Marder M.P. Condensed Matter Physics

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9. Electron-Electron Interactions

9.1 Introduction

Bloch's theory for periodic solids brilliantly solves the problem of a single electron

in a periodic potential. While engaged in the process of solving this problem, it is

easy to forget the severity of the approximations needed to reduce a real periodic

solid to a single-electron problem. The problem has two different sides. First, how

conceptually can interacting electrons be treated within a one-electron framework

at all? An answer to this question will be provided in Section 17.5 by Fermi liquid

theory. Yet Fermi liquid theory provides little practical guidance in constructing

the effective one-electron potential it shows may exist. The construction is instead

provided by a sequence of approximations, whose validity is in principle not quite

clear but in practice lies behind all attempts at realistic calculations. There is no

internally consistent test for the validity of these calculations, and sometimes they

fail rather badly. Even more often, however, they achieve detailed comparison

with experiment that is much better than might have been expected. The goal of

this chapter is to describe only those treatments of electron-electron interactions

leading to practical band structure calculations.

The Hamiltonian that one really should solve is Eq. (6.1), in which electrons

and nuclei all appear on an equal quantum-mechanical footing. A first simplifica-

tion is to remove the nuclei from the quantum mechanics problem. Because nuclei

are thousands of times more massive than electrons, they move that much more

slowly. Born and Oppenheimer (1927) suggested an approximation scheme that

is employed quite universally throughout condensed matter physics. So far as the

electrons are concerned, take the nuclei to be static, classical potentials, and solve

the electronic problem without worrying about the nuclei further. So far as the nu-

clei are concerned, the electrons are a rapidly moving shroud of charge that follows

them wherever they go. Because the motion of nuclei is accompanied by charge

redistribution, the energies involved in moving nuclei about depend upon the so-

lution of the electron problem, and the nuclei interact with complicated effective

potentials. All the ideas of different types of interatomic bonding arise from this

viewpoint and will be discussed further in Chapter 11.

The Born-Oppenheimer approximation may have to be abandoned whenever

the electrons and nuclei cannot be disentangled so neatly. The world is full of

phenomena where the approximation fails. For example, in striking a flint to create

a spark, mechanical motion excites electrons into a plasma that then emits light.

233

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

234

Chapter 9. Electron-Electron Interactions

Adopting the Born-Oppenheimer approximation, the electrons

Eq. (6.1)

solve

-h

N N

where Φ

an antisymmetric function of the immense number

of electrons

a solid. This problem is still intractable, even on the largest computers, for more

than several hundred electrons,

number that makes

impossible to investigate

large molecules

solids. The numerical problem grows exponentially with the

number of electrons; for macroscopic solids it will never be solved by brute force

on a conventional computer. Contemplating this situation in the first days of quan-

tum mechanics, Dirac wrote that "the underlying physical laws necessary for the

mathematical theory of a large part of physics and the whole of chemistry are thus

completely known, and the difficulty is only that the exact application of these laws

leads

equations much too complicated to be soluble" [Dirac (1929), p. 714].

The paradox of condensed matter physics is contained in this simultaneous cry of

triumph and despair. The fundamental equations are known, but fundamentally

unwilling in their original form to answer most physical questions.

9.2 Hartree and Hartree-Fock Equations

All the computational difficulty arises from the Coulomb interaction. Perhaps this

term may somehow be replaced by something more computationally tractable, such

as an effective electron-electron potential U

(r). A first guess at such an effective

potential in which to study the motion of electrons is that each electron moves in a

field produced by a sum over all the other electrons. Analogy with classical physics

suggests that the potential corresponding to electron-electron interactions could be

(r)= [dr>

-^Q,

(9.2)

J \r-r'\

where n is the number density of electrons

η(?)

\φ

]

(?)\

(9.3)

This guess can immediately be inserted into the Schrödinger equation, giving

-h

—

Vv + \U

Jr) + U

(r)]lp[ = £/?/>/.

derivation follows, but one can

(9.4)

2m simply write this down in the

spirit of

Eq.

(6.3).

Equation (9.4)

the Hartree equation, which lay behind the first systematic at-

tempts by Hartree (1928) to deduce atomic spectra from first principles.

Hartree and Hartree-Fock Equations 235

9.2.1 Variational Principle

In order to improve upon the Hartree equations, it is desirable to find a formal way

to derive them from Eq. (9.1). The best method is the variational principle derived

in Appendix B which states that wave functions solving Schrödinger's equation are

extrema of the functional

{Φ} = <Φ|Α|Φ>, (9.5)

subject to the constraint that (Φ|Φ) = 1. As discussed in Section

8.2.1,

approxi-

mate solutions of Schrödinger's equation are obtained by restricting the search for

extrema to a subset of all possible wave functions. The Hartree equations follow

from restricting one's search to wave functions Φ of the form

* = 1[Ψι(7ι), (9-6)

1=1

where the ψι are orthonormal. If one considers wave functions of this form, uses

Lagrange multipliers S

;

- to enforce the constraints (ißj\ipj) = 1, and demands

SFK δ

^E^7^W^(^) = 0

(9.7)

the Hartree equations (9.4) result almost immediately, apart from one small and

slightly annoying discrepancy. The formal calculation indicates that each electron

interacts not with the full charge density of the system, but with the charge density

minus the density due to the electron

itself.

The calculations are left as Problem 1,

but the more difficult case of Hartree-Fock will now be performed explicitly.

9.2.2 Hartree-Fock Equations

The central failing of the Hartree equation is that it does not recognize the Pauli

principle. The true many-body wave function must vanish whenever two electrons

occupy the same position, but the Hartree wave function cannot have this prop-

erty. The Pauli principle forces electrons in metals to occupy single-particle energy

states with energies on the order of 10 000 K, even in the ground state—such large

energies that this effect must be included in any calculation from the outset to have

any hope of realism.

Fock (1930) and Slater (1930) showed that the way to obey the Pauli prin-

ciple is to work within the space of antisymmetric wave functions. Absolutely

the simplest possible type of antisymmetric wave function is obtained by taking a

collection of orthonormal one-particle wave functions and antisymmetrizing them:

Φ(πσ, . . .

)

= -= νν-Ι)^,(ησι) · · · il>

)

The sum is over

(9.8)

w/\J\

—' all permutations

s of

. . . N. σ,

is a spin index.

236

Chapter 9. Electron-Electron Interactions

ψι(7\σ\)

Ψι

{ησ

)

Ψ\(τ

)

ψΝ{7\σ\) ipNihvi)

ΨΝ(7ΝΟ-Ν)

This type

wave function

called

Slater

determinant.

(9.9)

Because this wave function

not

simple product,

but

sum

products,

the

par-

ticles

are no

longer independent. Varying

single index,

for

example, causes

all

the particles

shift around. Thus

the

Pauli principle induces correlations among

the particles. Under these circumstances,

it is

impossible

to be

completely casual

about spin degrees

freedom, tossing them

in at the end of

the

calculation

as a

factor

two. Therefore,

the

spin index

σ,

taking values

±1 is

included

every

wave function.

long

as the

Hamiltonian does

not

involve

the

spin explicitly,

the

equations will

solved

giving

the

simple form

,_,

. ;

/-»

\ / \

The

spin function

χι is

either

the

"spin-up" fune-

WlVi^i)

Φΐνί)Χΐ\

ί)· tion,5i,

ffl

the

"spin-down" function <5_ι,

σ/

. (9.1Ü)

cumbersome

carry around

the

arguments

ψι,

adopt

the

shorthand

notation

To obtain

the

Hartree-Fock equations,

one

repeats

the

variational problem that

led

to the

Hartree equations,

but

now employing

the

wave function (9.8) rather than

(9.6).

The calculation

more difficult.

The

first

step

to take the expectation value

of the Hamiltonian with

the

wave function (9.8),

and the

next step

is to

require that

its functional derivative with respect

each

φι

vanish. This expectation value

can

be obtained

working explicitly with

the

wave function

Eq.

(9.8). However,

such computations

are

exactly what second quantization

was

designed

simplify.

Obtaining

the

Hartree-Fock equations

therefore

sensible starting point

gain

practice with second quantization.

As described

Appendix

given

complete

set of

states such

as ψι, the

Hamiltonian

(9.1) can be

rewritten

A=5]cÌc

ZÌ

(^(l)|^^V? + i/ion(ri)|#(l))

(9.12)

+ \

4Q-Q»(V/(l)#(2)|

//'/"/"

r\-r

#,»(1)#»(2)>,

or, denoting

{φι

( 1 ) |

(/1

make

the

notation even more compact,

2 a

W-k

= χ;

cjc

——v?

i/ion

)

|/'

(9.13)

+ 2

ήέι>έΐ'"έΐ"{1ΐ

ΙΙΊ'Ί'"

n-r

\l"l'").

Hartree and Hartree-Fock Equations 237

The product wave function Φ takes a very simple form in second quantized

notation. One can write it as

|Φ) = 111111 ... 10000. . .) (9.14)

which means that states I . . . N are occupied, and all other states are unoccupied.

The Hartree-Fock calculation proceeds by computing

(Φ|Λ|Φ) (9.15)

and choosing the functions ψι to make the expectation value as small as possible.

To compute the expectation value, first consider

(Φ|φ//|Φ).

(9.16)

When

C[>

acts upon Φ it gives 0 unless /' is one of the states obeying /' < N so that

it is occupied in Φ. The annihilation operator c/' removes this state from Φ, which

now can be written

s^ l'i *

// // /

c/ΙΦ) =

|11...

1011...

100...) (9_

)

When the creation operator c\ acts on (9.17), it had better create an electron

in state at /' again, or else when (Φ| acts from the left, the result will be zero. So

(9.16) is zero unless / = /', and the first term in Eq. (9.14) becomes

{if^-^ì

ÙUmi)

(9-18)

/^Occupied States

To compute the expectation value of the Coulomb operator in Eq. (9.13), con-

sider

<*|cjcj,c//«c///|*>. (9.19)

Unless /" and /'" are among the states occupied in |Φ), which is to say /" < N

and I < N Eq. (9.19) immediately gives zero. Furthermore / and /' must create

again the states that /" and /'" have just destroyed. There are two ways this can

happen:

and I = I or I = I and / = /. In the first case, using Eqs. (C.3c) and

(C.2b) one obtains from (9.19) the value +1. In the second case, use Eq. (C.3a) to

swap cj and

e],.

Thus one obtains from (9.19) the value

—

The conclusion is

(Φ |cJcJ/C;'//C/" jΦ) = δΐ'ΐδιι — δ['ΐδιι And / and/'must be occupied states (9.20)

which implies

1(Φ|

J]] c)clc

„c

,(lï\-^—

\1"1'")\Φ)

(9.21)

—ri

—

^ e

\ Σ

[<SH"<W"-<W'<$H<"'IF——ι\ι"ΐ")

(9·

)

— ri

—

ro\

J2{ll'\^

_^,\W)

(ll'\

_^.\l'l) (9.23)

,// i

i-^2| \r\-r

238 Chapter 9. Electron-Electron Interactions

Adding together Eqs. (9.23) and (9.18) gives

(Φ|:Κ|Ψ)

Σ/

αηΦΐ(ΐ)

-ft

(ί)

υ(7

)\Φ

(\)\

/■

+ 2 / dr\dr

r\ -r

Σ [iVi(i)i

i^(2)i

-v?(i)^(2M-(2)^(i;

(9.24)

The first of two terms in the double integral in (9.24) is called the Coulomb

integral. It is precisely the term that appeared in Eq. (9.3). The second is more

noteworthy; it is called the exchange

integral

and may be interpreted as saying that

particles 1 and 2 flip places in the course of interacting. Because of the antisym-

metry of the wave function, such an interaction comes in with a relative minus

sign.

Having found this expectation value, the next task is to vary the functional

(9.24) with respect to every single-particle function φ and require each variation to

vanish subject to the condition that the t/>'s be orthonormal. The real and imaginary

parts of φ are completely independent functions, so one must vary with respect to

them separately. Instead of doing it quite that way, it is legitimate and simpler to

treat φ and φ* as independent functions and vary with respect to φ* holding φ

constant. In order to restrict the search for wave functions to the space in which all

the ^'s are orthonormal, use the method of Lagrange multipliers to enforce the N

constraints Σ

σι

I άτ\φ*{\)φj{\) =

·,

and add

Σ^Σ [

αηΦΐΜΦ^ι)

σ,

(9.25)

to the functional before taking the functional derivative.

The result is that

(1)

υ(?

)φ

(1)

+φι(1)

j dr

- Σ

vv(i)

Σ /

\M2)\

\r\-r

„2

φ){2)φ

{2)

\f\ -hi

(9.26)

Of course, one can vary with respect to φί(τ) instead, and then take the complex

conjugate of the ensuing expression. The result is exactly Eq. (9.26), except that

E,j

is replaced by £^·. This proves that £,; = £^

(

so the matrix £

is Hermitian.

Therefore the Hartree-Fock equations can be simplified a bit further. Suppose in

Eq. (9.9) that the wave functions φι are replaced by

<Α·

Σ^·

(9.27)

Hartree and Hartree-Fock Equations 239

This replacement

equivalent

multiplying

the

matrix

-0's

by the

matrix

inside

the

determinant,

and

therefore

alters

the

wave function

only

mul-

tiplying

it by the

determinant

of W. If

one chooses

W to be a

unitary matrix,

its

determinant

one,

and Φ

takes exactly

the

same form

terms

the V'S

that

did

terms

the ^'s.

The

effect

this transformation upon,

for

example,

the

kinetic energy term

Jdrj2iï(™y

-h

~ _

φί(τσ)

^ΣΣ %rj(^)-^W

(ra)

ισ

)ϊ

^Σ^7^*(

?σ

2ντ2

°Jf

■Ψ](?σ)

-h

ψ*

(ra)

ipj

(?σ).

J<7

(9.28)

(9.29)

(9.30)

(9.31)

Similarly,

all of

the other terms

(Φ|ΙΚ|Φ)

are

invariant when

ψ is

turned

in for

ψ.

However,

the

Lagrange multipliers change

tfWZEijWwtpr

= E

ΨιϊινΨν,

IV IV

where

έ,,-Σ^Ϊ^

ÌV

(9.32)

(9.33)

the

matrix

in the

new

basis. Because

£ is

Hermitian, there exists

basis

which

£ is

diagonal;

one

may

well therefore just take

δ to be

diagonal

begin

with. Employing this simplification

and

carrying

out the

spin sums

in Eq.

(9.26)

gives finally

the

Hartree-Fock equation

£,·&(?)

2v72

■φί{Ψ)

+U(r)4>i(r)

ni2

*%(OI

XXJ

AK'

)

φ){7')φ^)

XiX

states

and

have

the

same spin

and zero otherwise

(9.34)

9.2.3 Numerical Implementation

Hartree-Fock calculations divide into

two

groups, restricted

and

unrestricted.

Re-

stricted Hartree-Fock calculations

are

ones

which there

is an

even number

electrons,

and one

assumes that wave functions divide into

two

groups: spin-up

240

Chapter 9. Electron-Electron Interactions

Figure

9.1.

(A) For two well-separated

atoms the ground state cannot be described by a

single spatial wave function multiplied by two different spin states. (B) But for the same

collection of particles assembled as an He atom, it can be.

functions and spin-down functions. The spatial part of a given spin-up electron is

exactly the same as the spatial wave function of some corresponding spin-down

electron. Unrestricted Hartree-Fock makes no such assumption. Suppose one has

two protons, very far separated, as in Figure 9.1. Each has its own private elec-

tron, and the ground state places an electron of definite spin on each atom.

contrast, if the same particles are placed in a helium atom, the ground state places

both electrons in the same spatial state, differing only in their spin. In the first case,

unrestricted Hartree-Fock is necessary, while in the second case, one can cut the

size of the computation in half by doing restricted Hartree-Fock.

Whether one attacks a restricted or an unrestricted problem, the Hartree-Fock

equations are

complicated set of nonlinear equations that can only be handled

numerically. A first instinct would be to describe the various functions on some

sort of cubic grid. This approach is used for atoms, but for molecules it is a terribly

inefficient idea. A better way to solve the Hartree-Fock equations for molecules is

to write each φι as a linear combination of basis functions that represent informed

guesses about the actual shape of the wave functions and that are easy to integrate.

The fewer basis functions one needs to represent electronic wave functions, and

the easier it is to integrate products of

the

basis functions, the better off one will be.

One set of examples of basis functions is provided by STONG. The STO stands

for Slater-type orbitals, while NG stands for N Gaussians. Experience has shown

that the electronic wave function at the /'th nucleus looks roughly like

The problem is that if there is more than one nucleus, one has to do integrals of the

form

[

dre-W-^e-W-^,

(9.35)

and these are numerically expensive integrals. A way to reduce the computational

effort is to find a least-square

fit

to these exponentials using Gaussians,

Ίι

ΣΑ

ιν

β-

^-«>')\

(9.36)

where all the coefficients are chosen to make

look as much like the desired ex-

ponential as possible. One develops a whole family of functions of this form,

71)72

··

IK,

(9.37)

Hartree and Hartree-Fock Equations 241

designed to look like best guesses about the actual shapes of the electronic wave

functions. The functions 71 ...

7A-

are supposed to provide a basis in which to

describe an arbitrary function, so K 3> N, but one hopes by choosing the basis

functions well to make

Λ^

as small as possible. The different 7's are not orthonor-

mal, in general. So one writes

Ψι

= Σ

i^k,

(9.38)

k=\

and by substituting this form into the Hartree-Fock equations finds a large nonlin-

ear matrix equation for the coefficients B^. One also must write \/\r\

—

?2|

as a

sum of products of basis functions, so that inside the Coulomb and exchange in-

tegrals one has products of four basis functions to integrate over. The size of the

matrix is K x K, which illustrates the importance of choosing as small a basis set

as possible. The equations are solved iteratively, starting with a guess for a set of

N wave functions. Using these, one calculates all of the Coulomb and exchange

integrals. The result is a linear K x K matrix equation, which is diagonalized us-

ing standard numerical routines developed for this purpose. After carrying out the

linear algebra, there is a new set of K wave functions. Choosing the N of lowest

energy, one again calculates all the Coulomb and exchange integrals, continuing

until, if all goes well, the calculation has converged. The most time-consuming

part of the process is the calculation of all the Coulomb and exchange integrals,

because there are of order K

of these (one has to do an integral of the product of

any four basis functions). Even if all the necessary integrals are done in advance

and stored in memory, just calling up the results to add them together is a K

pro-

cess.

These considerations provide a simple explanation for the fact that quantum

chemists use a huge portion of the world's supercomputer resources.

What sorts of results may be obtained by carrying out Hartree-Fock calcula-

tions? One first of all has an approximation to the ground-state wave function that

can be used to calculate such experimentally measurable quantities as the dipole

moment. Second of all, one has all of the excited states that were found while

diagonalizing the Hartree-Fock Hamiltonian, but not included in the ground state.

The lowest lying excited state provides an estimate of the ionization potential of

an atom or molecule. Third, one can calculate how the total energy of a molecule

varies with the external potential and in this way try to calculate the equilibrium

geometry of a molecule. Table 9.1 gives some representative examples of the re-

sulting accuracy compared with experiment.

Hartree-Fock does not do particularly well at computing dipole moments. It is

able to calculate only to within about 0.1 in atomic units and may even get the sign

wrong, as in the case of CO. Bond lengths come out better than dipole moments,

but ionization potentials are again obtained only at about a 10% level. In the case

N2,

Hartree-Fock incorrectly identifies the first excited state. The molecules in

this table have 10 electrons, and therefore they are rather simple test cases. Ones

conclusion must be that Hartree-Fock provides only a qualitative guide and is not

adequate for precise molecular calculations. To obtain better accuracy, chemists

242 Chapter 9. Electron-Electron Interactions

Table 9.1. Theory versus experiment for Hartree-Fock

Molecule CH

Q FH CO

Bond length (À): Hartree-Fock 2.048 1.890 1.776 1.696

Bond length (Â): experiment 2.050 1.912 1.809 1.733

Ionization potential (eV): Hartree-Fock 0.546 0.428 0.507 0.650

Ionization potential (eV): experiment 0.529 0.400 0.463 0.581

Dipole moment (e À): Hartree-Fock 0.653 0.785 0.764 -0.110

Dipole moment (e Â): experiment 0.579 0.728 0.716 0.044

Bond lengths, ionization potentials, and dipole moments, comparing Hartree-Fock,

and experiment. In the delicate case of CO, even the sign of the dipole moment is

incorrect. Source: Szabo and Ostlund (1982).

have moved to considerably more elaborate wave functions, including sums of

large numbers of determinental wave functions. The name given to effects that

cannot be calculated within Hartree-Fock is

correlation:

the tendency of electrons

to respond to the details of each others' presence above and beyond the implica-

tions of the Pauli principle and Hartree-Fock theory. Pines posed the problem of

correlation this way:

... we have neglected the effect of correlations in the position of the elec-

trons introduced by the Coulomb interactions. The correlations will tend

to keep the electrons apart, so that the energy of the system is further re-

duced This gain in energy will be designated the correlation energy

here.

Thus, we define the correlation energy as the difference between

the energy calculated in the Hartree-Fock approximation and that calcu-

lated using any better approximation ... the Coulomb interaction, which

on physical grounds makes an independent particle model appear un-

likely, on closer mathematical investigation renders it untenable in prac-

tice.

—Pines (1955), pp. 373-374

These observations end the brief view of the program of doing chemistry by

computer. Chemists have found it crucial to find approximations much better

than Hartree-Fock in order to obtain satisfactory accuracy, while the portion of

the physics community that carries out calculations of the electronic properties of

crystals by necessity contents itself with approximations that are worse. However,

the two communities have been converging toward a common approach based upon

density functional theory (Section 9.3).

9.2.4 Hartree-Fock Equations for Jellium

The Hartree-Fock equations can be solved exactly for jellium, a name that may

have come from a comment by Tonks and Langmuir

( 1929)

concerning plasmas