Local Density Approximations in Quantum Chemistry and Solid State Physics

848 Pages·1984·27.04 MB·English

by  Vedene H. Smith Jr. (auth.)

local Oensiw Approximations
in Ouantum
Chemist~ an~
State
Soli~ P~ysics
local Approximations
Densi~
in Ouantum
C~emistij an~
State
Soli~ P~ysics
EDITED BY
Jens
Pe~er Da~l
Technical University of Denmark
Lyngby, Denmark
AND
Jo~n Avery
University of Copenhagen
Copenhagen, Denmark
Springer Science+Business Media, LLC
Library of Congress Cataloglng In Publlcation Data
Symposium on Local Denslty Approximations in Quantum Chemistry and
Solid State Theory (1982: University of Copenhagen)
Local density approxlmations in quantum chemistry and solid state
physics.
lncludes blbliographlcal references and index.
1. Quantum chemistry-Congresses. 2. Solid state chemistry-Con
gresses. 3. Hartree-Fock approximation-Congresses. 1. Dahl, Jens Peder.
11. Avery, John, III. Title.
QD462.A1S95 1982 541.2'8 84-3482
ISBN 978-1-4899-2144-4
ISBN 978-1-4899-2144-4 ISBN 978-1-4899-2142-0 (eBook)
DOI 10.1007/978-1-4899-2142-0
Proceedings of a Symposium on Local Density Approxlmations In
Quantum Chemistry and Solid State Theory, held June 10-12,
1982, at the Unlversity of Copenhagen, Copenhagen, Denmark
© 1984 Springer Science+Business Media New York
Originally published by Plenum Press, New York in 1984
Softcover reprint of the hardcover 1s t edition 1984
Ali rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmltted,
in any form or by any means, electronic, mechanical, photocopylng, mlcrofllming,
recording, or otherwise, without written permlsslon from the Publisher
PREFACE
The· simplest picture of an atom, a molecule or a solid is the
picture of its distribution of charge. It is obtained by specifying
the positions of the atomic nuclei and by showing how the density,
p(E), of the electronic charge-cloud varies from place to place.
A much more detailed picture is provided by the many-electron
wavefunction. This quantity shows not only the arrangement of the
electrons with respect to the nuclei, but also the arrangement of the
electrons with respect to each other, and it allows the evaluation of
the total energy and other properties. The many-electron wavefunction
is in principle obtained by solving the many-electron Schrodinger
equation for the motion of the interacting electrons under the influ
ence of the nuclei, but in practice the equation is unsolvable, and
it is necessary to proceed by methods of approximation. Needless to
say, .such methods will as a rule depend on the complexity of the
system considered.
The present volume is concerned with an approximate scheme of a
very general type, in the sense that it is essentially independent of
the complexity of the system studied. The scheme reduces the many
electron Schrodinger equation to a one-electron Schrodinger equation,
and it assumes that the effective potential in this equation is the
sum of the potential from the nuclei, the classical Coulomb potential
from the electronic charge-cloud, and a so-called exchange-correla
tion potential. Moreover, it is assumed that the value of the ex
change-correlation potential at apointr only depends upon the value
of the electron density p at that point-(or perhaps upon the spin-up
and spin-down parts of p separately). The resulting one-electron
equation must be solved in an iterative fashion such that the scheme
becomes self-consistent.
This self-consistent theory, with its built-in local density
approximation, incorporates the electron density - and hence the
simple picture of an atom, a molecule or a solid - into the calcula
tion in a very direct way. Accordingly, it provides a simple concep
tual framework for the discussion of electronic properties, and it
has turned out to be a remarkably accurate theory, both in quantum
v
vi PREFACE
chemistry and solid state physics. Let us briefly comment on the
origin and development of the theory, often referred to as a density
functional theory.
In quantum chemistry, the main approach used for reducing the
many-electron Schrodinger equation to a one-electron equation has
been the Hartree-Fock approximation supplemented by configuration
interaction. However, if the Hartree-Fock approximation is applied
to the electrons in a metallic solid, it leads to a zero density of
states at the Fermi surface. Furthermore, configuration interaction
calculations to correct the Hartree-Fock approximation are almost im
possible to carry out in the case of a conducting solid. Hence, the
Hartree-Fock approach is seldom used in solid state theory.
In 1951, J.C. Slater introduced an alternative method for re
ducing the many-electron Schrodinger equation to an approximate one
electron equation. F. Bloch and P.A.M. Dirac had been able to show
that in the case of a uniform electron gas, the energy due to ex
change is proportional to the 4/3 power of p(r), and the exchange
potential is proportional to the cube root of-p(r). This fact led
Slater to propose a general approximate exchange-potential propor
tional to the cube root of the local electron density, the approxi
mation being made in the same spirit as the Thomas-Fermi approxima
tion.
In spite of the lack of rigorous theoretical justification for
using the Slater potential in cases where the density is non-uniform,
it has been used for many years with great success in solid state
theory. Because of this not well understood but nevertheless very
real success, there has recently been great interest in introducing
the Slater potential (and other exchange-correlation potentials de
rived from electron gas theory) into quantum chemistry.
The first calculations of this kind were made by Slater's
coworkers, who developed the multiple-scattering Xa method. In the
MSX~ method, which was modelled after existing solid state methods,
each atom in a molecule or cluster is surrounded by a sphere, within
which the potential is spherically averaged. In the region between
the spheres, the potential is approximated by a constant. Solutions
to the effective one-electron Schrodinger equation are found by
matching internal and external solutions at the boundaries.
An alternative method developed by later workers avoided the
problematical "muffin tin" approximation of the MSX~ method. Instead,
the effective one-electron Schrodinger equation was solved using a
variational LCAO method. Matrix elements of the effective one-elec
tron Hamiltonian were evaluated by Diophantine integration. Careful
calculations on small molecules using this method showed that the
PREFACE
Slater potential was able to give results which could compete favour
ably in accuracy with Hartree-Fock results.
Mqch interest has been focused on the problem of how to improve
the pl/3 potential. Many-body theory applied to a uniform electron
gas has yielded the energy as a function of density with both ex
change and correlation effects included, and other terms have been
added to correct for the effects of non-uniformity. Work along these
lines has led to the proposal of several improved exchange-correla
tion potentials.
The publication in 1964 of the Hohenberg-Kohn theorem (showing
that the ground-state energy of a system of electrons in an external
potential is a unique functional of the density) led to a great out
burst of interest in density functional theory, and recent work in
this field has yielded many important results.
The field of local density approximations and density functional
theory is currently a very rapidly developing and lively area of re
search. Nevertheless, many fundamental questions remain to be an
swered: For example, why do potentials derived from the theory of a
uniform electron gas work at all in cases where the electron density
is highly non-uniform? How can these potentials be systematically
corrected to account for effects of non-uniformity? What is the best
way to treat excited. states and transition probabilities? How should
relativistic systems be treated? These are only a few of the questions
on which attention is now focused.
Because of the excitement and rapid pace of development in the
field, we were happy to have had the opportunity to organize a sym
posium in Copenhagen devoted to local density approximations. The
symposium took place on June 10-12, 1982 as a satellite of the 4th
International Congress in Quantum Chemistry, which was held in Upp
sala a few days later. The present multi-author book includes both
lectures presented at the symposium and also invited articles from
a number of distinguished experts in the field who were unable to
attend the Copenhagen meeting. We are extremely grateful to the dis
tinguished scientists who contributed to this book and thus enabled
us to present a coherent and up-to-date picture of the field.
For practical reasons we have divided the book into two parts,
Fundamental Theory and Applications, but this division should not be
taken too literally. Important contributions to theory may be found
in the articles of part two, and likewise important applications may
be found in the articles of part one. We have included a few purely
experimental papers in order to stress the close connection to ex
perimental work on electron densities. As the last contribution we
have included an article by an ab-initioist, who serves as the
advocatus diaboli.
viii PREFACE
We give our warmest thanks to Annalise Andresen, Louise
Helweg-Mikkelsen, Janne Larsen, Astrid Schantz, and Katrine Winther
Petersen of Chemistry Department B, the Technical University of
Denmark, who typed all of the chapters of this book in photoready
form.
November, 1983 John Avery
Jens Peder Dahl
CONTENTS
FUNDAMENTAL THEORY
Density Functional Theory and Local Potential
Approximations from Momentum Space
Considerations • • • • • • • • • • • • 1
Vedene H. Smith, Jr.
Aspects of Density Functional Theory 21
Robert G. Parr
On the Calculation of Potentials from Densities 33
Scott H. Werden and Ernest R. Davidson
The Chemical Potential for Interacting Fermions
in a Harmonic Potential • • • • • • 43
Philip Phillips and Ernest R. Davidson
Density, Total Energy and Chemical Potential of
Atomic Ions and some Molecules • • • • • 53
Norman H. March and Renato Pucci
Some Aspects of the Role of Single-Particle
Density in Chemistry • • • • • • • • 75
B.M. Deb
The Mechanics of and an Equation for the
Electronic Charge Density • • • •• 129
R.F.W. Bader and Hanno Essen
The Qualitative Theory of the Current
Density in Molecules ••••• 145
J.A.N.F. Gomes
The Constrained Search Approach, Mappings to
External Potentials, and Virial-Like Theorems
for Electron-Density and One-Matrix
Energy-Functional Theories • • •• , • • • • 155
Mel Levy
ix
X CONTENTS
Self-Interaction Correction 173
John P. Perdew
Some Approaches to Density Functional Theory • • • • • . . • 207
Eugene S. Kryachko
Exchange-Correlation Energy Functionals
in the Density Functional Formalism 229
0. Gunnarsson and R.O. Jones
An Approach to Electron Correlation beyond
the Local Spin Density Approximation 245
L. Fritsche and H. Gollisch
Pair Correlation Energies and Local
Spin-Density Functionals ••• 263
A. Savin, H. Stoll, and H. Preuss
On the Extension of the Statistical Energy
Expression to Multi-Determinantal
Wave Functions • . • • • • • . • • • 273
Tom Ziegler
The Weizsacker Term, its Corrections and
Virial Partitioning in Density
Functional Theory • • . . • • 287
Eduardo V. Ludena
The Second Order Reduced Density Matrix
and the Chemimal Bond • • • . • . 303
William L. Luken and John C. Culberson
Characteristic Function Approach to
Density Matrix Calculations 323
Metin Demiralp
On the Systematic Assessment of Correlation
Effects in Local Density Models 333
J.R. Sabin and S.B. Trickey
Relativistic Density Functional Theory 353
E.K.U. Gross and R.M. Dreizler
A Phase Space Approach to Energy
Densities in Position Space 381
Michael Springborg and Jens Peder Dahl