Local Density Theory of POLARIZABILITY PHYSICS OF SOLIDS AND LIQUIDS Editorial Board: Jozef T. Devreese • University of Antwerp, Belgium Roger P. Evrard • University of Liege, Belgium Stig Lundqvist • Chalmers University of Technology, Sweden Gerald D. Mahan • University of Tennessee, USA Norman H. March • University of Oxford, England AMORPHOUS SOLIDS AND THE LIQUID STATE Edited by Norman H. March, Robert A. Street, and Mario P. Tosi CHEMICAL BONDS OUTSIDE METAL SURFACES Norman H. March CRYSTALLINE SEMICONDUCTING MATERIALS AND DEVICES Edited by Paul N. Butcher, Norman H. March, and Mario P. Tosi ELECTRON SPECTROSCOPY OF CRYSTALS V. V. Nemoshkalenko and V. G. Aleshin FRACTALS Jens Feder INTERACTION OF ATOMS AND MOLECULES WITH SOLID SURFACES Edited by V. Bortolani, N.H. March, and M.P. Tosi LOCAL DENSITY THEORY OF POLARIZABILITY Gerald D. Mahan and K. R. Subbaswamy MANY -PARTICLE PHYSICS, Second Edition Gerald D. Mahan ORDER AND CHAOS IN NONLINEAR PHYSICAL SYSTEMS Edited by Stig Lundqvist, Norman H. March, and Mario P. Tosi THE PHYSICS OF ACTINIDE COMPOUNDS Paul Erdos and John M. Robinson POLYMERS, LIQUID CRYSTALS, AND LOW-DIMENSIONAL SOLIDS Edited by Norman H. March and Mario P. Tosi THEORY OF THE INHOMOGENEOUS ELECTRON GAS Edited by Stig Lundqvist and Norman H. March A Continuation Order Plan is available for this series. A continuation order will bring delivery of each new volume immediately upon publication. Volumes are billed only upon actual ship ment. For further information please contact the publisher. Local Density Theory of POLARIZABI LITY Gerald D. Mahan University of Tennessee Knoxville, Tennessee and K. R. Subbaswamy University of Kentucky Lexington, Kentucky SPRINGER SCIENCE+BUSINESS MEDIA, LLC Llbrary of Congress Cataloglng-In-Publlcatlon Data Mahan, Gerald D. Local denslty theory of polarizablllty I Gerald D. Mahan and K.R. Subbaswamy. p. cm. -- (Physlcs of sollds and liqulds) Includes blbllographlcal references and Index, ISBN 978-1-4899-2488-9 ISBN 978-1-4899-2486-5 (eBook) DOI 10.1007/978-1-4899-2486-5 ,. Polarlzabl11ty (Electrlclty) 2. Solid state physlCS. 3. Solid state chemlstry. 4. Ionic crystals. 1. Subbaswamy, K. R. II. Tltle. III. Serles. QCI73.4.P64M34 1990 530.4·1--dc20 90-42552 CIP © 1990 Springer Science+Business Media New York Originally published by Plenum Press, New York in 1990 Softcover reprint oftbe hardcover lst edition 1990 AII rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording, or otherwise, without written permission from the Publisher Preface During the past decade the theoretical physics community has learned how to evaluate accurately polarizabilities and susceptibilities for many-electron systems such as atoms, solids, and liquids. The most accurate numerical technique employs a method often called the Time-Dependent Local Density Approximation, which is abbreviated TDLDA. The present volume is a review of recent research on the theory of po larizabilities and susceptibilities. Both authors have been doing these cal culations. However, this review surveys the entire field, summarizing the research of many contributors. The application of an external field, either ac or de, will induce a dipole moment which can be calculated and compared with experiment. For mod erately strong fields, both linear and nonlinear processes contribute to the moment. We cover topics such as polarizability, hyperpolarizability, pho toionization, phonons, and piezoelectricity. Density functional theory in the Local Density Approximation (LDA) has been shown to be a very accurate method for calculating ground state prop erties of electronic system. For static external fields, the induced moments are properties of the ground state. Then the calculation of the polarizability · is very accurate. For ac fields, the moment is not part of the ground state. However, the TDLDA methods are still very accurate. The book is divided into three parts. The first two chapters briefl.y discuss the LDA formalism: its history, development, and present status. Chapters 3 and 4 discuss the method of calculating linear and nonlinear polarizabilities for atoms. Most of the applications are for atoms and ions with closed shells. The last two chapters discuss the extension to soli~s, liquids, and surfaces. The latter subjects comprise most of the recent applications and research. One unique feature of the book is the discussion of numerical techniques. The formulas are first derived algebraically. Then the numerical steps for evaluating the formulas are discussed in detail. Our goal is to make the v vi PREFACE method accessible to advanced students and researchers. Enough detail is presented to enable them to write their own computer codes and to use the techniques to calculate accurate numbers for real systems. Historically two different, but equivalent numerical methods were developed for the polariz ability: the Green's function method and the modified Sternheimer method. We present both techniques. We have done both, and find they are numeri cally similar, in both the programing time and the CPU time. We thank Plenum Publishing Corporation for letting us write the manu script in I~.TEX, which permitted us to have control over the appearance and format of the final copy. One of us (KRS) is grateful to Drs. Gaetano Sena tore and Michael Johnson for fruitful collaboration on the subject matter of this book. We also thank our families for their patience and understanding during the long winter evenings of writing and rewriting. Jerry and Swamy Contents 1 Introduction 1 2 Formalism 9 2.1 Density Functional Theory 9 2.2 Local Density Approximation . 13 2.3 Linear and Nonlinear Response 18 2.3.1 Response Functions 19 2.3.2 Perturbation Theory .. 21 2.3.3 Static Perturbations .. 26 2.3.4 Self-Consistent Screening 29 2.4 Time-Dependent Potentials in LDA. 31 3 Computational Techniques 37 3.1 Exchange-Correlation Potentials 37 3.2 Kohn-Sham Eigenfunctions ... 45 3.3 Modified Sternheimer Approach . 52 3.4 Green's Function Method . 64 3.5 Self-Interaction Corrections 68 4 Atoms and Molecules 77 4.1 Linear Polarizability ................. . 77 4.1.1 Atoms and Positive Ions ........... . 81 4.1.2 Self-Interaction Correction and Negative Ions 88 4.1.3 Frequency Dependence . 92 4.1.4 Molecules .... 92 4.2 Photoionization . . . . . . . 93 4.2.1 Outgoing Waves .. 95 4.2.2 Numerical Methods 100 Vll viii CONTENTS 4.2.3 Numerical Results ........ . 103 4.2.4 Self-Interaction Correction .. 107 4.3 a(iu) : van der Waals Forces .. 109 4.3.1 a( iu) . . . . . . ..... 109 4.3.2 van der Waals Interaction 112 4.3.3 Axilrod-Teller Interaction 116 4.3.4 Multiatom Interactions 118 4.4 Hyperpolarizability .... . 118 4.4.1 Closed-Shell Atoms .. . 120 4.4.2 Negative Ions ..... . 126 4.4.3 Dipole-Quadrupole Susceptibility . 127 4.4.4 Frequency Dependence . . . . . . . 129 5 Ionic Solids 131 5.1 Spherical Solid Model 131 5.1.1 Metals . . . . 132 5.1.2 Ionic Solids .. 136 5.2 Dipole Polarizability . 143 5.3 Volume Dependence of Polarizability 150 5.4 Direct Phonon Calculations 157 5.4.1 Introduction . . .. 157 5.4.2 Frozen Phonons . . . 157 5.4.3 Sternheimer Method 159 5.4.4 Unified Molecular Dynamics. 160 5.5 Phonon Force Constants . 161 5.5.1 Introduction ...... . 161 5.5.2 Dynamical Matrix . . . . 162 5.5.3 Polarizable Ion Model . . 163 5.5.4 Deformation-Dipole Model 166 5.5.5 Indirect Ionic Interactions . . 174 5.6 Hyperpolarizability of Alkali Halides 182 5.6.1 Spherical Solid Model 184 5.6.2 Cubic Anisotropy .... 187 5.6.3 Diagonal Component. 192 5.6.4 Anisotropy Ratio . . 195 5.7 Raman Scattering Intensity . 203 5.7.1 Local Field ..... . 205 5.7.2 Field-Induced First-Order Scattering . 207 5.7.3 Second-Order Raman Scattering ... 211 CONTENTS ix 6 Other Solids and Simple Liquids 215 6.1 Semiconductors . . . 215 6.2 Metal Surfaces . . . 219 6.2.1 Static Fields 219 6.2.2 ac Fields . . . 222 6.2.3 Surface van der Waals Interaction 225 6.3 Small Metal Particles . 226 6.4 Rare Gas Solids . 229 6.5 Simple Liquids 230 Bibliography 237 Index 251