Marder M.P. Condensed Matter Physics

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Quantum Monte Carlo

253

In any case, compute g\ =

g(x\),

and accumulate the sum of go and g\ in a

variable G.

Return to step

and repeat M times, continuing to assemble the sum of g in

the variable G.

Finally, compute

G V

, 9:

(9.93)

G _ Σ£ι Si

° M M

An estimate of the convergence of the integral is provided by the standard error,

1 γ(α.-ο)2

ΔΡ = \ =—. Actually, this expression underestimates the (9.94)

\/~M V M — I error in the integral because results of suc-

cessive Monte Carlo steps are not completely

uncorrelated.

Not only averages, but any integral can be computed this way. To integrate f(x),

choose any probability distribution obeying Eq. (9.91) and write

J d"x f(x) = J d"x 7^n% =

JW/W)-

(9-95)

Problem 7 provides a simple example of how to use this method to compute

integrals.

9.4.2 Quantum Monte Carlo Methods

Starting with the ability to compute high-dimensional integrals, the quantum Monte

Carlo method sets down two paths,

variational

Monte Carlo and diffusion Monte

Carlo.

Diffusion Monte Carlo is the more powerful of the two techniques, but it is

quite complicated and will not be described here. Essentially the method involves

constructing random walks whose probability is determined by the potential en-

ergy terms of the Hamiltonian. However there are complications due to the fact

that wave functions change sign under interchange of particles. This fermion sign

problem means that as the random walkers walk along, the sum one would like to

accumulate gets both positive and negative contributions that nearly cancel, leading

to a horrible loss of numerical accuracy. See Foulkes et al. (2001) for a descrip-

tion of clever ideas used to evade this problem and details of how to implement

diffusion Monte Carlo.

Variational Monte Carlo is more straightforward. Let y\ be shorthand for the

space and spin variables Τ\σ\. Choose a trial wave function ty(y\ . . .y^) and com-

pute the ground state energy

„ ( d

y^*(y

{

. . .yM^iyx . . .y

)

t, = -—

—-. Integrals over y are shorthand for integrating

I d^y Φ*(7ι . · . .y/vì^CVl ■ · · J/v) over spatial variables rand summing over spin

variables σ.

(9.96)

254

Chapter 9. Electron-Electron Interactions

Varying parameters

in Φ, the

goal

is to

find

the

lowest possible energy,

and

there-

fore

estimate

of the

ground state energy.

The

Hamiltonian

might,

for ex-

ample,

be the

fundamental Hamiltonian

Eq. (9.1) for a

number

classical nuclei

interacting with quantum mechanical electrons.

The

nuclei

and

some valence elec-

trons might

replaced

pseudopotentials (Section 10.2.1).

The

calculations

might

carried

out in a

periodic system where each cell contains

electrons

and

M nuclei, and the whole many-body system repeats periodically to simulate a solid.

all

cases,

the

basic task

is to

come

with

good

variational wave function

possible,

and

then

compute

the

integrals

Eq. (9.96).

The first ingredient

conventional trial wave functions

is a

solution

of the

Hartree-Fock equations

Eq.

(9.34)

for the

Hamiltonian

one has

decided

solve.

This numerical problem is

not

simple, but techniques to address

are advanced and

described

Section 9.2.3. Call this wave function

^HF(JI · · · 9N)-

There would

be no sense using

<]/HF

itself

the trial wave function

Eq. (9.96) since the energy

that would come

out in the end

would have

to be the

energy

in the

Hartree-Fock

equations.

The

main defect

Hartree-Fock wave functions

that they

do not

account

for

correlations between electrons. Correlations

can be

included

using

wave functions

the

Jastrow

form, which means

Φ(?ι ...y

)

= ΦΗΡ(?Ι · ·

■

)e^>=>

x(?

H Σ

Β(ΑΛ)

(9.97)

The functions

and u are varied to make the energy

Eq. (9.96) as low as possible.

A constraint upon

u is

that

must obey

the

cusp conditions. Assume that

u is of

the form

u(y

yj) =

σ2

{ni).

Where

\r,

- rj\

(9.98)

Then

spins

are

opposite

(σ\ φ ai),

drtj

—

ÜQ

the Bohr

radius.

(9.99a)

while

they

are the

same

dnj

~ (9.99b)

2ao

These relations

are

derived

Problem

9.4.3 Physical Results

The most significant result obtained from Quantum Monte Carlo concerns

the be-

havior of jellium,

the

interacting electron

gas

described

Section 9.2.4. Calcula-

tions concerning this basic system go back to the earliest days

quantum mechan-

ics.

The

results

are

usually reported

terms

the electron separation

defined

Eq.

(6.30) over

the

Bohr radius ao. When r

/ao

large,

the

electron

gas is di-

lute,

while when

it is

small

the gas is

dense. Somewhat surprisingly,

the

problem

Kohn-Sham Equations

255

is easiest in the limit where the gas is very dense. In this limit, the Hartree-Fock

approximation is accurate and from Eq. (9.50) gives an energy per particle of

?^_A

5JL

(

^/4__Lì!(^y

2s.

mm)

N 5 2m 4π 5 2mal

4 ' r} 4π a

\ 4 ) r

For a high-density gas, the largest energy comes from the kinetic energy of elec-

trons pushed into high-energy states by the Pauli principle. Coulomb interactions

of electrons with each other cancel against the interaction with background charge,

so the next contribution comes the exchange energy, is negative, and is smaller by a

factor of

/üQ.

These results were confirmed by Gell-Mann and Brueckner (1957)

using methods of quantum field theory, and several additional terms in powers of

/ao were added to the sum.

However, there had long been indications that an expansion in powers of r

/ao

would not fully capture the behavior of jellium. Bloch (1929) performed further

calculations using Hartree-Fock theory that predicted that at a certain electron den-

sity, the electrons should become completely spin polarized and ferromagnetic.

The Hartree-Fock approximation was not thought to be reliable at the electron den-

sity at which this prediction was made, so it remained clouded in uncertainty. Soon

after, Wigner

( 1934)

examined the problem in the limit where the electron density

became very low, r

/ao was large, and concluded that electrons should behave like

classical particles and condense into a crystalline lattice, the Wigner

crystal.

Ob-

servation of such Wigner

crystals

in semiconductors with low conduction electron

densities is discussed by Field et al. (1988).

The electron densities where these transitions occur were first calculated by

Ceperley and Alder (1980) using Diffusion Monte Carlo. To find the energy of

electron gases, the calculations employed trial wavefunctions with Slater determi-

nants of plane waves times Jastrow functions as in Eq. (9.97). To find the energy of

Wigner crystals, the electron wave function was instead taken to be a Slater deter-

minant of Gaussian peaks centered at bcc lattice sites, again multiplied by a Jastrow

function. The transition from electron gas to Wigner crystal occurs when the best

wave function of the second type has lower energy than the first. The calculations

continue to be refined, and a phase diagram of jellium as a function of temperature

and electron density appears in Figure 9.3.

9.5 Kohn-Sham Equations

Quantum Monte Carlo calculates properties of solids exactly, but is far too time-

consuming to use in all but the simplest cases. Even the Hartree-Fock Eqs. (9.34)

are too slow to solve for systems with many atoms. Density functional theories

such as the Thomas-Fermi equation are suitably quick, but unacceptably inaccu-

rate.

The equation most frequently used in practice for large numerical calculations

in solids is a cross between the two and was introduced by Kohn and Sham (1965).

This theory constructs a density functional theory in such a way as to treat elec-

tron kinetic energies well, reproduce selected results from Quantum Monte Carlo

256

Chapter 9.

Electron—Electron

Interactions

Figure 9.3. Phase diagram for jellium, as deduced by Quantum Monte Carlo methods.

The onset of zero-temperature Wigner crystals at r

/ao =

106

is the most reliable re-

sult; other details of the diagram should probably be viewed as more tentative. [Source:

Càndido et al. (2004),

exactly, yet retain computational speed. In order to efficiently capture the kinetic

energy associated with electronic configurations, Kohn and Sham retreated slightly

from the plan of writing all material properties as functions of the electron density,

and proposed using instead a set of N single-electron wave functions

ψι (r)

as the

main ingredients, obtaining the density from them by

η(?) =

Σ\Ψι{7)\

(9.101)

ι=ι

In Kohn-Sham equations, the kinetic energy term of the energy functional is

2>] = Σ^ J

^/|

(9-102)

Apart from the kinetic energy, the energy functional has the same basic form as

the Thomas-Fermi theory, Eq. (9.77). Instead of varying with respect to charge

distribution, one now must vary with respect to the wave function, because one

knows the density as a functional of the wave function but does not know the wave

function as a functional of the density. Varying with respect to ψ* gives

~vViW

U(r) +

„, e

n{f) ô£

{n)

———■

+ —

\r —

r\ on

i>i(r)

= £iM'

)· (9-103)

Kohn-Sham Equations 257

Table 9.2. Hartree-Fock and LDA compared with experiment

Atom

LDA (KS)

-2.83

-7.33

-128.12

-525.85

Hartree-Fock

-2.86

-7.43

-128.55

-526.82

LDA (PZ)

-2.92

-7.50

-129.27

-528.39

Experiment

-2.90

-7.48

-128.94

-527.60

Total energies of atoms in Hartrees (27.2107 eV), comparing Hartree-Fock, the lo-

cal density approximation, LDA (KS) as first described by Kohn and Sham, the local

density approximation LDA (PZ) as improved by Perdew and Zunger (1981), and ex-

periment. Source: Tong and Sham (1966) and Perdew and Zunger (1981).

The class of approximations of the form (9.103) is referred to as the local density

approximation (LDA). If one uses the exchange potential derived in Eq. (9.72),

then

-|Vw(r)

„ , -<e

n(r") 2 ( * ,-Λ

1/3

U{r) + I df —A=7 -e

-nÇ"

\r—r\ \π

{7) =

{7).

(9.104)

Apart from a much disputed factor of 3/2 in front of

the

exchange-correlation term,

Eq. (9.104) was first written down by Slater (1951).

But the exchange-correlation functional Z

does not have to be approximated

by Eq. (9.72). In principle it could compensate for all approximations and be set up

to give exactly the right answer for interacting electron problems. However some

approximation must be adopted for practical computation. The main idea in prac-

tice is to use the exchange-correlation energy of jellium, whose exact ground-state

energy is known as described in Section 9.4.3. That is, the Kohn-Sham equations

can be set up to give the exact answer when the external potential U vanishes, and

once this is done they have proven wildly successful even after U is turned back

on.

The Kohn-Sham equations achieve reasonable correspondence with experi-

ment when applied to single atoms, as shown in Table 9.2. Calculations called

ab initio or first principles are usually based upon Eq. (9.103). The best LDA

calculations provide more accurate results than the Hartree-Fock approximation

and approach the accuracy demanded by quantum chemists; surveys comparing

computations in molecules with experimental results are provided by Curtiss et al.

(1998a,b). Many different types of approximations have been tried to bring den-

sity functional theory into as close correspondence with experiment as possible.

For example, the generalized gradient approximations (GGA) add extra derivative

terms.

See Martin (2004) for detailed descriptions of approximations currently in

use.

The pure research problem of painstakingly finding accurate solutions to the

electronic energy of jellium thus turned out to have much more practical impor-

tance than one might have expected from such a simplified model system. From

the quantum Monte Carlo calculations

jellium, it was possible to find the energy

258

Chapter 9. Electron-Electron Interactions

of the electron gas as a function of density. Simple functional fits to these results,

for example by Perdew and Zunger (1981), Lee et al. (1988), or Becke (1993),

produce the exchange-correlation functionals on which current density functional

codes are based. Papers containing these parameterizations win out over all recent

Nobel Prize-winning results, and are the most highly-cited papers in all of physics.

Problems

Hartree equations:

(a) Find the expectation value Fu of

Eq.

(9.1) in a state of form Eq. (9.6).

(b) Take the variation of

Fu,

subject to the constraint that each φ be normalized:

δψι(7) δφι(7)

Σ Ej j α7'φ*(7')φ

]

(7') = 0. (9.105)

that the Coulomb interaction term of the Hamiltonian becomes the same for

all wave functions. In this way, recover the Hartree equations. Notice that

demanding that each φ be normalized is sufficient to result in an orthonormal

set of functions.

Koopman's theorem:

(a) Evaluate the total energy of a collection of

electrons in jellium, obtaining

Eq. (9.49). You must add also the energy of the positive ionic background.

(b) Compare the result with the sum over / of the single particle energies £/.

energetic electron. Consider an atom with N electrons, and suppose it has

been solved in the Hartree-Fock approximation, producing wave functions φι

and energies £/. Suppose that when the electron of highest energy is removed,

the remaining N

— 1

wave functions continue to solve the Hartree-Fock equa-

tions,

and that the energies £/ do not change appreciably. Show that in this

approximation the ionization potential of the atom is £yy.

Screening: The idea of screening comes originally from electrolytic solu-

tions.

Imagine placing a charged ion into such a solution. At first the po-

tential due to the added ion extends its influence to the far reaches of the

system, dying off slowly as \/r. However, mobile ions nearby rapidly react

to the intruder, and the motions they make in response have the effect of al-

most completely canceling out its electric field, except within a characteristic

distance called the screening length. Because this phenomenon occurs gen-

erally for any assembly of charged particles, it can be studied in the context

of Thomas-Fermi theory, given by Eq. (9.77).

Problems

259

(a) Consider Eq. (9.77), omitting for simplicity the last term on the left-hand

side resulting from exchange. Suppose that

is the solution of this equation

when the potential U(r) vanishes. If now a small potential U(r) is added, find

the equation governing deviations Sn(r) of

the

density from perfect uniformity

to first order in U.

(b) Consider adding one extra electron to the uniform electron gas, and therefore

specialize to the case

U\r) =—. Yes, it is dubious to regard such a potential as (9.106)

T small, but the basic lesson is still correct.

Solve the linearized equation for δη by use of Fourier transforms. The answer

should be of the form

-r/i

δη . (9.107)

Identify the screening length ξ, and express it in terms of the Bohr radius and

the average volume per particle of the original uniform electron gas.

Thomas-Fermi theory: Use Thomas-Fermi theory to study an atom with

nuclear charge Z and Z electrons, so that the potential is

U{r) = -Ze

/r. (9.108)

(a) Rewrite Eq. (9.77), omitting the final term and substituting the function V(r)

for n(7) through

[~2mV]

3π

(b) Show that in the vicinity of the origin, V(r)

—>

—Ze

/r.

(9.109)

V(r) = X, r = DS, and b = — [-7-] Z ' . a

is the Bohr radius.

Show that for spherically symmetric solutions and s > 0 one obtains

1/2

3/2

1Η)

(d) Show that the boundary conditions on χ require χ(0) =

and

χ(οο) = 0.

260

Chapter 9. Electron-Electron Interactions

Thomas-Fermi numerics:

(a) Solve the two-point boundary problem in Eq. (9.111) numerically, and find

the function χ.

Figure 9.4. Solution of the

Thomas-Fermi equation.

2 4 6

Scaled distance s = r/b

(b) Find in particular the slope of χ at the origin.

—

Z electrons and a nucleus of charge

Ze, one obtains

"7;—

\M—7 — τ;— \M

■

Use the fact that the chemical potential u vanishes. (9.112)

oN oZ

(d) Show therefore that

n(r)

(9.113)

(e) Express the total energy £ in terms of the slope χ'(0) of χ at the origin, and

verify that the energy of the atom is

—

1.5375Z

Ry.

6. Variational estimates:

(a) Show that for any square integrable function,

Τ[ψ]= ( dr \Vip(r)\

j ί dr n

\ , (9.114)

where

and

^ = 3(π/2)

η(7) =

\φ(7)\

To carry out the demonstration, consider minimizing

/ Jr|V^(r)|

{fdrmr)\*}

6\l/3'

(9.115)

(9.116)

(9.117)

Problems

261

-ν

ψ-αψ

=0.

and show that ψ (assumed real) obeys a variational equation of the form

(9.118)

Find a formal expression for a. Verify by substitution that

,3a

is an arbitrary length.

(9.119)

solves Eq. (9.118) and that K

is given by Eq. (9.115).

(b) Use the inequality, valid when l/p+l/q = 1,

g{r)f(r)dx\ <

\m\

i(f)\

!/</

to show that

Τ[ψ] >K

ί drn

5/3

(7).

(9.120)

(9.121)

ground-state energy of the hydrogen atom from below by

—

12/5 Ry.

Integrals by Monte Carlo:

Write a routine to compute the volume of a unit sphere in n dimensions. That

is,

integrate the function

/<*)

{

For the probability

in Eq. (9.95), use

0>(*) =

< 1

else.

(9.122)

(9.123)

Take the constant e governing the size of Monte Carlo steps to be of order 1.

Use this routine to calculate the volume of a sphere within 1% in 1, 2, 3, and

4 dimensions. Monitor convergence of the integral through the standard error,

Eq. (9.94), which will however underestimate the error in the integral because

successive steps are not completely uncorrelated.

8. Cusp Conditions in Quantum Monte Carlo: Consider the local energy de-

fined for any wave function Ψ(?) and the Hamiltonian of

Eq.

(9.1) by

£/o(?)

φ(

-:ΚΦ(Γ)

(9.124)

(a) Show that if Φ(?) is the true ground state wave function, £/o has no singu-

larities, and in fact is constant.

262

Chapter 9. Electron-Electron Interactions

(b) Now take a wave function of the form Eq. (9.97) with u given by Eq. (9.98).

Assume that the Hartree-Fock wave function is nowhere singular, and that

X(y) has no singularities. Focusing on a pair of particles / and

write

Φ(?) =

-»(^/(3?

...^),

(9.125)

where / is defined to be absolutely everything in Φ except for the one term

involving w(r,-y) that has been singled out. Argue that singularities in the local

energy £/o can be eliminated by ensuring that there are no singularities in

1 ( H2

-«(ru)f

(9.126)

-"('■//)/

1 2m ' r

= 2^

(

'j)

+ ("'M)

2«'(^-)r

ψ - ^ j (9.127)

+ 2—e

^-^ + —.

2m r r

as r,j

—>

0. Assuming that lim

^o u'(r) and lim

^o u"(r) are finite, show that

the singularity in Eq. (9.126) can be eliminated by taking

lim u(r) = . iio is the Bohr

radius.

(9.128)

(d) If instead, however, particles / and j have the same spin, then / must be an

odd function of

r,-

and 7j. As r,-y

—>■

f^A-r

+ .... (9.129)

Show in this case that

lim u'(r) = — . a

is the Bohr

radius.

(9.130)

References

A. D. Becke (1993), Density-functional thermochemistry. III. The role of exact exchange, Journal of

Chemical Physics, 98, 5648-5652

Bloch

(

1929),

Note on the electron theory of ferromagnetism and electrical conductivity, Zeitschrift

für

Physik,

57, 545-555.

M. Born and J. R. Oppenheimer (1927), On the quantum theory of molecules, Annalen der

Physik,

84,

457^84.

J. Callaway and N. H. March (1984), Density functional methods: theory and applications, Solid

State Physics: Advances in Research and Applications, 38,

135-221.

L. Càndido, B. Bernu, and D. M. Ceperley (2004), Magnetic ordering of the three-dimensional Wig-

ner crystal, Physical Review B, 70, 094413/1-6.