Marder M.P. Condensed Matter Physics

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Hartree and Hartree-Fock Equations 243

that "when the electrons oscillated, the positive ions behave like a rigid jelly with

uniform density of positive charge ne." Jellium is just a collection of electrons, into

which ions are introduced as a spatially uniform background to maintain overall

charge neutrality. The Hartree-Fock equations are in this case solved by plane

waves. For N electrons in a volume V they take the form

Kinetic energy Interaction with ions

—h

N f e

2m V J \r—r2\

+Φι

(7)

iJMif-

- £

ΧιΧ

^(7)

Coulomb interaction Exchange interaction

Insert a plane wave of the form

„iki-r

(9.39)

(9.40)

The allowable set of k's as in Section 6.3 is given by imposing periodic boundary

conditions on the system, producing the usual conclusion that the density of k states

per volume and per spin is

1/(2π)

The kinetic energy term in Eq. (9.39) produces

-^Φι^)-

(9.41)

The interaction with the ions and the Coulomb interaction cancel against each other

because \φ]{τ2)\

— 1/"V. The only calculation remaining is the exchange interac-

tion, which becomes

e*J

f dfi

e'^i-^'

;

VV J V \r-r

= e\

N / S^ _, Changing the variable of integration to (9 43)

J V r'

XhXi

r' = r

-r.

' '

7=1

φι

y^ — — =i δ

χι χ

■

The Fourier transform of 1/r is Α-κ/k

. (9.44)

j=x

V\k-kj\

SÎ,

Assume that states are occupied up to a Fermi

wave vector and turn the sum over j into an (Q

(27r)3 1,2 I I,2 _ yi, Ί, integral. However, the restriction that χι =

\ ) K

-f K ΔΚ- Kj

cuts in

half

the

density of states in Section

6.3.

= e φι (r) ——- ί k

— kf J In < — > + 2kikf . Write k-k,= kk, cos

Θ,

integrate

2νΓ

*/ L

' ykp-ki) J

firstdcos0ythenk

(9.46)

244

Chapter 9. Electron-Electron Interactions

So plane waves solve the Hartree-Fock equations, and the energy of state / is

(9.47)

vP-iP- 1P

= ^-—k

F(k

2m π

where the Lindhard dielectric function F(x) is

F(x)

('-*>{^H

(9.48)

This expression has certain peculiarities. In particular, dE/dk is infinite at the

Fermi surface, which means that the group velocity of electrons at the Fermi sur-

face should be infinite, a conclusion that would have catastrophic consequences

for transport properties were it not both unphysical and incorrect. The source of

the difficulty is visible in Eq. (9.44); it arises from the divergence when ki

—>

kj,

which in turn arises from the very slow decay at long distances of the Coulomb

interaction

/1r

—

r'|. An effect that the Hartree-Fock approximation has missed is

screening. The effective interaction of two distant electrons actually falls off much

faster than \/\r

—

7*21

because the many electrons in between them adjust their po-

sitions to hide the distant electrons from one another, but Hartree-Fock misses this

fact because the nature of the approximation leads it to treat only two electrons at

a time. Screening is the subject of Problem 3 and is reviewed by Echenique et al.

(1990).

The energy of a collection of independent electrons is given by a sum of the

one-particle energies. This simple relation does not hold for the energies £/ of

Hartree-Fock theory. The sum of all £; counts terms resulting from the Coulomb

and exchange interactions twice. The correct expression for the total energy is

therefore, as shown in Problem 2,

tfk]

-k

7Γ

F from Eq. (9.48).

(9.49)

-N

!*-4

7Γ

The integration is straightforward, although (9.50)

care needs to be taken as k

—>

kp.

9.3 Density Functional Theory

While in the end it will produce formulae that look a great deal like Hartree-Fock,

density functional theory has a rather different flavor. In principle, it is an exact

account of the many-body wave-function, while in practice it spurs on varied types

of daring approximations.

The starting point of the theory is the observation of Hohenberg and Kohn

(1964) that electron density contains in principle all the information contained in

a many-electron wave function. The electronic density of a many-electron system

Density Functional Theory

245

at point 7 is defined

to be

n(r) =

(^\J2ô(r-R

)\^)

(9.51)

d?\...

**(r

h . . .

ΓΛ,)5(Γ-Γ,)Φ(ΓΙ

. . .

(9.52)

Hohenberg

and

Kohn pointed

out

that

one knows

the

density

the ground state

many-electron system,

one can

deduce from

it the

external potential

which

the electrons reside,

up to an

overall constant.

must

kept

mind that the only

ways

which two many-electron problems

can

differ are

in the

external potentials

and in the

number

electrons that reside

in the

potentials. According

this

result, both

these external parameters

are

determined

by the

electron density,

one

can say

that

the

density completely determines

the

many-body problem. This

statement

surprising, because

the

density

is a

real function

of a

single spatial

variable, while the complete quantum mechanical wave function needs

variables

for

its

description.

To prove

the

claim, suppose that

it is

false. Suppose that there exist

two ex-

ternal potentials

U\ (?)

and

(?)

that result

in the

same charge density. Call

the

Hamiltonians that result from them

3ΐι

and

let

the ground state wave func-

tions

for

the two Hamiltonians

be Φι and Φ

Assume that the ground states

the

two Hamiltonians

are

nondegenerate; this

technicality that will

avoided

completely

by an

improved version

the argument

to be

presented shortly. Then

the ground-state energy

"K\

realized only

by Φι. So

£ι

=(Φι|ίΚι|Ψι)

(Φ2ΙΉ1ΙΦ2) Because

is not the

ground (9.53)

state

"K\.

ε,

(Φ2|Ή

|Φ

<Φ

|(ΛΙ -Λ

)|Φ

)

(9.54)

< £

+ / drnir)

\U\ (?)

(?)}

Two

Hamiltonians with

the

same (9.55)

number

electrons

can

differ only

the potential.

However,

one can

equally well switch indices

and 2 to

obtain

£i +

n(?)

(?)

U\ (?)]

(9.56)

Adding Eqs. (9.55)

and

(9.56) gives

ει + ε

<ε

+ ε

, (9.57)

which

is a

contradiction. Therefore

the

potentials U\

and i/

must

be the

same.

result

these observations, one can imagine being handed many different

solid systems, told that they consist

electrons interacting with

one

another

via

Coulomb potentials, moving

potential U, and obeying Schrödinger's equation.

in each case one

given the charge density, as

function

space,

then

principle

one

can

deduce

U and

solve

for all

properties

the system. Thus

one can

think

246

Chapter 9. Electron-Electron Interactions

the ground-state energy £, kinetic energy T, and so on as being functionals of the

density, and can write the following, to indicate this fact:

Here n really means n(r), a function of space,

ε\η] = Τ\n] +

U\n}

In].

the

ener

gy*v

the

P°

tentialdue

(9.58)

L J L J ee

' to ions, and

Uee

is the Coulomb interaction

between electrons.

Hohenberg and Kohn next observed that if one can find the functional

[n],

then the true ground-state density n{r) minimizes it, subject only to the constraint

that

dr n(r) = N. (9.59)

This assertion is proved by noting that if

one

starts with the "wrong" density,

AÎ2,

for

Hamiltonian Ή\, then «2 should really be associated with a different Hamiltonian

Ä2,

which has ground-state wave function Φ2 and which does not minimize

<φ

|:Κι|Φ2> = ει[«2]. (9.60)

Only n\ minimizes

£1

[n{\,

and this

just what needed to be shown.

The most intriguing feature of this view of the many-body problem is that one

can write the energy functional £ as

E[n]

= j drn(r)U(r)+F

[n}, (9.61)

where

FHK

is the sum of kinetic and Coulomb energies:

K[n}

T[n]

[n}.

(9.62)

The functional

FHK

does not depend upon the potential U(r), and so it constitutes

a universal functional for all systems of N particles; if one only could find this

functional, it would solve all many-body problems for all external potentials U.

Second Derivation of Energy Functional. It is desirable to have a second demon-

stration of the existence of the energy functional £ for two reasons. First, the

demonstration given so far relies crucially upon assuming that ground states are

nondegenerate. It is worth knowing whether this requirement is a dull technicality

or a fatal flaw. Second, it might be possible to find a charge density n(r) that cannot

result from any conceivable potential U(r). In this case, the functional

£[n]

is not

yet even properly defined. Both these points are answered by defining a functional

F[n]

that is the minimum over all wave functions producing density n(r):

F[n]

= min (*|r +

|*).

(9.63)

This functional F can be defined even if there does not exist a potential U that

would produce density n for some quantum-mechanical ground state. The process

Density Functional Theory

247

of finding the ground state £o of a many-body system may therefore be carried out

in the following way:

= min(*|7

+ £/ + t/ee|1') (9.64)

Φ Minimize over all wave functions

•

r ·

/,τ,ΐτ

i H i H

Ι,ΤΛΙ

Ψ that produce density n, and

/ri

^c\

min[min(*|r + i7 + i/ee|*)J

then minimize over all densities. (9-65)

πιΐη(Φ|7' + £/

β|Φ)

+ [

U(r)n(r)d.

f>]

+ [

U{r)n(r)dr

ψ^„

min

Because the potential U depends (9.66)

only upon the density.

(9.67)

= mia E\n]. (9.68)

The problems

defining

in general and accommodating degenerate ground

states are therefore solved simultaneously.

In principle, what has been accomplished here is enormous. In principle, there

exists a universal functional

F[n]

that needs to be found once and for all. One adds

to it any particular set of

nuclei,

in the form of

the

potential £/(?), and then has only

to find the function n(r) that minimizes it in order to solve the full complexities of

Schrödinger's equation. As is often the case, the gap between accomplishments in

principle and in practice is also enormous. The functional F is a magical lookup ta-

ble that is supposed to solve all of quantum mechanics upon request. No one knows

the true F, and no one ever will, so it is replaced by various uncontrolled approx-

imations. The decades-long process of discarding approximations that disagree

with experiment and improving on the ones that are more successful has encoded

a great deal of accumulated physical insight within apparently simple functional

forms.

9.3.1 Thomas-Fermi Theory

The simplest approximation providing an explicit form for the functionals F

[n]

£[n]

is Thomas-Fermi theory. The basic idea of the theory is to find the energy of

electrons in a spatially uniform potential as a function of density. Then one uses

this function of the density locally even when the electrons are in the presence of

an external potential.

The problem of electrons interacting by

Coulomb interaction in

uniform

background (jellium) is unfortunately not possible to solve exactly, except in the

limit of high density. It was solved approximately with the Hartree-Fock approxi-

mation in Section 9.2.4. The strategy here will be to use the results of the Hartree-

Fock approximation to obtain an approximate account of any term in Schrödinger's

equation that cannot automatically be expressed in terms of density. The first term

of this sort is the kinetic energy; the kinetic energy of a single k state is H

/2m,

and one must sum over all available k, so that the total kinetic energy is

T = V [dk\—

See Eq. (6.15) for the definition of

[dk]

(9.69)

248

Chapter 9. Electron-Electron Interactions

^-=V— -(3ττψ\

/\ Using Eq. (6.29).

.70)

2m5ir

The electron-electron interaction results within the Hartree-Fock approxima-

tion

in two

terms.

The

first part,

the

classical piece,

is a

functional

electron

density automatically, because

equals

dr/

(9.71)

\r\

-r

The second piece, the exchange term, was evaluated in Eq. (9.50) for jellium, where

it was shown to

/3\'/

—TV

= —V— — ) en.

ust

rewrite

Λ?

and

kp in

terms

density. (9.72)

π 4 \πJ

The idea

Thomas-Fermi theory

is to

assume that

in a

system where

the

charge density

is not

uniform,

but

varies slowly, the kinetic energy

and

exchange

term will

given

the same expressions given above, but evaluated locally and

integrated over all space. The kinetic energy

T[n]

jdr^

-{3n

)

2/3

/\r), (9.73)

and the exchange energy

= - j

dr\ß\

n^(r).

(9.74)

So finally

first explicit density functional

£[n}

= J-ï

(3π

)

2/3

drn

{?)+

drn{r)U(r)

' /

ipm -far 1β)

Mfiffl.

<«5)

\r\

-r

\ J 4 \ιτ

Actually, this functional constitutes Thomas-Fermi theory only

the last term,

due to exchange,

simply omitted; with the exchange term included, the theory

called Thomas-Fermi-Dirac.

The conventional Thomas-Fermi-Dirac equation

obtained by writing down

the condition that the functional (9.75) be minimized by n, subject

course to the

constraint that the integral

be N. One has

δε

The

chemical potential

arises

as the La-

(9.76)

δη

Cf) grange multiplier enforcing the constraint that

density be conserved.

\y\r)

U(r)+

[α7

^-(

\\^(7)

μ.

(9.77)

/ J

—

\πJ

Density Functional Theory 249

Omitting the last term on the left-hand side of

Eq.

(9.77) gives the Thomas-Fermi

equation.

Thomas-Fermi theory is simple, but not particularly accurate, so its solution

is relegated to Problem 4. One result of solving Eq. (9.77) is that the energy of

an atom of nuclear charge Z is approximately

—

1.5375Z

Ry. For small atoms,

this result is large by a factor of two when compared with Hartree-Fock; even for

an atom as large as Xe (Z = 54), where one has better hope for an approxima-

tion based on slowly varying charge distributions, the equation is still in error by

20%.

The Thomas-Fermi-Dirac equation gives energies that deviate even further

from reality. Another disturbing feature of the equations is that Thomas-Fermi

and Thomas-Fermi-Dirac equations predict that atoms never bind into molecules;

the energy of a supposed molecule is always lowered by pushing the nuclei further

apart. This subject is discussed by Lieb (1981).

Thomas-Fermi theory smooths out the charge distribution, because it has no

way to know that electrons arrange themselves into separate shells. Thomas-

Fermi-Dirac is even less physical; it predicts that at some finite radius the charge

distribution drops instantaneously to zero. There have been attempts to develop

improved theories of this type by bringing in dependence upon gradients of the

charge distribution. The original Thomas-Fermi theory is most accurate for nearly

uniform charge distributions, so it is natural to work out the corrections that would

occur for an electron gas in a linearly varying potential, a quadratically varying

potential, and so on, using these results to construct an expansion in terms of gra-

dients of the density. However, none of the theories of this type has gained wide

usage.

9.3.2 Stability of Matter

A different type of application of density functional theory is to address the ques-

tion of the stability of matter. That is, why does the attraction between electrons

and nuclei not lead to a collapse in which the electrons crowd in upon the nuclei,

producing solids with an atom every 10^

m rather than every 10~

m? On this

point there is no need even to turn to experiment; common experience says that

elements and compounds are stable. However, it is interesting to find the features

of quantum mechanics that make the obvious possible.

To the obvious fact corresponds an obvious answer; the Heisenberg uncertainty

principle forbids electrons to come too close to nuclei. The momentum of an elec-

tron confined within a box of radius a must scale as h/a, so the kinetic energy must

scale as h

/ma

. However, the potential energy to be gained by coming close to

the nucleus only scales as

—e

ja, so the kinetic energy term seems to win. While

this argument is essentially correct, it does not provide the formal tools needed to

show that matter is stable. The tools need to provide a good estimate of the kinetic

energy of electrons based upon their density, and the Heisenberg uncertainty prin-

ciple does not do a good job of it. The precise statement of Heisenberg's principle

250 Chapter 9. Electron-Electron Interactions

is that

drh

\V^Y

drr

\^\

The root mean square momentum times the

. root mean square fluctuation in position of /n ηο\

— 4 ' a wave function must have a product greater ^ ' '

than

fi/2; ψ is normalized, and is assumed to

centered at the origin.

it can be used to bound electron kinetic energies T through

T[n}>

J dr r

n(r)

(9.79)

-$-

Figure 9.2. Profile of a wave function for which the Heisenberg uncertainty principle

provides a particularly poor estimate of

kinetic

energy.

However, (9.79) can be satisfied by wave functions looking like the one shown

in Figure 9.2. The broad base can be chosen to contain, say, half the wave function

probability and, by being made of width L, can turn Eq. (9.78) into no more than

the weak requirement that kinetic energy be greater than H

/mL

. In the meantime

if the central peak in Figure 9.2 is placed over a positive ion, it can lead to huge

amounts of potential energy that the weak estimate (9.79) does not counteract.

The vague feeling that a wave function shaped as in Figure 9.2 is cheating does

not remove the need to find a way to improve the argument that matter does not

collapse. Lieb showed that for a collection of yv electrons interacting with each

other and with nuclei, the kinetic energy of the electrons is bounded below by

T[n]>

W- 9.116

drn

5/3

(9.80)

2m (8TT)

Note the very strong resemblance of

this

exact result to the approximate expression

for the kinetic energy of a collection of fermions obtained by using Hartree-Fock

theory, as shown in Eq. (9.70). The density n appears raised to the 5/3 power in

both cases.

The arguments leading to Eq. (9.80) are rather elaborate, and those that follow

from it to prove the stability of matter are no simpler. Therefore, rather than devote

more attention to Eq. (9.80), this section will show how an identical but simpler

technique can be used to prove the stability of

atoms,

and in particular the hydrogen

atom.

Density Functional Theory 251

As shown

Problem

6, the

kinetic energy

any wave function

T\n}

= — Jdr\V^\

(9.81)

obeys

the

inequality

where

\n]>^f*

drn

(9.82)

ΛΓ

= 3(ΤΓ/2)

(9.83)

and with

the

density

given

η(7)

\ψ(7)\

(9.84)

Therefore,

for

the hydrogen atom,

the

energy

a wave function

ψ is

bounded

below

*-K, [

drn^-i

dr^à.

(9.85)

J J r

The variational equation that

can be

used

minimize this functional subject

to the

constraint

λ(1-

drn)=0

(9.86)

n I (7)

—

g /r + λ = 0

Here

λ is a

Lagrange multiplier enforcing the

(9 87)

2m '

constraint

(9.86).

There

also

the

constraint that

n has to be

positive.

One way to

enforce this

constraint

is to

define

n = σ

and

carry

out

variations with respect

to σ. One

for-

mally finds

this

way

that

n = 0 is an

acceptable local minimum

the functional

Eq. (9.87);

course

one

can't have

—

everywhere

Eq. (9.86)

violated,

but

instead

n{7)=

i{6m[e

/r-X}/(5K

)}

3/2

forr<e

(g gg)

[

else

One fixes

requiring

the

integral

to be 1, and one

finds that

3me

7Γ

i /-,

5ΡΕ

(

Τ>

,/3:

<989)

inserting this form

into Eq. (9.85) gives

lower bound

to the

energy

9me

, -Ko«

6me

12„

(2ττ

)

= --— = - -Ry.

(9.90)

\0h

5 h

The correct answer

is of

course that

the

ground-state energy

— 1

Ry, so the

esti-

mate

is not so bad.

252 Chapter 9. Electron-Electron Interactions

In this way, without explicit solutions of Schrödinger's equation, one can place

limits upon the total energy to be gained through compression in the presence of

Coulomb forces. Similar arguments prove the stability of matter using Eq. (9.80).

What is really crucial is the fact that since one is considering an assembly of elec-

trons,

the Pauli exclusion principle forbids them from occupying a single lowest-

energy state together. As a result, the kinetic energy penalty for compacting elec-

trons grows as w

rather than just as n. The arguments by Lieb (1976) and Lieb

et al. (1997) showing that many-electron systems are stable are rather elaborate,

but the basic flavor is contained in use of the inequality for the very simple case of

the hydrogen atom, and the end result is the same; matter does not implode because

it would cost too much in kinetic energy.

9.4 Quantum Monte Carlo

9.4.1 Integrals by Monte Carlo

Despite the intrinsic difficulties of solving Schrödinger's equation for many parti-

cles,

there has long been a tenacious effort to solve it with as few approximations

as possible. The most impressive progress has come from quantum Monte Carlo

methods. The starting point is the Monte Carlo method (Section 5.4.1), now ap-

plied to the computation of integrals in large-dimensional spaces.

The Monte Carlo idea can be used to compute the average of any function in a

probability distribution. Consider a function g(x) where x may be a vector in a n-

dimensional space. Suppose that the probability of being at point x is iP(x), where

to be a legitimate probability distribution

3>(x)

> 0; / d

x ?(x) = 1. (9.91)

The average of g is

g =

xg(x)nï)-

(9-92)

To compute this average, follow the same steps as in Section 5.4.1

Pick a starting value, xo, and compute

= g(xo)-

Pick a random vector Δχ, all of whose components range between — e and e,

where e is some constant.

Compute 7

(xo

+ Ax).

(a) If 7(xo + Ax?)/!P(jeo) >

then accept the move: 2\

—

+ Δχ.

(b) If ^(fo +

AX)/7(XQ) < 1, choose a random number p between 0 and 1.

i. If

iP(xo

Ax)/CP(Jco)

< p, then again accept the move so that x\

=XQ

AX.

ii.

Otherwise, reject the move and take 1\ =

XQ.