Optimized LCAO Method and the Electronic Structure of Extended Systems [Reprint 2021 ed.] 9783112483466, 9783112483459

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Helmut Eschrig Optimized LCAO Method and the Electronic Structure of Extended Systems

Optimized LCAO Method and the Electronic Structure of Extended Systems by Helmut Eschrig With 80 Figures

Akademie-Verlag Berlin 1988

Author: Dr. rar. nat. Helmut Eschrig Zentralinstitut

für Festkörperphysik und Werkstofforschung

der Akademie der Wissenschaften der DDR, Dresden

Reproduction of the original author's manuscript

ISBN 3-05-500253-9 Erschienen im Akademie-Verlag Berlin, DDR-108'6 Berlin, Leipziger Straße 3-4 (C) Akademie-Verlag Berlin Lizenznummer: 202

•

1988

100/438/88

Printed in the German Democratic

Republic

Gesamtherstellung : VEB Druckerei "Thomas Müntzer", 5820 Bad Langensalza LSV 1185 Bestellnummer: 763 671 9 04200

(9049)

5

Preface

Since

the

late

seventies,

the theory of the electronic

structure

of

ordered crystalline solids has been going through another period of vivid development

both

consequently

by

exploiting powerful

applying

computer

facilities

modern tools of many-body theory as

functional formalism and Green's function methods.

and

the

by

density

Precise and effective

numerical procedures of solving the one-particle SchrBdinger equation for given arrangements of atoms form a keystone of the present-day structure

theory.

successful versions,

There

approaches

to

exists this

quite goal

a as

variety KKR,

of

APW,

electronic

nearly their

equally

linearized

norm conserving pseudo-potentials and others, but at least the

results are preferably represented in the language of linear of atomic orbitals (LCAO) [100], electronic

combinations

Especially the interpretation of modern

structure theory in chemical terms [57] made LCAO representa-

tions

very-popular.

been

used

with

A non-orthogonal local orbital remarkable

success

to

deal

representation

with

has

electron-phonon

interaction and with phonon anomalies in metallic solids [117]. Having working author

taken on

part in the research activities of

metal

developed

the

theory for the past two decades the numerics for

a

Dresden

[120],

first-principles

the

group present

self-consistent

field LCAO method based on a special optimization procedure for the basis states and applied it successfully to many electronic structure problems. Providing the user with explicitly given wavefunctions, the advantages of a complete LCAO treatment aginst LCAO interpolation schemes are The

crucial

choice LCAO

of basis functions. treatment

depending

on

obvious.

problem for variational procedures like LCAO is the is the

In the version presented here

preceded by another variational structure type of the lattice only

proper

the

ordinary

parameter

adaption

and

not

on

the

potential. The of

present booklet gives a full description of the author's

version

an SCF-LCAO method including all numerical details needed to apply it

and considers the main physical ideas closely related to it.

It

further

provides the reader with complete data of SCF-LCAO band-structure results for all element metals up to the atomic number 30. component solids,

systems,

spin-polarized

and clusters,

which do exist,

similar way in subsequent volumes.

Extensions to

and relativistic

cases,

multi-

disordered

are planned to be presented in a

6 B e s i d e s the a l r e a d y m e n t i o n e d D r e s d e n g r o u p , the w a y of this a u t h o r electronic

by

m a n y p e o p l e . It is a p l e a s u r e to me t o take t h i s o p p o r t u n i t y to t h a n k

all

I.

p a r t i c u l a r l y p r o f e s s o r s P. Z i e s c h e , G . L e h m a n n , Y. M. K a g a n , M .

Kaganov,

d o n e in the Dresden were

t h e o r y has d i r e c t l y or i n d i r e c t l y b e e n

in

guided

of t h e m ,

structure

W.

A.

H a r r i s o n , and W . W e b e r . The w o r k p r e s e n t e d h e r e

Zentralinstitut

für

Festkörperphysik und

of the A c a d e m y of S c i e n c e s of G D R .

supported

Rossendorf.

by

Last

the

staff

not least,

complicated text with great

Dresden, April

1987

of

the

The

computer

was

Werkstofforschung

numerical station

calculations of

the

t h a n k s go to m y son M a t t h i a s who t y p e d

ZfK the

patience.

Helmut

Eschrig

7

CONTENT

1. Introduction 1.1. The adiabatic approximation

9 9

1.2. Density functional theory

11

1.3. Quasi-partiolea

20

2. The LCAO formalism

29

2.1. Atomic one-particle wavefunctions

30

2.2. The LCAO secular equations

38

2.3. The case of the bandstructure of a crystalline solid

41

2.4. Multi-centre integrals

46

3. Optimization of the basis 3.1. Parametrization of the basis states

53 53

3.2. Empty lattice tests

55

3.3. Optimum basis LCAO band calculation for real crystals

68

4. Wannier functions

73

4.1. LCAO interpolation

73

4.2. One-band Wannier functions

77

4.3. Multi-band Wannier functions

83

4.4. An example

87

4.5. The recursion method for resolvent operator matrix elements

90

4.6. The chemical pseudo-potential

96

5. The looal basis representations of the electron density 5.1. Symmetry 5.2. The total electron density

98 98 104

5.3. The case of a crystal lattice

106

5.4. Core, net, and overlap densities

109

8

Content

6. Simplex methods for k-spaoe integrations in d dimensions

113

6.1. Integrals containing one singular function

114

6.2. Integrals containing two singular functions

119

6.3. An example

124

6.4. Comparison with the proximity volume method

127

7. Potential oonstruotion and iteration

130

7.1. Principles of potential construction

130

7.2. The Hartree part of the potential

132

7.3. The exchange and correlation potential

135

7.4. Iteration of a high-dimensional non-linear vector equation

137

Appendix 1. Spherical harmonics and their transformations

141

Appendix 2. Some useful theorems on basis function expansions

144

Appendix 3. Results of DFT-LCAO band structure calculations for element metals

146

References

216

Subject i-ndex

220

Notation: Bold-face

type

A = [[A^j]].

is

used

for

vectors

type,

r,

it

technical reasons, by

0

text.

=

(R^, Rg, R^)

or

matrices

The absolute value (Euklidean norm) of a vector R is

roof " on a letter in ordinary type, face

R

means

H,

denotes an operator, on a bold-

the unit vector in the direction

of

the centre T of the Brlllouin zone (B.Z.) is

in bandstructure plots.

R. .A r.

For

denoted

All other notations are explained in

the

9

1. INTRODUCTION

The

present

quantum or

ions.

modern

The

ideas

and formal developments are

as they are related to the basic content of

following

the

definite

to

atoms

presented

this

in only

volume;

sketched.

Throughout

the whole text of

this

the

booklet

the non-relativistic description is observed in

argumentation notion.

as well as the

formal

order

developments

a to

within

a

Such a restriction is neither inherent the theoretical

constructions nor the technical tools; this

solving

introduction cannot be regarded as a comprehensive survey over

theories

restraint keep

of

This introduction sketches the ways those equations appear

theory.

inasmuch the

treatise deals basically with a special method

mechanical one-particle equations for systems composed of

practically everything treated in

volume may be generalized to the relativistlc cases of inner

shell

motion in heavy atoms, spin-orbit coupling and so on.

1.1. The adiabatio approximation The non-relativistic motion of a system of nuclei

a with masses M and a charges A a (in units of the electrostatic proton charge lei, the electrostatic charge differs from the usual electric charge by a factor 1/YiirF ) and of electrons with mass m may at least formally be obtained from the 1) Hami11 on i an 72 A A , H H, . = >" — 2 . + > — + tot *7T 2u a a?i' lH a -R a , I

K = > i

— 2

__ 1 + > i?i' Ir.-r.,I i l

(-i^)

H>

A

> ^ — = T + W + V. iTa Ir,-RI 7 i a

(1.2)

Here, R=(R1,...,Ra

Rn)

(1.3)

are the coordinates of the nuclei and r^, electrons. faster

Due

to

the

than the nuclei.

i = 1,...,N

are those of

large mass difference the electrons If the latter were fixed at

move

positions

the much

R,

the

l5 Atomic

units (a.u.) m = -ft = lei = 1

are used throughout.

unit is 1 a.u. = 1 Hartree = 2 Rydberg.

The

energy

10

1.

e l e c t r o n i c ground-state H. In the argument of

x

i

=

contains

i

( r

the

Hamiltonian depend

, s

i

" ^ ( x ^ R ) could be c a l c u l a t e d from the Hamiltonian

Vi,

)

( 1

electron

coordinates

spin

variables;

H does not contain the e l e c t r o n spins, in

[45],

applied

would

remain

infinitely

its

then be obtained from the Hl'(xi;R) parametrically coordinates

?

the

totally exchange

electronic

a l l the time

if

the

subsystem

nuclei

were would

equation (1.5)

depending on R.

I n t e g r a t i n g over a l l but one

-§(r;R)

spin v a r i a b l e s the

electronic

electronic

ground-

(9 being the number d e n s i t y i n our u n i t s )

: = >~

>H

...>11 l ^ ( ( r , s ) , x 2

x2

s

XN

xN ; R ) I 2 ,

(1.6)

e

e

abbreviation

1/2

f

,=-1/2

i s used.

t o be

The e l e c t r o n i c ground-state energy E ( R )

and summing over a l l

(r;R)

where the

in

= lt'(x1;R)E(R)

charge d e n s i t y

i s obtained

s t a t e s t h a t the

ground-state

s l o w l y moved.

the

The a d i a b a t i c theorem of Gell-Mann and Low

t o our s i t u a t i o n , in

though

'4)

i t s ground-state may

the v a r i a b l e s x^ and thus r e s u l t i n g

i n t e r a c t i o n of the e l e c t r o n s .

state

and

on them v i a the Pauli p r i n c i p l e demanding ^ ( x ^ R )

antisymmetric

Introduction

3

J

1

A g r e a t part of what f o l l o w s j u s t deals with the computation of

^ ( r ; R ) and E ( R ) . Within t h i s frame the motion of the n u c l e i would be obtained from

the

Hamiltonian

N V Hn = >— " —^ + > — n a 211 a

A A , -A a

+

E(H)

_

IR -R , a a

A f t e r e l i m i n a t i n g the t r i v i a l

(1.7)

motion of the c e n t r e of mass of the n u c l e i ,

( 1 . 7 ) y i e l d s v i b r a t i o n s around the minimum of U(R) = >" a?I'

"S A " , lRa-Ra,I

+ E(R).

(1.8)

11

1.2. Density functional theory Often

minimum configuration R = R q and the corresponding

the

potential

value U ( R o ) are of sufficient interest by themselves implying

structural

information and binding energies, the latter as differences between U ( R Q ) and the sum of the corresponding values for the separated constituents of the considered system. In

reality the nuclei move with a finite velocity,

frame

yet the

outlined

may be well founded as the zeroth order approximation of the Born-

Oppenheimer perturbation theory [11,10,94], The non-adiabatic coupling of the electronic and nuclear motion appears in the higher perturbation 1 / 2 . There are basically orders with respect to the small parameter (m/M ) a two cases where the adiabatic approximation breaks down. One is if the ground-state lowest

of

the electronic system is almost degenerate so that

electronic excitation energy is comparable with

energies.

This happens e.g.

predissociation

of

the

the

vibrational

in Jahn-Teller systems [64,63,23] or in the

molecules.

In the other case the lowest

excitation energy may be as large as one Rydberg,

electronic

but due to the lack of

screening the electron-nuclei interaction matrix element

diverges.

This

leads e.g. to the polaron mode in insulators [56]. In metals the electronic excitation spectrum has no gap.

Consequently

there are always electronic excitations near the Fermi surface subject to large non-adiabatic corrections. However, the phase space of these states is so small that the corrections do not substantially contribute to total energies [13] save again for metallic Jahn-Teller systems [42,43,29,118].

1.2. Density functional theory (DFT) The adiabatic approximation is only a first step simplifying the problem. The

resulting

differential considered

equation (1.5), equation

system,

in

10

as an eigenvalue problem for a 23 to 10 variables, depending

is still far from being subject to direct

partial on

the

numerical

calculations. There is, however, an indirect way attacing this task which starts there and

from is

the

the basic theorem of Hohenberg and Kohn [60]

stating

that

a one-to-one correspondence between the potentials V of ground-state electron densities (1.6) for a

provided H has a non-degenerate ground-state If.

number

Ng,

Hence, V may be regarded

as

a unique functional of the electron density 9(r).

as

the solution of (1.5),

the ground-state

given

(1.2)

On the other hand,

^(x.^) and consequently

any

ground-state property, particularly the ground—state energy E, are unique functional

of

functionals of ^

V.

This implies that those quantities are

also

unique

^(r) :

= n?[?]t E = E[ ? ] =

m^.HCsmq]).

(1.9)

12

1.

Introduction

The n o t a t i o n H [ ^ ] m e a n s t h a t we h a v e e x p r e s s e d V i n t e r m s of

Defining

t h e Hohenberg—Kohn

functional

p[§] = m < ? ] , [ T

(1 .10)

+ w] i f [ § ] )

and w r i t i n g V a s H v = >:: v(r i=1

1

(1.11)

),

w h e r e a g a i n v = v[^>] h o l d s ,

we may w r i t e

E [ ? ] = F[