ . . . . . .

,,,'~

o,

v

Fig. 3.4 The crystal structure of zircon. A separation of the primitive cell is shown and the atoms therein enumerated.

208

LAZAREV

F2 I

A 1

E

F2

I

I

I

Free

S i O 4" i o n

Zircon modes i

ii

Eg En Big A2u Alg Bzu B2g

I I "l

^

9

,,

,

?

,, ?

A1 u

9 I

I

1000

800

Fig. 3.5

I

600

I

400

I

200

c m -1

The experimentally identified zircon optical modes interrelated with ones

of the free SiO 44- ion. The more or less pure lattice modes are denoted by dotted lines. Question marks correspond to optically inactive internal modes. is allowed in hyper-Raman scattering. However, special attempts to identify it remained unsuccessful [29,30]). In contrast to the pulsation mode, Davydov's splitting of the polar F 2 mode (Eg-Eu or B lg-A2u) is much less significant, and the average frequency of its components nearly coincides with the frequency of that mode in a free SiO 4- ion. A similarity with the above one-dimensional model is thus quite evident, and it is strongly supported by the results of normal coordinate calculation for the zircon lattice. A closer inspection of the zircon structure shows (Fig. 3.6) that each oxygen atom in this lattice is surrounded by two zirconium cations and a number of oxygen atoms belonging to adjacent complex anions. (The numbering of atoms in Fig. 3.6 corresponds to one adopted in Fig. 3.4 when defining their equivalent to translation set.) Four atoms are positioned symmetrically to the mirror plane. The atoms Oll and O12 are 2.84A from 0 5 and the pair of 0 7 and 0 8 atoms are at 3.07A from it. A single oxygen-oxygen contact is unusually short: the 0 5 . . . 0 9 distance equals 2.494A (all structural information is obtained from ref. [31 ]), i.e., it is significantly shorter

INTERACTIONS OF LESS LOCALIZED ORIGIN

209

Z

~Y

X Fig. 3.6 The nearest surrounding of oxygen in the zircon lattice. Atomic numbers correspond to ones adopted in Fig. 3.4, tx and ~ specify two sets oftetrahedral angles whose interrelation (tx < 13) characterizes a compression of oxy-anion across 4 axis.

even than the lengths of the edges of the tetrahedral oxy-anion. The direction of this contact deviates from the directions of related Si-O bonds by only 27 ~ In these circumstances, the enhancement of the in-phase A 1g component of the SiO4 pulsation mode is predominantly determined by the elasticity of the external O...O contact, and its force constant may be estimated directly from the normal coordinate calculation. Similar phenomena have been found in various other zircon typ~ crystals, the smaller splittings in their spectra being in a nice correlation with the increased inter-anionic oxygenoxygen distances in these less densely packed lattices. Another example of unusually short interanionic oxygen-oxygen distances is met in tx-spodumene, LiAISi206, a monoclinic pyroxene whose spectrum has been partially discussed in the previous chapter. Its structure has been shown in Fig. 2.3, and the adopted numbering will be employed in subsequent considerations. Two non-bridging Si-O bonds

210

LAZAREV

in the anionic chain considerably differ in their lengths, 1.586A for Si-O13 and 1.638A for Si-O 7, and a qualitative description of the corresponding stretching modes proved to be more consistent in terms of their separate vibrations than in terms of the Vas and VsO-SiOgroup vibrations often adopted in the analysis of the spectra of chain silicates [32]. As the structural data [33] show, the 0 7 atom in the longer Si-O" bond participates in a number of interanionic oxygen-oxygen contacts at distances shorter than 3A: atomicdistance, A atomic numbers

distance, A

(multiplicity)

numbers

7,52.504 (xl)

7,16

2.697 (x2)

7,142.661 (x2)

7,15

2.705 (x2)

(multiplicity)

7,82.694 (xl) while the O13 atom in the shorter bond forms a single interanionic [13,14] contact 2.787A long. A difference in the inter-anionic contacts clearly reveals itself in the factor-group splitting of corresponding internal modes of the bond-stretching type which is shown in Fig. 3.7. Since each short Si-O13 and long Si-O7 bond enters the one-dimensional unit cell of the chain anion twice, two internal modes, v SiO" and v'SiO-, of each type of bonds would arise. These are denoted correspondingly as vSiO ~]) for the shorter and vSiO (2) for the longer bonds, their frequencies being determined unambiguoasly from the factor-group analysis of experimental bands in the crystal spectra. As is seen from Fig. 3.7, the SiO ~]) stretching frequencies exceed those of the VasSiOSi modes of the oxygen bridges in a chain while the SiO (-2) modes are below them. In contrast to most of the vSiO internal modes which are split in the factor-group of the crystal insignificantly, the splitting of the v SiO (-2) mode is more than 100 cm-1, i.e., it exceeds 10% of the magnitude of the internal mode unperturbed by the interanionic interaction. Again, it is seen from Fig. 3.7 that this splitting may be treated as a high-frequency shift of the in-phase (Ag) component of the intemal

INTERACTIONS OF LESS LOCALIZED ORIGIN

I

211

I I

vSiO<~) v.SiOSi

I

Ag Be

I , siosi

vSiO~)

internal modes v'SiO(1) v.'.SiOSi

v'SiO(~)

II 11~00

v~SiOSi

I 10~00

9{JO

B~ Au

I $00

7(JO

600 cm -1

Fig. 3.7 SiO stretching normal modes oftx-spodumene.

modes which allows the interrelation of the frequency enhancement with the contribution of the force constant of the unusually short oxygen-oxygen interanionic contact. In a more general case, however, various oxygen-oxygen interactions contribute equally to the internal modes and corresponding force constants are not easily estimated. Several normal coordinate calculations for the crystals containing complex oxy-anions were conducted by Iishi (see, e.g. [34,35]) who described the elastic properties of complex anions by molecular force constants in the UBFF model by combining them with pair interactions in a lattice, and thus estimating by the frequency fitting of the O...O force constants. Generalizing his estimations, Iishi proposed the potential function of the non-bonded oxygenoxygen interaction in a form V(R) = -AR-6+BR "9 with A=18 mdyn.A 7 and B=430 mdyn.A 10 [36]. In the particular case where R=3A, it corresponds to the force constant of the non-bonded O...O interaction equal to 0.103 mdyn/A. It seemed attractive to employ the molecular quantum mechanical computations in an attempt to deduce some first-principle estimations of the force constants of the nonbonded oxygen-oxygen interactions in oxide crystals. A system composed of two disiloxane molecules was designed [13 ], and corresponding calculations performed with the same I + d(O) basis set as in the previous subsection. In that system, the two-fold axes of the two

212

LAZAREV

disiloxane molecules coincided, their oxygen apices facing each other and the SiOSi planes being turned by 90 ~ The equilibrium geometry for that system could not be found, and a following approach was adopted in the estimation of the O...O force constant as a function of the O...O separation. At any arbitrarily fixed O...O distance, the oxygen atom of one of the molecules was shit'ted towards another oxygen atom by 0.01A, and the change of force on the second atom along the O...O direction was calculated by the quantum mechanical methods. The oxygen-oxygen force constant for a given distance was thus obtained simply as Af (O...O)/A1 (O...O). The results of those computations are shown in Fig. 3.8. A similarity of the magnitudes of the O...O force constants deduced in this way with ones obtained empirically from the normal coordinate calculations is clearly seen. For the sake of completeness, this force constant is compared with the results of its estimation as being determined by the Coulomb force in the oxygen-oxygen contact, 2(zdxYn)2/13. This estimation leads to the underestimation of the steepness of the O...O force constant dependence on the length of this contact. At distances larger than 3A, it overestimates this force constant. This discrepancy can be suspected to relate to the importance of polycentric interactions which implicitly enter into the results of quantum mechanical computation. As the results of the population analysis show, a shiR of any atom in a system changes the electron density distribution over the whole system. In particular, it means that considerations deduced fi'om quantum mechanical computation of molecular system are dependent on its total size. This is why some of the above estimations will be treated in subsequent sections repeatedly when discussing the results of quantum mechanical computations of larger systems. In conclusion, it should be mentioned that another first principle approach to the estimation of the contribution of non-bonded oxygen-oxygen interaction to the PF of a crystal exists. It proceeds from the local density functional method. Its employment in a form

INTERACTIONS OF LESS LOCALIZED ORIGIN

213

o

Ko... o, mdyn/A 0.2

0

0.1

2

0

I

I

I

I

2.8

3.0

3.2

3.4

o

lo... o, A Fig. 3.8

A dependence of the O...O non-bonded force constant upon the atomic

separation as deduced from quantum mechanical computation (1) and from Coulomb approximation (2). proposed in ref. [37] seems to be the most promising. This problem has briefly been reviewed in ref. [38] where other important references are given. II.

T E T R A M E T H O X Y S I L A N E AS A M O D E L OF T H E S I L I C O N O X Y G E N T E T R A H E D R O N IN A N E T W O R K OF P A R T I A L L Y C O V A L E N T BONDS A.

Experimental Data and Spectral Assignments

Since the earliest investigations of the Si(OCH3)4 molecule and related systems were reviewed in ref. [39], they attracted much attention as relatively simple molecules that are accessible to experimental investigation. The structure and properties of the siliconoxygen tetrahedron were believed to resemble ones of the tetrahedron incorporated in a spatially extended network of partially covalent bonds, e.g., in silicon dioxide. On the other hand, the ab initio geometry and force fields of the SiO4 tetrahedron in the SiO44- ion or the Si(OH)4 molecule discussed in the Chapter 1 (see also ref. [40] for the silicic acid) could not be checked by direct comparison with the corresponding experimental data.

214

LAZAREV

Thus, the Si(OCH3) 4 molecule appears to be one of simplest ~ystems containing an SiO 4 tetrahedron whose theoretical frequencies can be compared with the experimental data. The tetrahedral symmetry of the central part of the molecule was found to be reduced to the S4 point group of a whole system because of the rotations of methoxy groups around Si-O bonds [41 ]. A free rotation was proposed from vibrational spectroscopy data [42]. A restricted rotation was, however, assumed in the gas phase electron diffraction study, and all structural parameters were refined at a value of 64 ~ for the dihedral angle between the OSiO and SiOC planes [41 ]. The existence of rather rigorous selection rules, which has been found in a most comprehensive investigation of vibrational spectrum discussed below [43,44], supports this point of view. Two isotopically substituted species, Si(OCH3)4 and Si(OCD3) 4, have been investigated spectroscopically in refs. [43,44]. Their Raman spectra in the liquid phase are presented in Fig. 3.9 where the polarization properties of bands are clearly seen. In order to obtain a better resolution of partially superimposed bands in certain frequency intervals, the IR absorption was studied at liquid nitrogen temperature, the specimens being in a form of thin amorphous films precipitated from the vapor onto a cooled stainless steel mirror. Only the frequency interval corresponding to the skeletal modes and the OCH 3 bending vibrations is shown in Fig. 3.10. Most of the vibrational bands are easily assigned by analogy with the spectra of methyl silyl ether and other molecules containing the SiOCH 3 group, although some of them belonging to similar motions in various irreducible representations of the point group of tetramethoxysilane are hardly resolved experimentally. Among the Si(OC)4 skeletal modes, the ones of the bond stretching type are characterized by the strong coupling of the SiO4 modes with stretching modes of adjacent C-O bonds. It was found reasonable to describe these modes in terms of VasSiOC and vsSiOC vibrations, thus deducing their shapes from the corresponding modes of a single Si-O-C bridge [44]. This description enabled a better separation in the potential energy distribution (P.E.D.) deduced from the normal

Fig. 3.9 Raman spectra of liquid Si(OCH3)4 (upper curve) and Si(OCD3)4(lower) in perpendicular and parallel polarization.

216

LAZAREV

100 80 60 40 20 0 80 60 40 20 0 ,

14'00

Fig. 3.10

'

12'00

'

10'00

'

8()0

'

660 500

460

IR absorption of solid films at-190 ~ of Si(OCH3) 4 (upper) and

Si(OCD3) 4 (lower). coordinate computation. These Vas and vsSiOC modes remained, however, to be coupled with the bending and rocking modes of the CH 3 or CD 3 groups. A description of the skeletal modes in terms of the SiO4 symmetry modes of the bond stretching type coupled with C-O stretching modes and methyl group modes adopted in ref. [43] is preferable in discussing of the analogies with the spectra of silicates. Both approaches will be employed in a qualitative discussion of spectral assignments between 1200 and 600 cm "1. A group of strong bands between 1200 and 1100 cm "1 in the spectrum of Si(OCH3) 4 evidently corresponds to the VasSiOC modes and the rocking CH 3 vibrations which can be specified as in-plane p'CH 3 and out-of-plane p"CH 3 according to their orientation relative to the SiOC plane. Only the p'CH 3 modes can be coupled with the SiOC stretching modes. The existence of strong IR bands near 1100 and 1200 cm "1 and the partial polarization of Raman bands with approximately the same frequencies represent that coupling. Supposing that both the polarity of the vibrations and their polarization in the Raman spectrum originate from the bond stretch contribution, it is possible to assign the bands at higher frequencies mostly to the p'CH 3 modes (p"CH 3 modes are supposed to be weak both in IR and Raman and completely depolarized in the latter). Reversely, near 1100 cm -1 a contribution of the VasSiOC mode probably prevails.

INTERACTIONS OF LESS LOCALIZED ORIGIN

217

At the S4 point group of the molecule, four Si-O-C bridges generate three VasSiOC modes in the A, B and E irreducible representations. The strongest IR bands at 1082 and 1072 cm -1 evidently correspond to the B and E modes while the A mode is identified in the Raman at a slightly higher frequency (1115 cm "1) because of the polarized character of that band. Thus, a relation between VasSiOC frequencies, VasSiOC(A)> VasSiOC(B,E), which will play an important role in speculations concerning the force field, follows directly from the experimental data. In the spectrum of Si(OCD3) 4 all rocking CD 3 modes are shitted to 920-900 cm "1 and are free of coupling with the SiOC modes. However, the bending CD 3 modes are shitted into the 1150-1050 cm "1 frequency interval and begin to mix with the VasSiOC mode. Because of the peculiarities of their shapes, the "symmetrical" dsCD 3 modes can be expected to interact most significantly with the skeletal stretching modes, the IR intensity and polarization in the Raman being, as above, the indications of a predominant contribution. A polarization of the Raman band at 1152 cm -1, which is higher than the frequencies of most strong IR bands at 1107 and 1094 cm -1, allows the assertion that the relation VasSiOC(A)>VasSiOC(B,E ) holds in the spectrum of deuterated tetramethoxysilane as well. Four vsSiOC modes can be identified at lower frequencies. As with the VasSiOC modes, they belong to the A, B and E representations of the S4 point group. These modes can be, however, treated as the stretching type modes of the SiO 4 tetrahedron, being deduced from the A 1 pulsation mode in the T d point group and the triply degenerate F 2 mode, respectively. The latter can be assigned in the spectrum of tetramethoxysilane to the IR and Raman bands in the 845-820 cm -1 interval which corresponds to the B and E modes of S4 point group. A lowering of their frequencies by 5-6% upon deuteration represents the coupling with the vibrations of the methoxyl groups. The strongest and completely polarized Raman band at 642 cm -1 corresponds to the pulsation mode of the tetrahedron, its frequency being reduced to 611 cm "1 upon deuteration. Besides its coupling with the C-O stretching mode, this mode can be supposed to mix weakly with the methyl group motions. Numerous poorly resolved bands between 450 and

218

LAZAREV

300 cm -1 should be assigned to the modes originating from the F 2 and E bending modes of the tetrahedron since the SiOC bending modes are known to lie conventionally at lower frequencies [44]. Only the ~SSiOC mode in the A irreducible representation is identified because of the partial polarization of the Raman band near 200 cm -1. This polarization probably represents a contribution of the bond stretch in the SiOC bridge. The totality of the experimental IR and Raman frequencies of Si(OCH3) 4 and Si(OCD3) 4 is collected in Tables 3.1 and 3.2, respectively, and compared with the theoretically calculated normal mode frequencies. The frequencies of the C-H and C-D stretching vibrations are not included in these Tables.

B.

Quantum Mechanical Computation

Because of the relatively large size of the molecule, its electronic structure and equilibrium geometry were calculated in a sp-approximation, i.e., the basis set of type I (described in Chapter 1) was adopted. The molecular geometry was refined along all internal degrees of freedom, and the residual forces were less than 0.004 mdyn, with the exception of the torsions around Si-O bonds which were about 0.0002 mdyn. Although its symmetry is lowered in the molecule discussed, the local T d symmetry internal coordinates were used for the SiO4 tetrahedron. Local symmetry coordinates were also employed for the bending modes of the OCH 3 group. The internal degrees of freedom of each of the Si-OCH 3 fragments were described in a form similar to the one adopted in the treatment of the symmetrically planar H3SiOCH 3 molecule, i.e., neglecting the insignificantly rotated orientation of the CH 3 group relative to the SiOC plane in tetramethoxysilane. The mutual arrangement of those fragments was described by means of four SiOC bending coordinates and four torsions around the Si-O and C-O bonds introduced by the conventional method through the dihedral angles of OSiO]SiOC and SiOC]OCH, respectively.

INTERACTIONS OF LESS LOCALIZED ORIGIN

219

TABLE 3.1 Experimental and Calculated frequencies (cm -1) of Si(OCH3) 4.

Raman (liquid)

IR (- 190~

1478-

1474 sh 1463 m

-1466

sh s dp

1460 m dp

Symmetry

P.E.D., %

~ A,B,E

3x1492

70

f

2x1472

72d s, 26 d~s

1470

77d s, 22 d~s

J

1445 sh

B,E A A,B,E

t'-

vs

fOcalc(cm- 1)

A

3x1460 1212

d~s, 27d s

95 d~s 70r 22 vasSiOC

1196 m pp

1194

L B,E

2x1203

84r'

1161mdp

1163 vw ~ A,B L E

2x1185 1184

94r" 931~'

1115 s p

-

A

1118

78 vasSiOC, 20r'

1094 m dp

1082 vs

E

1111

91 vasSiOC, 8r'

-

1072 vs

B

1091

94 vasSiOC

846 m dp

843 vs

E

829

91 vasSiOC, 8d(F2)

820 sh m dp

827 s

B

819

92 vasSiOC, 6d(F2)

-

814s

-

642 vs p

648 w

A

623

100 vsSiOC

440 w pp?

428 m

B

445

82d(F2), 5d(E), 4 vsSiOC

406 w dp

413 s

E

411

62d(F2), 25SIOC, 11 vsSiOC

309 w pp?

312"

B

321

49d(E), 36SIOC, 7d(F2) 6 vsSiOC

-

-

A

282

86d(F2),2SiOC, 8xCO

202 m p

-

A

226

96SIOC, 2d(E)

-

-

E

172

59SIOC, 19d(F2) 17xCO

-

-

B

114

56SIOC, 38d(E)

-

-

*From the gas phase spectrum. REMARK: Calculated torsion frequencies 114, 107 and 104 em-1 for xCO and 53, 43 and 37 cm-1 for xSiO.

220

LAZAREV

TABLE 3.2 Experimental and Calculated frequencies (cm -1) of Si(OCD3) 4. Raman (liquid)

IR (-190 ~

Symmetry

152mp

1148 sh

A

1147

86 vasSiOC, 12ds, 4r'

lll8mdp

?

E

1129

47 vasSiOC, 47d s, 5 vsSiOC

B

1119

94ds, 4 vsSiOC

1107 vs

E

1115

49 vasSiOC, 47d s

1094 vs ~

A

1108

82d s 12 vasSiOC

L

B

1102

98 vasSiOC

1080 sh

A

1072

92 d~s

1070 sh

B,E

2x1071

94 d~s

A,B,E

3x1058

100d~s

-

1099 m dp

1071 sdp

-

917 spp

924

B,E S

L f

907 m dp

904 m

Ocalc (cm-1)

P.E.D., %

2x928

84r', 6 vsSiOC, 2 vsSiOC

A

918

90r', 4 vsSiOC, 4 vsSiOC

B A

908 907

98r" 98r"

E

905

98r"

798 m dp

801 s

E

793

79 vsSiOC, 9r', 6d s

786 sh dp

779 s

B

783

80 vsSiOC, 6r', 4d s

611 vsp

619 vw

A

591

96 vsSiOC

419wdp

418 w

B

435

81d(F2),5d(E), 4SiOC, 4 vsSiOC

391 wdp

395 m

E

394

62d(F2), 22SIOC, 12 vsSiOC

-

B

304

51 d(E), 32SIOC, 6d(F2) 6

365 sh? pp? 290 m dp

vsSiOC

184mp

-

A

275

87(E), 6xCO, 4xSiO

-

A

203

96SIOC

-

E

156

61SiOC, ! 8d(v2)

-

B

101

52SIOC, 37d(E)

REMARK: Calculated torsion frequencies 90, 80 and 76 cm -1 for xCO and 43, 36 and 32 cm -1 for xSiO.

INTERACTIONS OF LESS LOCALIZED ORIGIN

221

A totality of independent internal coordinates of tetramethoxysilane was thus introduced as follows 0', tx and 13 correspond here to the angles OSiO, OCH, and HCH, respectively)" Number

Description

1

SiO4(A1) = 1 (11 + 12 + 13 + 14)

2

SiO4(F~) = ~1 (11 + 12 - 13 - 14)

3

SiO4(F~') = ~1 (11-12 + 13-14)

4

SiO4(F~") = 1 (11 _ 12.13 + 14)

5-8

C-O

9-20

C-H

21

d(E') -- - ~1 (2Y12 + 2)'34- ~/13 -Y23- Y24- Y14)

22

d(E") = 1 ()'13 - Y23 + )'24 - )'14)

23

d(F:~) = ~ 2 (Y34-)'12)

24

d( F~' ) - ~ 2 0'24 - 713)

25

d(F~" ) = ~ 2 (Y23 - Y14)

26-29

SiOC (bending)

30-33

dsCH3 = ~ 6 E ( I ] i - ~ i ) i=l

3

34,36,38,40

r'CH3 = ~6- (2al - a2 - ct3)

35,37,39,41

r"CH 3

42,44,46,48

d~s CH3 = ~66 (2131 - 132 - 133)

43,45,47,49

d~s CH3 = ~ 2 (132 - 133)

50-53

SiO

54-57

CO

=

~22 ( a 2 - a3)

The adopted bond enumeration in the skeleton corresponds to one given in Fig. 3.11, which shows the molecule as viewed along the S 4 axis.

222

LAZAREV

C 5

8 o

Fig. 3.11 Molecular geometry of tetramethoxysilane. The symbols employed in the above listing are adopted in the discussion of the shapes of normal modes, P.E.D., and force constants. It should be noted that at S4 molecular symmetry, the F 2 components of the triply degenerate internal coordinates of the SiO4 tetrahedron correspond to the B irreducible representation of that point group, while the F6' and F6" components correspond to the degenerate representation E. Among the components of the E coordinate of a T d point group, the E' component falls into irreducible representation A and the E" component into representation B. In a conventional description of the intemal OCH 3 modes of the bending type as symmetric (ds), antisymmetric (das), and rocking (r) local coordinates (which can be symmetric (') or antisymmetric (") to the OCH plane), a record of the CH 3 or CD 3 group is often omitted (see, e.g., Tables 3.1 and 3.2). The equilibrium geometry parameters of tetramethoxysilane deduced from the energy minimization are presented in Table 3.3 in comparison with the results of computation for the H3SiOCH 3 and Si(OH)4 molecules and any experimental data where they are available. Although all theoretical magnitudes were obtained with the same basis set I, those

INTERACTIONS OF LESS LOCALIZED ORIGIN

223

Table 3.3 Theoretical optimized (re) and experimental molecular parameters (rg) of Si(OCH3)4 compared with the molecular parameters ofH3SiOCH 3 and Si(OH)4. H3SiOCH 3

Si(OCH3)4

re

rga

re

rgb

re

SiO

1.681

1.640

1.646

1.613

1.643

CO

1.438

1.418

1.435

1.414

CH

1.078

CH'

1.084

1.080

1.078 1.082

CH'

1.084

Kind of parameters Bond lengths (A)

t

t

Si(OH)4

1.120

1.083

Bond angles

OSiO (x2)

-

-

113.7

115.5

114.2

(degrees)

OSiO (x4)

-

-

107.4

106.0

107.2

SiOC

128.2

120.6

131.9

122.3

OCH

107.4 ]

OCH'

111.2

HCH'

109.1 ]

H'CH'

108.9

111 108

OSiOICOSi

-

SiOCHIOCH Net charges

107.4 } 110.9 109.2 } 108.9 83.8

5.4

Si

+ 1.175

O C

-

111 108 64

88.1

0

+2.144

+2.066

-0.872

-0.926

-0.964

-0.136

-0.120

aRef. [29] in Chapter 1. bRef. [41] in this Chapter. Note: see Chapter 1 for the difference between H and H'.

relating to the Si(OH)4 molecule differ slightly from the corresponding values in Table 1.6 (Chapter 1) due to a more complete relaxation of forces. These structures are included in the table in order to facilitate the analysis of changes in the SiO4 moiety of the Si(OH)4

224

LAZAREV

molecule when hydrogens replace the methyl group, and the changes in the structure of the SiOCH 3 fragment of the H3SiOCH3 molecule under the same H ~ OCH 3 substitution. The theoretical geometries reproduce (with sufficient accuracy the changes observed in the Si-O and C-O bond lengths upon the H ~ OCH 3 substitution in the Sill 3 group. These changes can be linked to significant growth of the calculated positive net charge of silicon in Si(OCH3)4. The calculated flattening of the SiO4 moiety along the S4 axis is nearly the sane in Si(OCH3) 4 and Si(OH)4, and it may relate to the similarity in the internal rotation around the Si-O bonds in both systems. The potential function governing internal rotation around the Si-O bonds was found to be rather flat. However, the energy minimum was obtained at a value of the dihedral angle between the OSiO and SiOC planes (0) close to 84~ (as in the structural investigation [41 ], these torsional angles were calculated with respect to the eclipsed periplanar configuration with D2d symmetry). It should be noted that rotations around the Si-O bonds are strongly correlated with the Si-O bond length and OSiO bond angle values. This can be rationalized in terms of oxygen lone-pair interactions [44], assuming that these interactions may strongly affect equilibrium values of the OSiO[SiOC dihedral angles and the OSiO angles in the SiO4 tetrahedron. Besides the deviations in the calculated bond lengths from their experimental values, the shortcomings of the adopted narrow basis set reveal themselves most significantly in the overestimated SiOC angles which may relate to the results of calculation of the internal rotation. A slightly unsymmetrical arrangement of the methyl group relative to the SiOC plane makes two C-H' trans-bonds that are not quite equivalent. This difference was, however, neglected in the normal coordinate calculation and the force constant assessment.

C.

Frequency Fitting and the Force Constant Evaluation

A comparison of the SiO4 fragment geometry in optimized Si(OCH3)4 and Si(OH)4 structures shows very small changes in the geometrical parameters of this group. The similarity of the SiO4 geometry allows us to assume that the same similarity exists between

INTERACTIONS OF LESS LOCALIZED ORIGIN

225

the force fields of the SiO4 moiety in these molecules. Thus, since the complete quantum mechanical force constant calculation for tetramethoxysilane would involve too much computer time, a set of force constants of this molecule was composed in the following way. The ab initio force field of the SiO4 fragment in the Si(OH)4 molecule (basis set I,

see Chapter 1 for discussion) was transferred to the Si(OCH3)4 molecule without any scaling. The empirically scaled force field of the SiOCH 3 group, which was calculated with the basis set II [44], was taken from the H3SiOCH 3 molecule (including such force constants as the ones describing the SiO/CO and SiO/SiOC interaction). The values of force constants describing rotation around the Si-O and C-O bonds were also, more or less, provisionally transferred from that molecule. The adoption of this force field for the Si(OCH3) 4 molecule implied that all interactions between the four OCH 3 groups were implemented through the interactions of localized origin inside the silicon-oxygen tetrahedron, and that any other interactions between these groups are negligible. The force field approximation deduced in this way was found to satisfactorily reproduce the experimental frequencies of SiO(CH3)4 and Si(OCD3)4 in the frequency areas above 1200 cm-1 and below 1000 cm -1. However, it failed to correctly reproduce the coupling of the VasSiOC vibrations with the rocking CH 3 or the bending CD 3 modes. In particular, the experimentally determined interrelation between the VasSiOC frequencies in various irreducible representations of the S4 point group could not be reproduced either by the initial approximation of the force field or by the variation of its parameters. At any assumed scale factor for the diagonal stretching SiO and CO force constants of the initial field, the calculated VasSiOC in the A representation remained lower than the corresponding frequencies in the E and B representations. In order to resolve this contradiction, some ab initio quantum mechanical computations of the force constants of the Si(OCH3)4 molecule were undertaken employing the set of independent internal coordinates described above. Only two lines in the force constant matrix of tetramethoxysilane, which corresponded to the A 1 and F 2 stretching SiO4 coordinates, were calculated (mdyn/A, mdyn,mdyn.A):

226

LAZAREV SiO 4 (A1)

SiO4(F2)

SiO4(A1)

7.161

-

SiO4(F~)

-

6.369

d(F2) -0.482

CO

~SSiOC

xSiO

-0.096

0.242

0.006

0.038

0.249

-0.017

Adopting the diagonal SiO4(A 1) force constant and assuming that the SiO4(F~', F:~") force constants coincide with the SiO4(F:~ ) force constant, it is possible to pass in the space of unsymmetrized internal coordinates, and to calculate the SiO stretch force constant fsio = r _ 6.567 and the interaction force constant of two Si-O bonds at a common Si atom "SiO 0.198. Further, using the above ab initio off-diagonal force constants SiO4(A1)/CO and SiO4(F~)/CO, the interaction force constants of the Si-O and C-O bonds at a common oxyCO - 0.009, and of the same bonds having no common atoms, "SiO r gen atom, fsio - -0.067, are deduced directly from the quantum mechanical computation. Since in the practice of normal coordinate calculations the interactions between nonadjacent bonds are believed to be negligibly small, the latter of the above quantities deserves special comment. Its non-vanishing magnitude indicates that the dynamic interactions in the tetramethoxysilane molecule are less localized than it was proposed in the originally designed approximation of the force field. In particular, the silicon-oxygen tetrahedron interacts with adjacent C-O bonds as a whole, and this interaction is not localized in a particular Si-O bond (in this connection, the considerations of Chapter 1 concerning the delocalized nature of the deformational polarization of the SiO 4 tetrahedron should be noted). The results of the ab initio calculation of a portion of the force field of tetramethoxysilane may be treated as an indication of how one should change its initial set of force constants in order to remove the contradictions between experimental and calculated frequencies. The probable errors of quantum mechanical computation with a restricted basis set should be taken in mind as well. It is known, for instance, that in a case of neglect of the polarizing functions of the oxygen atom, this computation underestimates the bond-bond interaction at this atom [44] and a negative sign of this force constant may be obtained, e.g., for the CO/CO force constant of dimethyl ether.

INTERACTIONS OF LESS LOCALIZED ORIGIN

227

Only a few parameters of the initial force field were varied in its refinement proceeding from the above considerations. These were the diagonal SiO 4 and CO stretching force constants and the constants of their interaction. In addition, the interaction force constants between the CO stretching and SiOC bending coordinates were adjusted, mainly, by fitting the identified ~SSiOC experimental frequency in the A representation near 200 cm "1. It should be noted that no difference between the Fi~(B) and the F:~', F:~"(E) components of the triply degenerate SiO 4 stretch was proposed in this firing because of an insufficient amount of experimental data and ab initio computed force constants. The results of the variation of the initial force field are shown in Table 3.4 where only a portion of the total force constant matrix is presented since other force constants do not differ from ones transferred from the scaled ab initio force field of methyl silyl ether. Calculated frequencies of Si(OCH3) 4 and Si(OCD3) 4 obtained with the refined force field (presented in Tables 3.1 and 3.2) are in reasonable agreement with the experimental data. In particular, the interrelation VasSiOC(A) > VasSiOC(B,E ) is reproduced. As it was already mentioned, the P.E.D. of the bond stretching skeletal modes is presented in terms of Vas and vsSiOC coordinates introduced as vCO-vSiO and vCO + vSiO combinations, respectively. In order to rationalize the results of the refinement of the initial force field of tetramethoxysilane, some of the force constants in Table 3.4 may be reduced to VFF in the unsymmetrized valence basis set: fsio = 6.02, "SiO eSiO = 0.36, fsioCO = 0.131 ' eCO' = -0.139, ~SiO where the latter quantity relates to the interaction between non-adjacent bonds. Its negative sign coincides with one deduced from the quantum mechanical computation while the absolute magnitude is twice as large. It should be noted that the non-vanishing interaction force constants of non-adjacent bonds are known to meet among the ab initio force fields of some aromatic compounds [45] where their appearance is tentatively interrelated with peculiarities of an electronic subsystem. I.e., it was supposed that such force constants represent the long-range interaction

TABLE 3.4 A portion o f the force constant matrix o f tetramethoxysilane.

0.178 4.384

0.430

0.178 0.430

0.178

0.430

0.178

0.430 Note: The underlined numbers correspond to the force constants whose magnitudes differ from the initial approximation of the force constant matrix.

INTERACTIONS OF LESS LOCALIZED ORIGIN

229

in a system originating from the electron delocalization. It has never been investigated with a sufficient degree of reliability in an estimation of the interaction force constants of nonadjacent bonds, if their existence may originate from interactions of another physical nature. This problem will be paid more attention in the following sections.

IlL

T H E D I S I L I C I C A C I D M O L E C U L E AS A M O D E L O F T H E FRAGMENT OF A SILICA NETWORK

A.

Electronic Structure and Equilibrium Geometry

The most often adopted approach to the ab initio quantum mechanical computation of solids in a cluster approximation consists of a treatment of a certain fragment of a considered lattice as a molecular-like system with broken peripheral bonds saturated by an appropriate number of hydrogen atoms. It makes the employment of simple and operative closed-shell SCP computational methods possible. In particular, several computations for the silicic acid molecule, Si(OH)4 (mentioned in Chapter 1) can be complemented by other references [46-48] where the PF of this molecule has been calculated in order to evaluate the force constants applicable to silica and silicates. Sometimes it was not discerned between the employment of the Si(OH)4 molecule as a source of information on the properties of the silicate ion, SiO 4, in a lattice or of the SiO 4 tetrahedron in a system with shared apices of tetrahedra. As a more consistent approach to the latter systems, the disilicic acid molecule, (OH)3SiOSi(OH)3 , has been repeatedly investigated by numerous authors [49-53]. Although various basis sets, from the minimal one up to 6-31G**, were employed in those computations, most of them were restricted to the calculation of the equilibrium geometry and of several force constants. Neither the complete force constant matrix has been ever presented, nor the force constants describing more or less distant interactions in this relatively large molecule discussed. It was decided, therefore, to repeat the quantum mechanical SCF computation of (OH)3SiOSi(OH) 3 paying most attention to its force field and, in particular, to its parameters which represent the interactions between moderately distant atoms. For the sake of

230

LAZAREV

economy, the basis set of the type I has been adopted (cf. Chapter 1) while excluding the oxygen atom in the Si-O-Si bridge, O br (whose basis set has been augmented by polarizing d-functions like in the I + d(O) basis set mentioned above), in order to avoid an unrealistic narrowing of the SiOSi angle. The force relaxation has been carded out up to residual forces less than 0.001 mdyn along any internal coordinate. Their non-redundant set was introduced in terms of local (approximate) symmetry coordinates, as it was explained above, for the geometrically related systems, and complemented by bending and torsion in the Si-O-Si bridge. A physically inconsistent equilibrium geometry was obtained in some earlier unrestricted optimizations.

The two Si(OH)3 groups were non-equivalent and the Si-O-Si

bridge was not symmetric. The total energy minimum with respect to the nuclear coordinates was, in our investigation, conducted with C 2 symmetry which enabled the equivalence of the terminal Si(OH)3 groups (and symmetric arrangement of the bridge) imposing no restrictions on the internal geometry of those groups. Their geometrical parameters are listed in Table 3.5 where some quantities characterizing the bonding and charge distribution are presented as well. The dihedral angles SiOSiIobrsio and obrsioISiOH which characterize the internal rotation around the obr-si and Si-O(H) bonds, respectively, are calculated from the more symmetric C2v geometry similar to the one obtained for disiloxane or dimethyl ether (see Chapter 1). It should be noted that three Si-O(H) and three O-H bonds are twisted in the same direction. The interaction between the lone pairs of the bridging oxygen and the valence pairs of the Si-O(H) bonds reveals itself in some lengthening of one of the bonds which is nearly in a

trans-position.

It can be deduced from the localization procedure that

the lone pairs and bonding pairs around the O br atom are arranged relative to the bonds of the Si-O-Si bridge similarly to their arrangement in disiloxane (cf. Chapter 1). In particular, the bonding density is shifted from the bond axes inside the SiOSi angle despite its reduced magnitude. The differences in the equilibrium bond lengths of the Si-O(Si) and Si-O(H)

INTERACTIONS OF LESS LOCALIZED ORIGIN

231

Table 3.5 Electronic structure and equilibrium geometry of the (OH)3SiOSi(OH)3 molecule

Numerical magnitudes

Parameters -Et~

29974.317

(eV)

Geometry

Net charges

reSiObr (A)

1.624

reSiO

1.649,

1.649,

1.652

reOH

0.954,

0.955,

0.956

ZSiOSi (degrees)

133.2

zobrsio

109.6,

111.8,

126.1

ZSiOH

129.4,

128.1,

108.5

/OSiO

107.4,

110.6,

156.1

/SiOSiIObrSiO

35.0,

84.0,

67.3

/obrsioISiOH

96.5,

83.8

Si

+2.127

obr

- 1.184

O

-0.958,

-0.963,

-0.973

H

+0.463,

+0.454,

+0.452

Overlap population

108.9

Si-Obr

0.358

Si-O

0.516,

0.507,

0.443

O-H

0.540,

0.540,

0.541

bonds and in the net charges of the corresponding oxygen atoms agree with the trends for simpler systems discussed in Chapter 1, and are apparently weakly dependent on the differences in the adopted oxygen wave functions. This large molecule composed of silicon and oxygen atoms can be employed in the estimation of the dynamic charge of the oxygen atom in the Si-O-Si bridge as it has been explained in the first section of this chapter. Both approaches originating from the applica-

232

LAZAREV

tion of eq. (3.7) or of eq. (3.8) have been tried. In the former, the dipole moment increment corresponding to a shift of O br along any Cartesian axis was calculated by quantum mechanical methods (Cartesian axes were oriented relative to the SiOSi plane like in the case of disiloxane) and zdY a13n was deduced from eq. (3.12). At negligibly small off-diagonal _dyn = -2.13e, Zyy _dyn = _1.0 le, Zzz _dyn -_ elements of that tensor, the diagonal elements were" Zxx -1.03e. The results thus resemble ones obtained for disiloxane, but the anisotropic tensor of the dynamic charge is smaller which may relate to the peculiarities of the geometry of the adopted molecular system (less anisotropic charge distribution). Another approach, based on the quantum mechanical computation of the electrostatic potential increments corresponding to the shifts of O br, has been employed as well. This approach allowed to judge how the dynamic charge elements calculated from eq. (3.13) depend from the remoteness of the point where the potential is determined from that atom. It is seen from Fig. 3.12 that the asymptotic limits of z dyn ctct (r) curves are not so far from the estimations of dynamic charge deduced from the dipole moment increments (horizontal lines). However, these limits are approached much less steeper than in the case of disioxane (cf. Fig. 3.2). It means, consequently, that a point ion approximation in the description of the electrostatic interaction becomes valid at distances approaching 4.5A, and probably an even a larger estimation would be obtained in the case of a similar calculation for a system of greater size. It can be concluded that the point ion the approximation in the description of the PF of a crystal would be adequate at the distances larger than 5A. The slowest approach of the zxxdyn component to its long-range limit, which has been found in both systems (cf. Figs. 3.2 and 3.12), should be emphasized. These findings relate to attempts to decompose the PF into the essentially shortrange contribution described by the ab initio force constants of a reasonably small molecular species and into the long range contribution calculated as a Coulomb interaction with the rest of a lattice in the point ion approximation. In its practical application, this approach will probably suffer from the inadequacy in the description of interatomic interactions at

INTERACTIONS OF LESS LOCALIZED ORIGIN

233

z, - e 2~

-

~• 2~

" X

1~

J

"

/ f

1o0

z~a~n

-

X

X

.+.._~.- +

x /

0.5 ]

-+

A~O - - - ' - - ' -

I

I

2

3

0

+-

_+ Z ~ ~n

0

0 Zzdzyn

I

I

4

5

0

r,A Fig. 3.12 A dependence of the dynamic charge elements of O br in H6Si207 upon the distance, r, as determined from eq. (3.13).

moderately long distances (from third to fourth coordination spheres), at which the point ion concept might be still too rough while the ab initio computation of corresponding interaction force constants in a space of intemal coordinates would need a laborious treatment of a very large molecular system. As it relates to silica and silicates, the corresponding effects can at least be partially estimated by the inspection of the ab initio force field of disilicic acid where the offdiagonal force constants interrelating the internal coordinates localized in adjacent tetrahedra are calculated directly by quantum mechanical methods.

234

LAZAREV B@

A b Initio F o r c e Field

Investigation

and

Intertetrahedral Interac-

tions The equilibrium geometry of (HO)3SiOSi(OH)3 is shown in Fig. 3.13 where the numbering of the bonds in the heavy atom skeleton is indicated. This numbering is employed to label the internal coordinates introduced for the description of the force field. A complete ab initio force constant calculation has been performed. However, only ones relating to the internal degrees of freedom of that skeleton will be discussed below. In order to avoid redundancy, the SiO 4 bending coordinates were introduced, as usual, in the form of local symmetry (C3v) coordinates. The total set of linearly independent internal coordinates of the skeleton was thus defined as follows: Number

Description

1,2

Si-O br = 11,12

3-8

Si-O

=

13, 14 ""18

9

~SSiOSi = ~512

10

dl0

11

dll

12

diE

_-

13

d13

_

14

d14

15-19

= _

-

_

-

1 ~ (t~45 + t~34 + 0t'35 - ~13 - 1~14 " ~15) 1

~

~

(2~13"~14"~15)

1

(~14"

~15)

1

(2~45" tx34- tx35)

1

~f~ (tx34" tx35)

dl 5"dl 9 similar combinations as in dl 0 - d l 4 for the angles in another tetrahedron

20

1;20

21

x21

=

1 x/3 (XI § 1;II § xIII)

=

1 x/3 (xIV + "rV 4- XVI)

where xI - XVi imply corresponding dihedral angles.

INTERACTIONS OF LESS LOCALIZED ORIGIN

235

O Fig. 3.13 Molecular geometry and Si-O bonds numbering in the disilicic acid molecule.

A corresponding portion of the force constant matrix is represented by Tables 3.6 and 3.7, the first describing the interactions inside one of the two SiO4 tetrahedra and the second containing the intertetrahedral interaction terms. The diagonal and interaction force constants of the obrsio 3 group in Table 3.6 are within values deduced for the corresponding force constants of other systems containing an SiO4 tetrahedron. The force constant of the SiOSi angle is, however, approximately enhanced by an order of value when compared with the corresponding force constant of disiloxane. Since the electronic structure of the Si-O-Si bridge remains essentially the same in both systems, including the character of the valence density distribution around O br, this enhancement may originate from the contribution of interaction between spatially extended Si(OH)3 groups. In other words, it can be suspected that enhanced the SiOSi bending force constant of (HO)3SiOSi(OH)3 does not represent the intrinsic property of that angle, and characterizes this system as a whole.

TABLE 3.6 The portion of the total H6Si207 force constant matrix which specifies the interactions inside of the obrsio3fragments (unscaled). Q\

No. of coordinates

1 siobr

3 SiO

4 SiO

5 SiO

10 dSiO,

11 dSiO,

12 dSiO,

6.431

0.224

0.193

0.244

-0.372

0.042

0.012

13 dSiO,

14 dSiO,

20 ssiobr

-

9 GSiOSi 0.343

TABLE 3.7 The portion of the total H6Si207force constant matrix which specifies the interactions inside of the obrsio3fragments (unsealed).

No. of coordinates

2 siobr

6 SiO

7

SiO

8 SiO

15 dSi0,

16 dSi0,

17

dSi04

18

dSi0,

19

dSi04

21 rsiobr

3

z n

238

LAZAREV

Several small (in their absolute value) force constants weakly affecting the vibrational frequencies are omitted in Tables 3.6 and 3.7. It is seen from Table 3.7, however, that a certain dynamic interaction exists between coordinates localized in two obrsio 3 groups including ones possessing no common atoms. Such interaction terms are otten set to zero in the empirical frequency fitting in molecular spectra by means of simplified versions of the GVFF model. Sometimes, however, their importance can be deduced directly from the experimental frequencies as it has been done in the above analysis of the spectrum of tetramethoxysilane. Since the interactions of this type are particular for relatively large molecular systems, an investigation of their origin deserves some attention if the subsequent employment of the force constants in the design of the force field of a crystal is intended. It is seen from Table 3.7 that there exist non-vanishing interactions between internal degrees of freedom of two SiO3 groups both of stretch/stretch and bend/bend types. An attempt to rationalize their magnitudes deduced from quantum mechanical computation in terms of some model approaches will be started from the simpler one relating to the bond/bond interactions of non-adjacent bonds. The siobr/siO(H) and SiO(H)/SiO(H) interaction force constants of non-adjacent bonds differ in their signs, the former being negative and the latter positive. It can be treated as an alteration of the sign of the SiO/SiO dynamic interaction depending on the order of the neighborhood of those bonds: negative interaction force constant for the secondorder neighboring bonds and positive for the third-order neighboring bonds. The possible origin of this trend will be discussed later. The absolute magnitudes of these interaction constants decrease in the same direction, and it would be of interest to investigate if they represent the Coulomb interactions between the stretchings of non-adjacent bonds. In order to simplify the problem, it is possible to calculate these force constants as originating from the dipole-dipole interactions. Two approaches have been tried when specifying the dipoles arising in the stretched Si-O bond. The first one describes them as originating from additional unit charges, which arise on both atoms. The another one takes into consideration the conclusions on the valence

INTERACTIONS OF LESS LOCALIZED ORIGIN

239

charge distribution in the Si-O bonds inferred from the ab initio computations and assumes the arising of point dipoles on the oxygen atoms in stretched bonds, their orientations being supposed to coincide with the bond directions. The results of these model calculations are compared with the ab initio interaction force constants below. Interaction force constants Interacting bonds

ab initio

charges at Si and O

point dipole at O

1/6"

-0.048

-0.106

-0.300

1/8

-0.074

-0.213

-0.239

1/7

-0.130

-0.338

-0.171

3/6

0.020

-0.068

0.037

3/7

0.029

-0.072

0.044

3/8

0.017

0.010

0.029

*Numbering of the bonds is according to Fig. 3.13. Both model approaches coincide in the correct reproduction of the sign of the interaction force constant for the second-order neighbor Si-O bonds, while only the model assuming additional atomic charges in the stretched bond succeeds in the reproduction of the sequence of the magnitudes of those force constants. However, only the approximation of point bond dipoles on oxygen atoms correctly reproduces the sign and relative magnitudes of the interaction force constants for the third-order neighbor Si-O bonds. Considering the sharp sensitivity of the results of the model calculations to the assumptions on the location and orientation of bond dipoles, they seem to substantiate the supposition predominantly on the Coulomb origin of the dynamic interaction between non-adjacent Si-O bonds. It is seen from Table 3.7 that the interactions of the bond stretching/angle bending type between adjacent SiO 4 tetrahedra practically do not affect the frequency spectrum of the H6Si20 7 molecule. The interactions between the bending deformations of the OSiO angles belonging to different tetrahedra are, however, more significant with their magnitudes approaching ones characterizing the interactions inside a tetrahedron (cf. Tables 3.7

240

LAZAREV

and 3.6). It would thus be instructive to discuss the intertetrahedral interactions of this type in more detail. Among the OSiO bending coordinates (Nos. 10-14) of one of tetrahedra in Table 3.7, the coordinates (Nos. 10-12) originate from the F 2 normal mode of the regular tetrahedron, and in another tetrahedron these correspond to coordinates 15-17. The components of the F 2 bending modes of the two tetrahedra are seen to interact dynamically in a larger extent than ones originating from their non-polar bending E modes (coordinates 13-14 and 1819, respectively). The signs of the interaction constants being the same for any pair of coordinates and their magnitudes not differing too much. The intertetrahedral interactions of the bending coordinates originating from the bending E modes are less significantly different with similar coordinates or with coordinates originating from the F 2 mode. In order to rationalize the ab initio intertetrahedral bend/bend interactions in terms of the GVFF model, it is possible to express them as the interactions between particular OSiO angles of two tetrahedra. If the local symmetry coordinates, s, are interrelated with the initial (redundant) set of internal coordinates, g, through the symmetrization matrix, C, as: s =

Cg, a back transformation of F s into an Fg matrix is defined as: Fg = C+FSC (adding of some zero rows and columns to the F s matrix is implied). A portion of the Fg matrix which corresponds to the intertetrahedral bend/bend interactions then looks out as follows: Coordinates

t~45

t~34

~35

~13

~14

~15

~78

0.1425

0.1965

0.1965

0.1272

0.1314

0.2059

ot67

0.1965

0.1845

0.1845

0.1802

0.1106

0.1437

r

0.1965

0.1845

0.1845

0.1802

0.2216

0.0327

1326

0.1272

0.1802

0.1802

0.2272

0.1811

0.1043

~27

0.1314

0.1106

0.2216

0.1811

0.1721

0.1832

~28

0.2059

0.1437

0.0327

0.1043

0.1832

0.3303 .

All angle/angle interaction force constants are thus positive, and their magnitudes are rather uniformly distributed over most of these interactions. It should be emphasized that

INTERACTIONS OF LESS LOCALIZED ORIGIN

241

there is no significant difference between the interactions of angles having a common atom and of angles separated by one or two bonds. An attempt to rationalize the magnitudes of these interaction constants in terms of some simple model approach (like as in the case of the stretch/stretch interactions) failed. The average over all types of interaction (tx/c~, 13/13, and ct/I]) magnitude of the interaction force constant (0.167 mdyn.A) should probably be slightly reduced in accordance with the expected result of the scaling procedure. The possibility of the arising of the OSiO/OSiO interactions between adjacent tetrahedra with values up to 0.15 mdyn.A are to be taken in mind when designing the GVFF models of silica and silicates. It can be concluded from the above discussion of the ab initio force field of the H6Si20 7 molecule that the dynamic interactions in a system of condensed SiO 4 tetrahedra are not so localized as it is otten assumed when specifying the set of force constants. A problem of separation of the Coulomb contribution to the interactions which cover third and fourth coordination spheres needs further investigation. Let us conclude this section with a brief discussion of the ab initio frequency spectrum of the Si20 7 moiety in the H6Si20 7 molecule. Since the experimental frequencies of this molecule are unknown, the quantum chemical force constants are more or less arbitrarily scaled by two coefficients, 0.8 for stretchings and 0.9 for bendings, as it has been done in Chapter 1 for the force constants ofH4SiO 4 obtained with a similar basis set. The following collection of theoretical Si20 7 frequencies is obtained with these scale factors (C 2 symmetry): B

1127

A

1028 ~E

B

1020

vasSiOSi

J

B

479

A

441

B

415

B

350

A

334

A

987 ~ E

B

973

A

951 }A

B

32g

B

821

A

317

J

vSiO 3 modes

SiO 4 bending modes

242 A

LAZAREV 711

vsSiOSi

B

276

A

237

A

227

B

152 SiO3 torsion

A

147

A

84 ~SSiOSi

Only a very approximate description of the shapes of the vibrations is given. For the bond-stretching modes, their origination from the F 2 modes of a C3V distorted SiO4 tetrahedron can be indicated (in-phase and out-of-phase combinations of corresponding movements in two joined tetrahedra). No similar separation of the SiO4 bending modes can be proposed. The VasSiOSi mode in the B symmetry species is coupled with the out-of-phase combination of the symmetric SiO 3 stretchings at 821 cm "1 and at a smaller Si-O(Si) force constant (or at smaller SiOSi angle) an "exchange" of the shapes of these modes is possible (in principle). The theoretical spectrum is in good agreement with the available experimental data on the spectra of silicates containing Si2 0 6- complex anions (see, e.g., [39]). These are characterized by the presence of a single band between 600 and 700 cm-1, moderate or weak in the IR, whose arising was treated as indicative to the existence of such groups in some unknown structure. Among numerous strong IR bands above 800 cm -1, a single band near 820-850 cm"1 is usually seen in the spectra ofpyrosilicates. This band evidently corresponds to the out-of-phase SiO3 symmetric stretch of the Si2 O6- ion (see, for example, the spectra of Ca3Si20 7 or Ca2ZnSi20 7 in ref. [39]). The OSiO bending modes in the spectra of silicates are usually coupled with lattice modes, and can hardly be identified unless a complete normal coordinate treatment of the crystal spectrum is performed. Unfortunately, a calculation of the lattice dynamics of any pyrosilicate with the bent geometry of the Si20 7 group was never attempted (using as the input set of its force constants their totality deduced directly from the scaling of the ab initio force field).

INTERACTIONS OF LESS LOCALIZED ORIGIN

243

The above estimations of non-locality in the ab initio force field of joined SiO4 tetrahedra will be taken into consideration in the next chapter when attempting to deduce the dynamic model of a silica network from the molecular force fields.

244

LAZAREV REFERENCES

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4.

K.B. Tolpygo, Usp. Fiz. Nauk, 74, 269 (1961).

5.

W. Cochran, Nature, 191, 60 (1961).

~

.

W. A. Harrison in "Electronic Structure and the Properties of Solids," Vol. 1., San Francisco, W. H. Freeman and Co. (1980). M. D. Ewbank, P. R. Newman, E. Ehrenfreud and W. A. Harrison, J. Phys. Chem. Solids, 44, 1157 (1983).

8.

Ch. Geisser and K. Htibner, Cryst. Res. Teehnol., 19, 1149 (1984).

9.

C. Falter, Phys. Rev. B, 32, 6510 (1985).

10.

R.M. Pick, M. H. Cohen and R. M. Martin, Phys. Rev. B, 1, 810 (1970).

11.

A. A. Maradudin in "Dynamical Properties of Solids," (G. K. Horton and A. A. Maradudin, eds.) Vol. 1. Crystalline Solids, Fundamentals, Amsterdam-N. Y.-OxfordTokyo, North-Holland (1974) p. 1.

12.

A. Komomicki and J. W. Mclver, J. Chem. Phys., 70, 2014 (1979).

13.

A. P. Mirgorodsky and B. F. Shchegolev in "Dynamical Theory and Physical Properties of Crystals (Russ.)," (A. N. Lazarev, ed.), St. Petersburg, Nauka Publ. (1992) p. 116.

14.

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

15.

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.

16.

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

17.

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

INTERACTIONS OF LESS LOCALIZED ORIGIN

245

18.

J.C. Decius, J. Chem. Phys., 49, 1387 (1968).

19.

R. Frech and J. C. Decius, J. Chem. Phys., 54, 2374 (1971).

20.

P.M. Ghosh, Solid State Commun., 19, 639 (1976).

21.

A. N. Lazarev, N. A. Majenov and A. P. Mirgorodsky, Izv. An. SSSR, Inorg. Materials (Russ.), 14, 2107 (1978).

22.

N. A. Majenov, A. P. Mirgorodsky and A. N. Lazarev, Izv. An. SSSR, Inorg. Materials (Russ.), 15, 495 (1979).

23.

N. A. Majenov and A. N. Lazarev, Izv. An. SSSR, Inorg. Materials (Russ.), 15, 505 (1979).

24.

A. N. Lazarev, A. P. Mirgorodsky and N. O. Zulumjan, in "Advances in Physics and Chemistry of Silicates (Russ.)," (M. M. Shulz, ed.), Leningrad, Naulca Publ. (1978) p. 29.

25.

Ju. P. Tsjashchenko, G. E. Krasnjansky and E. M. Verlan, Fiz. Tverd. Tela (Russ.), 20, 864 (1978).

26.

Ju. P. Tsjashchenko and G. E. Krasnjansky, Opt. Spektrosk, 47, 911 (1979).

27.

A. N. Lazarev, V. A. Kolesova, V. A. Ryjikov and I. S. Ignatyev, Chem. Phys., 3, 339(1983).

28.

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29.

M.B. Smimov and A. P. Mirgorodsky, p. 100 in ref. 17.

30.

A.P. Mirgorodsky, private communication.

31.

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32.

A.N. Lazarev, N. O. Zulumjan, V. F. Pavinich, B. Piriou and A. P. Mirgorodsky, p. 153 in ref. 17.

33.

J. R. Clark, D. E. Appelman, J. J. Papike, in "Pyroenes and Amphiboles: Crystal Chemistry and Phase Petrology," Special Paper Amer. Mineral Soc., No 2, 31 (1969).

34.

K. Iishi, Z. Kristallogr., 141, 31 (1973).

35.

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36.

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37.

R.E. Cohen, L. L. Boyer and M. J. Mehl, Phys. Chem. Miner., 14, 294 (1987).

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38.

M. B. Smimov in "Dynamical Theory and Physical Properties of Crystals (Russ.)," (A. N. Lazarev, ed.), St. Petersburg, Nauka Publ., (1992) p. 41.

39.

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

40.

A.C. Hess, P. F. McMillan and M. O'Keeffe, J. Phys. Chem., 92, 1785 (1988).

41.

L. H. Boonstra, F. C. Mijlhoff, G. Renes, A. Spelbos and I. Hargittai, J. Mol. Struet., 28, 129 (1975).

42.

L.W. Ipenburg and H. Gerding, Recl. Trav. Chim. Pays B, 91, 1245 (1972).

43.

I. S. Ignatyev, A. N. Lazarev, T. F. Tenisheva and B. F. Shchegolev, J. Mol. Struet., 244, 193 (1991).

44.

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

45.

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

46.

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

47.

W.B. De Almedia and P. J. O'Malley, Chem. Phys. Lett., 178, 483(1991).

48.

W.B. De Almedia and P. J. O'Malley, J. Mol. Struet., 246, 179 (1991).

49.

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

50.

M.D. Newton, M. O'Keeffe and G. V. Gibbs, Phys. Chem. Miner., 6, 305 (1980).

51.

M. O'Keeffe and P. F. McMillan, J. Phys. Chem., 90, 541 (1986).

52.

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53.

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CHAPTER 4

T HE A B INITIO M O L E C U L A R F O R C E C O N S T A N T S I N L A T T I C E DYNAMICS COMPUTATIONS

I.

Molecular Force Constants in Dynamical Model of cx-Quartz ............................. 248

A. Force Constant Sets and Other Dynamical Parameters ....................................... 248 B. Calculated Properties and their Comparison with Experiment ........................... 255 C. Phonon Frequency Dispersion ............................................................................. 269 D. A Representation of the Long-Range Coulomb Interaction in the Force Field Model Specified in Internal Coordinates .................................................... 271 II.

Ab Initio Force Constants of Molecular Species in Lattice Dynamics of The Quartz-Like Aluminum Phosphate ......................................................................... 279

A. Experimental Phonon Spectra and Band Assignment ......................................... 279 B. Related Molecular Systems and their Force Fields ............................................. 284 C. A Design of the Initial Approximation of the Force Field of Aluminum Phosphate ..............................................................................................................

293

D. An Extension and Modification of the Initial Force Field .................................. 301 III. Electrostatic Contribution to the Mechanical Modes of a More Polarizable Lattice: Pyroxene- Like Monoclinic Sodium Vanadate ....................................... 307

A. A Formulation of the Problem and Description of the Crystal ........................... 307 B. Experimental Data and Spectral Assignments ..................................................... 310 C. Normal Coordinate Calculation ........................................................................... 318 D. The Origin of the Transversal vasVOV Modes Softening .................................. 323 E.

Some Further Perspectives ................................................................................... 329

F.

Concluding Remarks ............................................................................................ 332

References ........................................................................................................................... 334

247

248

LAZAREV

M O L E C U L A R F O R C E CONSTANTS IN D Y N A M I C A L M O D E L OF

~-Quartz A.

Force Constant Sets and Other Dynamical Parameters

A calculation of dynamical properties of a-quartz with explicit treatment of the Coulomb contribution to potential function was mentioned in Chapter 2 and corresponding references can be found there. It included an adjustment of the balance between those forces and short-range interactions. Here we restrict ourselves to the calculations which adopt the force constant approach where a fulfillment of that balance is implicated by deftnition. Since the earlier attempt by Kleinman and Spitzer [1 ], such calculations were repeated by several authors. Among them, Shiro and Soi [2] was probably the first who calculated jointly phonon frequencies and macroscopic elastic constants of a-quartz. Mirgorodsky et al. [3,4] extended similar calculations to the quartz-like form of germanium dioxide. The importance of the IR intensity calculation for determining the reliability of calculated shapes of normal modes (eigenvectors) has been emphasized by Mirgorodsky and Lazarev [5] in their attempt to refine some of the off-diagonal force constants. Their approach has been adopted by Etchepare et al. [6,7,8] who repeated the calculation of a-quartz and extended it to the calculation of other polymorph forms of silicon dioxide and quartz-like form of A1PO4. Iishi [9] tried to investigate a variation of the force field of u-quartz under its transition to the 13-quartz form. An application of ab initio molecular force constants to the calculation of some mechanical properties of c~-quartz has been attempted by several authors as well [ 10,11,12]. In these investigations, however, the dynamical matrix of a crystal has not been designed and simplified calculations were restricted to the estimation of compressibility (some criticism of approaches adopted in these estimations may be found in ref. [ 13]). Thus, the validity of

LATTICE DYNAMICS COMPUTATIONS

249

adopted force constants could not be supported by calculation of vibrational frequencies, elastic constants and other parameters accessible to experimental determination. More recently, McMillan and Hess [14] discussed the vibrational spectra of or- and [3-quartz proceeding from their calculation with ab initio force constants of H6Si20 7 molecule. In our opinion, however, their force field contains too localized a nature and some of the important interaction constants are neglected, on the one hand and no scaling of the SCF force constants is provided, on the other. A more consistent approach proposed by Lazarev and Mirgorodsky [15] who compared their calculation with a wider set of experimental data in accordance of the generalized inverse vibrational problem (GIVP) outlined in the previous chapter, will be discussed below in comparison with the results of earlier calculations. The force constants of molecular species resembling some fragments of the quartz network and employed in the design of its force field are shown in Table 4.1. These are the scaled quantum chemical force constants of disiloxane and tetramethoxysilane (the force constants of the latter have been obtained by frequency fitting with due account of their ab initio calculation as explained above).

Of course, no scaling could be applied to non-

bonded oxygen-oxygen force constants deduced from quantum mechanical calculation for a non-existent model system. The ab initio molecular force constants were obtained in the space of linearly inde-

pendent local symmetry coordinates usually adopted in quantum chemical computations. These force constants can not be transferred directly into the force field of a crystal with another symmetry of corresponding fragments in a space group. Thus, the quantum mechanical force constants should be transformed to the space of non-symmetrized internal coordinates, which constitute a basis of the GVFF model applicable to any system. That space includes a larger number of coordinates which are fewer than the independent ones.

GI

TABLE 4.1 Ab initio force constants of some molecular systems and trial sets of force constants of a-quartz (mdydA, mdyn, mdyn.A).

Force constants

% (OSiO)

H6Si2O7

H4Si04

1.20 1

Molecular Systems H6Si20 (CH3)4Si04

1.180

a-quartz 2xH6Si20

I

11

111

Iv

1.OOO

1.OOO

1.OOO

1.OOO

The force constants relating to the Si-0-C bridge are given in square brackets. 'Scaled ab initio force constants of H3SiOCH3transferred into the force field of (CH3)4Si04with any variation.

0

ki <

LATTICE DYNAMICS COMPUTATIONS

251

A reverse transformation of the force field specified in the local symmetry coordinates to GVFF is possible only in frames of some additional assumptions. E.g., setting the interaction of OSiO angles with a common Si atom in tetramethoxysilane to zero, it is possible to determine the OSiO/OSiO interaction force constant for angles with a common bond. Analogously, setting to zero the OSiO/SiO angle bending/bond stretching interaction at the common Si atom, one can determine the OSiO/SiO interaction constant which corresponds to the bond on the side of that angle. Then, in order to reduce the number of force constants to be varied, it was reasonable to neglect in calculations for d-quartz the difference between the force constants of two non-equivalent sets of Si-O bonds and to assume the equality of force constants of all OSiO angles in a tetrahedron. Thus, the averaged magnitudes of force constants (both diagonal and off-diagonal) characterizing two sets of these angles in tetramethoxysilane for the S4 point group are given in Table 4.1. The unscaled ab initio force constants of other molecular systems relating to a-quartz, H4SiO 4 and H6Si207, evaluated by various authors are included in that table for completeness.

These force constants calculated by Hess et al. [16] and O'Keeffe and

McMillan [11], respectively, with rather flexible 6-31G atomic wave functions supplemented by polarizing d-functions have been transformed to the space of internal coordinates in a similar way. It is seen from Table 4.1 that the estimations of the SiO 4 force constants deduced from H4SiO 4 and (CH3)4SiO 4 do not differ considerably. However, among txvo different estimations of the Si-O stretching force constant, 5.36 mdyn/A in H6Si20 and ca. 6.0-6.3 mdyn/A in H4Si20, H4SiO4, and (CH3)4SiO 4, the lower value has been adopted in the description of the force field of quartz. It may seem dubious from the point of view of the bond length/force constant correlation discussed in Chapter 1 since the Si-O bond length in tetramethoxysilane is essentially the same as in a-quartz while in disiloxane it is signifi-

252

LAZAREV

cantly larger. A probable origin of this discrepancy will be clarified later when discussing how the long-range Coulomb interactions are implicitly represented by the force constants in the GVFF model. Most contradictory are the estimations of SiOSi bending force constant deduced from quantum mechanical computation for various molecular systems (Table 4.1). It is approximately by an order value larger in the disilicic acid molecule than in disiloxane which can not be explained either by the use of an unscaled magnitude in the former or by the difference in the adopted atomic wave functions. This problem has been partially discussed in previous chapter. It should be reminded that a supposition on the transferability of the molecular force constants to the force field of a crystal or of another molecule implies that some essentially local properties may be separated in the force field of multiatomic system. Rigorously speaking, it is not the case. When the force field includes off-diagonal terms, none of force constants represent the intrinsic property of any particular bond or angle being the characteristic of a whole system. This is, the more important, the weaker the corresponding internal coordinate is as itself. In other words, the intrinsic flexibility of the SiOSi angle, which reveals itself in the very small bending force constant of disiloxane, may be hidden in the force field of disilicic acid by the interactions between the two space-consuming terminal Si(OH)3 groups which are represented indirectly by the enhanced value of that force constant and some of the interaction constants. However, an unbiased estimation of the flexibility of the SiOSi angle in quartz deserves special attention. Among the molecular force constants in Table 4.1, two non-bonded O...O force constants are listed which correspond to the distances equal to the shortest intertetrahedral oxygen-oxygen distances in a-quartz. These force constants were deduced from the quan-

LATTICE DYNAMICS COMPUTATIONS

253

tum mechanical calculation for an artificially designed bimolecular system consisting of two disiloxane molecules as was explained in previous chapter. Various trial sets of force constants for a-quartz listed in Table 4.1 were combined from the force constants of related molecular systems and then tested in their ability to reproduce the dynamical properties of the crystal. The simplest one is Set I restricted to "internal" force constants of the Si-O-Si bridge and SiO4 tetrahedron which have been transferred from the scaled quantum mechanical force fields of disiloxane and tetramethoxysilane, respectively. One of initial values in this set, the OSiO bending force constant, has been reduced by about 15% on the basis of the calculation of dynamical properties of a-quartz which it affects predominantly (mostly, vibrational frequencies below 800 cm-1). A selection between two different values of the Si-O stretching force constant in molecular systems in favor of the smaller one has been already mentioned. Set I is transformed in Set II by adding a single interaction constant in order to correctly reproduce the wavenumbers of strong and weak IR bands belonging to highfrequency E modes. The idea of the probable importance of that interaction constant was deduced from inspection of the peculiarities of the force field of tetramethoxysilane and strongly supported by the analysis of the ab initio force field of disilicic acid in a previous chapter. Set III has been obtained from Set II by changing the SiOSi bending and SiOSi/SiO interaction force constants to values deduced by O'Keeffe and McMillan from their ab initio computation of H6Si20 7. This set was included into consideration to attempt to investigate the mechanical properties of a-quartz which would correspond to the supposition of relatively large intrinsic stiffness of the SiOSi angles. Another approach to the improvement of Set II is tested in Set IV which is complemented by the force constants of non-bonded oxygen-oxygen interaction at shortest dis-

254

LAZAREV

tances. A comparison of Sets III and IV helps to decide empirically, which scheme of local forces around the oxygen atom represents more accurately the mobility of the Si-O-Si bridges in u-quartz: more rigid SiOSi angle with no diagonal force constants preventing directly the oxygen out-of-plane displacement or a very flexible SiOSi angle in version IV with additional forces imposed on oxygen by the non-bonded interaction with other bridges in the framework. The polarization properties of a-quartz, i.e., IR intensities and piezoelectric constants, were calculated for each version of the force field mainly for the judgement on the plausibility of calculated LQ and L U vectors as it was explained in Chapter 2. Therefore, the simplest electro-optic model of a-quartz proposed by Kleinman and Spitzer [ 1] has been adopted and their parameters accepted without any variation. That model (which is similar to a simplified version of VCM discussed in Chapter 2) describes the polarization of a crystal as being determined by two processes. The first one is the shift of atomic charges in a strained lattice. Another one is the change of atomic charges induced by the bond length variation in any bond ensuing from a given atom. It represents a polarization of the bond due to the charge flow from one atom to another in the stretched bond. In a particular case of a-quartz two independent parameters are needed to calculate the polarization vector derivatives with respect to the shapes of the uniform deformations and normal coordinates. These are the oxygen charges in equilibrium position z~ and c o = 1 SiO "Oz0/cOlsio where z is the atomic charge and l is the bond length. They specify the oxygen charge at rest and its dependence on the bond elongation, their numerical values being fixed at z~) = -1.5e and c o = 3e (e is the electron charge). Corresponding values for o

o

the silicon atom are then determined by the electroneutrality condition as zsi = -2 z 0 and csi = -c 0. A comparison of the results of the IR intensity and the piezoelectric constant cal-

LATTICE DYNAMICS COMPUTATIONS

255

culation for a-quartz with various other electro-optic models are discussed by Mirgorodsky and Smimov [ 17].

B.

Calculated Properties and their Comparison with Experiment

All dynamical properties of quartz accessible to direct experimental determination have been calculated in ref. [ 15] with the above four sets of force constants (CRYME program was used in all computations). The structural parameters determined by Hazen et al. [18] have been adopted. The data on the microscopic structure of hydrostatic compression characterized by the derivatives of internal coordinates c ~ / ~ were deduced from the same paper using Ag magnitudes determined for Ap=8 GPa. The volume compressibility ~ and its components along the Cartesian axes, i.e., linear compressibilities (}-fxand (F(z, obtained in ref. [ 19] were taken into account. The experimental transverse optical frequencies and IR intensities of polar E and A 2 modes have been found in the paper by Spitzer and Kleinman [20]; these can be compared with the ones determined by Gervais and Piriou [21]. The Raman frequencies of the A1 modes have been taken from Dean et al. [22]. The experimental values of macroscopic elastic constants obtained by McSkimin et al. [19] and of piezoelectric constants determined by Bechmann [23] have been adopted in comparison with the calculated values. Comments to the controversy in the determination of their signs given by Barron et al. [24] were taken into consideration. These experimental values are compared with the calculated ones for various versions of the force field in Tables 4.2-4.4. As has been mentioned already, an overestimated local symmetry of the force field was assumed implicitly when simplifying it in order to reduce the number of independent force constants, C2v local symmetry for Si-O-Si bridge and T d for SiO4 tetrahedron, although the exact geometry was adopted for the calculation of

h,

TABLE 4.2 Vibrational spectrum: TO fiequencies (cm-1) and dipole moment derivatives (electron charge unit).

Experimental

A~

AI

m

Calculated

I E

VI

I1

I11

IV

w

IdP/dQI

w

l@/dQl

w

IdPjdQI

w

IdP/dQI

o

ldP/dQI

1162

0.177

1090

0.294

1165

0.131

1166

0.127

1166

0.134

1072

1.333

1117

1.278

1075

1.305

1076

1.294

1073

1.305

795

0.400

762

0.108

786

0.095

770

0.1 12

785

0.093

697

0.142

673

0.120

673

0.125

669

0.140

673

0.126

450

0.619

479

0.704

479

0.705

503

0.728

483

0.691

394

0.344

367

0.300

367

0.301

401

0.228

373

0.332

265

0.180

235

0.125

235

0.125

285

0.212

239

0.121

128

0.043

113

0.059

113

0.058

148

0.012

123

0.058

1080

1.343

1120

1.361

1080

1.361

1080

1.346

1076

1.361

778

0.373

742

0.145

766

0.138

770

0.009

765

0.138

495

0.61 1

496

0.614

496

0.613

567

0.63 1

498

0.605

364

0.456

377

0.529

377

0.533

379

0.568

380

0.542

1085

1123

1082

1079

1079

464

469

45 1

475

452

356

372

372

374

375

207

201

192

193

207

<

LATTICE DYNAMICS COMPUTATIONS

257

TABLE 4.3 Properties of a-quartz relative to homogeneous deformation

Experimental

Calculated I

II

III

IV

Elastic constants, GPa C11

86.80

52.5

52.5

118.8

85.3

C33

105.75

37.9

37.9

100.6

101.4

C44

58.20

52.2

41.9

54.5

52.3

C66

39.88

26.1

26.1

41.8

33.9

C12

7.04

35.2

17.5

C13

11.90

-11.8

- 11.8

44.6

25.5

C 14

- 18.00

-21.9

-21.9

-8.6

-13.2

0.35

0.35

Compressibility, GPa- 1.10-4 9-fx

87

228

288

48

83

9-(z

63

443

443

56

57

94*

237

1020

1020

154

223

Piezoelectric constants, esu.cm-2.104 el 1

5.130

3.207

3.286

6.679

4.533

el4

- 1.224

- 1.424

- 1.552

0.363

-0.245

+

kinetic energy. The numbering of atoms in a primitive cell employed in subsequent discussion is explained in Fig. 4.1. As is seen from Tables 4.2-4.4, the force constants of Set I are unsatisfactory from the point of view of the generalized formulation of IVP adopted in our approach. Though the deviations of the calculated frequencies do not exceed 10%, the errors in the calculated elastic constants are reaching about 100-300% and calculated c~/cTp values differ from the experimental ones nearly by an order of magnitude.

TABLE 4.4 Compressibilities of structural hgments in a-quartz lattice (&Gpa, DegreeIGpa).

Calculated Pressure derivatives of internal coordinatesa ‘I

Experimentalb -0.0001

I

I1

111

IV

+0.0018

+0.0018

-0.0018

-0.0016

=See Fig. 4.1 for numbering of atoms; 04 = LO4Si1O9(x2),a2= L04Si10.&x2),o3= L05Si109(xl),c4= L04Si10,(x1). bDeducedfrom data obtained in ref. [18] at 8 GPa.

LATTICE DYNAMICS COMPUTATIONS

259

7 5 1

9

4

3'

2"

4" t

2'

Z

8

IX Fig. 4.1

o-0

.-Si

The a-quartz network in projections on crystallographic planes. Bonded

and shortest non-bonded contacts are shown. The numbers of atoms in the neighboring primitive cells are primed.

Even in the description of the phonon spectrum this version leads to the wrong interrelation between IR intensities and frequencies of two high-frequency SiO stretching E modes as is seen from Table 4.2: the frequency of the very intensive mode is higher than that of the much weaker one, which is in evident contradiction with the experiment. The shapes of these modes differ considerably. One of them resembles the pulsation mode of a free SiO 4 tetrahedron and can not thus produce a significant change of dipole moment. Conversely, another one may be treated as originating from the x, y-components of the triply degenerate F 2 stretching mode of a free tetrahedron and is very polar. Their intensities are completely determined by the shapes of modes and can not be reversed by changing the electro-optic model or by the variation of its parameters [ 17]. The problem is thus, how to reverse the sequence of frequencies of these modes. The above difference in the IR intensities of two high-frequency E modes in a-quartz can be explained in another way by deducing them from the corresponding E 2 and E 1 modes of the more symmetrical 13-quartz where the former is IR inactive due to the more

260

LAZAREV

rigid selection rules. Raman data obtained by Bates and Quist [25] show that the spectrum of a-quartz transforms on heating smoothly in that of 13-quartz which should be correlated as:

a-quartz

f~-quartz

1162 E (weak in IR)

1173 E 2 (forbidden in IR)

1072 E (strong in IR)

1067 E 1 (active in IR)

This problem has been paid no sufficient attention although a wrong interrelation between IR intensities and frequencies of these modes was obtained in most calculations of the spectrum of quartz including those treating explicitly the Coulomb contribution to force constants. The same may be suspected in the recent calculation with ab initio molecular force constants by McMillan and Hess [14] judging by the shapes of the normal modes plotted in that paper and their comparison with E 2 and E 1 modes in 13-quartz (these authors did not calculate the IR intensities). The contradiction which is discussed has been removed earlier by Mirgorodsky and Lazarev [5] and then by Etchepare et al. [6] who proposed considerable enhancement of the SiO/SiO interaction force constant in the GVFF model for bonds having common silicon atom. Presently, however, that value seems dubious being about twice as large as that obtained directly by quantum mechanical computation for related molecular systems or certain specific effects of enhancement for the interaction force constant should be assumed to exist in crystal. A hint at another possibility to resolve this problem has been found in ref. [15] by inspection of the ab initio force field of tetramethoxysilane where the importance of SiO/CO' interaction between non-adjacent bonds follows both from frequency fitting and quantum mechanical computation. It is reasonable to assume the existence of a similar interaction between second-order neighbor Si-O bonds in a quartz network. This is done in Set II of the force constants. At fixed values of other force constants of Set I, this additional

LATTICE DYNAMICS COMPUTATIONS

261

interaction constant has been originally transferred from tetramethoxysilane and then its absolute magnitude has been slightly increased to obtain a better frequency and IR-intensity fit for high frequency E modes. This version of a force field exhausts the IVP solution for a-quartz in a standard approach since all spectroscopic quantities are reproduced reasonably well (Table 4.2). The underestimated IR intensities of A 2 and E modes near 800 cm -1 may originate from the inadequacy of the adopted electro-optic model, which assumes the polarizability only along the bond directions, and does not make doubtful the shapes of the modes. Sets I and II are, however, similar in their inability to reproduce the properties of a-quartz relative to uniform strain. As seen from Table 4.3, they lead to a wrong compressibility and its anisotropity and produce even qualitatively incorrect interrelations between various elastic constants including an erroneous sign of C13. Table 4.4 helps to decide which one of the structural fragments produces a wrong response to the external force and may indicate a way to the further refinement of the force field model. A qualitative description of the pattern of the quartz network compression follows from experimental c~/c~ magnitudes in Table 4.4 and the scheme of its structure on Fig. 4.1. The Si-O bond lengths remain practically unchanged with a slight trend to shortening. The SiO4 angular deformation consists mainly in the rotation of the plane of the 03 angle relative to the plane of 04. Both these angles do not change significantly whereas o 1 increases and a 2 decreases by about the same value. In the outline it resembles the shape of the doubly degenerate bending mode of a free tetrahedron. The tetrahedral edges in quartz change according to this scheme and differ from their equilibrium lengths by less than 2% (at 8 GPa). The parameters characterizing mutual orientation of the tetrahedra in a network are changed more markedly; the non-bonded 0 4 . . . 0 7 , distance with an equilibrium length

262

LAZAREV

of 3.335A reduces by 16% at 8 GPa and 0 4 . . . 0 8 (3.414A) by 13.5%. The Si-..Si distance in the Si-O-Si bridge shortens by 4.4% and the SiOSi angle reduces by 9%. The calculation with Sets I or II of the force constants leads to a drastically different microscopic pattern of tile structure variation under hydrostatic compression: all Si-O bonds elongate and thus the volume of each tetrahedron increases. Among the tetrahedral angles o 2 remains constant while o 4 increases considerably at the expense of a smaller decrease of o 1 and cr3. The compressibilities of all "intertetrahedral" coordinates are significantly overestimated by these force constants sets: from 5-7 times for ones characterizing Si...Si distance (SiOSi angle) or 04.-.0 8 distance up to nearly 10 times for 0 4 . . . 0 7, distance. Two possible directions of the further force field improvement follow from the above analysis. One of them consists in the introduction of additional force constants representing the stiffness of non-bonded O...O contacts in a lattice and another one in the enhancement of the stiffness of the Si-O-Si bridges. Let us investigate first the latter one. This has been done by Set III of the force constants (Table 4.1) which differs from Set II only by an increased SiOSi bending and the SiOSi/SiO bend/stretch interaction constants, both values being transferred for completeness from the ab initio force field ofH6Si207 calculated by O'Keeffe and McMillan [ 11 ]. This version of the force field really improves the results of the compressibility and elastic constants calculation (Table 4.3). A correct determination of the sign of the C13 elastic constant (in contrast of Sets I and II) should be mentioned in particular. As it concems the microscopic structure of hydrostatic compression, the experimental c~/dp values relating to the intertetrahedral coordinates (SiOSi angle and Si-..Si or O...O distances) are successfully reproduced. However, the description of the angular deformation of a tetrahedron under compression remains wrong.

LATTICE DYNAMICS COMPUTATIONS

263

Then, Set III of the force constants leads to an incorrect description of the anisotropy of the elastic properties by overestimating the C 11 elastic constant. As a result, the compressibility along x is half as large the experimental one and the volume compressibility appears to be underestimated as well. Moreover, this set partially worsens the results obtained with Sets I and II by incorrectly determining the sign of the piezoelectric constant el4 (cf. Table 4.3). As the more detailed analysis shows, both these drawbacks have a common origin connected with the incorrect determination of the contribution from the lowest E mode whose IR intensity is underestimated (a wrong shape of vibration?) while the frequency is overestimated. Besides, this version significantly deteriorates the results of the calculation of some other optical vibrations in the interval 450-500 cm-1. One of them also belongs to the E representation and thus affects the piezoelectricity. Another possibility to improve Set II has been implemented by adding the force constants of non-bonded intertetrahedral oxygen-oxygen interactions at distances less than 3.5A. Their importance in the adequate description of the lattice dynamics of quartz has already been discussed by Barron et al. [24]. Two sets of such distances (except the ones corresponding to the tetrahedron edges) exist in the quartz lattice and are taken into consideration when designing version IV of the force field. The initial values of the O...O force constants estimated by quantum mechanical computations for the model system as described above, were then refined by the fitting procedure. As Table 4.5 shows, the derivatives of the elastic constants with respect to the O..-O force constants considerably exceed any other ones. Only the SiOSi bending force constant affects the elastic constants in a comparable scale. The same relates to dependence of microscopic compressibilities (c~/cTpvalues for diverse g) upon various force constants, which were taken into account in the fitting procedure. The refined values of the O.--O force con-

264

LAZAREV

TABLE 4.5 Dependence of elastic constants of a-quartz on the force constant magnitude (version IV of the force field). Elastic constant derivatives with respect to force constants* C 11

C33

C44

C66

C12

C13

C14

Kq

0

0

0

0

0

0

0

I~

56

53

114

27

2

-34

-27

K5

183

136

51

61

61

129

53

h

2

2

0

0

2

2

0

H

0

0

2

0

0

0

0

a

3

0

12

3

-3

0

-5

A

29

27

-14

7

15

24

-7

l

-87

- 104

-46

-29

-29

66

35

d

3

3

-2

0

3

3

0

K'

335

1083

114

74

194

517

88

K"

858

367

226

241

376

476

231

Force Constants

*Force constants and elastic constants are implied to be expressed in units adopted in Tables 4.1 and 4.3, respectively. stants constitute together with unchanged force constants of Set II, the final Set IV which reproduces equally satisfactory the frequency spectrum, elastic constants and compressibility of a-quartz including the correct description of the anisotropy of these properties. It should be emphasized that version IV of the force field is unique in giving a rather good reproduction of the microscopic structure of compressibility: both the angular deformation of a tetrahedron and the variations of the intertetrahedral coordinates under compression are reproduced successfully.

The same relates to the polarization phenomena

excited by the lattice deformation: both the IR intensities and piezoelectric constants are reproduced semi-quantitatively without any attempt to refine the electro-optic model or its parameters (cf. their discussion in ref. [ 17]). It can be treated in accordance of the above

LATTICE DYNAMICS COMPUTATIONS

265

considerations (Chapter 2) as an additional confirmation of the compatibility of calculated LQ and L u vectors. Thus, the above comparison of the consequences deduced from Sets III

and IV of the force constants resolves, apparently, the difficulties in the separation of the contributions from the SiOSi bending and O...O stretching to dynamical properties of a-quartz mentioned earlier by Stixrude and Bukowinsky [26]. The force field model deduced above seems to be more adequate than the one obtained by McMillan and Hess [ 14] who also proceeded from the quantum mechanical force constants of molecular systems. In our opinion, the difficulties they met partially originate from the absence of experimental spectra for molecular species selected as the reference ones (H4SiO4 and H6Si207) which leads to the use of unscaled force constants. Secondly, some other shortcomings originate from a rather arbitrary oversimplification of the force field model: only a few ab initio force constants were adopted in their model and numerous off-diagonal force constants were neglected. As has been shown in previous chapter, a consistent analysis of the ab initio force field of H6Si207 clearly demonstrates the importance of the interaction between the second order neighbor Si-O bonds, although the corresponding force constant in our model has been deduced from the ab initio force field of tetramethoxysilane.

The inverse sequence of the two high-frequency E modes noted by

McMillan and Hess discussed in some detail above originate, evidently, from neglect of that interaction. It will be recalled that in the absence of this constant a correct sequence of those modes is obtained only at a considerably larger interaction constant for adjacent bonds than the quantum mechanical calculation for related molecular systems predict. The neglect of non-bonded interactions is an another shortcoming of that force field which is partially hidden by the overestimated values of the SiOSi bending and SiOSi/SiO interaction force constants, as has been shown when discussing Sets III and IV. That discussion proved in particular that a selection between two diverse estimations of the SiOSi

266

LAZAREV

bending force constant, both being deduced from the study of molecular systems, should be done in favor of the smaller value. In other words, it was concluded that the SiOSi angle in a-quartz is really very flexible as itself. A contradiction with the corresponding ab initio force constants in H6Si207 indicate, probably, a caution which is needed when the force constants of the "weakest" internal coordinates are transferred. Let us now discuss in more detail Set IV of the force constants and the influence of particular ones on the description of various dynamical properties. The force constants of this set can be divided into two groups according to the properties most dependent upon their values. As the inspection of the partial derivatives with respect to the force constants shows, among the eleven parameters of the force field only seven determine the most characteristic features in the frequency spectrum. Five of them (Kq, I ~ , h, a, 1) describe the internal force field of the SiO4 tetrahedron and two others correspond to the stretch/stretch interactions between adjacent tetrahedra (H, d). Another group of force constants affects mainly the properties of a-quartz relative to the homogeneous deformation. It is seen from Table 4.5 that the elastic constants are govemed predominantly by three diagonal force constants, K', K", and K6, representing the interactions between relatively distant atoms. A structural variation under compression is determined mainly by the two of them corresponding to non-bonded oxygen-oxygen interactions. It is possible to analyze their influence on some particular features of the structural variation using Fig. 4.1 for explanation. The contribution of any given force constant to some elastic constant may be deduced either by making use of Table 4.5 or by the calculation of that elastic constant with the above force constant set to zero. The three-dimensional a-quartz network is built up by helixes along the z axis, each coil containing six Si-O-Si bridges. A compression along the z axis shortens the pitch of a helix thus reducing the distances between the oxygen atoms of different coils. The shortest

LATTICE DYNAMICS COMPUTATIONS

267

of these distances is the 0 4 . . . 0 7, one. This is why the C33 elastic constant is determined in about 60% by the force constant K'. In difference of the 0 4 . . . 0 7 contact, which deviates from the z axis direction by only 21 o, the 04-..0 8 contact is inclined to it by 68 ~ and just the K" force constant contributes significantly to the stiffness of interhelical connections in the xy plane. Notwithstanding small absolute value of that force constant, it makes up about 40% of the magnitude of the C 11 elastic constant. The anisotropy of the elastic properties of a-quartz thus depends on our force field model for the interrelation between K' and K". It supports the treatment of this problem by Levien et al. [27] although the orientation of the O...O contacts is determined in that paper erroneously (cf. Fig. 2 in ref. [27]). A contribution of the K 5 force constant to the C ll and C33 elastic constants constitutes in our dynamical model only 12 and 8%, respectively. A comparison of the corresponding contributions from the K' and K" indicates that the properties of a-quartz relative to uniform strain are determined mainly by the non-bonded O...O interactions, while the characteristics of the very flexible SiOSi angles are of secondary importance. This conclusion is in contrast with opinions proposed by several other authors [8,28] who found that the explanation of mechanical properties and compressibility of a-quartz was mainly in the properties of the Si-O-Si bridge itself. It is worth adding that the above calculation shows the importance of the OSiO bending force constant (K~) for the correct description of the elastic properties of a-quartz. It is seen from Table 4.5 that the K' contribution into all elastic constants excluding the C 12 accounts for up to 10% and just the OSiO angle variations determine the negative sign of C 14. It follows from these considerations that the treatment of the SiO4 tetrahedra as rigid bodies for the calculation of the elastic properties of u-quartz adopted by some authors [10,11 ] is too oversimplified.

268

LAZAREV

The bend/stretch SiOSi/SiO interaction force constant deserves special comment. Although it influences, insignificantly, the vibrational frequencies, its choice determines a behavior of the Si-O bonds at the bending of Si-O-Si bridge. The physical meaning of this constant, when positive, may be rationalized as a force which endeavors to stretch the Si-O bonds when the SiOSi angle decreases. A guess concerning the positive sign of this interaction constant in most silicates may be deduced from the numerously discussed empirical correlations (the increase of the bond length in the Si-O-Si bridges with smaller angles). Quantum chemical force constant computations confirm its positive sign in molecular systems and non-vanishing value against K 5 which can be treated as an intrinsic property of a bridge. On the other hand, the shift of oxygen atoms due to the strain of a network composed by Si-O-Si bridges may depend upon their non-bonded interactions in addition to the internal properties of those bridges. It can be deduced from the comparison of Tables 4.4 and 4.1 that the trend in the Si-O bond length variation under compression may be reversed by means of the K' and K" force constants which try to shorten the bond when the SiOSi angle decreases. The small experimental change of the Si-O bond length in a-quartz under compression may be treated, thus, as a result of the approximate balance of two opposite effects. In a similar but less dense lattice of the quartz-like form of GeO 2 with longer O...O distances, the elongation of the Ge-O bonds under compression is found experimentally [28] which can be understood as a manifestation of the predominating influence of the force constant A.

Similar problems relating to the behavior of the oxygen bridges in

Si2N20 and Ge2N20 under compression have been discussed byMirgorodsky et al. [29].

LATTICE DYNAMICS COMPUTATIONS

269

C. Phonon Frequency Dispersion Set IV of the force constants of a-quartz has been adopted by M. B. Smimov in the computation of the phonon frequency dispersion along the main directions in the Brillouin zone [30]. All computations have been performed using the CRYME program. The results are shown in Figs. 4.2-4.4 as the eigenfrequency vs. the reduced wave vector curves. A notation of the irreducible representations of a finite wave vector coincides with the one employed by Elcombe [31 ] who calculated both the first time phonon dispersion in quartz in the RIM approach and with the simple Born von Kmxndn force field model. The results from the latter are represented by broken curves on Figs. 4.2-4.4 since no macroscopic polarization and transversal/longitudinal frequency difference is provided for by this set of molecular-type force constants. Those figures do not include rather flat optical branches above 1000 cm- 1. Although some optical branches differ considerably in the long wavelength limit, an unexpectedly close resemblance of the shapes of these vibrational branches deduced for two very dissimilar dynamical models is remarkable. It means, probably, that in more or less complicated crystal structures the shapes of the dispersion curves represent implicitly the mutual arrangement of various sublattices and are relatively weakly dependent on the peculiarities of the force field. Since the experimental study of those curves is often impeded by insufficient resolving power of neutron scattering spectroscopy, the above considerations concerning an extension of the IVY formulation by the introduction of the properties of a crystal relative to the homogeneous deformation are additionally validated. The longitudinal branches corresponding to Set IV of the force constants can be, in principle, calculated at any assigned scheme of the lattice polarization caused by its distortion (e.g., one adopted in the IR intensity computation). An application of a simple analytic

270

LAZAREV

c m -~

800

600

400

200

II

o

Io~ol

1.o

Fig. 4.2 Phonon dispersion curves along 010 (no symmetry of a finite wave vector).

cm'l

800 _

_

_ ~ -_ = -I--_-_ _-_ m

600

400

200 -

0

[~oo]

T~

1.0

0

[~00]

T2

1.0

Fig. 4.3 Phonon dispersion curves along 100. Broken curves correspond to Born von Kfirmfin force field in ref. [31 ] whose notation of irreducible representations of the wave vector group is adopted.

LATTICE DYNAMICS COMPUTATIONS

cm-I

271

800

600

400

_

200

. _

...

- "

,r

. o Fig. 4.4

[oo~l

A1 1.0

,..f/f 0

..... [oo~l

A2 1.0

o

[00;1

A 3 1.0

Phonon dispersion curves along 001. Broken curves correspond to Born

von Kfirmfin force field in ref. [31 ] whose notation of irreducible representations of the wave vector group is adopted.

approximation of slowly convergent lattice sums proposed by Smimov [32] seems to be very promising from the point of view for the promotion of this rather time-consuming procedure.

Do A Representation of the Long-Range Coulomb Interaction in the Force Field Model Specified in Internal Coordinates It seems amazing that all the experimental dynamical properties of quartz, which is certainly a semi-ionic crystal, are successfully reproduced by the set of force constants transferred from the force fields of molecular species which lack long-range interactions. Although it has been shown in a previous chapter that in the force fields of those molecules the Coulomb interactions between distant atoms are represented implicitly by some offdiagonal force constants, this problem deserves a more complete analysis. This will be attempted below by being restricted to the force constant approach which assumes a refusal of the explicit presentation of the potential function in a sense of eqs. 2.44 or 2.84 (Chapter 2);

272

LAZAREV

whose adoption would need a laborious searching of a balance of forces of various origins at any atom. A following semi-quantitative approach for the evaluation of the Coulomb contributions to the vibrational frequencies of a crystal can be proposed. The eigenvectors of the normal modes (shapes of vibrations) obtained from the calculation with the adopted set of force constants are assumed to be the correct ones. Then, by adopting any suitable notion for the charge distribution in a crystal, it is possible to calculate the Coulomb contribution to any element of a diagonalized dynamical matrix as if it were deduced from the potential function specified explicitly by eq. 2.44 (Chapter 2) (rigorously speaking, this approach is sufficiently exact only for a case where there is a small Coulomb addition to the dynamical matrix). The results of such calculations, with the eigenvectors corresponding to the Set IV of the force constants of ref. [15], are presented in Table 4.6, the Coulomb contributions being determined in the RIM approximation, i.e., at fixed atomic charges adopted in the calculation of IR intensities (z0 = - 1.5e). As has been emphasized, the absolute Coulomb contributions in Table 4.6 are of low significance (apart from other considerations, the lattice would be hardly stabilized in a sense of eq. 2.34 (Chapter 2) because of the overestimation of atomic charges). However, a discussion of their sign and their relative magnitudes seem to be instructive if the letters are believed to be proportional to the real Coulomb contributions to the P.E.D. of normal modes. A systematic alteration of the signs and magnitudes of the Coulomb contributions from large and negative ones for the high-frequency modes to smaller and positive ones for the low-frequency modes, is most remarkable. It is easily explained by remembering that the high-frequency modes correspond mainly to the bond stretching type deformations of

LATTICE DYNAMICS COMPUTATIONS

273

TABLE 4.6 Coulomb contribution to dynamical matrix of a-quartz calculated with eigenvectors of its mechanical approximation at z 0 = -1.5e.

Symmetry Species E

A2

A1

COcalccm "1

Dtota1"10-6 cm -2

DCoulomb"10-6 cm -2

1167

1.3619

-0.7199

1072

1.1492

-1.2419

784

0.6146

0.0092

674

0.4543

0.0353

483

0.2333

0.6983

373

0.1391

0.3769

239

0.0571

0.3620

123

0.0151

0.2895

1075

1.1556

-1.2680

765

0.5852

-0.0794

497

0.2470

0.7743

380

0.1444

0.4645

1078

1.1621

-1.3225

451

0.2034

0.4141

376

0.1414

0.3996

207

0.0428

0.2142

the quartz network. The Coulomb contribution to these modes originates in the first place from the interaction between neighboring, oppositely charged atoms whose second derivative is negative and thus tends to destabilize these modes. Reversely, the low-frequency modes represent the angle-bending deformations and the Coulomb contribution originates from the interaction between the peripheral atoms of valence angles with the same charge. The second derivative of their Coulomb interaction energy is positive and enhances the elasticity of such modes. Then, it should be noticed that in the polar E and A 2 symmetry species, the larger the absolute magnitudes of the Coulomb contributions are to the elasticity of the transversal

274

LAZAREV

modes, the higher their IR intensities. It originates from the same mechanism of the microscopic polarization contributing to the elasticity of the transversal mode and macroscopic field which controls the elasticity of the longitudinal mode. It is seen from Table 3.7, however, that the Coulomb contribution to the elasticity of the non-polar A 1 mode may be equally large and depends on the shape of the normal mode. Hence, the local electrostatic fields created by the deformation of a complex lattice, which cancel each other by eliminating the total polarization of the primitive cell, can contribute significantly to the frequencies of non-polar modes. A problem of separating the effects originating from the local Coulomb interaction and true long-range interaction which determines the macroscopic polarization is thus of interest and attention to this problem will be paid in the subsequent section. It has been argued into the Coulomb origin of the next-neighbor stretch/stretch interaction force constant of some related molecules (see previous chapter). As has been shown in a treatment of the phonon spectrum of a-quartz, this exact type of force constant ensures a correct sequence of frequencies for weaker and stronger IR high-frequency E modes. A deeper model consideration presented below helps to recognize a possible Coulomb contribution to other force constants affecting the bond-stretching modes of a crystal. Let's consider the simplest model of a one-dimensional diatonic AB crystal [33]. The problem to solve is to what degree the Coulomb interaction in the chain contributes to the force constants of its stretching motions defined in a space of internal coordinates. The chain consists of alternating A and B atoms with opposite charges z ~ which are interconnected by the perfect springs determining the force constant fAB (no tensions in equilibrium or interaction at longer distance is implied). Restricting this to the simplest approximation of constant atomic charges (RIM) and denoting r ~ the equilibrium interatomic distance, one obtains the following interrelations between the Coulomb addition to the stretching force

LATTICE DYNAMICS COMPUTATIONS

275

constant K and Coulomb interaction force constants of adjacent bonds hi, bonds separated by one other bond h2, bonds separated by two bonds h 3, etc.: K (r~

=-1.644.8

h 1 (r~

=

h2

r~

h 3 (r~

0.1580

0.0387

=

0.0142

etc. The original diagonal force constant matrix of the chain acquires thus a form fAB-K

hl

h2

h3

fAB-K

h1

h2

fAB-K

h1 fAB-K

A numerical calculation with these force constants shows that it is possible to reproduce in the GVFF model the results of the exact (Ewald summation) evaluation of the Coulomb contribution to the stretching frequencies of the chain within 1% if the interactions of adjacent bonds and over one and two bonds are considered. A restriction to the in-

276

LAZAREV

teraction over one bond (K, h 1, h2) reduces the accuracy to 2%. The interrelation between various GVFF parameters deduced for the above particular case, h 1 ~ -0.1K, h 2 ~ 0.03K, h 3 ~ 0.01K, may be useful in a rough estimation of the numerous additional force constants arising in the computation for a real three-dimensional lattice. A negative sign of the Coulomb contribution to the diagonal bond stretching force constant and alternating signs of the off-diagonal force constants of various order are most remarkable. It should be emphasized that in a given approach a total Coulomb contribution to the parameters of GVFF is treated without discerning one originating from the interaction of the nearest atoms and the interaction in the rest of the crystal. Since the former is taken into account implicitly in the force constants transferred from some related molecule, there is some ambiguity in a rigorous application of a given approach. Another origin of ambiguity arises from the multiplicity of additional force constants to be introduced in a real threedimensional case. The above effects are easily recognized in the bond stretching type force constants of quartz deduced by Lazarev and Mirgorodsky from the molecular force constants [15]. Judging by the force constant/bond length correlation which is known to exist in silicates (see Chapter 1), their diagonal Si-O force constant seems to be significantly underestimated. As has been noted above, the average Si-O bond length in a-quartz nearly coincides with one in tetramethoxysilane while a smaller Si-O force constant of disiloxane with a larger bond length is preferred in their force field model. Adopting that correlation, it is possible to estimate the Coulomb softening of bond-stretching modes in a-quartz through the -K parameter of the above model treatment which constitutes in this particular case about 15% of the "mechanical" diagonal force constant. Then, the interaction force constant of the non-adjacent bonds, d(SiO/Si'O'), originally estimated by quantum mechanical computation of the related molecular system, has

LATTICE DYNAMICS COMPUTATIONS

277

been increased in the force field of ref. [15] (the absolute magnitude is implied). It may originate either from larger Coulomb contribution to that force constant in a crystal or from the fixed magnitude of the interaction force constants of adjacent bonds whose positive values can be enhanced by the crystal field. No equally simple model treatment of the Coulomb contribution to the anglebending force constants has been proposed. Moreover, from the point of view of the above considerations on the influence of the Coulomb interaction on these force constants (which is supported numerically by the data of Table 4.6), a reduced molecular OSiO force constant in the force field of a-quartz adopted in ref. [15] may seem confusing. A following explanation of this apparent contradiction can be proposed, however, proceeding from the quantum mechanical computation of the force field of H6Si207 molecule in the previous chapter. It has been shown that there exists a dynamical interaction between the OSiO angles of adjacent SiO4 tetrahedra and that the sign and magnitudes of those force constants can be rationalized in terms of their Coulomb interaction, in particular, in terms of the dipole forces arising in distorted molecule. These interactions have been shown to be satisfactorily reproduced in the GVFF model by the OSiO/OSiO interaction force constants. No interactions of this type were introduced in the force field of a-quartz proposed by Lazarev and Mirgorodsky. It could be thus suspected that a neglect of these interactions was due to the underestimation of the diagonal OSiO force constant in their model. In order to verify this supposition, their final Set IV of the force constants has been complemented by the bend/bend interactions in adjacent tetrahedra and the diagonal OSiO force constant was simultaneously enhanced to 1.1 mdyn.A, which is closer to its magnitude deduced from the scaled ab initio force fields of related molecular systems.

278

LAZAREV

It should be reminded now that the force field of a-quartz has been specified in a space of redundant internal coordinates. Because of that redundancy, no change in the frequencies or other properties would arise in a case of the introduction of the same interaction between a given OSiO angle and six angles of the neighboring tetrahedron (a summation over variations of those six angles at any deformations necessarily produces zero). In order to avoid ambiguity from the introduction of various interaction constants, only ones corresponding to the OSiO angles having common atom and ones separated by two Si-O bonds have been taken into consideration and the same magnitude O.1 mdyn.A adopted. Keeping all other force constants the same, as in version IV of the force field of a-quartz, one obtains the following frequencies for the optical modes and elastic constants: Frequencies

E

1170

A2

1071

A1

1077

1069

793

468

827

501

330

695

372

205

464 362 259 134 Elastic constants

C 11

93.2

C 12

15.7

C33

105.5

C13

22.4

C44

61.6

C14

-17.9

C66

38.7

A comparison of these values with the experimental magnitudes (Tables 4.2 and 4.3) definitely shows an improvement of their fit attained at the cost of the introduction of one more parameter in the Set IV and a small but physically meaningful variation of the another

LATTICE DYNAMICS COMPUTATIONS

279

one. Although no further variation of the force field has been tried, this example shows, probably, that just the account for the Coulomb interaction in a lattice may improve the resuits of the transfer of ab initio molecular force constants into its dynamical model and that it can be represented at least partially in a space of internal coordinates.

II.

A B I N I T I O F O R C E C O N S T A N T S O F M O L E C U L A R S P E C I E S IN

LATTICE DYNAMICS OF THE QUARTZ-LIKE ALUMINUM PHOSPHATE

A.

Experimental Phonon Spectra and Band Assignment

There exist a variety of ABO 4 compounds whose structm'e resembles one of the silicon dioxide and a-quartz in particular. Some of them have been studied spectroscopically [34] and their relation to the spectrum of SiO2 discussed. However, a sufficiently complete set of experimental IR and Raman data obtained in polarized radiation at suitably adjusted single crystal specimens are presently available only for c~-berlinite, A1PO4 with the a-quartz structure, while its crystallographic parameters were determined in ref. [35]. A subsequent discussion will be restricted to the various approaches to lattice dynamics of this crystal including ones proceeding from the ab initio quantum mechanical force fields of molecular species which are, in difference of the simpler case of a-quartz, inaccessible to experimental investigation. It is hoped that this discussion will be useful in judging the perspectives of the proposed approach in a more general case. A comprehensive single-crystal Raman investigation of ~t-berlinite has been originally attempted by Scott [36] who studied some anharmonic interactions relating to the behavior of low-frequency weak mode responsible for the or-13 transition as well [37]. A misassignment of the most high-frequency A 1 mode was criticized by Mirgorodsky [38] who resolved some experimental difficulties originating frown the nearly coincident strong

280

LAZAREV

A1 and E modes near 1100 cm- 1 and tentatively identified that weak A 1 band at 1173 cm- 1. Some later investigators [39] were probably unaware of his conSiderations and repeated the original assignment. Single-crystal IR reflection investigation has been carried out originally in ref. [40] and most of the optical phonons in the polar symmetry species, A 2 with

Ellc and E with Elc,

identified and the oscillators strength determined. TO and LO frequencies have been compared with corresponding Raman data obtained at suitable polarization conditions and some "oblique" phonons were identified. A subsequent IR and Raman investigation of AIPO4 [39,41 ] confirmed most of the assignments proposed in the list of fundamentals given originally in ref. [42] and complemented or refined it in some detail. A set of these fundamentals, which will be compared with the results of the normal coordinate calculation below, is combined in a most plausible way from all of the above data, a disputable assignment of high-frequency A 1 mode being solved in favor of the version argued for by Mirgorodsky [33]. At the P3121-D 4 space group with three A1PO4 formula units in a primitive cell, a opt representation of long-wave optical phonons is written as" F q=0 = 17E + 8A 1 +9A 2. ~le crystal structure is similar to the one of quartz with Al and P regularly substituted in the Si positions and the corresponding doubling of the c lattice parameter. The crystal structure is shown in the projection of the plane normal to the optical axis in Fig. 4.5 where a possible selection of a primitive cell is indicated and the atoms therein are enumerated for convenience of further disc assion. Since the earliest attempts to interpret the optical spectrum of a-berlinite and to calculate its normal modes [8,36], its strong resemblance to the spectrum of a-quartz was widely exploited. Taking into consideration nearly the same masses of tetrahedral atoms

LATTICE DYNAMICS COMPUTATIONS

281

14'

J

<

J

J

J

5

J

Y

J

..•13;'

L _ J 14

9/~

7'

17

l'~x4

O-P 9 -Ai 14'

6~ J

J

J

o--0

J I"~P) 4' 15'

,513' ?,

X Fig. 4.5

A1PO 4 ot-berlinite structure as viewed along three-fold axis. A possible

separation of the primitive cell is shown. The numbers of atoms of the primitive cell repeated by translation are primed. Continuous lines correspond to P-O bonds and broken lines represent A1-O bonds.

and supposing a similarity of force fields in the PO 4 and A10 4 structural units, it is possible to rationalize the optical modes of cz-berlinite as originating from the optical modes of a-quartz in F(0,0,0) and A(O,O,Tt/c) points of its Brillouin zone, the optical activity of phonons of the A point being deduced from the "folding" of the zone at a doubled lattice vector c. Providing for the same symmetry of F and A points and a transformation of acoustic A phonons into the optical ones in F point (zone folding), a following correlation of the spectra of the two crystals can be observed:

282

LAZAREV AIPO 4

SiO 2

~

J

~

4

A

F Opt ~ ' , A I ~ 4 A

2

t

1

A Opt

A2 ~ - - - - - - - - - - - - - - - - - ~ ~ ~ A 2 ~

E

2

}

A ac~

This approach implies that the spectral intensities of e~-berlinite modes corresponding to the A modes of a-quartz should be insignificant in comparison with F modes, since the intensities of the formers are determined entirely by the inequivalence of Al and P atoms substituted in the Si positions. Additionally, an attempt has been undertaken to deduce an improvement of the equalization of the A1 and P charges in et-berlinite from this approach and its comparison with the spectrum of a-quartz [36]. Knowledge on the frequencies of a-quartz in A points in this approach was taken from the experimental inelastic neutron scattering data for low-frequency modes and calculated magnitudes for the more highfrequency ones. This approach proved to be useful, in particular, in the estimation of probable frequencies of some experimentally, unidentified optical modes. Really, two originally unidentified, weak IR bands in the low-frequency part of the spectrum have been found later [39] in the vicinity of their predicted positions. This approach corresponds to the description of the shapes of the normal modes to B prolongation of the stretching ones as symmetric and antisymmetric POAI vibrations (cf. with similar description of the SiOC modes in tetramethoxysilane in the previous chapter). E.g., six of the vasPOAl modes originating from six vasSiOSi F-modes of a-quartz are assigned to strong A1, A 2 and two E bonds near 1100 cm-1 while six other vasPOAl modes

LATTICE DYNAMICS COMPUTATIONS

283

deduced from the A-modes of a-quartz are attributed to weaker bands in the same irreducible representations at slightly higher frequencies. Similar considerations are applicable to vsPOA1 modes between 750 and 550 cm-1 if the shitt of one of them (in A 1 representation) to lower frequencies caused by the existence of closed loops of the P-O-AI bridges in the network is taken into account [40]. The normal coordinate calculation and P.E.D. analysis shows, however, that this approach is more applicable to bending modes where the OPO and OAIO deformations are strongly coupled indeed. In the high-frequency area where the bond stretching modes are lying, P.E.D. calculations reveal much more separation of the P-O and A1-O contributions in particular modes which originated from considerable difference in their force constants [40]. It seems more reasonable thus to deduce the qualitative considerations concerning their shapes and frequencies from another approach which interrelates them with the normal modes of free PO 4 and AIO4 tetrahedra, respectively. Formally, this approach proceeds from a well known factor-group analysis which interrelates the internal modes of a free comply ion with the lattice modes by taking into account its site symmetry and a number of non-equivalent ions in a primitive cell. In the case of a-berlinite, twelve stretching type modes of the PO 4 (or AIO4) tetrahedra in a cell are interrelated with ones of a free ion as: Free ion Td

Site- group

Factor- group

symmetry

C2(x ) in a lattice

D 3 of a lattice

F~ --------___

B

E A2

A

I

~

A1

284

LAZAREV

Each component of the triply degenerate mode of a free tetrahedron antisymmetric to the two-fold axis of a lattice generates one E and one A 2 mode of a crystal. The component symmetric to that axis and the totally symmetric mode of a free tetrahedron each generate one E and one A 1 mode. The triply degenerate mode is thus represented by two E, one A 2 and one A 1 bands in the crystal spectrum while one E and one A 1 bands are to be deduced from the pulsation of a free tetrahedron. Their recognition in the calculated and experimental spectra will be discussed below. Obviously, the two above approaches represent implicitly two different notions on the chemical structure and bonding in A1PO4, the former emphasizing its network nature with supposed more or less uniform valence charge distribution and the latter representing it as an "island" structure, i.e., as an aluminum salt of phosphoric acid. As the earlier attempts to rationalize its vibrational properties in terms of these external approaches show, none of them is an exhausting one. A reinvestigation of this problem would be of interest in a case of accessibility for more definite estimations of the dynamical parameters adopted at the initial step of the frequency fit. Like as in a case of a-quartz, such estimations may be deduced from the ab initio computation of suitable molecular systems.

B.

Related Molecular Systems and their Force Fields

In the earlier attempt to calculate the long-wave optical vibrations of ot-berlinite [40], it was shown that their resemblance to the spectrum of a-quartz originated mainly from the kinematic similarity of two crystals (similar spatial distribution of atomic masses) and could be reproduced by computation without any supposition of the force constants "averaged" over the PO 4 and A10 4 tetrahedra as it was proposed by Scott [36] (the same relates to the "averaging" of the charge distribution proposed in that paper). The high-frequency optical modes were shown to relate to the normal modes of a free PO 4 tetrahedron and their calcu-

LATTICE DYNAMICS COMPUTATIONS

285

lated shapes discussed in some detail including their graphical presentation in ref. [40]. Similar considerations have been developed later in a more qualitative form in ref. [41 ]. It means that an "island" approach to the description of the lattice dynamics of A1PO4 in a sense defined above is really able to reproduce the peculiarities of its phonon spectrum. The force field description adopted by Lazarev et al. [40] was specified in terms of GVFF with the initial magnitudes for its force constants estimated very roughly from the normal coordinate calculations for several phosphates and aluminates in ref. [43]. Relatively large interaction terms were found, however, to be necessarily introduced into the force constant matrix in order to reproduce the experimental frequency distribution in the spectrum. Having no idea of their physical nature and attempting to reduce the number of independably adjusted parameters, these were introduced quite formally as the interaction between the internal modes of three non-equivalent PO 4 tetrahedra relative to translation. A success of that approach is in the description of eigenfrequencies (and of lR intensities at appropriately the adopted description of the polarization properties of the crystal) which can be treated as an indirect indication in favor of repeating such calculations with a more consistent estimation of the force constants of local interactions deduced from the ab initio determination of the force fields of suitable molecular systems and a more physically

clear conjectures on the influence of the interactions of a less localized origin. It has been done in ref. [44], its main statements being reproduced and discussed below. In searching the molecular prototypes of aluminum phosphate, a system resembling the disilicic acid molecule, (HO)3POAI(OH)3, was originally designed. No equilibrium geometry could be found, however, because of its trend to decompose into H3PO 4 and AI(OH)3. Therefore, the molecular force constant computation was restricted to monotetrahedral species isoelectronic with H4SiO 4, H4PO ~ and H4A10 ~ ions, and the properties of the bonds in the P-O-AI bridges were believed to be represented to a certain extent

286

LAZAREV

by those of P-O-H and AI-O-H, respectively (cf. with Chapter 1 for comparison of the Si-O-Si and Si-O-H bridges). For the sake of economy, the quantum mechanical SCF computations were performed with the atomic basis set labeled above as Set I, i.e., a splitvalence type set with no polarization functions on any atom. Some limitations of its applicability to the geometry and force constant computations have been mentioned in preceding chapters and certain consequences of the nonzero total charges of adopted systems discussed. The results of a total geometry optimization are presented in Table 4.7 in comparison with corresponding data for H4SiO4. The residual forces in the weakest internal coordinates (torsional) were less that 0.003 mdyn. The deepest energy minimum has been found in the S4 point group while a more shallow one has been met at the higher D2d symmetry. Two conformers slightly differ in the geometry of the central TO 4 unit: it is flattened relative to the axis at S4 symmetry and elongated at D2d. Only the more stable former one is treated below and its parameters are represented in Table 4.7. Using the calculated d0/reTO relation from that table as a characteristic of the T-O bond polarity, it is possible to conclude that its enhancement in a series H4AIO~, H4SiO4, H4PO ~ (21.6, 25.9, 31.8%) represents the systematic decrease of the bond polarity which reveals itself in the removal of valence density front the oxygen nucleus. The net charge on oxygen decreases in the same succession. The AO composition of the bond LMO brings out the decrease of the 2s,2p-oxygen contribution and the increase of the 3s,3p contribution of the central atom. Some of data presented in Table 4.7 can be compared with the results of computations with more extended basis sets. A geometry optimization of AIH4A10~ in the S4 point group with 6-31 G* basis set [16] predicts similar structure with the AI-O bond length increased by 0.01A. The computations for H4PO ~ with the same basis set hardly can be

LATTICE DYNAMICS COMPUTATIONS

287

TABLE 4.7 Spatial and electronic structure of molecular systems adopted to model the fragments of the aluminum phosphate lattice.

Properties of a system -Etotal, eV

reTO, A

reOH

-

H4A10 4

+

H4SiO4

H4PO 4

14745.788

16018.260

17410.810

1.761

1.648

1.590

0.960

0.945

0.969

115.0

114.2

113.9

/OTO (x4)

106.8

107.2

107.3

/TOH

118.3

127.5

128.0

89.6

88.2

90.8

+1.470

+2.049

+2.404

O

-0.971

-0.964

-0.894

H

+0.354

+0.452

+0.543

Overlap Population T-O

0.444

0.471

0.466

O-H

0.493

0.539

0.512

do, A

0.380

0.427

0.505

0.740

0.745

0.766

ZOTO (x2), degrees

ZTOHIOTO Net charge

Localization LMO TO:

r

LMO LP:

do r

LMO OH:

ZLPIOILP, degree

uAo

0.329, 0.315

0.31 O, 0.311

0.302, 0.298

0.734, 0.724

0.723, 0.715

0.717, 0.721

0.517

0.499

0.471

0.728

0.707

0.700

117.0

113.2

112.5

compared, however, since the optimization in ref. [45] was restricted to the D2d point group. The calculated P-O bond length (1.524A) may rise by several thousandth of an A, according to our estimation, when passing to the S4 point group. It can be concluded thus

288

LAZAREV

that in a case of a calculation with a nearly two-exponential level basis set the equilibrium P-O bond length from Table 4.1, would be shorter by at least 0.06A. The internal, non-redundant coordinates adopted in the force constant calculation were the same as the ones described in detail in Chapter 3 for the central Si(OC)4 unit of tetramethoxysilane. Only the ones representing the internal degrees of freedom for the TO 4 group at S4 symmetry will be treated below. These nine symmetry coordinates, g l, g2. . . . g9, were specified on page 221 (Chapter 3) as the approximate T d local symmetry coordinates interrelated with the S4 irreducible representations as A 1 m A, E ' ~

A, E " m B,

F~ m B, F~',F:~" m E. These interrelations determine the introduction of non-zero interaction terms in the ab initio force constant calculation which would vanish at the exact T d symmetry for the TO 4 unit: gl g5 (in A representation), g2g6 and g6g7 (in B representation), g3g9 = g4g8 and g4g9 = g3g8 (in E representation). A calculation of the force constants in ref. [44] has been performed by numerical differentiation of the forces arising at the shifts from equilibrium along gl, g2, g5, g6, g7, g9. The diagonal and off-diagonal force constants of the g8 coordinate have been determined from the symmetry considerations using the corresponding data for g9. For the sake of economy, no shifts along the g3 and g4 were treated assuming rather arbitrarily that their diagonal force constants were coincident with the one found for g2 which belongs to another irreducible representation of the S4 point group (these magnitudes are given in brackets in Table 4.8 where the ab initio force constants in the symmetry coordinates are presented). It was attempted to justify a value for this oversimplified approach by taking into consideration the further use of the ab initio force constants of molecular systems in the design of the force field model of aluminum phosphate [44]. PO 4 and A10 4 tetrahedra po-

LATTICE DYNAMICS COMPUTATIONS

289

TABLE 4.8 Unscaled ab initio force constants in symmetry coordinates (mdyn/A, mdyn, mdyn.A) ofA10 4 (upper number) and PO4 (lower number) tetrahedra in H4TO 4 molecular ions.

gl

g2

g3

g4

g5

g6

g9

g8

g7

4.672

0.000

0.000

0.000

0.012

0.000

0.000

0.000

0.000

8.341

0.000

0.000

0.000

0.020

0.000

0.000

0.000

0.000

3.996

0.000

0.000

0.000

-0.110

0.000

0.000

-0.243

7.940

0.000

0.000

0.000

0.287

0.000

0.000

-0.319

(3.996)

0.000

0.000

0.000

-0.138

0.289

0.000

(7.940)

0.000

0.000

0.000

0.265

0.427

0.000

(3996)

0.000

0.000

-0.289

0.138

0.000

(7.940)

0.000

0.000

-0.427

0.265

0.000

0.501

0.000

0.000

0.000

0.000

0.742

0.000

0.000

0.000

0.000

0.715

0.000

0.000

-0.003

1.068

0.000

0.000

0.066

0.883

0.000

0.000

1.329

0.000

0.000

0.883

0.000

1.329

0.000 0.986 1.351

Note: A sequence in numbering of the symmetrycoordinates g7, gs, g9 is inverted, as compared with ref. [44], in order to reduce it in agreement with one adopted in Chapter 3 for tetra methoxysilane. ssess in the lattice a single symmetry element, two-fold axis. The four-fold set of bonds in a tetrahedron is split correspondingly into two two-fold sets with slightly different bond lengths and, probably, different force constants as well. It is impossible to estimate their difference (or a difference between three non-equivalent stretch/stretch interaction constants) from the computation of a system with S4 symmetry and four equivalent T-O bonds.

290

LAZAREV

A supposition of similar dynamical properties in two sets of bonds in any TO 4 tetrahedron which has been successfully employed in the treatment of a-quartz, reduces the number of independent parameters, diagonal stretch and stretch/stretch force constants. Taking into account a certain ambiguity in the subsequent scaling procedure (see below), a knowledge of the force constants for the gl symmetry coordinate and any of the components of the triply degenerate (at Td) stretching coordinate is sufficient to obtain an estimation of these two parameters. Relating to the angle bending coordinates, two two-fold sets of the OTO angles of a tetrahedron with C 2 symmetry in a lattice can be combined in the four-fold set of angles at S4 symmetry and two different angle-angle sets in one two-fold set. It can be noted that the experimental OA10 or OPO angles in a-berlinite give some evidence in favor of this simplification. With regard to their listing on page 229, six tetrahedral angles of their redundant set are denoted hereafter as a and [3 for the two-fold and four-fold sets, respectively (these are easily interrelated since the bond numbering adopted to label them remains the same). The ab initio force constants determined in Table 4.8 from the independent symme-

try coordinates can be employed in various ways to deduce the stretch~end interaction constants in the redundant space of internal ("natural") coordinates. Let us inspect now the structure of the force constant matrix for any TO4 group with S4 symmetry specified in intemal coordinates (Table 4.9). Proceeding from the ab initio determined gigs, g2g6, g3g9, g4g9, and g2g7 interaction constants in symmetry coordinates (Table 4.8), it is possible to deduce five equations to calculate the force constants al, bl, a', b', a", b" in Table 4.9.

LATTICE DYNAMICS COMPUTATIONS

291

TABLE 4.9 A structure of the force constant matrix in internal (redundant) coordinates for the TO4 group possessing $4 symmetry.

ll

12

13

14

t~12

~34

~13

~24

~14

~23

Kq

h

h

h

a1

b1

a'

b'

a"

b"

i%

h

h

a1

b1

b'

a'

b"

a"

1%

h

b1

aI

a"

b"

b'

a'

1%

b1

aI

b"

a"

a'

b'

Kct

m

11

11

12

12

I%

12

12

11

11

KI3

ml

13

14

KI3

14

13

KI3

m2 K[3

The ab initio force constants in intemal (unsymmetrized) coordinates presented in

Tables 4.10 and 4.11 were determined in ref. [44] by assuming b' = b" = 0 and averaging some numerical values as a' ~ a", a 1, b 1. This force field obeys the S 4 symmetry of A10 4 and PO 4 tetrahedra. As explained above, the stretching force constants Kq and h were determined using the interrelations for a regular tetrahedron fglgl = Kq + 3h and fgzg2 = h. Bend/bend interaction force constants were deduced from interrelations fg7g7 = (K a -

ml)/2' fg9g9 = (K[3 " m2)/2' fg6g7 = (4K[ 3 - 2ml + 2m2 - 413 - 411)/4 at suppositions on m = m 1 = m 2 = 0 and 11 = 12 = 13 = 14 = 1 which made possible a unique determination of three magnitudes, Ka, KI3, 1. The force fields of AIO 4 and PO 4 tetrahedra in Tables 4.10 and 4.11 could be transferred directly into the force field model of a-berlinite under the condition of coincidence of their S 4 axes with the two-fold axes of those tetrahedra in the lattice.

292

LAZAREV

TABLE 4.10 Ab initio unscaled force constants ofA10 4 group transformed to the internal (redundant) coordinates 11

12

13

14

12

34

13

24

14

23

4.165

0.169

0.169

0.169

0.193

0.022

0.204

0.000

0.204

0.000

4.165

0.169

0.169

0.193

0.022

0.000

0.204

0.000

0.204

4.165

0.169

0.022

0.193

0.204

0.000

0.000

0.204

4.165

0.022

0.193

0.000

0.204

0.204

0.000

1.972

0.000

0.526

0.526

0.526

0.526

1.972

0.526

0.526

0.526

0.526

1.766

0.000

0.526

0.526

1.766

0.526

0.526

1.766

0.000 1.766

TABLE 4.11 Ab initio unscaled force constants of the PO 4 group transformed to intemal (redundant) coordinates 11 8.040

12

13

14

12

34

13

24

14

23

0.100

0.100

0.100

0.272

0.046

0.301

0.000

0.301

0.000

8.040

0.100

0.100

0.272

0.046

0.000

0.301

0.000

0.301

8.040

0.100

0.046

0.272

0.301

0.000

0.000

0.301

8.040

0.046

0.272

0.000

0.301

0.301

0.000

2.702

0.000

0.795

0.795

0.795

0.795

2.702

0.795

0.795

0.795

0.795

2.658

0.000

0.795

0.795

2.658

0.795

0.795

2.658

0.000 2.658

LATTICE DYNAMICS COMPUTATIONS

293

However, an application of these ab initio force fields calculated in the SCF approximation to the frequency computation is conditioned by the determination of scaling factors which would correct empirically the shortcomings of the adopted approach. With regard to the above case of a-quartz, no experimental frequencies are available for molecular ions treated in the quantum mechanical computation and about 20% uncertainty in the absolute magnitudes of the ab initio force constants may be supposed. Besides, the plausible changes of the molecular force constants of A10 4 and PO4 tetrahedra are to be taken into consideration when applied to the description of the force field for a condensed system. A probable valence charge redistribution should be investigated when discussing the applicability of the molecular force constants to the A1PO4 crystal with significantly modified bonding assumed by some authors. Some other problems originating from the long-range interactions in a lattice will be treated below.

CQ A Design of the Initial Approximation of the Force Field of

Aluminum Phosphate Since no direct determination of the scaling factors for the force constants of a molecular species is possible in a present case, some care is needed when they are transferred into the force field model of a crystal. Specific considerations should be probably taken into account. Some preliminary estimation of the scaling factors can be deduced from the knowledge of the atomic basis set adopted from the quantum mechanical computation. However, some complications are involved in a case of computations performed for charged systems. As has been already discussed in Chapter 1, an excessive negative charge on the H4A10 ~ ion should destabilize it, this effect being underestimated in the calculation with a restricted basis set. Conversely, a deficient electronic charge on H4PO ~ may overstabilize this ion by the corresponding enhancement of the force constants.

294

LAZAREV

Certain considerations of another origin, which do not relate to peculiarities of the adopted quantum mechanical method, may influence the scaling factor estimation indirectly. As has been mentioned above, the valence charge distribution can vary when passing from the free aluminate and phosphate ions to the condensed system of alternating AIO4 and PO4 tetrahedra in the crystal and there exist some controversy in the suppositions on the direction of that redistribution. According to an earlier guess by Scott [36], an averaging of the valence charges in P-O and AI-O bonds may occur while the normal coordinate calculations in ref. [40] and more qualitative considerations in ref. [40] were treated in favor of the opposite direction of the charge redistribution with strengthened P-O and weakened AI-O bonds. A comparison of the experimental bond lengths in c~-berlinite gives evidence for the latter point of view. The same relates to the probable changes in these bonds under a transition from molecular ions to the crystal. A comparison of the calculated bond lengths in those ions with the experimental ones may be treated as representing the total influence of both origins and being thus applicable to the estimation of the scaling factors. The calculated equilibrium P-O bond length in H4PO ~ is 0.075A larger than the average bond length in the crystal (two sets, 1.519 and 1.512A). The theoretical A1-O bond length in H4A10 ~ ion exceeds, however, the average bond length in the crystal (1.746 and 1.737A in two sets) by only 0.02A. This difference is in contradiction with the one expected from the preferential influence of the over-stabilized nature of the former molecular cluster and under-stabilized nature of the latter. It can be treated, tentatively, as originating from the more covalent bonding in the PO 4 tetrahedron and less covalent bonding in the A10 4 tetrahedron in ot-berlinite than in the above molecular clusters thus supporting a description of this crystal as an aluminum salt of phosphoric acid. A problem of the scaling factor estimation for the ab initio force constants of the PO4 and A10 4 tetrahedra specifically directed towards their dynamical properties in the

LATTICE DYNAMICS COMPUTATIONS

295

AIPO4 crystal has been solved in ref. [44] accordance of the GIVP approach with a joint treatment of the experimental phonon frequencies and macroscopic elastic properties. Like in the case of a-quartz, a coincidence of calculated and experimental IR intensities was treated as a justification of the calculated shapes of the normal modes. The same electrooptic model was employed with parameters (in the electron charge units) z~) = -1.25, Z p o

.__

3.0, Z~l = 2.0 and cpo = 3.0, CAlO = 0.5 (signs of the c parameters were determined as characterizing the oxygen charge variation in a stretched bond with phosphorus or aluminum). These parameters were adjusted at the first steps of the frequency fitting and not varied in further calculation. A trial set of force constants of the crystal was composed from the ab initio force constants of the AIO 4 and PO 4 tetrahedra in Tables 4.10 and 4.11 which constituted the force constants No. 1-28 in the force field of the crystal (Table 4.12). Since no ab initio estimations of the dynamical properties of the P-O-AI bridges were available, the POAI force constants of two non-equivalent sets of bridges in the crystal and stretch/stretch or stretcbfloend interaction constants were assumed to coincide with the ones of quartz (Si-O-Si bridges) excluding the AIO/AIOP interaction which was neglected. These force constants (No. 29-34) complemented the local force constants of the tetrahedra thus enabling a definite description of any deformation of the crystal. Despite the fact that the force constants No. 1-34 ensured non-vanishing frequencies of all phonon modes, that set did not permit even a rough reproduction of the macroscopic elastic constants of cz-berlinite.

Hence, the force constants No. 35-37 of non-bonded

oxygen-oxygen interactions in the lattice at distances shorter than 3.5A were introduced since these force constants were known to affect, significantly, the elastic properties of related quartz crystal. Their numerical magnitudes were estimated initially proceeding from

296

LAZAREV TABLE 4.12 Internal coordinates and force constants of ot-berlinite.

No. of force constant 1

Type of force constants (interacting coordinates*) K4,8(PO)

2

K4,16(PO)

3

K8,4,11(OPO)

Equilibrium value of coordinate, A, ~ 1.512 1.529 108.63

"1 ~

7.236

7.236

l

2.432

2.432

2.392

2.392

J

4

K16,4,13(OPO)

108.81

J

5

K11,4,13(OPO)

109.25

1

6

K11,4,16(OPO) PO ** hpo

110.45

J

7

-

10

4,16 (PO/OPO) a13,4,16 4,11 (PO/OPO) a8,4,11 a4,16 (PO/OPO)

11

4,16 (PO/OPO) a16,4,8

12

14

b4'16 11,4,8 (PO/OPO) b4'l 1 (PO/OPO) 16,4,13 OPO ** 10PO

-

15

KI,10(A10)

1.756

16

KI,17(A10)

1.732

17

K7,1,10(OA10)

112.30

18

K14,1,17(OAIO)

112.07

19

K17,1,10(OA10)

109.17

20

K14,1,10(OAIO)

107.10

21 22

h A10 ** AIO al'10 7,1,10(A10/OA10)

23

al,17 (A10/OAIO)

8 9

13

11,4,16

17,1,14

Force constant sets I II

0.090

0.090

"

t J

0.245

0.245

"

l J

0.271

0.271

-

l J

0.041

0.041

} } }

0.716

0.716

2.499

2.499

1.183

-

1.060

-

0.101 0.116

0.101

LATTICE DYNAMICS COMPUTATIONS

297

TABLE 4.12 (continued). No. of force constant

Type of force constants (interacting coordinates*)

Equilibrium value of coordinate, A, ~

Force constant sets I II

24

_1,10 a17,1,10

25

_ 1,10 ( A I O / O A I O ) a14,1,10

26

b 1,10 (AIO/OAIO) 14,1,17

27

b 1,17 (AIO/OAIO) 7,1,10

28

10AIO ** OAIO

0.316

29

u,12, 4 , 88 (PO/OA1)

0.300

0.200

30

u"'3,13 4,13 (PO/OA1)

0.300

0.400

31

K4,11,3(POAI)

142.57

32

K6,17,1(POAI)

142.94

33

6 (PO/POAI) a4:16,2

34

1,10 (AIO/AIOP) a5,10,1

35

K8,7(O..-O)

36

0.122

0.013

}

0.060

0.060

0.200

0.200

3.315

0.040

0.040

K15,16(O'"O)

3.362

0.020

0.020

37

K9,10(O...O)

3.411

0.010

0.010

38

d AIO PO (through PO)

-0.500

-0.300

39

d A10 PO (through A10)

-0.100

-0.050

40

K12,15(OOedge of AIO4)

2.798

41

K12,16(OOedge of AIO4)

2.834

42

K14,17(OOedge of AIO4)

2.874

43

K8,12(OOedge of A104)

2.899

*The adopted bond numeration correspondsto Fig. 4.5. **Interactions of any bonds having a commonatom or of any angles having a common bond.

0.400

298

LAZAREV

the corresponding data for a-quartz and their ab initio determination for molecular systems in the previous Chapter. (Note that none of these force constants correspond to the tetrahedron edges whose properties are exhaustively specified by their internal force constants.) A complete set of experimental data on the long-wavelength phonon frequencies and IR intensities of a-berlinite is represented diagrammatically by Fig. 4.6. The results of their calculation with an initial version of the force field deduced from ab initio determination of local interactions in tetrahedral groups and partially complemented, as explained above, are represented by a similarly designed Fig. 4.7. These frequencies (and elastic constants) were obtained by adoption of the scaling factors 0.9 and 0.6 for the ab initio force constants of PO 4 and AIO4 tetrahedra, respectively. This considerable difference is far beyond the uncertainty of their determination and evidence, as follows from the above considerations, for a relative strengthening of PO 4 and a moderation of A10 4 in the crystal. Moreover, this resuit may indicate that the H4A10 ~ ion is not a good approximation to the description of the dynamical properties of the A104 tetrahedron in the case of its interaction with adjacent PO 4 tetrahedra. More attention to this problem will be given below. Relative to the frequency reproduction by the initial version of the force field, the most important contradiction to the experimental spectrum of a-berlinite was observed in the high frequency area between 1200 and 1000 cm-1 (cf. Figs. 4.6 and 4.7). Instead of the very intensive IR-active A 2 and E modes near 1100 cm -1 (and nearly coinciding A 1 mode very strong in Raman) and the weaker IR and Raman bands at higher frequencies, the calculated spectrum in this area consists of a narrow group of A 2 and E modes in the vicinity of 1200 cm -1, some of them being very intensive in the IR, and one of the A 1 modes. The remaining single A 1 mode and single E mode which is very weak in the IR are predicted to have considerably smaller frequencies (below 1000 cm -1) although they also belong to the P-O stretching modes, as indicated by the inspection of their calculated shapes.

LATTICE DYNAMICS COMPUTATIONS

A1

I I I I

A2

I I I I

299

I I I I

I I I I

I I I I

I I I I

I I I I

E I I

I

I

I

II

_

II

I

600

1000

Fig. 4.6

I

I

I

200

cm "1

The experimental spectrum of transversal optical modes in tx-berlinite.

The heights of continuous vertical lines corresponding to the polar modes are proportional to the IR intensities (S~ magnitudes). Broken lines correspond to Raman data.

A1

I I I I

I I I I

I I I I

I I I I

I I I I

I I I I

I I I I

I I I I

A2 I

E II I

1000

I

I

600

i ,I

, I

,,

,, I

200

I

c m "1

Fig. 4.7 The theoretical spectrum of a-berlinite with initial version of the force field deduced from ab initio molecular force constants (the same notations as in Fig. 4.6).

300

LAZAREV

It was deduced from a closer inspection of the calculated shapes for the P-O stretching modes that those possessing their frequencies near 1200 cm-1 originated from the F 2 modes of three non-equivalent PO 4 tetraheda in the primitive cell, their total number, 3E, 2A 2, 1A 1, being in correspondence with the point group/site group/factor group correlation chart presented above. Obviously, their negligible frequency differences represented a weak coupling of similar distortions in different tetrahedra at the adopted force field approximation. On the other hand, one A 1 and one E mode near 1000 cm-1 in the calculated spectrum could be similarly deduced from three pulsation vibrations of the PO4 tetrahedra in variously interrelated phases. Their nearly coinciding frequencies represent extremely weak coupling. A reminder of the strong difference in polarity of the two above groups of modes was then involved into subsequent speculations. Contrary to the lone E mode near 1000 cm -1 in the calculated spectrum almost vanishing in the IR, most of the E and A 2 modes near 1200 cm-1 were very IR-intensive. This difference did not relate to the peculiarities of the adopted electro-optic model and could be easily explained by the very polar nature of the F 2 stretching mode of a free tetrahedron and the non-polar nature of its totally symmetric mode whose sh'4pes resembled corresponding modes in the calculated crystal spectrum. In other words, it could be suspected that the two types of internal modes of intrinsically the same PO 4 tetrahedra as in a molecular system deviate from the experimental frequencies of the crystal due to the difference in their sensitivity to the electrostatic field of a lattice. On the other hand, it has been stated above that a considerable difference in the scaling factors for the AIO4 and PO 4 species in molecular systems may originate from the changes in their intrinsic properties under transition to a condensed system of interdependent tetrahedra. In the case of a predominating influence of this effect, the force constants of the molecular prototypes would not be the best initial approximation to the force field of a

LATTICE DYNAMICS COMPUTATIONS

301

crystal. Both of the above possibilities were investigated for two versions of the force field of tx-berlinite, which will be discussed in some detail in the next sub-section.

D.

An Extension and Modification of the Initial Force Field

It has been shown earlier that a deficiency of a "molecular" force field originating from the neglection of the Coulomb long-range interaction in a lattice can be offset, at least, partially, by a certain extension of a set of force constants. In particular, a microscopic electrostatic field created by the atomic displacements on the bond-stretching type modes may be represented by adding some stretch/stretch interactions of non-adjacent bonds. This possibility has been tried in version I of the force field for ct-berlinite in ref. [44]. Only the second order stretch/stretch interactions have been taken into consideration, i.e., the interactions of bonds over a bond. These are the PO/AIO' interactions in a given lattice, their negative sign being predicted by physical meaning as explained above. In principle, a number of interactions of this kind can be introduced since two equivalent sets of P-O and A1-O bonds exist in tx-berlinite. For the sake of definiteness, only two types of PO/A10' interaction force constants were discerned, depending on whether the P-O or AI-O bond separated the interacting bonds. These two interaction constants will be referred to as the force constants No. 38 and 39, respectively (Table 4.12). An adjustment of their magnitudes permits us to close into each of the other calculated frequencies of the modes originating from the F 2 and A 1 stretching modes of a free PO 4 tetrahedron. As is seen from Fig. 4.8, all of these modes come together near 1100 cm-1 which is much closer to the experimental frequency distribution (Fig. 4.6) than could be achieved with the initial version of the force field. A more detailed comparison of frequencies calculated with Set I of the force constants and experimental ones is given in Tables

302

LAZAREV

AI

I I I I

I I I I

I I I I

I I I I

I I II II II

I I I I

I I I I

A2 E I i

i

1000 Fig. 4.8

i

i

600

,I i

, , i

200

!

c m -~

The calculated spectrum of et-berlinite, Set I of the force constants (the

same notations as in Fig. 4.6).

4.13-4.15. The IR intensities, treated as implicit characteristics of the shapes of normal vibrations, are compared as well. The macroscopic mechanical and electromechanical prop erties of the crystal deduced fi'om this version of the force field at fixed magnitudes from the electro-optic parameters are presented in Table 4.16. Their experimental values are taken from refs. [46,47]. No further improvement to the frequency fit has been attempted, e.g., by introduction of the third-order neighboring stretch/stretch interactions. Instead, another possible origin of the deviation of the calculated spectrum from the experimental one has been investigated. It should be reminded that the most significant residual discrepancies of the frequency calculation with Set I of the force constants have been met in the area corresponding to the stretching type AI-O vibrations. On the other hand, it has already been mentioned that an unusually small scaling factor (at the given type of the atomic basis set) deduced for the quantum mechanical force constants of the A104 tetrahedron could represent a dissimi-

LATTICE DYNAMICS COMPUTATIONS

303

TABLE 4.13 Experimental and calculated frequencies of a-berlinite, A 1 modes.

coexp, Raman

coexp, Raman

Calculated I

[41,42]

II

Calculated

[41,42]

I

II

1173

1153

1139

437.5

453

454

1106

1132

1119

334

433

320

727

775

758

218

205

231

459

521

498

1 160

182

151

TABLE 4.14 Experimental and calculated frequencies of ot-berlinite, A 2 modes.

Experimental IR [39,42] cot, cm'l

Calculated I S 89

cot, cm'l

II [dP/dQ[

cot, cm'l

]dP/dQI

1158

0.291

1144

0.299

1145

0.382

1101

1.277

1139

0.754

1121

0.766

688

0.399

725

0.897

746

0.604

680

0.192

691

0.394

681

0.598

494

0.671

610

0.463

526

0.071

450

0.371

533

0.390

476

0.710

273

0.295

346

0.303

230

0.323

140

-

172

0.019

134

0.089

48

-

47

0.001

47

0.015

*A quantity S whose magnitude is compared with the theoretical dipole moment derivatives interrelates with Sexp deduced from the IR reflection as S=Sexp(nC2f~m/e2) where c is the light velocity, f) is the primitive cell volume, m is the atomic mass unit adopted in the normalization of the shapes of the normal modes and e is the electron charge adopted to specify the electro-optic model parameters.

304

LAZAREV

TABLE 4.15 Experimental and calculated frequencies of a-berlinite, E modes. Calculated

Experimental IR and Raman [39-42] oat, cm- 1 S 89

I

II

Ot, cm-1

IdP/dQI

oat, cm-1

IdP/dQI

1228

0.125

1174

0.211

1175

0.297

1198

0.143

1147

0.405

1140

0.272

1130

0.378

1122

0.169

1077

0.050

1101

1.184

1142

0.631

1129

0.753

747

0.113

762

0.810

779

0.473

704

0.404

728

0.301

737

0.399

648

0.043

678

0.453

685

0.487

607

0.341

514

0.205

565 465

0.615

578

0.478

502

0.735

442

0.094

553

0.257

459

0.027

418

0.119

422

0.083

428

0.025

375

0.333

397

0.199

337

0.262

305

286

0.069

288

0.075

193

226

0.099

189

0.139

161

182

0.021

145

0.155

114

131

0.045

116

0.045

114

0.019

105

0.098

104 * See the footnoteto Table 4.14.

larity of its chemical constitution (valence charge distribution) in the molecular ion and in the crystal. Hence, its force field deduced ab initio for the former could not be a best approximation of its local elastic properties in the latter. Supposing that aluminum in a-berlinite can be imagined more realistically as a monoatomic A13+ cation bounded to PO 3+ complex anions, it is reasonable to investigate some other initial approximations of the force field around it. In terms of the force constant

LATTICE DYNAMICS COMPUTATIONS

305

TABLE 4.16 Macroscopic elastic and piezoelectric properties of a-berlinite. Calculated

Experimental [46,47] I

II

Elastic Constants, GPa Cll

64.88 87.14

C33 C44

43.12

71.25

67.66

86.59

86.95

56.40

40.70

C66

27.95

30.92

25.22

C12

8.98

9.41

17.21

C13

14.60

14.28

18.48

C14

-12.14

-14.66

-8.79

Piezoelectric constants, esu'cm -2"104 ell

6.6*

2.28

2.94

el4

-4.5*

-0.75

-0.93

*From the experimental piezomoduli of ref. [47], eik = dijCjk. approach, it could be done most simply by describing the force field of the A10 4 tetrahedron exclusively by the force constants of the two-body interactions along A1-O bonds andO...O edges and neglecting all other parameters introduced in the force constant Set I. In other aspects this set, which will be referred to as Set II, does not differ significantly from Set I. As is seen from Table 4.12, the number of parameters describing the force field of the AIO 4 tetrahedron is significantly reduced in Set II at the cost of the introduction of the force constants No. 40-43 whose magnitudes are assumed to coincide and some variation of the interaction force constants of the non-adjacent P-O and A1-O bonds. The diagonal A1-O and off-diagonal AIO/A10 force constants in A10 4 are taken the same as in Set I. A comparison of the experimental and calculated spectra presented diagrammatically by Figs. 4.6 and 4.9, respectively, shows that Set II of the force constants ensures a better

306

LAZAREV

A1

II

I

I I

1

I

I

II

I

I

I

I

I

II

I

I

I

I

I

I

I

A2 I

E II, |

I |

1000

Fig. 4.9

|

!

I,

i |

600

I I,I |

200

!

c m "l

The calculated spectrum of et-berlinite, Set II of the force constants (the

same notations as in Fig. 4.6).

reproduction of the experimental frequencies below 800 cm -1 (a more detailed comparison of both spectra are shown in Tables 4.13-4.15). Some worsening of the macroscopic elastic constants occurs, however, on the transition to Set II (Table 4.16). A considerable underestimation of the piezoelectric constants, which is inherent to both versions of the force field, may originate from the inadequacy of adopted electrooptical model. This problem has not been investigated in more detail (a complexity of the model representing the piezoelectricity even in a simpler case of ~x-quartz has been shown by Mirgorodsky and Smimov [ 17]). Although no further refinement of the above force constant sets has been attempted, the results of the calculation of the dynamical properties of ot-berlinite, with a model whose parameters are estimated by ab initio quantum mechanical computation for suitable rr.,)lecular clusters, seem promising. The successfulness of this approach, however, has been restricted by a complication of the initial model which would represent the influence of long-range interactions in a lattice (projected on the space of internal coordinates) and by a

LATTICE DYNAMICS COMPUTATIONS

307

possible dissimilarity in the valence charge distribution in the adopted molecular system and in the considered crystal.

III.

ELECTROSTATIC CONTRIBUTION TO THE MECHANICAL

MODES OF A M O R E P O L A R I Z A B L E L A T T I C E : P Y R O X E N E L I K E M O N O C L I N I C S O D I U M VANADATE

A.

A Formulation of the Problem and Description of the Crystal

It is more or less self-evident that, in general, an attempt to reproduce the deformational properties of a crystal by adoption of the force constants of some molecular system would be the least efficient method of choice; the less localized is a response of the lattice to a local distortion. Since this non-locality originates from the slowly decreasing electrostatic interaction due to the decreasing distance, it relates to the non-locality of the electric response and may be anticipated to enhance due to the increase of the polarizability of ions (or bonds) constituting the crystal. It would thus be of interest to extend the above lattice dynamical computations of oxides with force constants transferred from suitably selected molecules to the transition elemental oxides with larger dielectric permeability. However, no sets of molecular force constants, either determined from the experimental spectra or deduced from ab initio calculation, were accessible for the crystals of this type. Therefore, an example which will be treated below, relates to the transition oxide isostructural with a well known non-transition oxide whose force constants have been determined by frequency fitting. A supposition of the similarity of the structure of their force constant matrices (interrelations between stretching and bending force constants and between diagonal and off-diagonal ones) can be assumed rather arbitrarily as an initial step. Monoclinic sodium vanadate NaVO 3 is known to possess a pyroxene-like structure [48]. It is isostructural with diopside and tx-spodumene whose spectra have been discussed

308

LAZAREV

in Chapters 2 and 3. The crystal belongs to the same C2/c - C 6h space group and contains two Na2V206 formula units in the primitive cell. The latter consists of the fragments of two anionic chains [V206]oo extending along the c axis and interrelated with each other symmetrically through the inversion center. It means, in particular, that the internal vibrational modes of these chains produce duplicated long-wavelength crystal modes, one being symmetric and another antisymmetric to inversion. These chains possess a single nontrivial symmetry element, a glide plane along the c axis, and thus their intemal modes can be classified in the irreducible representations of a one-dimensional space group [49] whose factor-group is isomorphous with the C s point group. The chains are composed by distorted VO 4 tetrahedra, each sharing two apices with adjacent tetrahedra. The elementary translation of each chain consists of two tetrahedra, or, otherwise, of two V-O-V bridges. In contrast to similar silicates, the bridging V-O(V) bonds are significantly (more than by 0.2A) longer than unshared the V-O- bonds which should probably be manifested in their force constants. No remarkably short interchain oxygen-oxygen distances exist in this lattice. Four Na + cations occupy the two two-fold sites in the primitive cell. As is shown in Fig. 4.10, both the non-bridging (O') and bridging (O) oxygen atoms participate in the formation of their coordination polyhedra. The numbering of non-equivalent atoms relative to translation is given as well and a possible separation of the primitive cell indicated. Before any detailed treatment of the spectrum is made, it is clear that the peculiarities of the structure and bonding favor an approach to the qualitative description of the bond-stretching bands of complex anions applied above to the chain silicates. Really, both the expected large difference in the force constants and the enhanced mass of the tetrahedral atom favor a better separation of the stretching modes localized mainly in the V-O-V

LATTICE DYNAMICS COMPUTATIONS

309

c

t

r

O 9

Fig. 4.10

r

oV

OO

oNa

Sodium vanadate lattice in projections on crystallographic planes. The

atoms of the set non-equivalent relative to translation are enumerated.

bridges, or reversely, in the V< O- lateral groups. Since two symmetrically independent V-O" bonds in the latters do not differ significantly, there is probably no reason to treat their stretching modes separately as it has been done in the case of a-spodumene with two different types of lateral Si-O- bonds. Treating the anionic chain as a one-dimensional crystal with C s symmetry and taking into account the existence of two V-O-V bridges and two V< ~ - lateral groups in its period of identity, it is thus possible to classify its stretching modes as symmetrical (Vs) and antisymmetrical (Vas) modes of those three-atomic units and to discem the ones occurring in phase and out of phase in two non-equivalent units relative to translation. Among eight bond-stretching modes of the chain, four correspond to in-phase motion and belong to the A' irreducible representation of the C s group" Vas,vsO-VO" and Vas,vsVOV. Four out-ofphase modes belong to A" representation and are referred to as V~s,v~O'VO- and V~s, v~ VOV. Each of these modes is split in the crystal spectrum as A'---~Ag + B u and A"--+Bg + A u where one Davydov's component is IR-active and another Raman-active.

310

LAZAREV

Before any normal coordinate calculation is done, the frequency interrelation Vas,vsO'VO" > vasVOV > vsVOV can be predicted from the expected difference in force constants (and shapes of modes in the bridge). An insignificant frequency difference between in-phase and out-of-phase O-VO- modes of a chain can be expected because of the spatial separation of the lateral groups. These preliminary considerations were helpful in the experimental study and band assignment in the spectrum of this rather complicated lattice possessing 57 long-wave optical modes. It should be emphasized that the vanadate lattice is more loosely packed than ones of similar silicates which reveals itself, e.g., in larger interchain oxygen-oxygen distances. A more perfect localization of intemal modes of any chain can be thus expected with negligible interchain coupling except those determined by the long-range interaction.

B.

Experimental Data and Spectral Assignments

The representation of long-wave optical vibrations of the crystal decomposes as opt F q= 0 = 14(10)Ag + 13(10)A u + 16(10)Bg + 14(10)B u where the number of internal modes of the complex anions is given in brackets. As was discussed in Chapter 2, the IR spectrum of the B u species is very informative on the shapes of the normal modes in the case of the experimental determination of the orientation of transition dipoles relative to the axes of the monoclinic plane. A series of reflection spectra from the ac plane at a nearly normal incidence has been obtained down to 85 cm -1, the specimen being rotated relative to the plane of the light polarization as specified by the q~ angle on Fig. 4.11. Two pairs of spectra, obtained at mutually orthogonal orientations of the electric vector E, ~p= 0, 90 ~ and 60, 150~ were employed in the oscillator fit by means of eq. (2.101) of Chapter 2 for the dielectric permeability. The existence of a very strong oscillator polarized along the c axis is remarkable. A totality of Au oscillators has been de-

LATTICE DYNAMICS COMPUTATIONS

311

R,% 80 60

s E"~'

4020

........... I

0

I

-

-

"

~

I

~

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

80 60 40 20

C

0

--

- " = -I

I

I

I

I

so q 60 4o

20

~

0

80 60 40 20 0

O~ a

'

1100 Fig. 4.11

~1

-

1000

I

I

I

900

800

700

I

I

600 500 CO,cm -1

I

I

I

I

400

300

200

100

A series of the reflection spectra at nearly normal incidence from the

plane of monoclinic angle. The orientation of the electric vector is shown. Dotted curves represent the result of oscillator fit.

termined using the conventional approach (eq. (2.100) in Chapter 2) from the single reflection spectrum obtained at the light polarization along the two-fold axis (axis b). It is represented by Fig. 4.12. The optimization of the coincidence for the calculated spectra (points on Figs. 4.11 and 4.12) with the experimental ones including the optimization of the parameters of highoO

oo

frequency dielectric permeability ellipsoid, main magnitudes e33,~;11 and angle 13 which specified the orientation of e~~ relative to the c axis (this angle was counted in the same

312

LAZAREV

R, % 100$060 40 20 0 "1100

I

I

I

I

I

I

I

i

I

I

1000

900

800

700

600

500

400

300

200

100

CO, c m "1

Fig. 4.12

A reflection spectrum from the plane parallel to b axis at Ellb. Dotted

curve represent the results of oscillator fit.

direction as q~). The orientation of the third axis for the ellipsoid was fixed by the symmetry O0

conditions, the E22 magnitude being determined from the A u spectrum. The oscillator strengths were determined in the low damping approximation. Among the 14 expected Bu phonon oscillators, only 13 have been identified because of the probable small strength of the missing one. The frequency of the weak band near 580 cm-1 in the A u spectrum has been estimated from the IR absorption spectrum for the isostructural LiVO 3. Two other weak A u phonon oscillators have not been identified. The ellipsoid of the retraction indices has been determined from the optical measurements in a good accordance with the above high frequency dielectric permeability estimation and adopted to define the geometrical conditions of the 90~ Raman scattering. The tensor of the polarizability derivatives appears as Ag [CrUx La~

Bg

0 ayy '

a~l

0

axe

0

a~xy

0

0

r azz

0

ayr z

00, ay z

In order to separate the contributions from the particular components of these tensors, the following scattering geometries have been adopted z(xy)x, z(xz)x, z(yy)x, z(yz)x, y(zz)x, z(xx)y. Some contamination of the spectra by the lines forbidden in this particular geome-

LATTICE DYNAMICS COMPUTATIONS

313

try has been found, however, in numerous cases which originates from the high birefringence (ng = 2.04 and np= 1.77) and probable frequency dispersion of the axes of the refraction indices ellipsoid. Despite these difficulties, special cautions enabled us to substantiate with confidence the existence of nearly coinciding modes in the Ag and Bg representations between 270 and 220 cm -1 (252.4 cm -1 in Ag and 251.0 cm -1 in Bg or 261.1 cm -1 in Ag and 260.3 cm -1 in Bg). Some spectra have been recorded at liquid nitrogen temperature in order to obtain a better resolution. In total, all 14 of the expected one-phonon transitions in species have been identified. Among the 16 predicted Bg modes one remained unidentified. A complete set of experimental data has been presented in refs. [50-52]. The most important parameters of the optical phonons in sodium vanadate will be listed below (Tables 4.17 through 4.19) when being compared with the results of normal coordinate calculation.

The nearly exhausting knowledge of the zone-center frequencies and of the

transition dipoles in polar modes of this low-symmetry crystal partially compensates the lack of data on its properties relative to the homogeneous deformation. Before any normal coordinate calculation, a tentative assignment of the bondstretching type internal modes of the chain anions is possible by analogy with the spectra of structurally similar silicates. The above qualitative considerations concerning the expected sequence of frequencies which correspond to the various types of movements in three atomic units composing the chain and symmetry considerations are helpful in this preliminary analysis of the high-frequency portion of the spectrum. It is simplified in the present case by the existence of the frequency gap between ca. 500 and 400 cm -1 separating the intemal modes of the bond-stretching type from the angle-bending ones and lattice modes. Figure 4.13 shows the imaginary scheme for repairing the spectrum of a sole chain from its

314

LAZAREV

TABLE 4.17 Force constants of sodium vanadate, mdyn/A, mdyn, mdyn.A.

Internal coordinate or

Equilibrium

Force

Internal coordinate or

Equilibrium

Force

the type of interaction

value

constant

the type of interaction

value

constant

Kvo-(1,5) =

1.653A

6.13

Koo(9,11) =

3.286A

0.15

= Kvo-(1,15)

1.631

6.13

= KOO(9,10)

3.314

0.15

KVO(1,9) =

1.806

3.54

KO-VO-(5,1,15)

109.1 ~

0.88

= KVO (1,12)

1.802

3.54

Ko-vo(5,1,12) =

111.7

0.71

KNaO'(13,17 )

2.314

0.24

= Ko-vo(5,1,9)

111.1

0.71

KNaO'(7,17) =

2.381

0.21

KO-VO(15,1,9) =

110.2

0.65

= KNaO-(5,19)

2.409

0.21

= KO-VO (15,1,12)

105.3

0.65

= KNaO- (5,17)

2.397

0.21

KOVO(9,1,12)

109.3

0.41

KNaO'(13,19 )

2.519

0.18

KVOV(1,12,4)

140.6

0.12

KNaO(9,19)

2.611

0.12

VO hvo(V)

-

0.12

KO-O-(5,15)

3.140

0.06

VO(o) HOV

-

0.41

Ko-o-(7,16)

3.268

0.06

VOh v o - (V)

-

0.71

0.02

VO h v o - (V)

-

0.24

-

0.12

Ko-o-(5,13)

3.273

Ko-o-(8,16)

3.274

0.06

OVO avo

KO-O-(7,14)

3.352

0.12

AVOV VO

-

0.06

KO-O-(5,7)

3.380

0.06

a O 0 VO

-

0.06

KO-O-(15,16)

3.421

0.02

O-VO IO-VO

-

0.06

KOO- (11,15)

3.430

0.06

OVO mo-vo-

-

0.06

Note: the internal coordinates are specified by the atomic numbers in Fig. 4.10.

TABLE 4.18 Experimental and calculated optical modes, Ag and B, species

*0 = 180 - cp, i.e., the angles Which specify the dipole orientation are counted from the c to the a axis in their obtuse angle, as it was adopted above in other monoclinic crystals.

Table 4.19 Experimental and calculated optical modes, Bgand A, species.

LATTICE DYNAMICS COMPUTATIONS

A~ x

317

I I I

I

I

I

I

i I

I

I

vsO'VO" v~O'VO" v~O-VO- v~O-VO- v~sVOV

~,

Bg

'

I I

i

I

i

I

1000

II

gVOV

I

I I

Au

I

v~VOV

v~VOV

I

900

I

I I

I

I

I

800

700

600

O3, c m

I

I

I

500

400

-1

Fig. 4.13 A reconstruction of the frequency spectrum of a sole anionic chain from the experimental spectrum of sodium vanadate.

experimental frequencies in the crystal spectrum. It should be emphasized that the proposed assignments are supported both by their correspondence to the symmetry requirements and by the directions of dipoles in the B u representation relative to the crystal axes of the monoclinic plane. The exceptionally strong IR band with a dipole along the c axis is most remarkable and definitely corresponds to the vasVOV mode with in-phase displacement of all the bridging oxygen atoms along the chain axis. This displacement occurs in opposite directions in two types of the chains, non-equivalent, relative to the translation and interrelated through the inversion, their dipoles thus being put together. The lattice mode of the Ag symmetry which corresponds to the in-phase displacement of a similar type in both sets of chains is easily recognized at nearly the same frequency, its vanishing total dipole being determined by the mutual cancellation of two chain dipoles.

318

LAZAREV

The unexpectedly low frequency for the vasVOV mode of the chain deserves, however, special discussion. This mode has been identified in the range of frequencies approaching the interval where the vsVOV mode is lying (Fig. 4.13). The simple kinematic considerations allow us to expect a much larger frequency difference between the v s and vasVOV modes like it has really been met in a case of structurally similar silicates. Moreover, the V~s VOV mode, which differs only by the phase interrelation in two of the subsequent V-O-V bridges of the chain, is confidently identified above 800 cm -1, which agrees well with the expected magnitude of the V~s VOV and vasVOV frequencies. There is no evident physical reason for a significant frequency difference for these two types of vibrational motion in the anionic chain. However, since the assignments seem quite unambiguous, the significantly different V~s VOV and vasWOV frequencies are to be explained.

In other words, a mechanism

should be proposed which couples the energy profiles of the oxygen displacement along the chain axis in the two neighboring V-O-V bridges interrelated symmetrically by the glide plane.

C.

Normal Coordinate Calculation [52,53]

Unfortunately, no experimentally deduced or theoretically evaluated force constants of molecular species containing VO 4 tetrahedra are yet known. On the other hand, a lack of data on the properties of sodium vanadate relative to the homogeneous deformation (elastic constants, microscopic structure of hydrostatic compression) excluded a possibility of the application of the GIVP philosophy in a total volume in order to determine the force constants. The normal coordinate treatment has thus been restricted to the calculation of the frequencies of the long-wave optical modes and of their shapes whose reliability could be checked by the calculation of the IR intensities and directional properties of vibrating di-

LATTICE DYNAMICS COMPUTATIONS

319

poles, as explained above. A relatively low lattice symmetry is even favorable to realize this opportunity with respect to the B u phonon modes with symmetrically unrestricted dipole orientations in the

ac

plane. It reduces considerably the danger of mixing the assign-

ment of numerous modes belonging to the same irreducible representation when searching empirically for the best frequency fit. In order to avoid an ambiguity of the electro-optic parameters determination, a simpie model, rather successfully employed in previous sections, has been adopted, It operates with the parameters of two types, namely, constant atomic charges in equilibrium, z ~ and additional charges, c, which arise (with opposite signs) at both ends of a stretched bond. Since this mechanism of polarization is supposed to be intrinsic to partially covalent bonds, it was considered only for V-O bonds discerning two chemically different types of these bonds, lateral V-O" and bridging V-O(V) ones, by different c magnitudes.

Correspond-

ingly, only two different magnitudes of the equilibrium oxygen charges were treated, neglecting a larger number of crystallographically, non-equivalent sets of these atoms in sodium vanadate lattice. The total set of the electro-optic parameters consists thus of six quantities (the static charges being interrelated through the electroneutrality condition). Their magnitudes (electron charge units), z~r = +2.2, z Na = +0.9, z ~3 = - 1.1, z ~3- = - 1.0, cVO(V) = +2.4, cvo- = + 1.0, have been estimated at the earliest steps of the normal coordinate analysis and fixed in the process of a further frequency fit. A large difference between the c parameters of the two types of V-O bonds may represent implicitly a very anisotropic nature of the deformational polarizability of anionic chains. That set of electro-optic parameters enables a satisfactory reproduction of the IRintensities and dipole orientations of the bond-stretching modes of complex anions in the crystal spectrum. Its poorer efficiency in the low-frequency part of the spectrum may origi-

320

LAZAREV

nate both from the shortcomings of the adopted model of the polarization mechanism in application to the angle-bending deformations and from erroneous determination of the shapes of numerous modes rather than the density positioned in that area. The initial set of force constants for the complex anion has been designed taking into consideration the VFF model for the VO 3- ion in the zircon-like YVO4 [52] (the original Raman data and ones of ref. [54] were employed jointly with the results of the IR-reflection investigation in ref. [55]). That force field has been deduced proposing a similarity of the structure of force constant matrices in isostructural YVO 4 and YPO 4. The force constants of VO 3- gave a rough estimation of the diagonal V-O stretching force constant at a particular bond length, OVO bend and some interaction constants. The force field of sodium vanadate has been designed as a VFF of complex anions, which includes a restricted number of important interaction constants representing the stretch/stretch or bend/bend interaction at a common central atom and stretcl~end interaction at a cOmmon bond. Purely diagonal sub-matrix, composed by Na...O force constants in the coordination polyhedron of sodium and by non-bonded oxygen-oxygen force constants in all interanionic 0 . . . 0 distances shorter than 3.5A, has been adopted to describe the force field of the lattice, i.e., cation-complex anion interaction and interanionic vibrational coupling. A participation of the bridging oxygen atoms of the chains in those interchain contacts should be noted. Some of the initial magnitudes for the force constants have then been refined, sometimes, proceeding from their influence on the calculated IR intensities of the polar modes. For example, the interaction constant h VOv o - has been enhanced up to 0.7 mdyn/A in order to obtain a better reproduction of the interrelation between IR intensities of the A u crystal modes originating from V~s and v~O-VO" modes of a single anionic chain. That enhancement improves simultaneously a reproduction of the experimental dipole amplitudes

LATTICE DYNAMICS COMPUTATIONS

321

and orientations of the B u modes which correspond to Vas and vsO-VO- internal degrees of freedom of the chain. However, since the earliest steps of the normal coordinate analysis, it has been found that it is impossible to reproduce the experimental tot magnitudes in the high-frequency part of the spectrum in frames of conventional structure of the force constant matrix (the most frequently met interrelations between diagonals and off-diagonal elements are implied). Really, it is difficult to propose a physical mechanism which would couple the in-phase and out-of-phase vasVOV motions (Vas and V~s VOV) of neighboring V-O-V bridges of the same chain so strongly that their frequency difference would exceed 25% of the average magnitude. The mechanism should be dependent on the interchain origin since the interchain coupling (Davydov's splitting of the chain modes) is insignificant as is seen from the scheme on Fig. 4.13. Since the B u and Ag vasVOV frequencies were surprisingly low (shifted downwards from their expected positions nearly to proximity of v s and v~ VOV modes), it was decided to repeat the frequency fitting at the excluded two above frequencies from the optimization procedure. At this condition, a fairly well agreement between all other calculated and experimental frequencies was readily obtained. The results shown diagrammatically on Fig. 4.14 confirm a satisfactory reproduction of the dipole amplitudes and orientations in the polar modes. A more detailed comparison of the experimental and calculated spectra is given in Tables 4.18 and 4.19. As is seen from these Tables or Fig. 4.14, this conditional force constant optimization predicts vasVOV frequencies in B u and Ag representations to be even slightly higher than the ones of V~s VOV which have been found experimentally between 800 and 850 cm -1. It should be emphasized that the predicted B u frequency nearly approaches the ex-

BUC" ,I , I a

/ -

I

I

-\

X

B

xf

Bu / -

+

1

X

XB

1

CI

II

r.

X

X

X

*

7-//

,.

Ir r

..

/..-"t,//\ # .

I

z r.

v, cm-l Fig. 4.14 Calculated (upper) and experimental (lower) optical TO frequencies of sodium vanadate. The heights of vertical lines corresponding to the itequencies of polar modes represent their IR intensities, the frequencies of those modes with nearly zero intensities being denoted by crosses. Arrows represent the orientations of B, dipoles.

LATTICE DYNAMICS COMPUTATIONS

323

perimental LO frequency of corresponding IR band (r 1 = 888 cm-1). The origin of this similarity will be discussed in some detail later. Then, it is important for subsequent considerations that, unless the calculated ~t frequencies of Ag and B u vasVOV modes are considerably overestimated, their eigenvectors (shapes of vibrations) can be correctly determined. It follows from the successful reproduction of the huge IR intensity (and dipole orientation) ofB u by means of the calculated eigenvector.

D.

The Origin of the Transversal vasVOV Modes Softening

It follows from the above consideration, that among the numerous long-wave optical modes predicted by the physically consistent mechanical model of the lattice, two modes are significantly softened in the real spectrum of the crystal. Both modes are of the bond stretching type, i.e. their elasticities are governed mainly by the interaction between neighboring oppositely charged atoms. The electrostatic interaction between such atoms has been shown to weaken their bond-stretching frequency (cf. previous sections). It is therefore reasonable to investigate, if there exist some peculiarity of the Coulomb contribution to those modes which would be able to explain their unusually large softening. Supposing the shapes of these normal modes determined correctly by means of the adopted mechanical model, it is possible to calculate the Coulomb contribution to the dynamical matrix of sodium vanadate at any assumed charge distribution like it has been done earlier in a case of a-quartz. Again, as it has been already mentioned, the calculated DCoul magnitudes are not, rigorously speaking, the Coulomb contributions to the dynamical matrix deduced from the potential energy decomposition in the form (2.44) of Chapter 2. More correctly, these magnitudes can be treated as being proportional to the partial derivatives of the frequency squares determined in the force constant approach with respect to corresponding DCoul elements.

324

LAZAREV

At first step, the D C~ magnitudes have been calculated in the RIM approximation with adoption of static atomic charges, z ~ determined from the IR intensity fit in the above normal coordinate calculation. Similarly like in a case of a-quartz, the calculated DCoul magnitudes vary more or less smoothly in a sequence of the eigenfrequency, diminishing from large and negative ones to smaller in absolute value and then to positive ones in the low-frequency part of the spectrum. As explained above, this trend originates from a reduction of the contribution of the bond stretch and increasing contribution of the angle bending to these shapes of the normal modes. The results of the calculation presented in Table 4.20 include only the data relating to the higher-frequency bond-stretching modes of chain anions in the crystal. Among the DCoul magnitudes in Table 4.20, the ones corresponding to the modes, whose "mechanical" frequencies are most considerably softened in the experimental crystal spectrum, are really larger than for any other mode. Their difference from other D Coul magnitudes seems, however, far insufficient to explain a very specific softening of those two modes. Talking into account a relatively high and anisotropic e ~176 of sodium vanadate, an important contribution of atomic or bond polarizability to the mechanism of that softening could be expected. It is thus desirable to pass from the RIM to the PIM approximation in the D C~ calculation in order to clarify, whether the high polarizability may define the particular character of softening. The refractometry data are used conventionally in order to estimate the atomic polarizabilities in the crystals [56]. On the other hand, the atomic polarizabilities oti can be deduced from the IR intensity calculation in the PIM approach which adopts the same "mechanical" eigenvectors and complements a polarization produced by the shifts of atomic charges by the contribution of their own polarization (cf. Chapter 2). This approach is,

TABLE 4.20 Coulombic contributions to the c matrix calculated with eigenvectors of its mechanical approximation (bond-stretchingtype modes). -

Force constant method Symmetry representation

w

yP,cm-I

--

-

D'O~' calculation (. two bonds all ai = 0 ai# 0 -0.2135 -0.2273 -0.2764 -0.2783 -0.3072 -0.3327 -0.3213 -0.3296 -0.3920

-0.2955 -0.3601 -0.5569 -0.5496 -0.5047 -0.5805 -0.543 1 -0.5860 -1.8088

~ m -in~RIM ) and PIM one bond ai#O solo = 0 -0.1564 -0.1584 -0.2735 -0.2737 -0.2581 -0.2883 -0.2579 -0.2854 -0.9189

-0.1547 -0.1562 -0.2506 -0.2506 -0.2564 -0.2846 -0.2560 -0.2820 -0.4566

326

LAZAREV

however, complicated by the non-additivity of calculated St magnitudes relative to the contributions of particular a i. Proceeding from the initial refractometric estimations, it is possible to reproduce, approximately, the intensities of the strongest IR bands of sodium vanadate by adoption of the following set of atomic polarizabilities" a v = 0.2A3, a 0 = 1.8A3, eto- = 0.5A3 where a high polarizability of the bridging oxygen is remarkable (the polarizability of Na + cation is set to zero). The results of the IR intensity calculation with this set are given in Table 4.21. The significantly underestimated magnitudes of e ~176 components originate from intrinsic shortcomings of the PIM approach (otherwise, the IR intensities would be drastically overestimated). It should be emphasized that the large polarizability of the bridging oxygen atom is directly interrelated in this approach with the intensity of the most strong IR band in the B u representation. Unfortunately, a very plausible strongly anisotropic character of atomic polarizability in the chain remained uninvestigated. Most of calculated negative D c~

magnitudes are enhanced when passing from the

RIM to the PIN approximation. However, the negative contributions to the two modes in question, vasVOV in B u and Ag representations, are enhanced much more then the contribution to any other mode and their particular sensitivity to the polarizability of complex anion becomes quite evident. Let us emphasize once more that a similarity in the softening of the microscopically polar B u mode and of the dipole-less Ag mode, whose dipoles in the two crystallographical sets of chains cancel each other in the total polarization of the crystal, proves the essentially localized nature of that softening intrinsic to a sole chain. Even more drastically enhanced local field softening of these modes can be expected in a case of

LATTICE DYNAMICS COMPUTATIONS

327

TABLE 4.21 IR intensities of the most polar modes of sodium vanadate calculated in PIM approach.

Symmetry representation

Experimental cot, cm- 1

St. 10-5 , cm -2

Calculated cot, cm- 1

St. 10 -5 , cm -2

Bu

960.4

0.20

953

0.79

Au

950.1

1.69

948

2.15

Bu

914.3

3.52

914

1.96

Au

913.2

0.17

919

0.57

Au

814.7

4.05

828

1.62

Bu

635.1

13.90

848

15.30

Bu

478.1

0.17

478

0.12

Au

374.0

0.21

362

0.36

Bu

333.9

0.91

341

0.39

Notes: 1. Calculated frequencies obtained by the force constant approach. 2. Calculated dielectric permeability at adopted atomic polarizabilities: exx~176 = 1.63, eyy~176 = 1.61, oo =3.20, exx oo = 3.05, ezzoo= 1.89. The magnitudes deduced from the IR reflection fitting are: eyy oo = 3.78 (13=4.8~ ~zz a calculation with the explicitly introduced anisotropy of bond or atomic polarizability along the chain. On the other hand, similar cancellation of the polarizations in two subsequent V-O-V bridges of the same chain in their out-of-phase V~s VOV movement may be treated as a tentative explanation of the absence of significant softening of that mode (irrespective of the phase relation between such modes in two sets of chains). It follows from there considerations that, although the microscopic mechanism of the vasVOV softening is essentially 1ocalized inside the chain, it is created by Q fragment of the chain extending over at least its period of identity which contains two V-O-V bridges.

328

LAZAREV

The next to last column in Table 4.20 presents the results of the D Coul calculation with the charges and polarizabilities of vanadium and oxygen in one of two sets of the anionic chains set to zero, the Na + charges being reduced twice to keep the electroneutrality condition. In this case, the D coul magnitudes are seen to be about one half of the initial ones, their interrelations being approximately the same including the extraordinary strong softening contribution to vasVOV. This numerical experiment substantiates the intra-chain origin of that softening, although it does not restrict its mechanism to any definite piece of the chain. In other words, the microscopic electrostatic field of the lattice acts upon a macroscopic quantity, the frequency of the long-wave optical mode vasVOV (irrespective of the phase relation of these movements in two crystallographic sets of the chains), as it would be affected by an additional negative force constant. The latter can be specified generally as K I+K2•

2 where K I=K2 represent the equal contributions from two sets of chains and

K1,2 is some interaction constant. The above numerical experiment shows that setting K 2 to zero (vanishing of the field created by one set of the chains) reduces the softening effect nearly twice and thus K1,2 is negligibly small. That conclusion follows as well from the similarity of the DCoul magnitudes for any pair of crystal modes differing only by the phase relation between similar internal motions in two sets of the chains. Further, one more numerical experiment can be carried out. Let in the last model, with charged and polarizable atoms only in one of the two sets of chains, the polarizability of one of the non-equivalent oxygen atoms (0 9 and O 10) is set to zero. As is seen from the last column in Table 4.20, none of the calculated D Coul magnitudes change considerably except the ones corresponding to vasVOV. D C~ magnitudes for these two modes, both "n the B u and Ag representations, do decrease drastically thus emphasizing the importance of

LATTICE DYNAMICS COMPUTATIONS

329

the existence of extended chains of polarizable bonds for the origination of such unusually strong electrostatic softening for these types of motions. As has been shown in previous sections, the influence of the microscopic electrostatic field on the bond-stretching vibrations is reproduced in frames of the GVFF model if the interactions of the non-adjacent bonds are taken into consideration. A numerical analysis shows that the frequencies of all 16 V-O stretching modes of sodium vanadate (8 modes of a single chain), including those two vasVOV modes, are successfully reproduced within 20 cm-1 at minor changes of other modes if a sole interaction constant is added to their initial set. It corresponds to the interaction of the two next-neighbor V-O bonds in the chain, d V'O' v o . It should be negative according to the above model consideration. At slightly varied KVO and h VO VO (V) values of the initial set, 3.06 and 0.30 mdyn/A, respectively, the best aV'O' frequency fit is obtained with u VO = -0.3 mdyn/A. It is proven thus that even in a case of tremendously strong (and very selective with respect to the shape of vibration) electrostatic softening of come bond-stretching modes, this effect can be at least formally reproduced in the force constant approach to the description of the force field of a crystal in a space of internal coordinates. Unfortunately, the additionally introduced interaction constants (and varied constants of the initial set) are not deduced directly from the charge distribution and polarizability of the lattice,

E.

Some Further Perspectives

A solution of the problem of long-range Coulomb interaction in a crystal contributing to its force constant matrix specified in a space of internal coordinates was searched above in terms of its influence on the elements of that matrix estimated initially as the corresponding force constants of suitable molecular systems. There exists, however, an approach to the evaluation of the eleotrostatic contribution to the polar optic modes which

330

LAZAREV

interrelates it with the macroscopic polarization induced by those modes. It makes possible, at least in principle, to separate, proceeding directly from the experimentally accessible quantities, a part of the Coulomb contribution which is intrinsic only to the polar modes and to deduce their imaginable frequencies at the excluded contribution of that origin. Those imaginable frequencies were shown to be compatible with the experimental frequencies of the non-polar modes in a sense that their totality would be determined by the interactions of any origin at distances shorter than the elementary translation a and vanishing outside the primitive cell not more slowly than 1/a 5. In a simplest case of a one-mode diatomic cubic crystal such an imaginable frequency is expressed [57] as:

where all quantities in the right-hand sides are experimentally determined. (Note that at sufficiently high e ~176 the coo magnitude may approach col instead of proximity to cot which is conventionally implied.) A problem of the influence of the Coulomb interaction in a lattice on its stability and phonon spectrum was subject to extensive theoretical study in a series of papers [57-59] which treated it in terms of dipole-dipole interactions between point dipoles arising on the atoms shitted from their equilibrium positions (or of changes of those dipoles which do not vanish in equilibrium in a case of pyroelectric crystals). In general, that approach decomposes an additional contribution to the force constant matrix of a crystal which originates from the dipole-dipole interaction in two different parts. The first one is determined by the mutual rearrangement of sub-lattices, each composed by the atom equivalent to translation ( the number of such sub-lattices being equal to the number of atoms in a primitive cell). It contributes to the force constant matrices for both polar and non-polar modes. This part of the dipole-dipole interaction has been shown

LATTICE DYNAMICS COMPUTATIONS

331

to vanish rapidly outside the primitive cell. It can thus be included into the force constants representing the short-range interactions in a lattice of various origins. The second part of the dipole-dipole interaction is determined by the dipole moment variation of a primitive cell which is intrinsic solely to the polar modes. This part of the dipole-dipole interaction can be thus interrelated with the macroscopic polarization of a crystal (IR intensity). It can be calculated for any polar irreducible representation proceeding from the macroscopic symmetry of a crystal (the symmetry of its Bravais lattice) which determines the tensor of inter-cell dipole-dipole interaction, the known e ~176 tensor, and the set of oscillator strengths of all modes belonging to a given representation. It is proposed [60] to treat a set of coo frequencies of polar modes calculated in this way from their experimental cot frequencies and other experimentally accessible quantities jointly with a set of experimental frequencies of non-polar modes as an imaginable totality of the normal modes of a crystal with removed long-range inter-cell interaction, it is possible, probably, to describe this totality of normal modes by means of the force constant matfix whose elements can be transferred from the force constant matrix of a molecule of the size similar to the size of a primitive cell. A procedure of the reinstatement of the shapes (eingenvectors) of the TO polar modes proceeding from the calculation with molecular type force constants shapes of corresponding w o modes is provided [60]. This approach seems to be promising in a more rigorous separation of the contributions to the force field of a crystal composed by charged and polarizable atoms, which originate from the long-range interaction related to the macroscopic polarization mechanism, and of contributions of the short-range origin including those slowly decreasing with distance within the primitive cell. The latter contributions can be assumed to resemble the dipole-dipole contributions to the force constants of chemically related molecules whose importance has been emphasized in Chapter 3.

332

LAZAREV

There exists still no practice of the application of this approach to particular crystals and the conditions of its compatibility with ab initio force constants of suitably designed molecular systems have never been investigated.

F.

Concluding Remarks

An approach to the use of molecular force constants, including those deduced from ab initio quantum mechanical computation, in the calculation of photon spectra and other

dynamical properties of crystals, which has been outlined and exemplified in this chapter, is directed mainly to the application in material science. It is not a unique one and various other theoretical approaches are developed presently for the satae aim. For example, in the related problem of the structure and properties of silicon nitride, even a very elaborate first principle approach found their application [61 ]. On the other hand, the computation of the force constants of chemically related molecules at the SCF level have been applied to that problem as well [62]. In difference of our approach, the ab initio molecular force constants have not been involved into the lattice dynamics computa-

tion directly. Instead, these have been employed in the evaluation of the parameters of a model potential function of the crystal which has been presented as a sum of potential functions corresponding to interactions of various origins. It seems that our approach treated above has some adva,atages, which are: the more important, the more complicated is the structure of a crystal in question, originating just from the direct introduction of suitably selected ab initio molecular force constants into the lattice dynamics problem. Another advantage originates, in the author's opinion, from a wide use of the internal coordinates which both simplifies a transfer of molecular force constants and a joint treatment of the phonon spectra and the properties of a crystal relative to the homogeneous deformation. Being restricted to its treatment as a semi-empirical ap-

LATTICE DYNAMICS COMPUTATIONS

333

proach, which provides for an empirical refinement of some parameters, it can be recommended as a method to design a physically consistent and chemically meaningful initial approximation of the dynamical model of a crystal suitable for the further improvement in extent determined by the amount of experimental data or for the prediction of some properties not easily accessible to investigation.

334

LAZAREV

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AUTHOR

INDEX

Numbers in brackets are reference numbers and indicate that an author's work is referred to although his name is not cited in the text. Underlined numbers give the page on which the complete reference is listed.

----

Benedek, K. G., 100 [32], 184

A-----

Beyerlein, R. A., 160 [115], 188

Abarenkov, I. V., 97 [26], 184

Billy, S., 160 [115], 188

Agranovich, V. M., 166 [123], 188 Almenningen, A., 20 [28], 79

Bilz, H., 94 [18, 22], 100 [32], 101 [39], 108 [49], 183, 184, 185

Altona, C., 5 [9], 7 [9], 19 [24], 78, 79

Bin, Xu, 302 [46, 47], 305 [46, 47], 336

Andreath, P., 255 [19], 334

Binkley, J. S., 10 [19], 11 [19, 22], 56 [55], 78, 79, 80

Appelman, D. E., 210 [33], 245

Birman, J. L., 121 [59], 185

Armbruster, A., 320 [55], 336

Blom, C. E., 5 [9], 7 [9], 19 [24], 78, 79

Axe, J. D., 144 [100], 187

Blukis, U., 20 [27], 79 Bochtarev, V. A., 139 [92, 93], 187

B Balkansky, M., [68], 186

Boggs, J. E., 5 [8], 9 [8], 10 [8], 11, 13 [20], 19 [20], 36 [20], 78

125 [68], 126 [68], 127

Boonstra, L. H., 214 [41 ], 223 [41 ], 224 [41 ], 246

Baraton, M. I., 128 [79], 142 [79], 160 [79], 268 [29], 186, 335

Born, M., 84, 88 [1], 88 [1], 89 [1], 91 [1], 92 [1], 143, 164, 183

Barron, T. H. K., 122 [64], 169 [125], 255, 263, .185, 188, 335 Barsch, G.R., 122 [63], 125 [63], 185

Boyer, L. L., 100 [34], 110 [34], 120 [57, 58], 171,213 [37], 184, 185, 245

Bastiansen, B., 20 [28], 79

Bradley, C.A., 163 [ 116], 188

Bates, I. B., 255 [24], 263 [24], 335

Brand, H., 35 [39], 79

Batsanov, S. S., 324 [56], 336

Brown, G. E., 68 [63], 81

Baur, W. H., 62, 80

Bruesh, P., 94 [19], 184

Beagley, B., 20 [29], 223 [29], 79

Buchanan, M., 101 [39], 184

Bechmann, R., 255, 335

Bukowinski, M. S. T., 265, 335

Belousov, M. V., 135 [83, 85], 186

Bulanin, M. O., 46 [53], 49 [53], 64 [53], 123 [67], 133 [67], 136 [67], 138 [67], 140 [67], 203 [17], 206 [28], 208 [29], 210

Belov, N. V., 61, 80 337

338

AUTHOR INDEX

[32], 280 [40], 283 [40], 284 [40], 285 [40], 294 [40], 304 [40], 80, 186, 244, 245, 335

Crawford, B., 164 [119], 188

Bums, G., 148 [109], 188

Cummins, H.Z., 166 [123], 188

Busing, W. R., 95 [24], 169 [127], 174 [134], 184, 188, 189

Curtiss, L. A., 35 [39], 79

Cross, P. C., 128 [70], 186

Bussmann-Halder, A., 100 [32] 184

Cuseo, R., 280 [41], 285 [41], 303 [41], 304 [41], 335

Butikova, I. K., 72 [67], 81

Cyvin, S.J., 128 [72], 186

------- C ~----

Califano, S., 4 [5], 8 [5], 46 [5], 128 [71], 129 [71], 145 [71], 161 [71], 164, 78, 186 Camassel, J., 280 [39, 41], 282 [39], 285 [41], 303 [39, 41], 304 [39, 41], 335 Cartz, L., 160 [115], 267 [28], 268 [28], 188,335 Catlow, C. R.A., 111 [52], 173 [133], 192 [2], 185, 189, 244 Catti, M., 170, 188 Chalvet, O., 28 [37], 50 [37], 79 Chandrasekhar, J., 38 [45], 80 Chaves, A., 320 [54], 336 Chisler, A.E., 139 [88], 313 [50, 51, 52], 318 [52, 53], 320 [52], 187, 336

Dandel, R., 28 [37], 50 [37], 79 De Almedia, W. B., 229 [47, 48], 246 Dean, K. J., 255, 335 Deb, B. M., 3 [3], 78 Decius, J. C., 128 [70], 204 [18, 19], 186, 245 Defrees, D. J., 11 [22], 79 Dent Glasser, L. S., 72, 81 Dewar, M. J. S., 65 [62], 80 Dick, B. G., 98 [29], 101, 184 Diner, S., 28 [37], 50 [37], 79 Dolin, S. P., 65 [61], 248 [13], 80, 334

Clark, J. R., 210 [33], 245

Domeng~s, B., 287 [45], 336

Clark, T., 38 [45], 80,

Dovesi, R., 229 [52], 246

Claus, R., 135 [84], 186

Dultz, W., 279 [34], 335

Clementi, E., 36, 80

Dunning, T. H., 10 [17], 78

Cochran, W., 84 [5, 12], 88 [12], 94 [20], 106 [48], 192 [5], 183, 184, 185,244

Durig, J. R., 9 [15], 227 [45], 78, 246

Cohen, M. H., 196 [10], 244 Cohen, M. L., 332 [61], 336 Cohen, R. E., 184, 245 Cowley, R. A., [122], 183, 188

100, 110 [34], 213 [37], 84 [12], 88 [12], 166

E Ehrenfreud, E., 192 [7], 200 [7], 244 Elcombe, M. M., 121,269, 270 [31], 271 [31], 185,335 Eljashevich, M.A., 128, 186

AUTHOR INDEX

Etchepare, J, 248, 260, 267 [8], 280 [8], 334 Ewbank, M. D., 192 [7], 200 [7], 244 Ewing, W., 20 [28], 79

~ Falter,

F------

C., 192 [9], 244

Farmer, V. C., 139 [94], 142 [94], 187 Feldkamp, L.A., 102 [46], 185

339

Gliss, B., 94 [22], 184 Goddard III, W. A., 332 [62], 336 Gordon, M. S., 10 [19], 11 [19, 22], 78, 79 Gordon, R. G., 100 [31], 110, 111 [31], 184, 185 Gormack, A.N., 173 [133], 189 Gorre, G., 121 [61], 185 Goullet, A., 280 [39, 41], 282 [39], 285 [41], 303 [39, 41], 304 [39, 41], 335

Felsche, J., 72 [65, 66, 68], 74 [68], 81

Grigor'eva, L. Z., 128 [80], 142 [80], 166 [80], 170 [80], 174 [80], 186

Fillips, J. C., 100 [35], 184

Grzegorzevsky, O.A., 156 [ 112, 113], 188

Finger, C. W., 255 [18], 258 [18], 334

Guissoni, M., 164 [120], 188

Fisher, K., 101 [39], 108 [49], 184, 185 Flerova, S.A., 156 [ 112, 113], 188 Fletcher, R., 4 [4], 78 Fogarasi, G., 5 [8], 6 [13], 7 [13], 9 [8, 15], 10 [8], 227 [45], 78, 246

Haberkom, R., 101 [39], 108 [49], 184, 185 nandke, M., 42 [52], 43, 44 [52], 80

Francl, M.M., 11 [22], 79

Hanke, W., 94 [22], 184

Frech, R., 204 [19], 245

Hardy, J. R., 94 [17], 97 [17], 98, 120 [57], 192 [1 ], 183, 184, 185,244

Freeman, C.M., 111 [52], 185 Freeman, J. M., 20 [29], 223 [29], 79

Hargittai, I., 214 [41], 223 [41], 224 [41], 246 Harrison, W.A., 192 [6,7], 200 [6,7], 244

------

G

--~-

Geisser, Ch., 192 [8], 244 Gerding, H., 214 [42], 246

Hazen, R. M., 157 [114], 255, 258 [18], 188,334 Hedberg, K., 20 [28], 79

Gerrat, J., 5 [6], 78

Hehre, W. J., 10 [19], 11 [19, 22], 56 [55], 78,79,80

Gervais, F., 255, 334

Hemley, R. J., 255 [18], 258 [18], 334

Ghosh, P. M., 204 [20], 245

Henry, N. F. M., 177 [135], 189

Gibbs, G. V., 27, 68 [63], 208 [31], 229 [46, 49, 50, 53], 248 [10, 12] 267 [10], 287 [45], 79, 81,245, 246, 334, 336

Herman, F., 94 [23], 184

Glidewell, C., 20 [29], 223 [29], 79

Hess, A. C., 213 [40], 249, 251,260, 265, 286 [16], 246, 334

340

Hipps, K. W., 6 [12], 78 Hirano, H., 147 [ 104], 187 Hohenberg, P., 111 [53], 112, 185 Hong, Wang, 302 [46, 47], 305 [46, 47], 336 Horton, G. K., 84 [6, 7, 8], 88 [13], 94 [17, 21, 22], 97 [17], 106, 114 [21], 192 [1], 196 [11], 183, 184,244 Hu, Li, 302 [47], 305 [47], 336 Huang, C. C., 122 [64], 185 nuang, K., 84, 88 [1], 88 [1], 89 [1], 91 [1], 92 [1], 143, 164, 255 [24], 263 [24], 183,335 Hiibner, K., 192 [8], 244 Husinaga, S., 10 [16], 78 Huttner, G., 72 [68], 74 [68], 81

AUTHOR INDEX

~

J

~

Jackson, M.D., 100 [31], 111 [31], 184 Jamieson, P. B., 72, 81 Jeffery, J. W., 72 [69], 81 Jianru, Han, 302 [46, 47], 305 [46, 47], 336 Jorgensen, J.D., 160 [115], 267 [28], 268 [28], 188, 335 Jurchenko, E.N., 187

-------

K

147 [ 102], 149 [102],

~

Kaplan, P., 248 [7], 334 Kasai, P. H., 20 [27], 79 Kieffer, S. W., 157 [114], 182 [136], 188, 189 Kim, Y. S., 110, 185

Ignatyev, I. S., 6 [11], 7 [11], 13 [11], 19 [11, 25], 20 [11, 30, 31], 21 [11], 22 [11], 32 [11, 25, 38], 36 [42, 43], 38 [43], 42 [50], 44 [43], 46 [38, 50], 47, 50 [43], 57 [43], 64 [50], 128 [76], 130 [76], 133 [76, 82], 135 [76], 140 [76], 148 [76], 204 [27], 214 [43, 44], 216 [43], 218 [44], 224 [44], 225 [44], 226 [44], 385 [43], 78, 79, 80, 186, 245,246, 335

Klanjsek, M., 149 [110], 188

Iishi, K., 122 [65], 138 [87], 211 [34, 35, 36], 248, 185, 186, 245, 334

Komomicki, A., 11 [21], 173, 196 [12], 78, 189, 244

Ikeda, T., 147 [103], 156 [103], 187

Koput, J., 19 [26], 33 [26], 35 [26], 79

Iler, R. K., 35 [40], 79

Korol, E.N., 102 [43], 184

Imazu, I., 147 [103], 156 [103], 187

Krasnjansky, G. E., 204 [25, 26], 245

Ipatova, I. P., 84 [4], 105 [4], 183

Krebs, B., 42 [51], 44 [51], 80

Ipenburg, L. W., 214 [42], 246

Kress, W., 94 [18], 183

Isobe, M., 307 [48], 336

Kudzin, A. Ju., 156 [ 112, 113], 188

Iton, L. E., 35 [39], 79

Kunc, K., 109 [50], 125 [68], 126 [68], 127 [68], 185, 186

Iwai, S., 307 [48], 336

Kleinman, D. A., 140 [95], 187, 248, 254, 255,334 Kohn, W., 111 [53, 54], 112, 185 Kolesova, V. A., 148 [108], 204 [27], 187, 245

Kvyatkovskii, O. E., 330 [57, 58, 59], 331 [60], 336

AUTHOR INDEX

L

Lasaga, A. C., 229 [53], 248 [12], 246, 334 Lax, M., 100 [30], 184 Lazarev, A. N., 27 [36], 32 [38], 35 [41], 36 [42, 43], 38 [43], 42 [41, 50], 44 [43], 46 [38, 50, 53], 49 [53], 50 [43], 57 [43], 61 [56, 57, 65], 64 [50, 53, 56], 65, 84 [9, 10, 11], 116 [9, 10, 55, 56], 119 [56], 120 [9], 123 [9, 10, 56, 66, 67], 124 [9, 10, 56], 125 [9, 10, 11, 56], 128 [56, 76], 129 [81], 130 [76], 133 [67, 76, 82], 135 [76, 86], 136 [67], 138 [67, 86], 139 [88, 89, 94], 140 [9, 66, 67, 76], 142 [56, 81, 94, 97, 98], 144 [56], 146 [56], 147 [106], 148 [76, 107, 108], 166 [124], 170 [124], 174 [97, 174], 197 [13], 199 [14, 15], 200 [14, 15], 201 [16], 202 [14, 16], 203 [16, 17], 204 [16, 21, 22, 23, 24, 27], 206 [21, 28], 208 [29], 210 [13,14, 32], 211 [13], 213 [38, 39], 214 [43], 216 [43], 242 [39] 248 [3, 13], 249, 255 [15, 17], 259 [17], 260 [5, 15], 264 [17], 272 [15], 276, 277 [15], 280 [40, 42], 283 [40], 284 [40], 285 [40, 43], 294 [40], 303 [42], 304 [40, 42], 306 [17], 308 [49], 313 [50, 51, 52], 318 [52, 53], 320 [52], 79, 80, 183, 185, 186, 187, 188, .244, 245,246, .334, 335, 336

341

-----

M

-----

Mackrodt, W. C., 192 [2], 244 Majenov, N. A., 204 [21, 22, 23], 206 28], 280 [40, 42], 283 [40], 284 [40], [40, 44], 288 [44], 289 [44], 291 [44], [40], 295 [44], 301 [44], 303 [42], 304 42], 245, 335

[21, 285 294 [40,

Makerenko, I. P., 248 [3], 334 Maksimov, V. G., 330 [58], 336 Malik, K. M. A., 72 [69], 81 Malrieu, J. P., 28, 50 [37], 79 Mao, H. K., 255 [18], 258 [18], 334 Maradudin, A. A., 84 [4, 6, 7, 8], 88 [13], 94 [17, 21, 22], 97 [17], 105 [4], 106, 114 [21], 166 [123], 192 [1], 196 [11], 183, 184, 188, 244 Martin, R. M., 91 [16], 93, 100 [36], 109 [50], 196 [10], 183, 184, 185, 244 Marumo, F., 307 [48], 336 Mathieu, J. P., 89 [14], 183

LeCalve, N., 149 [110], 188

Matsui, M., 95 [24], 169 [127] 184, 188

Lehmann, G., 279 [34], 335

Matsumura, S., 147 [104], 187

Leibfried, G., 84 [2, 3], 158 [2], 169 [2, 3], 183

Mclntyre, 72 [65, 66], 81

Levanyuk, A. P., 166 [ 123], 188 Levien, L., 267, 335 Liebau, F., 13 [23], 29 [23], 79 Liu, A. Y., 332 [61 ], 336 Lonsdale, K., 177 [135], 189 Ludwig, W., 84 [3], 169 [3], 183 Lundqvist, S. O., 101, 184 Lurio, A., 148 [109], 188

Mclver, J. W., 173, 196 [12], 189, 244 McMillan, P. F., 213 [40], 229 [51], 248 [11], 249, 251 [11, 16], 260, 262, 265, 267 [11], 286 [16], 246, 334 McSkimin, H. J., 255, 334 Meagher, E. P., 27 [33], 79 Mehl, M.J., 100 [34], 110 [34], 213 [37], !84, 245 Merian, M., 248 [6, 7, 8], 260 [6], 267 [8], 280 [8], .334 Merten, L., 135 [84], 186 Meyer, W., 8 [14], 78

342

AUTHOR INDEX

Mijlhoff, F. C., 214 [41], 223 [41 ], 224 [41 ], 246

Nye, J. F., 89 [15], 183

Miller, P. B., 144 [100], 187 Mills, I. M., 5 [6], 53 [54], 78, 80, 164 [121], 188 Mirgorodsky, A. P., 27, 42 [50], 46 [50], 61 [56, 57], 64 [50, 56], 84 [9, 10], 116 [9, 10, 56], 119 [56], 120 [9], 123 [9, 10, 56, 66], 124 [9, 10, 56], 125 [9, 10, 56], 128 [56, 76, 78, 79, 80], 130 [76], 133 [76], 135 [76, 86], 138 [86], 139 [89], 140 [9, 66, 76], 142 [56, 78, 79, 80, 98], 143 [78], 144 [56], 145 [78], 146 [56], 148 [76, 107, 108], 149 [111], 157 [78], 160 [79], 166 [80], 168, 170 [80, 128], 174, 197,199114, 15], 200 [14, 15], 201 [16], 202 [14, 16], 203 [16], 204 [16, 21, 22, 24], 206 [21, 28], 208 [29, 30], 210 [13,14, 32], 211 [13], 248, 249, 255 [15, 17], 259 [17], 260 [5,15], 264 [17], 268, 272 [15], 276, 277 [15], 279, 280 [40, 42], 283 [40], 284 [40], 285 [40, 43], 294 [40], 303 [42], 304 [17, 40, 42], 313 [50], 79, 80, 183, 185, 186, 187, 188, 244, 245, 334, 335, 336 Miyazawa, T., 128 [75], 130 [75], 144 [101], 186, 187 Moiseenko, V., 149 [110], 188 Montroll, E. W., 84 [4], 105 [4], 183 Muller, A., 42 [51 ], 44 [51 ], 80 Murtagh, A., 173, 189

Oberhammer, H., 11, 13 [20], 19 [20], 36 [20], 78 O'Keeffe, M., 27 [32], 213 [40],229 [50, 51], 248 [10, 11], 251 [11, 161, 262, 267 [10, 11], 286 [16], 287 [45], 79, 246, 334, 336 O'Malley, P. J., 229 [47, 48], 246 Orel, B., 149 [110], 188 Overend, J., 164 [119], 188 Overhauser, A. W., 98 [29], 101, 184

~

p

~ _ _ _

Pang, F., 5 [8], 9 [8], 10 [8], 78 Papike, J. J., 210 [33], 245 Parker, S. C., 173, 189 Pascual, J., 280 [39, 41], 282 [39], 285 [41], 303 [39, 41], 304 [39, 41], 335 Pastemak, A., [24], 185,335

122 [64], 255 [24], 263

Pauling, L., 61, 80 ------

N

-------

Pavinich, V. F., 135 [83, 85, 86], 138 [86], 139 [88, 92], 210 [32], 186, 187, 245

Nakagawa, I., 121 [60], 185

Person, W. B., 128 [73], 186

Navrotsky, A., 27 [32], 157 [114], 182 [136], 79, 188, 189

Pick, R. M., 196 [ 10], 244

Newman, P.R., 192 [7], 200 [7], 244 Newton, M. D., 27, 229 [49, 50], 248 [10] 267 [10], 79, 246, 334

Pietro, W.J., 10 [19], 11 [19, 22], 78, 79 Piriou, B., 210 [32], 245,255, 334 Pisani, C., 229 [52], 246

Nicholas, J. B., 35 [39], 79

Pletcher, R., 171 [ 130], 189

Nlyers, R. J., 20 [27], 79

Pople, J.A., 10 [19], 11 [19, 22], 56 [55],

Novak, A., 149 [110], 188 Nusimovici, M. A., 121 [59, 61], 125 [68], 126 [68], 127 [68], 185, 186

78,79,80

Porto, S. P. S., 320 [54], 336 Poshusta, R. D., 6 [12], 78

AUTHOR INDEX

343

Poulet, H., 89 [14], 18___33

Schwendeman, R. H., 5 [10], 78

Prewitt, C. T., 267 [27], 335

Scott, J. F., 279 [34, 36, 37], 280 [36], 282 [36], 284, 294, 335

Prima, A. M., 128, 130 [74], 186 Pulay, P., 2, 3[2, 3], 5 [7, 8], 6 [13], 7 [13], 9 [8, 15], 10 [8], 11,227 [45], 78, 246

Q Quiliehini, M., 279 [34], 335 Quintard, P., 128 [79], 142 [79], 160 [79], 268 [29], 186, 335 Quist, A. S., 255 [24], 263 [24], 335

- ~ -

R-----

Sham, L. J., 94 [21], 106, 111 [54], 114 [21], 184, 185, Shehegolev, B. F., 36 [42, 43], 38 [43], 44 [43], 50 [43], 57, 65 [61], 197, 210 [13], 211 [13], 214 [43], 216 [43], 248 [13], 285 [44], 288 [44], 289 [44], 291 [44], 295 [44], 301 [44], 313 [50], 80, 244, 246, 334, 335,336 Sheldrick, G. M., 20 [29], 223 [29], 7._99 Shepelev, Ju. I., 72 [67], 8__!1 Sherman, W. F., 255 [22], 335 Shimanouchi, T., 128 [75], 130 [75], 164 [118], 186, 188

Rankin, D. W. H., 20 [29], 223 [29], 79

Shiro, Y., 128 [77], 143, 144 [101], 168, 248, 186, 187, 334

Renes, G., 214 [41], 223 [41], 224 [41], 2..46

Shulz, M. M., 204 [24], 245

Ribbe, P. H., 208 [31 ], 245

Shuxia, Shan, 302 [47], 305 [47], 336

Robiette, A. G., 20 [29], 223 [29], 79

Sidah, H. A. A., 302 [46], 305 [46], 336

Robinson, K., 208 [31], 245

Siebert, H., 42, 43 [48], 44 [49], 80

Roetti, C., 36, 229 [52], 80, 246

Singh, R.K., 102 [42], 184

Royle, R. L., 111 [52], 185

Smetankine, L., 248 [6], 260 [6], 334

Rustagi, K. C., 101 [38], 184

Smimov, M. B., 27, 36 [43], 38 [43], 44 [43], 50 [43], 57 [43], 61 [56, 57], 64 [56], 84 [9, 10], 116 [9, 10, 55, 56], 119 [56], 120, 123 [9, 10, 56], 124 [9, 10, 56], 125 [9, 10, 56], 128 [56, 78, 80], 129 [81], 139 [93], 140 [9], 142 [56, 78, 80, 81, 96, 97], 143 [78], 144 [56], 145 [78], 146 [56], 147 [102], 149 [102], 157 [78], 164 [117], 166 [80], 168, 170 [80, 128], 174, 199 [14, 15], 200 [14, 15], 201 [16], 202 [14, 16], 203 [16], 204 [16], 208 [29], 210 [14], 213 [38], 248 [13], 255,259 [17], 264 [17], 269 [30], 271, 274 [33], 280, 285 [44], 288 [44], 289 [44], 291 [44], 295 [44], 301 [44], 306 [17], 313 [52], 318 [52, 53], 320 [52], 79, 80, ..183, 185, 186, 187, 188, 244, 245,246, 334, 335, 336

Ryjikov, V. A., 147 [102], 149 [102, 111], 204 [27], 187, 188, 245

S Sargent, W.H., 173, 189 Sauder, G. A., 302 [46], 305 [46], 336 Sauer, J., 41 [46], 80 Schmid, R. L., 72 [65, 66, 68], 74 [68], 81 Schr/Sder, U., 100 [33], 184 Schwarzenbach, D., 279 [35], 335

Smolin, Ju. I., 72 [67], 81

344

AUTHOR INDEX

Soi, J., 248, 334 Solntzeva, L. S., 148 [108], 187

~--U~ Urey, H. C., 163 [116], 188

Spelbos, A., 214 [41 ], 223 [41 ], 224 [41 ], 246 Spitzer, W. G., 140 [95], 248, 254, 255, 187, 334 Sputznagel, G. W., 38 [45], 80 Srinivasa, S. R., 160 [115], 188 Stengel, M. O., 139 [91 ], 187

Veillard, A., 10 [18], 78 Verlan, E. M., 204 [25], 245 Verma, M. P., 102 [42], 184 Vodopjanova, V. P., 139 [88, 89, 90], 187

Stepanov, B. I., 128 [69, 74], 130 [74], 186

Volkenstein, M.V., 128 [69], 186

Stixrude, L., 267, 335

V611enkle, H., 147 [105], 187

Striefler, M.E., 122 [63], 125 [63], 185

Volmyanskii, M., 149 [110], 188

Swanson, D. K., 27 [33], 79

Volnjansky, M. D., 156 [ 112, 113], 188

Szigeti, B., 106 [47], 192 [3], 185,244

~ W ~ - T Tagantsev, A. K., 96 [5], 184 Tarte, P., 42 [47], 43 [47], 44 [47], 80

Wallis, R. F., 100 [30],169 [125], 184, 188

Weber, W., 100 [37], 101 [38], 108 [49], 184, 185

Tenisheva, T. F., 6 [11], 7 [11], 13 [11], 19 [11], 20 [11], 21 [11], 22 [11], 32 [11, 38], 46 [38], 47, 133 [82], 214 [43, 44], 216 [43], 218 [44], 224 [44], 225 [44], 226 [44], 78, 79, 186, 246

Weidner, D. J., 267 [27], 335

Thiel, W., 65 [62], 80

Wemer, H.-J., 8 [ 14], 78

Thurston, R. N., 255 [ 19], 334

Wilkinson, G. R., 255 [22], 335

Tolpygo, K. B., 98, 102, 106 [27], 192 [4], 184, 185,244

Wilson, E. B., 128, 186

Weiss, G. H., 84 [4], 105 [4], 183 Wendel, J. A., 332 [62], 336

Worlton, T. G., 160 [115], 188

Tomisaka, T., 138 [87], 186 Traetteberg, M., 20 [28], 79 Tsjashchenko, Ju. P., 204 [25, 26], 245 Tsuboi, M., 128 [75], 130 [75], 186 Tupizin, Ju. F., 97 [26], 184

~X-~Xiling, Liu, 302 [47], 305 [47], 336

AUTHOR INDEX

-----

Z-----

Zallen, R., 169 [126], 188 Zerbi, G., 128 [73],164 [120], 186, 188 Zulumjan, N. O., 135 [86], 138 [86], 204 [24], 210 [32], 186, 245

345

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SUBJECT INDEX

-----

A

-----

---~-

D

~---

Davydov splitting, 203,208

tx-berlinite, A1PO4, 279, 284, 295,296, 303

deformable dipole model, 98 a-quartz, 123,258, 273 deformable shell, 100 ct-spodumene, LiAISi20 6, 307

209, 211, dielectric properties, 84

AI(OH)3, 285

disilicic acid molecule, 229

AIPO4, tx-berlinite, 279, 284, 295,296, 303

disiloxane, 198, 232 dynamic charge, 196

aluminum phosphate, 279, 293

dynamical theory of crystal lattice, 84

atom-atom potential, 96 E

atomic wave functions, 10 ------

B

effective bond dipole, 51

~

effective charge tensor, 106

fl-quartz, 249

elastic constants, 154, 264, 305

breathing shell, 100

elastic properties, 84

C

electric response function, 103

Cartesian forces, 4 ~

correlation field, 204

F

~

-

factor-group splitting, 203

Coulomb interaction, 96, 97, 111, 123, 329

ferroelastic transition, 170

crystal mechanics program, 174

force constant determination, 5 force constant evaluation, 224 347

348

SUBJECT INDEX

force constants, 202 - - - -

H

lithium metagermanate, Li2GeO 3, 147 long-range Coulomb interaction, 271

- - - - -

(HO)3SiOSi(OH)3, 234

-~-

M----

molecular force constants, 248

H3PO4, 285

111, 116,

H3SiOCH 3, 218, 222 N

H3SiOH, 16, 18 Na2B(OH)4CI, teepleite, 204 H3SiONa, 16 Na2V20 6, 308 H3SiOSiH 3, 16, 21, 23 ~ - -

O

- - - -

H3SiOX type, 12 (OH)3SiOSi(OH)3, 231 H3XOXH 3, 18

~ p ~ _ _

H4SiO 4, 251

paratellurite, TeO 2, 170

H6Si20 7, 233, 251

phonon frequency dispersion, 269 piezoelectric effect, 127

internal coordinates, 129, 161 PO4 tetrahedra, 295 inverse vibrational problem, 157 point ion concept, 192 ionic-covalent crystals, 114 polarizable ion model (PIM), 97 IR intensities, 7, 8, 134, 139, 260, 326 potential function refinement, 171

L protonated silicate ions, 64 lattice stability, 128

Q_____

lattice strain, 141 quantum mechanical computation, 55, 218

Li2GeO 3, 147 LiAISi20 6, tx-spodumene, 307

209, 211,

SUBJECT INDEX

-------

R

~

rigid ion model (RIM), 96

349

tetramethylsilane molecule, Si(CH3) 4, 126 thermal expansion, 170

scaling factors, 6 Si2N20, silicon oxynitride, 159 Si(CH3)4, tetramethylsilane molecule, 126 Si(OCH3) 4, 214, 216, 219 Si(OH)4, 66 silicon carbide, 121, 126 silicon dioxide, 121, 192 silicon oxynitride, Si2N20, 159 silicon-oxygen tetrahedron, 48, 213 Si-O bond, 12 SiO 4- oxyanion, 35, 36, 44, 58, 66 Si-O-Si bridge, 32 sodium vanadate, 307, 309, 314

~

T

~

teepleite, Na2B(OH)4CI, 204 TeO2, paratellurite, 170 tetrahedral AB 4 molecular system, 53 tetramethoxysilane, 213, 261

thortveitite, 139 ~

Z

~

-

zircon crystal, ZrSiO4, 206 ZrSiO4, zircon crystal, 206

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