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Recent prog ress in the f ield of electron correlat ion

G .

Senatore

Dipartimento di Fisica Teorica, Universita di Trieste,1-34014Grignano (TS), Italy

N. H. March

Department of TheoreticalChemistry,University ofOxford,Oxford, OX13UB, E ngland

Electron correlation plays an important role in determining the properties of physical systems, from

atoms and molecu les to condensed phases. Recent theoretical progress in the field has proven possible us

ing both analytical methods and numerical many-body treatments, for realistic systems as well as for

simplified models. Within the models, one may mention the jellium and the Hubbard. The jellium model,

while providing a simple, rough approximation to conduction electrons in metals, also constitutes a key

ingredient in the treatment of electrons in condensed phases within density-functional formalism. The

Hubbard- and the related Heisenberg-model Hamiltonians, on the other hand, are designed to treat situa

tions in which very strong correlations tend to bring about site localization of

electrons.

The character of

the interactions in these lattice mo dels allows for a local treatment of correlations. This is achieved by the

use of projection techniques that were first proposed by G utzwiller for the m ulticenter problem, being the

natural extension of the Coulson-Fischer treatment of the H

2

molecule. Much work in this area is analyt

ic or semianalytic and requires approximations. Howev er, a full many-body treatment of both realistic

and simplified models is possible by resorting to numerical simulations, i.e., to the so-called quantum

Mon te Carlo method. This method, which can be implemented in a number of ways, has been applied to

atoms, molecu les, and solids. In spite of continuing progress, technical problems still remain. Thus one

may mention the fermion sign problem and the increase in computational time with the nuclear charge in

atomic and related situations. Still, this method pro vides, to date, one of the most accurate ways to calcu

late correlation energies, both in atomic and in multicenter problems.

CONTENTS

I. INTRODUCTION

I. Introduction 445

II. Homogeneous Electron Assembly 446

A. Low-density Wigner electron crystal 446

B.

Density-functional theory of many-electron system 447

C. Local-density approximations to exchange and corre

lation 448

D .

Density-functional treatment of Wigner crystalliza

tion 449

III.

Localized Versus Molecular-Orbital Theories of Electrons

in Multicenter Problems: Gutzwiller Variational

Method 450

A. Coulson-Fischer wave function with asymm etric or-

bitals, for H

2

molecule 450

B.

Gutzwiller's variational method 451

C.

Local approach to correlation in molecules 452

D. Gutzwiller's variational treatment of the Hubbard

model 454

1. Exact analytic results in one dimension 455

2. Num erical results 457

E.

Resonating valence-bond states 459

IV. Quantum Monte Carlo Calculation of Correlation Energy 460

A. Diffusion Mon te Carlo method 461

B.

Green's-function Mon te Carlo technique 462

C. Quantum Monte Carlo technique with fermions:

Fixed-node approximation and nodal relaxation 464

D .

Mon te Carlo computer experiments on phase transi

tions in uniform interacting-electron assembly 466

E.

Calculation of correlation energy for small molecules 468

F. Auxiliary-field quantum Monte Carlo method and

Hubbard model 471

V. Summary and Future Directions 475

Appendix A: Model of Two-Electron Homopolar Molecule 476

Appendix B: Positivity of the Static Green's Function

G(R,R') 477

References 477

In this review, we shall consider some aspects of elec

tron correlation in molecules and solids on which recent

progress has proven possible. It is natu ral enough to

start with the homogeneous electron fluid: i .e., the so-

called jellium model of a metal, in which electrons in

teracting Coulombically move in a nonresponsive uni

form, positive neutralizing background charge. Here, the

ground-state energy is simply a function of the me an den

sity p

0

==3/477r

5

3

ao;

a

0

=ii

2

/m e

2

.

The pair correlations in

this system are by now relatively well understood, as is

the transition from a strongly correlated electron liquid

to an insulating Wigner electron crystal at a suitable low

value of the density, i.e., at large enough r

s

. This is a

spectacular manifestation of the effects of correlation.

Only in the last decade have quantum Monte Carlo

(QMC) simulations largely settled the critical value of r

s

for this transition, placing it at about 100. Some atten

tion will be given to such QMC results in Sec. IV of the

present article, where c orrelation energies for small mole

cules,

as well as numerical solutions of Hubbard and

Heisenberg H amiltonian s, will also be referred to.

The jellium model affords the basis for the so-called

theory of the inhomogeneous electron gas, which had its

origin in the Thom as-Ferm i method. This latter theory

was formally completed by the Hohenberg-Kohn

theorem (1964), which states that th e ground-state energy

of an inhomogeneous electron gas in a unique functional

of its electron density p( r) . How ever, the functional is

not yet known. Ther e are fairly recent reviews both on

Reviews of M odern Physics, Vol. 66, No. 2, April 1994 0034-6861/94/66(2)/445(35)/$08.50 1994The American Physical Society 44 5

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Senatore and March: Recent progressinelectron correlation

t he e l e c t ron gas (Singwi and Tos i , 1981 ; I c h im a ru ,1982)

a n d on the so-ca l led densi ty- func t ion a l the ory (Bamz ai

a n d Deb, 1981;G h o s h and Deb, 1982;L u n d q v i s t and

M a r c h , 1983; C a l l a w a y and M a r c h , 1984; J o n e s and

G u n n a r s s o n , 1 9 89 ; P a r r andY a n g , 1989 ). Th us we sha l l

keep th is d iscussion re la t ive ly short , focusing

on the way

W igne r e l e c t ron c ry s t a l l i z a t ion can be t r e a t e d w i th in the

de ns i ty - func t iona l f r a me w ork .

It is

r e l e va n t

in

t h i s c on

t e x t to s t ress at the ou t se t t ha t , w h i l e de ns i ty - func t iona l

the o ry r e ve a l s the way in w h ic h e l e c t ron c o r re l a t i on

e n te r s thec a l c u l a t i on of thes ingle -p ar t ic le densi ty , th ere

is no m e a n s w i t h i n the fo rma l f r a m e w ork for c a l c u l a t i ng

the r e qu i re d func t iona l de sc r ib ing c o r re l a t i on .

Se c t ion III is c onc e rne d w i th themu l t i c e n te r p rob le m s

of molecules and so l ids, and the qua l i t a t i ve a spe c t s of

e l e c t ron c o r re l a t i on are first stressed by s u m m a r i z i n g the

i de a be h ind the C ou l son -F i sc h e r (1949) w a ve func t ion for

t he g round s t a t e of the H

2

mo le c u le . Th i s i de a is t he n

t a k e n up in r e l a t i on to the w o r k of G u t z w i l l e r

(1963,

1964,

1965) on s t rong ly c o r re l a t e d e l e c t rons in n a r r o w

e n e r g y b a n d s . In t h i s t r e a tme n t , e l e c t ron in t e ra c t ionsare

t r e a t e d bym e a n s of t h e H u b b a r d U, w h i c h ,by definition,

is the e ne rgy re qu i re d to p l a c e two e l e c t rons w i th a n t i -

para l le l sp in on the sa me s i t e . As in the C ou l son -F i sc he r

t r e a t m e n t , one r e duc e s the w e igh t of the ionic

c on f igu ra t ions t ha t areo b t a i n e d by e x p a n d i n g out a s in

g l e S l a t e r de t e rmina n t ofBloch wave func t ion s.

B e c a use of the c u r re n t e xc i t e me n t c onc e rn ing supe r

c o n d u c t i v i t y , the r e l e va nc e of the H u b b a r d H a m i l t o n i a n

in p rov id ing a qua n t i t a t i ve ba s i s for the d e v e l o p m e n t of

Pa u l ing ' s i de a s on the r e sona n t va l e nc e -bond the o ry of

m e t a l s is b r ie f ly sum ma r i z e d a l so w i th in the c o n t e x t of

G u t z w i l l e r ' s m e t h o d .

To th i s s t a ge in thea r t i c l e , thema in the o re t i c a l de ve l

o p m e n t

is via

l a rge ly a na ly t i c a l me th ods . H ow e ve r ,

as

m e n t i o n e d a l r e a d y , i m p o r t a n t p r o g r e s s in the c a l c u l a t i on

o f c o r re l a t i on e ne rgy in m u l t i c e n t e r p r o b l e m s has r e su l t

ed from the d e v e l o p m e n t of q u a n t u m c o m p u t e r s i m u l a

t i on , p ione e re d

in

ge ne ra l t e rms

by

A n de rso n (1975 ,

1976, 1980)anda pp l i e d to j e l l i um byC e pe r l e y and A lde r

(1980 ). Im por t a n t he re is to i n p u t a t r ia l wave func t ion ,

t r a nsc e nd ing w he re ve r poss ib l e a s ingle Sla te r de te r

minant . Brie f ly , one fo rm of the QMC m e t h o d s t a r t s

from the t ime -de p e nde n t Sc h ro d inge r e qua t ion or the

e qu iva l e n t B loc h e qua t ion in ima g ina ry t ime , in c on t ra s t

t o the me th ods u se d e l se w he re in the a r t i c l e . The a l te r

na t ive a pp roa c h w orks d i r e c t ly w i th the ma ny-pa r t i c l e

G re e n ' s func t ion , de pe nd ing on alle l e c t ron ic c oo rd in a t e s

a n d

on an

e n e r g y p a r a m e t e r .

The

s t r e n g t h s

and the

w e a kne sse s of t h es e m e t h o d s areassessed . As wel las the

j e l l i um re su l t s a l r e a dy me n t ione d , the pa ra l l e l de ve lop

m e n t in c a l c u l a t i ng c o r re l a t i on e ne rg i e s in sma l l mo le

cules will also be de sc r ibe d . At the t i m e of w r i t i ng ,

t h o u g h t h e r e ares t i ll some t e c hn ic a l p rob le ms , t h i s c om

p u t e r s i m u l a t i o n a p p r o a c h p r o v i d e s a ve ry a c c u ra t eway

of c a l c u l a t i ng e l e c t ron ic c o r re l a t i on e ne rg i e s in m u l t i -

c e n te r p rob le m s . Some p romis ing d i r e c t ions for fu ture

w o r k aresugge s t e d in c o n c l u d i n g the a r t ic le .

II .

HOMOGENEOUS ELECTRON ASSEMBLY

W e now t u r n to thed i sc uss ion of c o r re l a t i on in a m o d

e l a p p r o p r i a t e to s imp le me ta l s . It willbeuseful to focus

first on the e le c t ron a sse mbly fo rm e d by the c o n d u c t i o n

e l e c t rons in a meta l l ikeNa or K. In these cases, there is

a w ho le body

of

e v ide nc e tha t de m ons t r a t e s

the

w e a kne ss

of

the

e l e c t ron - ion in t e ra c t ion . The re fo re

the

je l l ium

mode l , i n t roduc e d in i t i a l l y by Somme r fe ld l ong ago,

w h e r e the ionic la t t ice iss m e a r e d outi n to a un i fo rm ne u

t r a l i z i n g b a c k g r o u n d of pos i t i ve c ha rge in w h i c h the

c o r re l a t e d e l e c t ron ic mo t ion t a ke s p l a c e , a f fo rds a va lu

a b le s t a r t i ng po in t .

E le c t ron c o r re l a t i on can be t r e a t e d qua n t i t a t i ve ly in

t h i s j e l l i um m ode l by n u m e r i c a l m e a n s (see, e.g., Sec.

I V . D )

at all

de ns i t i e s, inc lud ing me ta l l i c one s . A pp rox i

m a t e t r e a t m e n t s , on the o t h e r h a n d , are s imp le r in the

two l imi t ing cases of ve ry h igh and very low densi t ies .

T h o u g h at first the pro blem lo oks to ta l ly d i ffe rent f rom

the mo le c u la r c a se (seeespec ia l ly Sec.I I I .A ) , t h e r e is, in

fac t , a c lose para l le l in the se nse t ha t in the h igh -de ns i ty

l imi t a de loc a l i z e d p i c tu re (cf. mo le c u la r o rb i t a l s ) is

c o r re c t w h i l e , in the low-de nsi ty l imi t , e lec t ron loca l iza

t i on (cf. va l e nc e bond the o ry ) is a ga in i nduc e d by s t rong

C ou lomb re pu l s ion be tw e e n e l e c t rons .

In thefo l lowing we sha l l focus espec ia l ly on the r e g ime

of e x t re me low de ns i t ie s , w h e re theeffects of c o r re l a t i on

a re domina n t , l e a v ing a s ide the r a n g e of h igh and m e t a l

l ic densi t ies (see, e.g., S ingw i and Tos i , 1981). He re we

sha l l just re ca l l tha t at h igh densi t ies , i.e., for r

5

- > 0 , the

k ine t i c e ne rgy domina te s the po te n t i a l e ne rgy and an

i nde pe nde n t -pa r t i c l e de sc r ip t ion p rov ide s a r e a sona b le

s t a r t i ng po in t to de sc r ibe the e l e c t ron a sse mbly . In fact,

a s ing l e p l a ne -w a ve de t e rmina n t of sp in orbi ta ls is a

H a r t r e e - F o c k s o l u t i o n for the j e l l i um mod e l , y i e ld ing a

homoge ne ous de ns i ty d i s t r i bu t ion ( se e ,e.g.,M a r c h et al.,

1967)and an energy

N

2 .21 0.916

Ry (2.1)

M a n y - b o d y p e r t u r b a t i o n on s u c h a s t a t e , in wh ich e lec

t r o n s are fu lly de loca l ized , y ie lds an a pp ro x im a te e s t i

m a t e of the c o r re l a t i on e ne rgy E

c

(G e l l -Ma nn and

Bru eckn er , 1957), def ined as the d iffe rence be tw een the

t rue g round -s t a t e e ne rgy E

0

and its H a r t r e e - F o c k ap

proximationE

HF

, *-

e

>E

c

=

^O~^UF-

A. Low-density Wigner electron crystal

T u r n i n g to the e x t re me low -de ns i ty l im i t r

s

>oo,

W igne r (1934 , 1938)p o i n t e d out t h a t the a bove de loc a l

i z e d p i c tu re b roke dow n c omple t e ly , ando n c e the p o t e n

t i a l e ne rgy be c a me l a rge c ompa re d w i th thek ine t ic ener

gy , the e l e c t rons w ou ld the n w a n t to avoid each o the r

m a x i m a l l y . He s tr e sse d tha t t h i s s i t ua t ion w ou ld be

a c h ie ve d by e l e c t rons be c oming loc a l iz e don thes i tesof a

l a t t i c e . He a rgue d tha t one m us t f ind the s table la t t iceby

m i n i m i z i n g the M a d e l u n g e n e r g y . Of the la t t ices so far

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Senatore and M arch: Recent progress in electron correlation

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examined, the bcc lattice has the lowest Madelung term.

This yields, for the electron bcc Wigner crystal, the ener

gy

1.792

r

E

hm =

r

c

- *oo

N

Ry ,

(2.2)

which shows, by comparison with the Hartree-Fock

plane-wave result (2.1), that this latter approximation is

no longer of any physical utility, the energy being too

high by a factor of about 2.

As the repulsive coupling between the electrons is re

laxed, that is, r

s

reduced below the range of validity of

Eq. (2.2), the electrons will vibrate about the bcc lattice

sites.

Since the bcc lattice has a Wigner-Seitz cell of high

symmetry, it is a useful first approximation to neglect the

(multipole) fields of the other cells in considering the vi

bration of an electron in its own cell. Then the po tential

energy V(r), in which the electron vibrates, is created

solely by the uniform positive background in its own cell,

which, in the spherical approxim ation, gives

e

2

r

2

V(r) =

J ~ J

+ const

2r /a$

(2.3)

with the ground-state isotropic harmonic-oscillator wave

function as

tf(r)

=

3/4

e x p ( - l a r

2

) , a = ( r , a

0

) ~

3 / 2

. (2.4)

This Wigner oscillator leads to a kinetic energy per elec

tron in the low-density limit as

lim =-

N 2r,

3/2

Ry ,

(2.5)

This is qualitatively different from the independent elec

tron result (2.21/r

2

)Ry. In fact, an obvio us effect of

switching on the electron-electron interaction away from

th e r

s

)limit is to prom ote electrons outside the Ferm i

sphere of radius k

F y

leaving holes inside (cf. Fig. 3). This

creation of electron-hole pairs obviously increases the

kinetic energy. Thu s, if one writes

T/NK/r

2

,

the in

crease in kinetic energy appears as an increase in K(r

s

).

In the limit r

s

oo, the creation of particle-hole pairs is

so prolific that K diverges as r]

n

and the Ferm i sphere

picture breaks down completely. The calculation yield

ing Eq. (2.5) is an Einstein-type model, whereas one

should, of course, treat the vibrational modes of the bcc

Wigner crystal by collective phono n theory. The

coefficient 3 in Eq. (2.5) is then re duce d to 2.66 (Carr ,

1961;Coldwell-Horsfall and M arad udin, 1963).

The Fermi distribution is then profoundly altered by

the creation of particle-hole pairs, and another way of

seeing this is to take the Fourier transform of the Gauss

ian orbital (2.4), which, of course, yields also a Gaussian

mom entum distribution. At most, rem nants of a Ferm i

surface remain in the low-density limit [in one dimension,

this statement is made quantitative by Holas and March

(1991)].

2.0

1.5

1.0

0.5

K\\\)y

\ un

1 1 1

FIG. 1. Pair-correlation function g{r) vs x r/r

s

a

0

, for

different densities: (i) Wigner form for

r

s

=

100;

(ii) Wigner form

for

r

s

= 10;(iii) Fermi hole, correct in the limit

r

s

-*0.

A measure of the order present in a many-body system

is afforded by the so-called pair-correlation function g (r).

This gives the probability of finding pairs of electrons at

distance r. In the Wigner-crystal regime, and within the

approximate framework of electrons behaving like Ein

stein oscillators, it is a straightforward matter to con

struct g(r). Resu lts for different values of r

s

are shown

in Fig. 1, taken from the work of March and Young

(1959).

The tendency of electrons to avoid each other

with increasing r

s

is apparent, as well as the greater

amount of order present in the state in which the elec

trons are localized in Gaussian orbitals in contrast to

that in which they are in plane waves.

B. Density-functional theory of many-electron system

One approach to incorporate, approximately, of

course, electron correlation is afforded by the density-

functional theo ry. However, because of related recent re

views (Lundqvist and Ma rch, 1983; Callaway and M arch ,

1984;

Jones and Gunnarsson, 1989) and books (see, e.g.,

Parr and Yang, 1989), our treatment will be very

brief,

merely to establish the notation and recall a few facts.

The theory has developed from the pioneering work of

Thomas (1926) and Fermi (1928) and Dirac (1930), with a

later paper by Slater (1951) also being very important in

developing the present form of the density-functional

theory. The step that was lacking, namely, the proof tha t

the ground-state energy of a many-electron system is

indeed a unique functional of the electron density, was

taken by Hohenberg and Kohn (1964), who supplied the

proof for a nondegenerate ground state.

It has proven helpful in developing the theory to

separate the energy functional into a number of parts,

closely paralleling the energy principle of the original

Thomas-Fermi theory (cf. March, 1975).

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448 Senatore and March: Recent progress in electron correlation

Thus one w r i t e s t he t o t a l e ne rgy E [p] as a sum of an

inde pe nde n t -pa r t i c l e k ine t i c -e ne rgy func t iona l

T

s

[p],

e l e c t ro s t a t i c po t e n t i a l - e ne rgy t e rms , a nd the n a c on t r ibu

t i on

E

xc

[p

]f rom e xc ha ng e a nd c o r re l a t i on w h ic h a re u se

fu l ly considered toge ther . Expl ic i t ly , we wri te

E[p]=T

s

[p]+fp(T)v{T)dT

+ j*^dTd*

+

E

xe [

p]

9

(2.6)

w h e r e

V(T)

is the exte rna l potent ia l ac t ing on the e lec

t rons (Ze

2

/r in an a tom ).

A c c o rd ing to H ohe nbe rg a nd K ohn (1964 ) , E[p] is a

min imum a t t he e qu i l i b r ium de ns i ty d i s t r i bu t ion in t he

g ive n e x t e rna l po t e n t i a l

V(T).

Th us one min imiz e s t h i s

to t a l e ne rgy w i th r e spe c t t o t he g round -s t a t e e l e c t ron

densi ty p(r) , subjec t only to the condi t ion tha t the e lec

t ron de ns i ty sa t i s fy t he no rma l i z a t ion c ond i t i on

fp(T)dr = N ,

(2.7)

for a system with N e l e c t rons .

W i th the i n t roduc t ion o f a La g ra nge mu l t ip l e r \i,

which has the s ignif icance of the chemica l potent ia l of

the e lec t ron c loud of the a tom, molecule , or so l id be ing

c ons ide re d , t he Eu le r e qua t ion o f t he va r i a t i ona l p rob le m

re a ds

sr

a

(2)

evidently representing both electrons on nu

cleus a, etc. Tha t Eq. (3.3) is incorrect as the internu-

clear distance R gets large compared with the size a

0

of

th e Is hydrogen orbitals is quite clear; the molecule dis

sociates into two neutral H atoms, just as described by

the original Heitler-London wave function (3.1).

A. Coulson-Fischer wave function

with asymmetric orbitals, for H

2

molecule

An im portan t clarification of the role of electron c orre

lation in molecules came with the work of Coulson and

Fischer (1949) on H

2

. They asked the question as to what

was the best admixture of covalent and ionic states at

each internuclear distance R, by contemplating asym

metric molecular orbitals (f>

a

-\-X(f>

b

, A,

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Senatore and March: Recent progress in electron correlation

451

overcorrelate d. The spin-density and charge-d ensity

waves are less easily classified, with the question of

under- or overcorrelated depending on the strength of

the interaction.

Falicov and Harris construct from spin- and charge-

dens ity-wave s tates , which have broken symmetry, sym

metriz ed versions. The se sym metric al states were always

found in their work to be slightly undercorrelated. This

work has been extended by Huang et al. (1976).

B. Gutzwiller's variational method

As mentioned above , a poss ible way to account for

correlation of antiparallel electrons in a multicenter

problem is to partly project out from a given uncorrelat-

ed wave function those components corresponding to

doub le occupied (ionic) s ites. In the simple case of H

2

,

this was most s imply done by Coulson and Fischer (1949)

as set out in Sec. I I I . A . The general ization of such an ap

proach to situations with an arbitrary number of centers

is due to Gutzw iller (1963, 1965). Here we shall just out

l ine the key points of Gutzwil ler 's approach, which has

been reviewed by Vollha rdt (1984). In the next two sec

t ions ,

we shall discuss in some detail the application of

the Gutzwiller variational method to treat the correlation

in molecules , on the one hand, and the Hubbard Hamil-

tonia n, on the other. Th e latter has received renew ed at

tention in connection with the exc it ing discovery of

high-7"

c

superconductivity (Bednorz and Miiller, 1986).

The starting point of Gutzwiller's variational approach

is the uncorrelated wave function, for the problem under

consid eration . This is constru cted, for a regular lattice

with N s ites and one Wannier orbital f> per site, from the

Bloc h wave s

^ i^

1

*)'

^ ( r j ^ ^ ^ e x p f z k R / ^ r - R , . ) . (3.5)

Us ing the second-quantization formalism, the uncorre lat

ed ground-state wave function can be written as

> o > = n

< T

n k i > >

where

a \

a

is the creation operato r of an electron in the

Bloch wave *

k

and with spin projection a, |0.) is the vac

uum state , and K and Q are sets of points in reciprocal

space, which in general may be delimited by different

Ferm i surfaces. If one denot es the creation operator of

an electron in the Wannier orbital f> at the site / by aj

a

,

4 = 2 e x p ( - / k R , . ) a L , 0.7)

and the corresponding number operator by n

ia

, the

Gutzwil ler wa ve function can be written as

l * > = I I [ l - ( l - * ) i t ' / i ] l * o > = *

2 >

l * o > . (3 .8)

i

Above , ^

=

2 /

r t

/ f

w

/ l *

s t n e

operator that counts the

num ber of doub le occu pied sites. Clearly, in the wav e

function 4>, the comp onents containing double o ccupied

sites are reduced by a fractional amount 1 g (0< g

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2 e(k) .

(3.13)

tu rn d i s t r i bu t ion

n

ka

a t t he Fe r mi su rfa c e ;

d=D/N,

w a ve 4^ ,

w i t h

D

t he a ve ra ge numbe r o f doub le oc c up ie d s i t e s ; a nd

_

=

_ 1 _

and the zero of energy has been chosen so tha t

In Eq. (3.12),

e(k)

is the energ y e igenvalue of the Bloch ^ k e (k ) =

0;

i .e .,

t

ti

=0. Th e G A simp ly y ie lds

(3.12)

(k) = - J : 2 ^ P [ - ' W R / - ^ ) ] >

ij

V

\k\=n*M*o>

i = 0

w he re the

P

i

' s a re projec t io n op era tor s defined by

' ( = 1 1 ( 1 - ^ ) .

and

(3.18)

(3.19)

(3.20)

(3.21)

Fo r 1 = 0 , Eq . (3 .18) y i e lds t he o r ig ina l G u tz w i l l e r a ns a t z .

H ow e ve r , fo r nonz e ro J , de ns i ty c o r re l a t i ons be tw e e n

diffe rent gr id points a re a lso taken in to account , up to the

7 th ne ighbo r ing po in t s . N o t i c e t ha t , i n t he l a t t e r c a se ,

one is corre la t ing a lso the mot ion of para l le l -sp in e lec

t rons on d i ffe rent gr id points . The reason for th is i s tha t ,

t hou gh the unc o r re l a t e d w a ve func t ion | O

0

) i s ant isym-

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Senatore and March: Recent progress in electron correlation 453

metrized, the Paul i exclusion princip le i s completely

effective only at very short distances.

The problem of a finer grid can also be dealt with in

quite a simp le way (Stollhoff and Fuld e, 1978). Instead o f

directly making the grid finer, one can alternatively

choose to keep as grid points the atomic posi t ion in the

molecule , whi le using at each s i te a number of basis-set

funct ions with varying degrees of local izat ion , possib ly

also off-center. Th us the opera tor a}

a

creates an electron

in the basis-set function i rather than in one of the

self-

consistent one-particle orbitals in terms of which 14>

0

) is

constructed . Here , the index i labeling the creation

operator stands, in fact , for the pair (z ,a) , where a indi

cates e i ther one of the original basis-set funct ions or a

sui table new combinat ion; i t might be, for instance, an s-p

hybrid.

The effect of taking into account off-site correlation

through the use of the variat ional wave funct ion given in

Eq. (3.18) with l^0 has been investiga ted by Stollhoff

and Fulde (1977) by studying the H

6

model of Mattheiss

(1961) . This mode l has the advantage that one know s the

exact solut ion , whic h was obtained numerical ly. The

model assumes s ix s i tes equal ly spaced on a ring, with

on e s atomic orbi tal at each s i te . From these, one con

structs a Wannier orbi tal , in terms of which the Hamil -

tonian reads

ijkl

(3.22)

ijcr

Bloch waves are then bui l t from the Wannier orbi tal , and

from these the uncorrelated Hartree-Fock ground state

|

0

) i s obtained. The correlat ion energy of the system

was studied by Stollhoff and Fulde for various values of

the first-neighbor distance R, with the variat ional wave

functio n of Eq. (3.18). T hey found that the simpler

Gu tzwi l l er an satz , i . e. , 7 = 0 , on l y y i e l d s ab ou t 50% of

the exact correlat ion energy at the exact equi l ibrium dis

tan ce , R = 1.8 a.u. On the other hand, the s i tuat ion im

proves substant ial ly by going to I = 1, and even more for

7 = 2 .

In the lat ter case, one recovers more than 95 % of

the correlat ion energy for any distance. The s i tuat ion i s

i l lustrated in som e detai l in Fig. 2. I t should also be no

ticed that a further increase of I from 2 to 3 does not pro

d u ce an yth in g n ew, s i nce 7 = 3 corresp on d s to the l arges t

distance of the atoms in the ring.

Stol lhoff and Fulde also invest igated the accuracy of a

simplified version of Eq. (3.18) obtained by l inearization,

| * > =

1 = 0

j

l*0>

(3.23)

in order to reduce com putat iona l complexi ty . With th is

simplified variational ansatz, they found that at the equi

l ibrium distance about 90% of the correlat ion energy was

s t il l recovered wi th 7 = 2 . How ever , th e agreem en t wi th

the exact resul ts becomes worse at larger d istances , in

contrast with the systematic improvement found with the

full variational wave function of Eq. (3.18).

1.0

0.8

0.6 h

0.4 h

0.2

I 1=1*

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some small atoms and molecules , taking into account

intra-atom ic correlations as well. To this end, the radial

correlation was treated by considering basis-set functions

o f s type with different degrees of localiza tion. On the

other hand, angular correlations were introduced by us

ing appropriate hybrid functions. In particular, s-p mix

ing was used to get sets of tetragonal hybrids, and

d

and

/ function s were used to obtain hybrids with h exago nal

and octagonal symmetry, respective ly . Within the ir sim

ple scheme they were able to recover 93% of the experi

mental correlation energy for He and 90% for H

2

. Thus ,

starting from the Hartree-Fock uncorrelated ground

state, in both cases the results of the local approach

based on a Gutzwiller-like ansatz yielded a fraction of the

correlation en ergy, very clo se (a few percen t of difference)

to the one obtained in configuration-interaction (CI) cal

culation s with the same basis sets . Similar results were

also obtained for He

2

and Be.

A very detailed study of Ne and CH

4

along these lines

(Stollhoff and Fulde, 1980) shows that also in more com

plex situations the local approach is capable of account

ing for the correlation energy obtainable with CI calcula

t ions ,

within a few percent. One should notice , though,

that the agreement with experiment of the correlation en

ergy yielded by CI calculations with reasonable basis sets

tends to worsen with increas ing complexity of the sys tem

studied.

More recently , Oles

et al.

(1986) proposed another

computational scheme for the treatment of correlations

in more complex molecules containing C, N, and H

atom s. Wh ile the interato mic correlations are still treat

ed within a local approach based on semiempirical self-

consistent-field calculations, intra-atomic correlations are

dealt with by means of a different and simpler scheme.

Good agreement is found with experimental results ,

discrepancies being within a few percent. Th ey have also

shown that simple algebraic parametrizations are possi

ble for the various contributions to the correlation ener

gy. In particular, interato mic correlation s are found to

depend only on bond lengths , whereas intra-atomic

correlations are found to be determined for a given atom

by its total charge and fraction of/? electrons. A further

study, on the determination of optimal local functions for

the calculation of the correlation energy within the local

approach, has also recently appeared (Dieterich and

Fulde, 1987).

In the foregoing, we have been concerned with the ap

plication of a local approach to the calculation of correla

t ion in molecules . Yet we should l ike to mention here an

important development along these l ines concerning ex

tended system s. Th e above local appro ach, in fact, has

been suitably modified (Horsch and Fulde, 1979) to treat

also short-range and long-range correlation in the ground

state of solids. A first attempt to calcula te in this man ner

the correlation contribution to the ground-state energy of

diamond (Kie l et aL, 1982) suffered the limitation of a

poorly converged uncorre lated ground state . Wh en an

uncorrelated ground state of good quality is used, howev

er ,

the local approach performs remarkably well, repro

ducing the electronic contribution to the binding energy

of diamond with an accuracy of about 2% (Stollhoff and

Bohnen, 1988).

D. Gutzwiller's variational treatment of the

Hubbard model

As we have mentioned above , the Gutzwil ler variat ion

al method was originally applied to the study of the so-

cal led Hubbard model , character ized by the Hamiltonian

of Eq. (3.10). In spite of the apparent simplicity of such a

Hamiltonian, the relevant variational calculation is of

considerable difficulty. Therefore Gutzw iller solved the

problem in an approx imate ma nner. It has been only re

cently, after some decades, that a renewed interest in this

problem has brought about a certain amount of work

leading to the exact solution of the variational problem,

by direct numerical evaluation, by analytical means, or

by the Monte Carlo method.

Kaplan, Horsch, and Fulde began a s tudy in 1982 to

assess the accuracy of the Gutzwiller variational wave

function (GVW) in describing the ground state of the

single-band Hubbard model with only nearest-neighbor

hopping, in the atomic limit, i .e ., the limit in which

U>

oo. In this case there is no variation to be taken. In

fact, g = 0 , and the Gutzwil ler wave function reads

l^

0

o> = II[

1

- " / T a ] l

< I >

o> 0.24)

i

They were able to perform, for the half-filled-band situa

t ion , the direct numerical evaluation of the spin-spin

correlation function,

qi

= (sfsf

+l

) ,

(3.25)

for a num ber of small regular rings. The actual calcu la

tions were done for rings of 6, 10, 14, and 18 sites. By ex

trapolating from the results for finite rings, they found

that in the thermodynamic limit (number of s ites N going

to infinity) q

x

0 .147 4, th us be ing within 0 .2% from

the exact result, i .e ., within the error bars of their calcu

lat ion. For q

2

y instead, the discrepancy with the exact re

sult was of about 7% . They also noticed that the spin-

spin correlations that they obtained reproduced as well

qualitative features of the exact ones. He nce for large U,

at least in one dimension, Gutzwiller's wave function de

scribes the short-range spin-spin correlations in an accu

rate mann er. W e should notice here that, in the atom ic

limit and for the half-filled-band case, the exact ground

state of the ID Hubbard model is known to be the same

as for the antiferromagnetic Heisenberg chain, for which

q

x

was exactly evaluated by Bonner and Fisher (1964),

an d q

2

by Takahashi (1977). In fact, in the large-U limit,

the Hubbard Hamiltonian goes , to leading order in t/U,

into the Heisenberg Hamiltonian with anti ferromagnetic

coupl ing,

H=J X ( s

r

s , - | ) , (3 .2 6)

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455

where / is proportional to

t

2

/U

and

t

is the hopping en

ergy. We shall return to this point in some detail.

Kaplan, Hor sch, and Fulde also studied the energy for

large but finite

U.

In this case one has to start from the

full variational wave function (3.8) and consider an ex

pansion of the energy in powers of the parameter g, to be

determined variationally. They found that the leading

term of such an expansion was of the form

E

=

~Nat

2

/U,

with

a

depending on the kinetic-energy

operator and on the zero- and first-order wave functions

obtained from the expansion of the variational wave

function (3.8) in powers of g. N ote tha t the zero-order

wave function, which was given in Eq. (3.24), has no sites

with double occupa ncy, as is clear by inspection. Simi

larly, the

first-order

one has on average only one site dou

bly occupied. They found that, at variance with the

spin-spin correlations, the coefficient

a

yielded by the

GVW was in gross disagreement with the known exact

value. It is now known, from the exact solution of the

Hubbard model in one dimension with the GVW

(Metzner and Vollhardt, 1987), that

a

is not a constant,

but vanishes as l/ln(

U/t)

in the limit considered by Ka

plan,

Horsch, and Fulde. These authors argued that the

unsatisfactory result for the energy yielded by the GVW

was due to the incorrect description of correlations be

tween doubly occupied sites and empty sites, or holes, as

we shall call them in the present contex t. Therefore they

considered a modification of the GVW containing a

second variational parameter to improve the treatment of

such correlations. In this manner they were able to

reproduce the exact value of a within 1%.

A further step toward the assessment of the reliability

of the GVW was then taken by Horsch and Kaplan

(1983).

They extended the ca lculation for finite rings to

other values of

.N,

corresponding to open-shell systems,

and also performed calculations for much larger systems

with

N

up to 100 by using the Monte Carlo method to

evaluate the relevant averages. This second step was par

ticularly important. First, it fully confirmed the con

clusions previously obtained from the exact calculations

for small rings. Secondly, it was the first application of

the Monte Carlo method to this problem. Before turning

to the presentation of more recent studies on the Hub

bard model, with a variational Monte Carlo technique

based on the GVW, we shall summarize below the exact

results that have been recently obtained in one dimension

for arbitrary band filling and interaction stre ngths.

1.

Exact analytic results in one dimension

Analytic results for the ground-state energy and

momentum distribution function for the Hubbard model

with on-site repulsion have been recently obtained by

Metzner and Vollhardt (1987) for the paramagnetic situa

tion.

The key point of their attack on the problem at

hand lies in a new approach to the calculation of the ex

pectation values of relevant operators on the GVW .

The calculation of the ground-state energy appears to

require the evaluation of the expectation values of both

kinetic and potential energy for arbitrary values of the

variational parameter g, 0

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456


* * * * *

i r

+ 1

n

w

+ 1

/ ( m + l )

(3.31)

With the above equa t ion for the coeff ic ients c

m

, th e series

in Eq. (3 .27) i s exac t ly summed to

(3.32)

^

n

g

\-G

2

1

In -^r + G

2

-l

G

2

w i th

G

2

=l

n +ng

2

. W e st ress tha t , as i s c lear f rom

Eq . (3 .32) , for the ha l f- f il led b an d fo r wh ich

n

= 1 the

c o r re l a t i on e ne rgy

U d

is found to be nonana lyt ic in g , be

c a use o f t he p re se nc e o f t he t e rm ln ( l / g ) . Fo r s t rong

corre la t ions (g()) , one f inds

d

= g

2

l n ( l / g ) . M o r e o v e r ,

t he doub le oc c upa nc y d is nev er vanis hin g for finite

corr e la t io ns, in con tras t wi th the G A resu l t of Eq. (3 .15) .

This seems to be the case a lso in two and three d imen

s ions w he n the G V W i s e xa c t ly ha nd le d (Y oko ya ma a nd

Shiba, 1987a).

A s imi l a r , t hough more c ompl i c a t e d , a na ly s i s a l l ow s

the c a l c u l a t ion o f t he mom e n t um d i s t r i bu t ion func t ion

for the same si tua t ion considered immedia te ly above , i .e . ,

n^=n^=n/2. W e sha l l con tent ourse lves here wi th

quo t ing ju s t t he r e su l t fo r t he d i sc on t inu i ty

q

in the

m o m e n t u m d i s t r ib u t i o n f u n ct i on n

ka

,

q = G-

1

[(G+g)Al+g)]

2

(3.33)

The overa l l shape of

n

ka

for hal f filling an d at d ifferen t

va lues of g is i l lust ra ted in Fig . 3 . I t shou ld be note d tha t

fo r n = 1 the a bove e qua t ion re du c e s t o q = 4 g / ( l + g )

2

,

w h ic h i s t he r e su l t o f G u tz w i l l e r ' s a pp rox ima t ion (G A ) .

T h i s , how e ve r , doe s no t imp ly tha t , fo r t h i s pa r t i c u l a r

va lue of f i l l ing , the k ine t ic energy coinc ides wi th tha t of

G A , s inc e g ha s t o be de t e rm ine d by min im iz ing the t o t a l

energy, and the equa t ions for the potent ia l energy a re

different in the two cases.

By combining Eq. (3 .32) wi th the resul t for the

mo me n tu m d i s t r i bu t ion func tion , one ma y c a l c u l a t e

E

0

(g )

i n one d ime ns ion fo r t he pa ra m a gne t i c g rou nd

sta te . M inim iza t io n wi th respec t to g y ie lds the energy

for g iven va lues of

t, U, n.

F o r

n

= 1 , t h e c o m p a r i s o n

w i th t he r e su l t o f G u tz w i l l e r ' s a pp rox ima te t r e a tme n t

and the exac t resul t of Lieb and Wu (1968) shows tha t

the G V W , w he n e xa c t ly ha nd le d , g ive s a n e ne rgy tha t i s

i n t e rme d ia t e be tw e e n the o the r tw o . Th i s i s i l l u s t r a t e d in

Fig . 4 . In par t icula r , spec ia l iz ing to the case n =

1

a nd

t / / f, wi th only nearest -ne ighbor hopping, one f inds

i f

0

= 4 r / 7 r , a n d

E/N = -(4/7rnt

z

/U)(lnU)

- i

(3.34)

w h e r e

U=U/\

Q

\.

We not e , wi th re fe rence to the nu

me r i c a l r e su l t s o f K a p la n et al. d iscussed ear l ie r , tha t the

e ne rgy i s c e r t a in ly p ropo r t iona l t o t

2

/U , bu t t he p rop o r

t iona l i ty fac tor goes logar i thmica l ly to zero wi th U/t.

Thus, for the ha l f- f i l led-band case , the GVW gives a re

sul t tha t i s qua l i ta t ive ly d i ffe rent f rom t he exac t re sul t .

From Eq. (3 .33) i t i s c lear tha t a d iscont inui ty in the

mome n tum d i s t r i bu t ion r e ma ins a t a ny f in i t e

U ,

w he re a s

the e xa c t so lu t ion o f t he H ubba rd mode l o f L ie b a nd W u

(1968) g ives an insula t ing system for any nonzero

U

(see

a l so Fe r ra z

et al.,

1978) . I t would seem from these con

s ide ra t ions t ha t t he G V W i s no t a pa r t i c u l a r ly good a n -

sa t z fo r t he g round s t a t e o f t he H ub ba r d mode l . H ow e v

er , we have a l ready seen from the resul ts of Kaplan et al.

t ha t i t pe r fo rms muc h be t t e r i n t he c a l c u l a t i on o f sp in -

spin corre la t ions than for the energy.

U s ing t e c hn ique s s imi l a r t o t hose e mp loye d in t he e n

e rgy c a l c u l a t i ons , G e bha rd a nd V o l lha rd t (1987 ) ha ve

c a l c u l a t e d , fo r t he pa ra ma gne t i c g round s t a t e , a numbe r

of corre la t ion func t ions,

Cf

Y

=N~

l

2-

(3.35)

w he re X ^ F ^ S f , ; , # ; , # ; . H er e n^n^ + ri; a nd

H

t

= 1

n

t

| ) ( 1

rifi)

i s t he numb e r ope ra to r for t h e

ho le s . I n a dd i t i on ,

X=N~

l

^

t

X

i7

a nd the Fou r i e r

t ransforms of the func t ions def ined in Eq. (3 .35) a re s im-

E/ t

c

0 r

- 0 . 2

- 0 . 4

- 0 . 6

- 0 . 8

- i . o

- 1 . 2

U / t

2 4 6 8 10

i i i I

-~

i

f r

. /

/

r /

i I I I I .

12 14 16

i l i

_

e x a c t

G V W

G V W + G A

i i i

18

i

2 0

j

\

\

\

FIG. 3. Momentum distribution (n

k

) for the one-dimensional

Hubbard model. The results obtained by using the Gutzwiller

variational wave function are shown for several values of the

correlation parameter g in the case of a half-filled band

(n

T

=n

i = \

). From Me tzner and Vollhardt (1987).

FIG. 4. Ground-state energy E for the one-dimensional Hub

bard model with

n)=ni = j

as a function of

U .

The results for

E, as calculated with the Gutzwiller variational wave function

(GVW), are compared with the result of the Gutzwiller approxi

mation (GA). From Metzner and Vollhardt (1987).

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457

ply denoted

by

C

XY

(q).

The

results

for the

spin-spin

correlation function

q

}

=

Cf

s

obtained byGe bhar dtand

Vollhardt fully confirm thenumerical calculations ofq

j

by Kaplan

et al.

However , with the analytical solutionit

is possibletoestablish that there is ,infact,adifferenceof

0 .2% be twe e n the exact and the GVWresults in the

atomic l imit , and that such a differenceis notdu eton u

merical uncertainties.

In

additio n, the features

of

the ex

act q

x

found in numerical invest igation (Betsuyakuand

Yok ota, 1986; Kaplan et al., 1987)o n the Heisenberg an-

t i ferromagnetic chain are reasonably reproduced, though

the agreement worsensatlarger distances, as already sug

gestedbyKaplan et al. (1982). The comparison betw een

the GVW result and the exact result for the hole-hole

corre lat ion function

C

HH

{q) in the

atom ic limit also

s hows

an

overal l agreement, wh ich tends

to

impr ove

as

the number

of

holes tends

to

zero. Final ly ,

we

notice

that Gebhardt and Vol lhardt a lso comment onthe corre

lat ions between holes anddoub ly occupied sites. Their

conclus ion is that these kinds of correlations arenot de

scribed particularly well by the GVW and that this might

wel lbe thereasonfor thelogarith mic singularity found

in the energy for

g

^0.

2.

Numerical results

Re c e nt ly , anumbe r ofnumerical invest igations on the

Hubbard model with theGV W we r e c onduc te d w i ththe

helpofthe Mon te Carlo metho d. One should dis t inguish

between two kinds of invest igations . In one case the

original Gutzwil ler program is implemented; i.e.,

ground-s tate p roperties ofthe Hubbard H amiltonianare

variational ly calculated with the wave function of Eq.

(3.8),forarbitraryn,t, Uand dimens ional i ty . In theo th

er case, following

the

observation

of

Kaplan

et al.

(1982)

that theG V W forg= 0 g ives anaccurate d escriptiono f

spin correlation

in the

Heisenb erg antiferrom agnetic

c ha in ,

one

wor ks

on the

effective H am iltonia n

to

whic h

the Hubbard

one

reduces

in the

l imit

U>oo,

thus

re

stricting onet othe s trong-coupl ing s i tuation.

Yokoyama and Shiba (1987a) have sys tematical ly s tud

ie d theHubbard mod el , fo l lowing thefirst ofthe above-

mention ed approaches . Thu s they have performed cal

culations inone , tw o, and three dimens ions over thefull

range of coupl ing

U/t

and for both half-filled andnbn-

half-filled bands. Hereweshall not enter into th e tech ni

cal details of the ir variational M onte Carlo technique,

with which they performed calculations with up to 216

sites (6 X 6 X 6

in

three dimension s). W e shall merely giv e

a brief reviewoftheir ma in results .

In onedimens ion and forn= 1 , the y have c a lc u la te d

E

kin

/t andd =E

pot

/U as function s ofg between0 an d1.

F orag iven valueof U, one can then construct from such

functions thetotal energy as afunction ofg and findits

m i n i m u m by inspection. They have also calculatedthe

mome ntum di s tr ibut ion . We merely n ote that in this

case

the

exact solution

is

available and that ac cordin g

to

M e tz ne r

and

Vollha rdt (1987)

the

agreement with the ir

analytic results is exce l lent . In the l imit in whic h

/*oo, however, they have notbeen able to isolatethe

logarithmic correction to the quadratic dependen ce t

2

/U

of theenergy, but it would have been surprising other

wise. Similar calcula tions were performed

for the

test

case

n

= 0 . 8 4 , t a k en

as

representative

of a

band less than

half filled.

The

results

for the

energy

are in

fair agree

ment with

the

exact result, even

if

they observe that

im

provements upon the GVW are cal led forifone wishesto

obtain better agreement. Theneed to take into account

intersite correlation is a lso mentioned, a point thathas

already been stressed by Stollhoff and Fulde (1977),as we

discussedatsom e length earlier.

In two dimens ions only thecase

n

=

1

on a square lat

tice wascons idered. They found, for the total energy,

that thevariational results andthose of the Gutzwil ler

approximation are much c loser than they wereino nedi

me ns ion , as can be seen byc ompar ing the 2D casere

ported inF ig . 5 with the IDcaseinF ig .4. T hi sis in ac

cord with the expectation that the GA s hould be c ome

more accurate inhigher dimens ions . How ever , compar

ison with the Hartree-Fock anti ferromagnetic energy

shows that the discrepancies with this are s izable for

large U,whereas they w ere muc h smallerin oned ime n

sion.

The calculations in three dimens ions were performed

fo r a cubic lattice andagainfor a half-filled band. The

resultsfor thetotal energy as a function of the interac

tion strength U/t ares imilar to those found in two di

mensions . Furthermore , in going from two to threedi

mensions , ch anges qual i tat ive ly s imilar tothose observed

in going from one totwo dimen s ions are apparent. Thus

the agreement between variat ional and approximate

treatments, i .e ., with theG A, improve further , whi lefor

large U the discrepancy with the anti ferromagnetic

Hartree-Fock energy, which

is

lower, rem ains sizable.

o

- 0 . 5 -

MEOI

- 1 . 0 -

U / t U

c

* 12.97

5

10 I 15

I T |

V M C ( N - ) ^ > ^

HF > ^ l ^ - " ~ ~ ~ " ^ ~

GA

^ ^ ^ ' ' ^ ^

| 2D

FIG. 5. Normalized ground-state energy for the two-

dimensional Hubbard model

as a

function

of

U: VMC, varia

tional Monte Carlo with the Gutzwiller variational wave func

tion (extrapolated

at an

infinite number

of

sites,

N=oo

);

HF,

antiferromagnetic Hartree-Fock; GA,Gutzwiller approxima

tion. The

half-filled-band case (n

f=

ni

=

~)

is

shown. From

Yokoyama and Shiba (1987a).


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458


This suggests that, in two and three dimensions, one

should perform the variat ional calculations us ing a GVW

based on the Hartree-Fock ground state of the antiferro-

magnetic system, rather than on that of the paramagnetic

one.

To introduce the second of the approaches mentioned

above, we shall briefly summarize the derivation of the

effective Hamiltonian which can be obtained from the

Hubbard one for s trong coupl ings U/t. We shall follow

the very s traightforward method given by Gros et al.

(1987a), but reference should also be made to Castellani

et ah (1979) and to Hirsch (1985). One can start from

the Hubbard Hamiltonian given in Eq. (3.10) and rear

range the kinetic energy T, specializing to the case of

nearest-neighbor hopping, to read

H = T

h

+T

d

+ T^+.V , (3.36)

with

T

h

=~ t

2 ( l - / , - , ) 4 ^ ( l - "

;

; - a )

>

(3.37)

(ij),a

T

d

= -t X n

u

_

a

ala

io

n

u

_

0

, (3.38)

Uj),a

-t 2 {\-n

u

-

G

)al

a 3 9 )

where T

h

an d T

d

are the kinetic energies describing the

propagation of holes and doubly occupied sites, respec

tively, in their Hubbard bands, and T

m i x

is clearly a mix

ing term that couples such bands . V is , of course, the

usual on-site repulsion energy, V=U^

i

n

n

n

n

. The

next step is to apply to the Hamiltonian a suitable uni

tary transformation,

H

eff

= e

iS

He~

iS

=H +i[S,H]+ , (3.40)

such that, in lowest order in t/U,

T

mix

vanishe s on th e

right-hand side of Eq. (3.40). This can be acco mp lished

by choos ing S such that /[S, T

h

+

T

d

~\-V]=

T

mix

,

i.e.,

, v (n\T

mb i

\m ) / .

S = 2 \n) ., ,

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Senatore and M arch: Recent progress in electron correlation 459

Vollhardt, 1987), that the small difference between the

exact result for this quantity and that obtained from the

infinite-U Gutzwil ler wave function is beyond the numer

ical uncertainty of the Mo nte Carlo calculation. The in

vestigation of the kinetic energy as a function of the

filling, on the other hand, shows good agreement with the

exact result of both M onte Carlo and GA results . Gros ,

Joynt, and Rice also discuss the fact that for large

U

the

hole-hole corre lat ion function can be put into correspon

dence with that of a system of spinless noninteracting fer-

mio ns. This allows for an assessmen t of the quality of

their results . The y find fair agreemen t, even if som e qual

itative features of the exact result, related to the presence

of a sharp Fermi surface, are missing. Mo reover , an

enhancement with respect to the exact result of correla

tions at short distances is found . Rega rding the total en

ergy, for large U and values of the filling very close to 1,

the accuracy of the Monte Carlo result is dominated by

the kinetic -energy contr ibution, with an accuracy of

about 6%. In fact, the potential energy, being deter

mined by q

{

, has a mu ch better accuracy, i .e . , 0 .2% . Fi

nally, they find that when excited-state Gutzwiller wave

functions are cons idered, the accuracy of G A is mu ch re

duced. For the cases in whic h direct compa rison with

the analytical results of Metzner and Vollhardt (1987)

and Gebhard and Vollhardt (1987) was possible, excellent

agreement was found.

One of the reasons for the revived interest in the Hub

bard Hamiltonian is its possible relevance in the under

s tanding of the mechanism underlying high-temperature

superconductivi ty , as suggested by Anderson (1987) . In

particular, investigations in the strong correlation regime

are called for. The imp ortan ce of the schem e, briefly out

lined above, for dealing with the large coupling situation

is then apparent. A n investigation in this direction wa s

made recently by Yokoyama and Shima (1987b), within

the effective Hamiltonian app roach. They used the

effective Hamiltonian of Eq. (3.39) and considered both a

paramagnetic and an antiferromagnetic ground state.

This can be done by s imply chan ging the type of un corre

c t e d wave func tion | O

0

) , from which the Gutzwil ler

wav e function of Eq. (3.24) is built. One and two dim en

sions were considered . Th e purpo se of the study was to

character ize the competit ion between the two types of

ground states. Before briefly summarizing their findings,

i t should be mentioned that they have also shown that

the simple infinite-U Gutzwil ler wave function consti

tutes one of the many possible realizations of Anderson's

resonating valen ce-bon d state or singlet state. In one di

mension, they find that for both half-filled and non-half-

filled bands, the singlet state is stable against the Neel

(antiferromagnetic) state. The situation is quite different

in tw o dimen sions. For the half-filled case the singlet

state is found to be unstable against the Neel state.

Whereas the results for the energy and magnetization are

in reasonable accord with estimates obtained with

different approaches, they question whether the same sit

uation would be found if next-nearest hopping were to be

0.3

t / u

0.2

0.1

S q u a r e L a t t i c e

1 . . .

-T

r - i

S i n g l e t

i. i i /

i i

1

1

1

\

\

\

\

\

\

\

\

y^^ F e r r o

i |

A F

\

A 1

v

0.5

1.0

FIG. 6. Phase diagram of the two-dimensional square-lattice

Hubbard mod el, as inferred from variational M onte C arlo using

a Gutzwiller-type wave function (Yokoyama and Shiba, 1987b).

Here

n

e

=n^+ni

is the band filling.

include d. M ovin g from the half-filled band (ft = 1 ) , at

very large U, the ferromagnetic state becomes favorable

in energy. Ho wev er, by decreasing at som e point

(ft = 0 .6 1) the s inglet state takes over again. On the oth

er hand, if U is decreased, the Neel state appears to again

decrease in energy. Fro m these results they draw a quali

tative phase diagram as show n in Fig. 6. This shou ld be

considered meaningful only in the small t/U region,

where the effective Hamiltonian approach is appropriate.

Final ly , we should mention here the work of Gros ,

Joynt, and Rice (1987b). Using the effective Hamiltonian

approach, they investigate the stability of generalized

Gutzwil ler wave functions against Cooper pair ing, in the

large-

U

limit and in two dimens ions . This is done by

evaluating the binding energy of two holes in the varia

t ional wave function. As they note , no real attempt w as

made to optim ize the ir wave function. They f ind that the

paramagnetic or singlet state is stable against s-wave

pairing but unstable against d-wave pairing. On the oth

er hand, the antiferromagnetic state is stable against both

kinds of pairings. The y discuss a possible pairing mech a

nism and the relevance of their results for high-r

c

super

conductors .

E. Resonating valence-bo nd states

In the foregoing we have seen how the Hubbard

Hamiltonianfor large values of the coupl ing U

can

be transformed into an antiferromagnetic Heisenberg

Hamiltonian by means of a suitable unitary transforma

t ion .

In this connection and also in re lat ion to the high

ly superconductivity, it seems appropriate here to briefly

review the mathematical formulation due to Anderson

(1973) of the concept of the resonating valence-bond

(RVB) states first put forward by Pauling (1949).

Rev. Mo d. Phys . , Vo l . 66 , No. 2 , Apr il 1 994

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460


In discussing theground-state properties of

the

tri

angular two-dimensional Heisenberg antiferromagnetfor

S=j,

Anderson (1973; see, also, Fazekas and Anderson,

1974) proposed that

at

least forthis system, and perhaps

also

in

other cases, theground state might be theanalog

of the precise singlet

in

theBethe (1931) solution

of

the

linear antiferromagnetic chain.

In

fact,

the

zero-order

energy

of a

state consisting purely

of

nearest-neighbor

singlet pairsis more nearly realistic than thatofthe Neel

state.

For electrons

on a

lattice, one canthink

of a

singlet

bondorpairasthe state formed when two electrons with

opposite spin

are

localized

on two

distinct sites.

A

resonating valence-bond state

is a

coherent superposition

of such singlet bonds;

its

energy

is

further lowered

as a

result

of

the matrix elements connecting

the

different

valence-bond configurations.

Heuristically, valence bonds can beviewed as real-

space Cooper pairs that repel

one

another,ajoint effect

of

the

Pauli principle

and the

Coulomb interaction.

When there

is

one

electron

per

site , charge fluctuations

are suppressed, leading

to

aninsulating state. However,

as one moves away from half filling, current can flow;

the

system becomes superconducting

as the

valence bonds

Bose condense.

Anderson (1987), while stressing thedifficulty ofmak

ing quantitative calculations with RVBstates, in fact

gives a suggestive representation ofthem byexploiting

the Gutzwiller-type projection technique.

Clearly, adelocalized ormobile valence bond canbe

writtenas

I < * > R V B > = J V * V

/2

IO>

(3.50)

bl\-

1

1

VN

2 iW+r i 10 >

I

J

lo),

(3.46)

where b\

is the

creation operator

for a

valence-bond

state with lattice vector

r;

a}

a

>

the

single-electron

creation operator; andN,thetotal number ofsites. A

distribution ofbond lengths canbeobtained bysumming

b I

overrwith appropriate weights. One then gets

a

new

creation operator,

*

t=

2>=(*>V

/2

|o>,

(3.49)

andby (b)projecting out thedouble occupanc ywhich

would otherwise

be

presentwith

an

infinite-

U

Gutzwiller projection operatorP

d

=J|/ (1 ~

n

i \

n

n)>

One can then show that the RVB state written above can

be obtained with simple manipulations from a standard

BCS statebyprojecting onthe state withNparticles and

projecting out,at the same time, the double occupancy:

*

RVB

/

~PN/2P,

N/ 2

r

d

1

Ck

Vl+c

k

Vl+c

k

a

k\

a

-k\

| 0>.

(3.51)

Baskaran et aL(1987) have subsequently shown that

by treating thelarge-U Hamiltonian

of

Eq. (3.44) with

a

mean-field (Hartree-Fock) approximation,

one

obtains

precisely

a

BCS-type Hamiltonian whi chfor half

fillingyields the RVB state heuristically introduced

above. Infact,

one

also finds that 2^

c

fc

=

0, |cr |== 1

and

thec

k

change sign across what they call a pseudo-Fermi

surface. We shall not deal here with the natureofexcita

tions from sucha RVB state, referring the reader instead

to the original papers.

IV. QUANTUM MONTE CARLO CALCULATION

OF CORRELATION ENERGY

Until quite recently, two systematic methods have pro

vided themain routes to thecalculation of correlation

energies formany-electron systems

at

zero temperature,

namely, configuration interaction

CI) and

many-body

perturbation theory. Relatively recent overviewsofthese

two approaches areavailable [see,e.g.,Shavitt et aL

(1977) and Wilson (1981)], and therefore weshall not

go

over that ground

in

thepresent article.

In

thelast

ten

years, however, progress

in

thecalculation

of

electronic

correlation

has

also been made

by

using

a

completely

different approach,

the

so-called quantum Monte Carlo

(QMC) method.

The goalof QMC [see, e.g., Ceperley (1981), Reynolds

et

aL (1982), Ceperley

and

Alder (1984),

and

Kalos

(1984)]

is to

obtain theexact ground-state wave function

of a many-body system by numerically solving the

Schrodinger equation inoneofits equivalent forms. In

practice, this

is

achieved

by

means

of

iterative algo

rithms, which propagate thewave function from

a

suit

able starting guesstothe exact ground-state value. Thus ,

in thediffusion Monte Carlo (DMC) method, oneiscon

cerned directly with theevolution

in

imaginary time

of

the wave function, which corresponds

to a

diffusion pro

cess

in

configuration space.

In the

Green s-function

Monte Carlo (GFMC) technique, on theother hand,a

time-integrated form of the Green s function orresolvent

is used topropagate the wave function. Infact, ineither

case thesampling ofanappropriate Green s functionis

required, andthisisaccomplished bymeansofsuitable

random-walk algorithms.

Another implementation oftheQMC (Blankenbecler

et

al.

9

1981; Scalapino andSugar, 1981; Koonin et aL,

1982) emphasizes the roleofthe imaginary-t ime propaga-

R e v .Mod. Phys., Vol. 66, No. 2, April 1994

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461

to r e~P more directly. For long times 3(small temp era

tures r = l//?), the imaginary-time propagator is dom

inated by the ground-state energy

0

. Thus E

0

can be

suitably extracted from the knowledge of the partition

function Z Txe~

H

or, equally well, from that of the

expectation value of the imaginary-time propagator on a

state which is not orthog onal to the ground state. In ei

ther case, the key simplification in the calculations comes

from the mapping of the problem of interacting particles

onto that of independent particles. This is accomplished

at the expense of introducing auxiliary external fields

(Koonin

et al.

y

1982; Hirsc h, 1983; Sugiyama and Koo -

nin, 1986) having their own probability density.

In the following, we first review the main features of

the QMC method for a many-body system in the DMC

and GFMC implementations, discussing also the compli

cations that arise from antisymmetry when dealing with

systems of fermions, which is the case of interest here.

Some applicat ions of the QMC methodto the study of

the homogeneous electron assembly and small

molecu les are then briefly summa rized. Finally, we

give the basic ideas underlying the auxiliary external field

technique and review some of the recent results on the

two-dimensional Hubbard model.

A. Diffusion Monte Carlo method

The Schrodinger equation in imaginary time for an as

sembly ofN identical particles of massm interacting with

a potential V(R) reads

= [-DV

2

+V(R)-E

T

](R,t) , (4.1)

where D =fi/2m, R is the 3iV-dimensional vector speci

fying the coordinates of the particles, t is the imaginary

time in units of72, and the constant E

T

represents a suit

able shift of the zero of energy . Th e (imaginary) time

evolution of an arbitrary trial wave function is easily ob

tained from its expansion in terms of the eigenfunctions

^(R ) of the Hamil tonian

H

as

R,t)=

y

N

i

exp[-(E

i

-E

T

)t](f>

i

(R ) . (4.2)

Here ,E

t

is the energy eigenvalue corresponding to 0,( R) ,

and the coefficients

N

t

are fixed by the initial condition s,

i.e., by the chosen trial wave function. Clearly, for long

times one finds

(R,t)=N

0

exp[-(E

0

-E

T

)t](f)

0

(R ) , (4.3)

provided

N

0

^0.

Moreov er, if

E

T

is adjusted to be the

tru ground-state energy E

Q

, asymptotically one obtains

a steady-state solution, corresponding to the ground-state

wave function

0

. Thu s the problem of determining the

ground-state eigenfunction of the Hamiltonian H is

equivalent to that of solving Eq. (4.1) with the appropri

ate boundary conditions.

It is not difficult to reco gnize in Eq . (4.1) two equa tions

that are well known in physics, though combined togeth

er. In fact, if only the term w ith the Laplacian were

present on the right-hand side of Eq. (4.1), one would

have a diffusion equation, such as the equation describing

Brownian motion (see, for instance, van Kampen, 1981).

On the other hand, by retaining only the term

[V(R)E

T

](f>(R,t)

9

one would obtain a rate equation,

that is, an equation describing branching processes such

as radioactive decay or birth and death processes in a

population.

A convenient manner ofsimulating Eq . (4.1) is as fol

lows. Let ^(R) .be the wave funct ion at / = 0 . One can

generate an ensemble of systems (points in the 3JV-

dimensional space representing electronic configuration)

distributed with density ^

r

( R ) . Hence the t ime evolu

tion of the wave function will correspond to the motion

in configuration space of such systems or

walkers,

as

determined by Eq. (4.1). In particular, the Laplacian and

energy terms in Eq. (4.1) will cause, respectively, random

diffusion and branching (deletion or duplication) of the

walkers describing the wave function. This schem atiza-

tion of the Schrodinger equation is particularly appealing

from the prac tical point of view. How ever, a density is

non-negative. Thus the same should hold for the wave

function. This would seem to limit the applicability of

such a scheme to the ground state of a bosonic system.

For the sake of simplicity, we shall accept this restriction

for a while and deal with the complications associated

with Ferm i statistics later.

Solving the Schrodinger equation in its form (4.1) by

random-walk processes with branching is not particularly

efficient. In fact, the branch ing term can becom e very

large whenever the interaction potential V(R) does so,

causing large fluctuations in the numb er of walk ers. This

slows down the convergence toward the ground state. A

more efficient co mpu tational schem e is obtained by in tro

ducing the so-called importance sampling. This amounts

to considering the evolution equation for the probability

distribution f(R,t) = R,t)ip

T

R)> rather than directly

for the wave function . By using Eq. (4.1) and t he

definition of

f(R>t),

one obtains with a little algebra

- ^ ^ =-DV

2

f + [E

L

(R)-E

T

]f+DV[fE

Q

(R)] .

(4.4)

Above,E

L

{R)=Htp

T

/tp

T

defines th e local energyassoci

ated with the trial wave function, and

F

Q

(R) = V\n\if;

T

(R)\

2

= 2Vit>

T

(R)/ip

T

(R ) (4.5)

is the quantum trial force, which drifts the walkers away

from regions where ^(R ) is small. Obviously, when deal

ing with Eq. (4.4), the walkers are to be drawn from the

probability distribution /. With a judicious choice of rp

T

,

one can make E

L

(R ) a smoo th function close to E

T

throughout the configuration space and thus reduce

branc hing. In this respect it is essential that ^

r

( R )

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462 Senatore and March: Recent progress in electron correlation

reproduce the correct cusp behavior as any two particles

approach each other, so as to exactly cancel the infinities

originating from

V(K).

That

F

Q

(R )

isa force acting on

the walkers becomes clear by comparing Eq. (4.4) with

the Fokker-Planek (FP) equation (see, e.g., van Kampen,

1981). It turns out, with the rate term neglected, that the

term containing

FQ

has to be identified with the so-called

drift term

of the FP equation, and consequently

FQ

must

be identified with the external force acting on the walk

ers. We stress again that

an

imp ortant feature

of

this

force is todrift the w alkers away from regionsoflow

proba bility. In fact, from its definition it is app aren t tha t

the force becomes highly repulsiveinregions where the

trial wave function becomes small, diverging where the

latter vanishes. Thus the trial wave function determines

the probability with which different regions of

configuration space are sampled. Thereforeit isimpor

tant that

rp

T

bea good approximationto

T

(R)

0

(R)exp[-(E

0

-E

T

)t],

4.6)

which becomesa steady-state solution whenE

T

is adjust

ed toE

0

.

The differential equation (4.4) for the probability distri

butionf(R,t) can be recast in integra l form as

f(R,t +t )= fdR f(R

f

t )K(R ,R,t) , 4.7)

where the Green's function

K(R',R,t)

is a solutionof

Eq. 4.4)with the boundary condi tion J( R' ,R ,0)

= 8(R R') and is simply related to the Green 's

funct ion G(R',R,f)

for Eq.

(4.1),

K(R\R,t)

= ^

r

( R ' ) C / ( R ' , R , r ) ^

1

(R ). The advantage of the above

equation is that for short timestit is possible to w rite ap

proximate simple expressions for K{R >R,t) (Moskowitz

et

a/., 1982b; Rey nolds

et

a/., 1982). The time evolution

of f(R,t) for finitet can then be obtained by successive

iterationsofEq. (4.7), startin g from the initial distribu

t ion / ( R ,0 ) and usinga small t ime step. Ineach time

iteration, advantage can be takenofthe positivity of

K

for short times so as to interpret it as a transition density.

A suitable modified random-walk algorithm can then be

devised, which allows the integration of

E q.

(4.7) during a

small time interval. The essential steps of such an algo

rithm are as follows. The walkers represen ting the initial

distribution are allowed to diffuse randomly and drift un

der the action of the quantum force

FQ

. After the new

positions have been reached, each walkerisdeletedor

placed in the

n ew generation

in an appropriate number of

copies, depending on the sizeof the local energyatthe

old and at the new position relative to the reference ener

gyE

T

. Finally, the num ber of walkers in the new genera

tion isrenormalized to the initial population. Thein-

terested reader may finda detailed description of sucha

random-walk algorithm in Reynolds

et al.

(1982).


Once the long-time probability distribution has been

obtained and made stationary bya suitable shiftofthe

constant E

T

, on e canestimate equilibrium quantities.

For the ground-state energyE

0

,inparticular, by using

the fact tha t

H

is Herm itian, one finds

i

The sumonthe right-hand sideofthe above equation

runs over the positions of all the walkers representing the

equilibrium probability distribution.

It should be stressed that th e short-time approximation

to the Green's function introducesa systematic error in

the compu tations. How ever, this error should decrease

withadecrease in the time step. Moreov er, in the calcu

lation by Reynolds et ah (1982), an acceptance/rejection

step was used which, within thecalculational scheme

outlined above, should correct forthe ap proxim ate na

ture of

G:

B. Green's-function Monte Carlo technique

We have seen that in the diffusion M onte Carlo

method onestar ts from theSchrodinger equationin

imaginary time in order to construct an integral equation

which can then be solved iteratively. Such an equation

contains a time-dependent Gree n's function. In the

Green's-function M onte Ca rlo technique, one proceeds in

a similar fashion, starting, however, from the

Schrodinger eigenvalue equation. The resulting integral

equation,aswe sha ll see, differs from Eq . (4.7) m ainly

throug h the absence of t ime. In fact,itcould be directly

obtained from Eq. (4.7)by a straightforward timein-

tegration (Ceperley and A lder, 1984). Her e, however, we

choose to follow a noth er r oute (Ceperley andKalos,

1979; Kalos, 1984) which in the present context is some

what more instructive.

The Schrodinger eigenvalue equation for aniV-body

system, describedbythe Ham iltonianH introducedin

Eq . (4.1), is

H R)^E t> R)

. 4.9)

Let us assume that the potential

V(R)

appearing inHis

such that the spectrum ofHis bounded from below,E

0

being the lowest eigenvalue, i.e., the ground-state energy.

This is

a

natural assumption for

a

physical system in the

nonre lativistic limit. One can choose

a

positive constant

V

0

such that

E

0

+

V

0

>

0, and rew rite Eq . (4.9) as

(H +

V

Q

)(R)

= (E +

V

0

)(R). (4.10)

The resolvent G(R,R'), i .e. , the Green's function for Eq.

(4.10), is then defined by

(H +

F

0

) G ( R , R ' ) = 8 ( R - R ' ) (4.1 1)

D uetothe manner inwhich V

0

has been chosen,it is

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463

clear that the operatorH+

V

Q

is positive definite. This is

immediate in the energy representation. The same,of

course, holds forthe inverse oper ator

G =\/(H + V

0

).

Actual ly,a muc h m ore strict inequality involving G can

be established, namely,

G ( R , R ' ) = < R | G | R ' ) > 0 f o r a l l R , R '

(4.12)

which is of central importance for the calculational

scheme that we are about to describe. A brief sk etch of

how one can prove the inequality (4.12) isgiven in A p

pendix B. Here, we merely note for future reference that

G is integrable. Therefore, because of the above inequali

ty ,it can be regarded asa probability,or more precisely

as atransition density.

A space integration

of

Eq. (4.10), after multiplying by

G(R,R') and utilizing Eq. (4.11), yields the desired

in

tegral equation,

0 ( R )=(+ K

o

) / d R ' G ( R , R ' ) < ( R ' )

(4.13)

In the above equation one ha s to solve simultaneously for

E and 0(R). However,

if

one has

a

trial wave function

$

T

{R )

and, consequently, a trial energy

E

T

>

it is tempt

ing to try solving Eq. (4.13) by iteration . One wou ld gen

erate a sequence of wave functions, according to

0

(R), say, at the positions

{R,-j.

Then,

a new set of configurations {Rj\ is generated at ran

dom, with each one having a conditional density

2i

r(E + V

0

)G(R

i9

1

R

,

/

).

This density determines the

number of walkers corresponding to each new

configuration. Finally,

a

renormalization

of the new

walker population to the initial size yields the new gen

eration. The above series of steps corresponds to the first

iteration ofEq. (4.14). Th e successive iteration s canbe

performed in the same manne r, by only changing the first

step.

In the nth iteration, in fact, one hasto take as the

initial population ofwalkers th e new generation ofthe

previous iteration.

It is clear from Eq. (4.17) that the change in size of the

walker population isdetermined by the trial energy

E

T

.

Depending on whether this is largeror smaller than the

true ground-state energyE

0

, the population will asymp

totically grow or decline. This suggestsa way of estimat

ing the ground-state energy (Ceperley and Kalos, 1979).

In fact, from the space integration of Eq. (4.17) it follows

that asymptotically

EnE

T

-\- Vrs

N

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