GROUND-STATE PROPERTIES OF 3He↑ AND D↑ WITHIN THE METHOD OF CORRELATED BASIS FUNCTIONS

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GROUND-STATE PROPERTIES OF 3He ↑ AND D ↑ WITHIN THE METHOD OF CORRELATED BASIS

FUNCTIONS

J. Clark, E. Krotscheck, R. Panoff

To cite this version:

J. Clark, E. Krotscheck, R. Panoff. GROUND-STATE PROPERTIES OF 3He ↑ AND D ↑ WITHIN

THE METHOD OF CORRELATED BASIS FUNCTIONS. Journal de Physique Colloques, 1980, 41

(C7), pp.C7-197-C7-212. �10.1051/jphyscol:1980733�. �jpa-00220170�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6ment au n o 7 , Tome 41, juiZZet 1980, page ~ 7 - 1 9 7

GROUND-STATE PROPERTIES OF 3 ~ e t AND D t WITHIN THE METHOD OF CORRELATED BAS1 S FUNCTIONS J.W. Clark, E. ~ r o t s c h e c k * and R.M. Panoff

McDonneZl Center for t h e Space Sciences and Department of Physics, Washington University S t . Louis, Missouri 63130, USA.

*

Department of Physics, S t a t e University o f New York, Stony Brook, New York 11794, USA.

Resume

.-

On $ t i 1 is e des techniques d ' e q u a t i o n s - i n t e g r a l e s avancees

,

comportant des resommations de chaines dans 'la t h e o r i e HNC, pour en d @ d u i r e le s p r o p r i 6 t 6 s des modPles v a r i a t i o n n e l s de Jastrow pour l e s e t a t s fondamentaux de 3 ~ e t , de 3 ~ e non p o l a r i s e e t des deux especes de D+. On donne l e s r e s u l t a t s pour l ' e n e r g i e en f o n c t i o n de l a densite, pour l a f o n c t i o n de s t r u c t u r e du l i q u i d e e t (dans c e r t a i n s cas) pour l a masse e f f e c t i v e , l a s u s c e p t i b i l i t & magnetique e t l e s elements de ma- t r i c e de formation de p a i r e s . Les r e s u l t a t s i n d i q u e n t q u a i l sera necessaire, p a r t i c u l i e r e m e n t pour 3He non p o l a r i s e , d ' a l l e r p l u s l o i n que l e modele de Jastrow pour o b t e n i r une d e s c r i p t i o n micros- copique q u a n t i t a t i v e de ces s y s t h e s . On d e c r i t l e s e f f o r t s p r e l i m i n a i r e s q u i a n t @t@ f a i t s en vue d ' i n c o r p o r e r des c o r r e l a t i o n s dependant de l ' i m p u l s i o n e t du s p i n ( e t p l u s g&n@ralement d ' a u t r e s c o r r e l a t i o n s non inclusesdans l e modele de Jastrow)

a

l ' a i d e de l a methode des f o n c t i o n s de base correlees, (CBF)

.

A b s t r a c t

.-

Advanced i n t e g r a l - e q u a t i o n techniques, i n c l u d i n g Fenni hypernetted-chain resummation, a r e used t o d e r i v e t h e p r o p e r t i e s o f Jastrow v a r i a t i o n a l models o f t h e ground s t a t e s o f 3Het, u n p o l a r i z e d 3He and two species o f Dt. R e s u l t s a r e r e p o r t e d f o r t h e energy as a f u n c t i o n o f d e n s i t y , f o r t h e l i q u i d s t r u c t u r e f u n c t i o n and ( i n some cases) f o r t h e e f f e c t i v e mass, magnetic s u s c e p t i b i l i t y and p a i r i n g m a t r i x elements. The r e s u l t s i n d i c a t e t h a t i t w i l l be necessary, p a r t i c u l a r l y f o r u n p o l a r i z e d 3 ~ e , t o go beyond t h e Jastrow model t o achieve a q u a n t i t a t i v e microscopic account o f these systems. P r e l i m i n a r y e f f o r t s toward t h e i n c o r p o r a t i o n o f momentum- dependent, spin-dependent and o t h e r non-Jastrow c o r r e l a t i o n s by means o f t h e method o f c o r r e l a t e d b a s i s f u n c t i o n s a r e described.

1. I n t r o d u c t i o n . -

In

t h i s paper we e x p l o r e t h e Jastrow ansatz. Nevertheless, t h e v a r i o u s theo- p r o p e r t i e s o f a Jastrow v a r i a t i o n a l model o f t h e r e t i c a l r e s u l t s i n d i c a t e t h a t t h e Jastrow model ground s t a t e s o f t h e f o l l o w i n g h i g h l y quanta1 s t i l l c o n t a i n s much o f t h e c o r r e c t p h y s i c s o f systems: (a) o r d i n a r y , s p i n - s a t u r a t e d He, ( b ) 3 s t r o n g l y - c o r r e l a t e d Fermi systems even a t r e l a t i v e l y f u l l y s p i n - p o l a r i z e d He and ( c ) two analogous 3 h i g h d e n s i t y , p r o v i d i n g a u s e f u l vantage p o i n t species o f s p i n - p o l a r i z e d D. The p r e d i c t i o n s o f f r o m which more q u a n t i t a t i v e d e s c r i p t i o n s may be t h e model f o r such q u a n t i t i e s as t h e ground-state sought.

energy, t h e l i q u i d s t r u c t u r e f u n c t i o n , t h e s i n g l e - The search f o r more r e a l i s t i c and more com- p a r t i c l e energies and e f f e c t i v e mass, t h e magnetic prehensive t h e o r i e s may, f o r example, be c a r r i e d s u s c e p t i b i l i t y and t h e e f f e c t i v e p a i r i n g m a t r i x o u t w i t h i n t h e framework o f t h e method o f c o r r e - elements a r e presented as f u n c t i o n s o f density. l a t e d b a s i s f u n c t i o n s /9-11,7,8/ (CBFX e i t h e r by The techniques o f Fermi hypernetted-chain t h e o r y means o f (i) nonorthogonal CBF p e r t u r b a t i o n t h e o r y /I-8/ a r e a p p l i e d t o t h e e v a l u a t i o n o f these /I?-13/ o r v i a (ii) a more powerful scheme which q u a n t i t i e s . Comparing w i t h experimental r e s u l t s imp1 ements t h e coup1 ed-cl u s t e r ( o r exponential-S) on r e a l , s p i n - s a t u r a t e d l i q u i d He, 3 it i s found procedure i n t h e CBF c o n t e x t /14/. We r e p o r t here t h a t f o r t h i s system t h e Jastrow d e s c r i p t i o n has on some features o f t h e beginnings o f a t h e o r y c e r t a i n c l e a r d e f i c i e n c i e s , which a r e a s s o c i a t e d f o l l o w i n g p a t h ( i ) (which may b e looked upon as a w i t h t h e absence of momentum-dependent ( o r back- r e n o r m a l i z e d v e r s i o i l o f an e a r l i e r approach o f Woo f l o w ) and spin-dependent c o r r e l a t i o n s f r o m t h e /13/ and Tan and Feenberg /15/). As an a l t e r n a t i v e

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980733

C 7 - 1 9 8 JOURNAL DE PHYSIQUE

t o CBF theory, one may of course elaborate on t h e v a r i a t i o n a l description, incorporating backflow /16/, spin-dependent /17,18/ and other non-Jastrow c o r r e l a t i o n s d i r e c t l y i n t o t h e ground-state t r i a l function.

2. The Jastrow Model.- We focus our a t t e n t i o n on t h e Jastrow model of the ground s t a t e of a Fermi f l u i d , which i s based on a t r i a l wave function of t h e form

In t h e present applications, Qo i s the ground-state wave function of t h e noninteracting Fermi gas a t density r, = vk:/67r2, where v i s the s i n g l e - p a r t i c l e level degeneracy, i . e . , t h e number of p a r t i c l e s allowed i n a given c e l l of k-space. Correlations due t o i n t e r a c t i o n s a r e introduced by means of t h e operator F, which, i n t h e Jastrow case, i s taken a s t h e superposition

of r e a l , state-independent two-body c o r r e l a t i o n functions. I d e a l l y , the c o r r e l a t i o n function f ( r ) should be determined (optimally) by solving the v a r i a t i o n a l problem

6E[f]/6f = 0

,

(3)

where t h e extremum of

~ [ f ] = <H> = < Y ~ ~ H ~ Y ~ > / < Y ~ ~ Y ~ > ( 4 ) is supposed t o be a minimum. In practice, t h e Euler equation (3) and t h e expectation value (4) must be approximated /19,20/ ( o r the variational problem may be solved straightaway f o r a functional approximating ( 4 ) /21/). A l e s s ambitious and more widely practiced approach involves minimization of an approximate energy functional w i t h respect t o t h e parameters in a s u i t a b l y chosen a n a l y t i c form f o r f ( r ) .

The exact energy functional (4)--in partic- u l a r , i t s k i n e t i c energy portion--can be expressed i n several d i f f e r e n t (but s t r i c t l y equivalent)

forms r e l a t e d by integral i d e n t i t i e s . The most widely used a r e t h e Pandharipande-Bethe (PB), Jackson-Feenberg (JF) and Cl ark-Westhaus (CW) forms Detailed expressions may be found i n /1/ o r / 4 / . These expressions give cH> i n terms of the two- and t h r e e - p a r t i c l e radial d i s t r i b u t i o n functions g ( r l Z ) , g 3 ( r I 2 , r 1 3,r23) and portions of them involving t h e

- 3

S l a t e r function t ( k F r ) = 3 ( k F r ) ( s i n k F r

-

kFr coskFr), and of course a l s o in terms of t h e assumed two- p a r t i c l e potential v(r1 2)

,

t h e c o r r e l a t i o n , function f ( r 1 2 ) and t h e p a r t i c l e mass m. In general, i f approximate versions of g ( r 1 2 ) and g3(123) a r e i n s e r t e d , t h e t h r e e expressions <H>pB, <H>JF and

<H>CW w i l l not agree. The discrepancies give some measure of t h e inconsistency of t h e approximate evaluation of t h e d i s t r i b u t i o n functions--in par- t i c u l a r , a measure of t h e v i o l a t i o n of t h e BBGRY r e l a t i o n between g and g3 /22/.

Construction and a n a l y s i s of approximations t o t h e d i s t r i b u t i o n functions (and therewith t h e energy expectation value) a r e f a c i l i t a t e d by a configuration-space graphical scheme,analogous t o t h a t used widely i n t h e s t a t i s t i c a l theory of clas- s i c a l f l u i d s /23,24/. The building blocks f o r g (and g3) a r e (sub)diagrams with two external (labeled) points, say i j . In t h e present case of Fermi s t a t i s t i c s such diagrams may contain two kinds of 1 ine elements, namely dynamical c o r r e l a t i o n l i n e s cortesponding t o f a c t o r s f ( r ) 2

-

1 and s t a t i s - t i c a l c o r r e l a t i o n 1 in e s (exchange I i n e s ) correspon- ding t o S l a t e r f a c t o r s $(kFr). Fermi i j diagrams a r e therefore c l a s s i f i e d not only according t o t h e i r topological s t r u c t u r e (nodal a s opposed t o non-nodal /23,1/) but a l s o according t o whether a p a i r of exchange l i n e s (one e n t e r i n g , one leaving) i s present a t n e i t h e r ("dd" diagrams),

one

("de"

diagrams) o r ("ee" diagrams) of t h e external points / I / . The sums of a l l contributing non-nodal

One popular scheme f o r c a l c u l a t i n g t h e key diagrams of t h e indicated exchange c l a s s e s a r e

q u a n t i t i e s X i s t h e Fermi hypernetted-chain XY

procedure of Fantoni and Rosati /3/ (FR-FHNC).

denoted, respectively, X d d ( r ) y Xde(r) and Xee(r) (where r

r Ic.-C. 1).

^{I t} is convenient a l s o to i n -

J

troduce a fourth exchange c l a s s , comprised of i j The X a r e i n t h e i r turn represented in coordindte

XY 2

space a s simple functions of f ( r ) , R(kFr), t h e nodal-diagram sums N and t h e elementary-diagram

XY

(sub)diagrams i n which t h e r e i s a continuous exchange-line path running from i t o j ( o r vice

versa), possibly through intermediate p a r t i c l e s , sums E : XY

x d d ( r ) = r d d ( r )

-

^{Ndd(r) 9}

but no return path closing t h e exchange loop.

These a r e c a l l e d "cc" diagrams; excluding the graph consisting of a s i n g l e a l i n e joining i and j, t h e sum of a l l contributing non-nodal

xcc(r) = -v-lrdd(r)[~(kFr)-v~cc(r)~

+ r d d ( r ) l E c c ( r ) ⁹ (8) wherein

r d d ( r ) 2 (r)exp[Ndd(r) + Edd(r)]

-

1

-

^{( 9 )}

These equations express t h e formation of t h e non- diagrams of t h i s category is denoted Xcc(r).

The compound-graphical o b j e c t s X ( r ) , with

XY

xy E (dd,de,eel, a r e found t o be the c e n t r a l in- gredients of t h e diagrammatic a n a l y s i s , i n the sense t h a t once they a r e known, t h e d i s t r i b u t i o n

functions ( o r more conveniently t h e corresponding nodal X q u a n t i t i e s by p a r a l l e l connection of

"simple" (nodal o r elementary) diagrams. (An s t a t i c s t r u c t u r e functions) can be calculated from

exact, closed formulas /2,6,1/. For example, t h e elementary diagram i s a non-nodal diagram which s t r u c t u r e function

~ ( k ) = 1 + p/[g(r)-l]eib'C d_r can be obtained from

i s topologically i r r e d u c i b l e . )

In p r i n c i p l e , equations (6) (together w i t h an analogous construction of t h e s t r u c t u r e func- t i o n S3 corresponding t o g 3 ) , equations (8) and equations ( 7 ) s u f f i c e t o determine t h e Jastrow d i s t r i b u t i o n functions g and g3 ( o r S and S3) where we have absorbed a f a c t o r p in forming t h e

(dimensionless) t i l d e Fourier transforms, i x y ( k )

= pIXxy(r)exp(ik:~)dy, e t c .

TheFourier transforms of the sums ofallowed

exactly f o r a given c o r r e l a t i o n function f ( r ) . Indeed, solution of t h i s s e t of equations would generate t h e necessary ingredients f o r exact computation of the energy expectation value i n nodal diagrams of the various exchange c l a s s e s a r e

a l s o simple function of t h e

2

( k ) (and of z ( k ) ) :

XY

any of i t s forms. However, these equations do not a c t u a l l y supply a closed solution of the problem, because t h e E a r e given only a s i n f i n i t e s e r i e s .

XY

In p r a c t i c e , therefore, successive approximations iee(k) = S(k)

-

2Sd (k) + 1

-

^{iee(k) +}

fdd

^{( k ) ,}

i c c ( k ) = icc(k)[v-ll(k)

-

fcc(k)l/[l

-

i c c ( k ) l

.

⁽⁷⁾

These a r e t h e Fermi chain equations, which build

a r e defined by feeding i n successively more com- p l icated s e t s of elementary diagrams. The simplest approximation, explored by Fantoni and Rosati /3/, Zabol itzky /4/ and o t h e r s , c o n s i s t s of s e t t i n g a1 1 t h e chain (nodal) functions N xy by s e r i e s connec-

--

t i o n of non-nodal diagrams

.

-

^{the Exy = 0.} This approximation i s termed FHNC/O.

C 7 - 2 0 0 JOURNAL DE PHYSIQUE

I n t h e n e x t step, elementary diagrams w i t h f o u r - p o i n t b a s i c s t r u c t u r e s a r e incorporated, g i v i n g t h e FHNC/4 approximation; then f i v e - p o i n t b a s i c s t r u c t u r e s a r e t o be i n c l u d e d (FHNC/5); and so on.

(For more p r e c i s e s p e c i f i c a t i o n , see, f o r example, r e f e r e n c e /I/. )

I t t u r n s o u t t h a t such a scheme v i o l a t e s ( a t e v e r y stage) c e r t a i n long-wavelength asymptotic p r o p e r t i e s /5/ of t h e q u a n t i t i e s i d e ( k ) and i e ( k ) , name 1 y

where S (k) i s t h e t w o - p a r t i c l e s t r u c t u r e f u n c t i o n F

f o r t h e n o n i n t e r a c t i n g system. These p r o p e r t i e s a r e r e f l e c t i o n s o f t h e Paul i e x c l u s i o n p r i n c i p l e , i.e., t h e a n t i s y m n e t r y of t h e (Jastrow) wave f u n c t i o n . They may be e s t a b l i s h e d r i g o r o u s l y f o r t h e f u l l Xde, Xee; they a l s o h o l d i n approximation schemes i n which s u i t a b l y chosen f i n i t e subclasses o f elementary diagrams a r e i n c l u d e d ( f o r example, i n t h e s o - c a l l e d KR-FHNC// approximations /1,6/).

Normally, f o r reasonable f, v i o l a t i o n o f (10) by t h e FR-FHNC/O approximation ( o r h i g h e r approxi- mations w i t h i n t h e FHNC/ scheme) w i l l n o t have any s e r i o u s e f f e c t on an e v a l u a t i o n o f t h e ground-state energy, s i n c e i t i s , predominantly, t h e long-range b e h a v i o r o f t h e i n g r e d i e n t s Xde(r), Xee(r) which i s a t issue. However, o t h e r q u a n t i t i e s o f physical i n t e r e s t (e.g. S(k) a t small k ) a r e more s e n s i t i v e t o P a u l i v i o l a t i o n s . Moreover, when we t u r n t o t h e v a r i a t i o n a l problem (3), and t h e i m p l i c a t i o n s o f i t s behavior f o r t h e s t a b i l i t y o f many-body s t a t e s , i t becomes e s p e c i a l l y d e s i r a b l e t o t a k e p r o p e r c a r e o f t h e asymptotic c o n d i t i o n s (10).

To t h i s end, we s h a l l make use o f t h e FHNC/C approximation devised by one o f us /6/. L e t t h e FR-FHNC/O approximations t o

gde, gee,

as g i v e n by equations (81, be designated

, : : P 2;:;

then t h e

e f f e c t s o f t h e o m i t t e d elementary diagrams, r e - q u i r e d t o ensure ( l o ) , a r e e s t i m a t e d by u s i n g i n - stead t h e " c o r r e c t e d " approximations

I n a l l cases i n which t h e r e l e v a n t elementary diagrams have a c t u a l l y been c a l c u l a t e d , t h i s e s t i m a t e has proven t o be a c c u r a t e t o w i t h i n a few p e r c e n t /6/. The FHNC/C a p p r o x i m a t i o n t o t h e con- s t r u c t i o n o f t h e d i s t r i b u t i o n ' f u n c t i o n s c o n s i s t s i n m o d i f y i n g t h e FHNC/O scheme s i m p l y by t h e use o f (1 1 ) f o r jde and

gee

i n t h e c h a i n equations (7)

(and

only

t h e r e i n ) .

The E u l e r e q u a t i o n (3) may be s u b j e c t e d t o a g r a p h i c a l a n a l y s i s which para1 l e l s t h a t a l r e a d y executed f o r t h e d i s t r i b u t i o n f u n c t i o n s e n t e r i n g

<H>. Indeed, ( 3 ) assumes t h e f o r m

-(fi2/4m)02g(rl + g c (r) =

o ,

⁽¹²⁾

where g

'

( r ) i s a g e n e r a l i z e d t w o - p a r t i c l e d i s t r i - b u t i o n f u n c t i o n which may b e f o r m a l l y c o n s t r u c t e d by a process o f g r a p h i c a l d i f f e r e n t i a t i o n , i n d i - cated t h r o u g h o u t by a prime, a p p l i e d t o t h e d i a - grammatic r e p r e s e n t a t i o n o f g ( r ) /5/. We s h a l l n o t e n t e r i n t o t h e d e t a i l s here, b u t i t i s im p o r t a n t t o n o t e t h a t t h e e x p l i c i t d e f i n i t i o n f o r g t ( r ) depends on t h e e x p r e s s i o n

(PB,

JF, CW) adopted f o r t h e k i n e t i c energy p o r t i o n o f <H>, and ac- c o r d i n g l y so do t h e new s o r t s o f l i n e elements i n t r o d u c e d upon g r a p h i c a l d i f f e r e n t i a t i o n o f g.

The o p t i m i z a t i o n s c a r r i e d o u t i n t h e p r e s e n t s t u d y a r e based on t h e JF form f o r <H>. I n t h a t case, each g ' diagram w i l l c o n t a i n ( i ) a s i n g l e e f f e c t i v e i n t e r a c t i o n l i n e r e p r e s e n t i n g vJF(r) = v ( r )

-

('ti2/2m)~2~nf, o r ( i i ) a s i n g l e d i f f e r e n t i a t e d

2 2

exchange l i n e r e p r e s e n t i n g (3 /2m)v R(kFr) o r ( i i i ) a s i n g l e connected p a i r o f d i f f e r e n t i a t e d exchange 1 in e s r e p r e s e n t i n g (TI 2 /2m)viR(kFri j)

*vik(kF rik), p l u s a s s o r t e d f -1 and R l i n e s . 2

It i s more convenient t o work i n F o u r i e r space, t h e Eul e r e q u a t i o n beconii ng

w(k) 5 (45 2 2 k /4m)[S(k)-l]

+

S 1 ( k ) = 0

,

(13) where S' ( k ) = p f g ' (r)exp(i_k-_r)d_r i s th e g e n e r a l i z e d s t r u c t u r e f a c t o r corresponding t o t h e g e n e r a l i z e d d i s t r i b u t i o n f u n c t i o n g l ( r ) . By diagrammatic a n a l y s i s - - o r f o r m a l l y by f u n c t i o n a l and g r a p h i c a l d i f f e r e n t i a t i o n - - w e may e s t a b l i s h a r i g o r o u s de- composition of S1(k) analogous t o (6), i n terms o f S(k), Sd(k) and t h e primed c o u n t e r p a r t s i i d ( k ) , i i e ( k ) and i i e ( k ) o f t h e non-nodal q u a n t i t i e s i d d ( k ) , i d e ( k ) and dee(k). Equations analogous t o (7), d e r i v e d i n a s i m i l a r manner, g i v e 5 ~ ~ ( k ) , R i e ( k ) , R;,(k) and i i c ( k ) i n terms o f t h e primed i ' s and t h e v a r i o u s unprimed q u a n t i t i e s a l r e a d y introduced. F i n a l l y , by g r a p h i c a l d i f f e r e n t i a t i o n o f ( 8 ) , we o b t a i n a s e t o f 1 in e a r equations f o r t h e key q u a n t i t i e s X ' (These a r e c a l l e d t h e FHNC-

xy '

p r i m e equations, o r s i m p l y t h e prime equations.) We t h u s a r r i v e a t an a r r a y o f coupled equations, namely (6)-(9) and t h e i r primed c o u n t e r p a r t s ( 6 ' ) - ( 9 ' ) , which, i n c o n j u n c t i o n w i t h t h e E u l e r e q u a t i o n (73), determine i n p r i n c i p l e t h e o p t i m a l c o r r e l a - t i o n f u n c t i o n f (r) as w e l l as t h e corresponding t w o - p a r t i c l e s t r u c t u r e f u n c t i o n S (k)

,

and t h e r e - a f t e r f h e o p t i m a l Jastrow d i s t r i b u t i o n f u n c t i o n s and energy e x p e c t a t i o n value. Again, however, t h e problem o f t h e elementary diagrams, appearing i n t h e i n f i n i t e s e r i e s E and E '

,

must be faced i n

xy XY

a c t u a l c a l c u l a t i o n . Schemes f o r t h e i n c o r p o r a t i o n o f E' diagrams r u n p a r a l l e l t o t h o s e o f o r d i n a r y

XY FHNC theory.

Corresponding t o ( l o ) , t h e primed E ' s a r e r e s p o n s i b l e f o r t h e maintenance o f t h e r i g o r o u s long-wave1 ength p r o p e r t i e s

As i n d i c a t e d e a r l i e r , i t i s d e s i r a b l e t o observe

such p r o p e r t i e s when f o r m u l a t i n g t h e E u l e r problem.

A c c o r d i n g l y we extend t h e FHNC/C p r e s c r i p t i o n f o r c o r r e c t i n g t h e FHNC/O approximation, supplementing

Means f o r r e a l i z i n g a p r a c t i c a l numerical t r e a t m e n t o f t h e E u l e r e q u a t i o n (13) w i l l be de- s c r i b e d i n a separate a r t i c l e . We remark t h a t J.

Owen has a l r e a d y p u b l i s h e d work a l o n g s i m i l a r l i n e s /19/.

3. Beyond t h e Jastrow Description.-The Jastrow model i s expected t o d e s c r i b e r a t h e r w e l l some aspects o f t h e s t r o n g s p a t i a l c o r r e l a t i o n s among p a r t i c l e s i n a quantum f l u i d . However, s i n c e t h e assumed c o r r e l a t i o n o p e r a t o r i s s t a t e independent

(meaning i t i n v o l v e s o n l y t h e r. .) and c o y t a i n s

1 J

o n l y two-body f a c t o r s , such i m p o r t a n t phenomena as b a c k f l o w and s p i n - d e n s i t y f l u c t u a t i o n s w i l l be

i n a c c e s s i b l e t o t h i s model. One would l i k e t o have some way o f i n c o r p o r a t i n g state-dependent (as w e l l as d i r e c t three-body, four-body,

. .

^.)

c o r r e l a t i o n s i n t o t h e t h e o r y w i t h o u t g i v i n g up t h e s u c c e s s f u l aspects o f t h e Jastrow approach.

Systematic procedures f o r c o r r e c t i n g t h e Jastrow model may be f o r m u l a t e d w i t h i n t h e method o f cor- r e l a t e d b a s i s f u n c t i o n s /9-11,7,8,14/ (CBF).

D i s c u s s i o n o f t h i s method w i l l be c o n f i n e d t o t h e aspects needed f o r a general understanding o f the r e s u l t s t o be presented i n t h e f o l l o w i n g s e c t i o n s .

We extend c o n s i d e r a t i o n from a s i n g l e J a s t r o w - c o r r e l a t e d wave f u n c t i o n (1 ) - (2) t o a b a s i s o f such f u n c t i o n s ,

$,,,=

12

f(rij)@,,,

,

i < j

n o r m a l i z e d t o u n i t y by

Imm

fII(,,@<, 2 (r. .)I@,,,>

3 J

t h e

cP,

c o n s t i t u t i n g a complete orthonormal s e t o f

C 7 - 2 0 2 J O U R N A L DE PHYSIQUE

Fermi-gas energy eigenfunctions. The label m r e l i a b l e ( l e a s t s e n s i t i v e t o e r r o r s , p a r t i c u l a r l y i d e n t i f i e s t h e c o l l e c t i o n of plane-wave s i n g l e - i n t h e g++ g3 connection / 2 2 / ) anlong t h e t h r e e p a r t i c l e s t a t e s occupied i n t h e Fermi-gas function choices PB, JF, CW. In t h e CW case, the single-

@,,,,

with m = o denoting t h e f i l l e d Fermi sea. In p a r t i c l e energies a r e found t o have a very simple terms of the c o r r e l a t e d matrix elements Hmn = s t r u c t u r e i n terms of t h e q u a n t i t i e s already in-

<+mmJHl+n> and Nmn =

<qrn

of the Hamil tonian troduced i n optimal FHNC theory, namely and u n i t y , a perturbation expansion f o r the ground- fi2k2

.-.

~ ( k ) =

7

+ i i C ( k ) / [ l

-

XCC(k)l + Uo

,

s t a t e energy E may be generated, (18)

2 where Uo i s a constant, independent of k. The JF

E = HO0

- ^I

^Hmo

^-

^HooNmo

^I

⁺

^{... ,}

(16) r e s u l t i s a b i t more e l a b o r a t e , containing addi- m o Hmm

-

t h e higher terms involving more and more f a c t o r s of Hmn

-

HooNmn o r Nmn (l-6,,,). I t i s seen t h a t t h e leading term is j u s t the Jastrow energy expectation value, H o o = < H > . Here we s h a l l concentrate on t h e next term, t h e negative semi-definite second-order perturbation correction t o Hoo. W further r e s t r i c t e a t t e n t i o n to m l a b e l s which d i f f e r from t h e Fermi sea o in exactly two s i n g l e - p a r t i c l e o r b i t a l s .

(Therefore we include only the e f f e c t s of "cor- r e l a t e d two-particle-two-hole s t a t e s " . ) The re- sul t i n g correction i s denoted 6 ~ ( ~ ' ~ ) .

Two of us /7/ have c a r r i e d out an extensive diagrammatic a n a l y s i s of t h e q u a n t i t i e s Nmn

,

Hmn

-

HooNmn and Hmm

-

Hnn f o r choices of

m,n

i n - (2,2) cluding those needed f o r t h e evaluation of 6E

.

Denoting by m l , m2 and o l , o2 t h e o r b i t a l s i n which m and o d i f f e r , t h e r e l e v a n t r e s u l t s a r e conve- n i e n t l y expressed i n t h e forms

where the E ' S a r e i n t e r p r e t e d a s s i n g l e - p a r t i c l e energies and V(12) is a (non-local) e f f e c t i v e i n t e r a c t i o n . The derivations of reference /7/were based on t h e Clark-Westhaus (CW) form f o r t h e CBF matrix elements Hmn, because of the formal simplic- i t y of t h a t choice. In the meantime, t h e analysis has been extended t o t h e Jackson-Feenberg (JF) form, which we regard a s generally t h e most

t i onal terms

where C

l F

stands f o r Fourier transform, and a f u r t h e r (small ) contribution sE3] a r i s i n g from t h e three-body p a r t of t h e Jackson-Feenberg k i n e t i c energy operator. The s i n g l e - p a r t i c l e energy ~ ( k ) determines an e f f e c t i v e mass a t t h e Fermi surface via

fi2kF/iW= [di(k)/dkIkF

.

(20)

We s h a l l take m* t o be t h e e f f e c t i v e mass predicted by the Jastrow model.

The non-local e f f e c t i v e - i n t e r a c t i o n operator has the s t r u c t u r e

<mlm21 V(12)I 0 1 0 2 - 0 2 0 1 ~ = <mlm2 lW(12) 10102-0201> + + $ ~ ( m , ) + E ( ~ ~ ) - E ( o ~ 1 )-s(02)1-

-<mlm21N(12)lo102-0201>

,

(21

where W(12) and N(12) a r e again non-local operators, In d e t a i l , t h e l a t t e r operators a r e r a t h e r compli- cated; however, ~ ( 1 2 ) may be determined from N(12) by t h e graphical d i f f e r e n t i a t i o n process. The matrix elements of N(12) a r e found t o contain f a c t o r i z a b l e diagrams, implying t h e s t r u c t u r e

<mlm2 IN(12) lo102-0201~

<mlm21N (12)

B

10102

-

^0201>

-

~ ~ ~ - ~ c c ~ ~ l ~ ~ ~ ~ - ~ c c ~

'

~ 2 ~ ~ ~ ~ - ~ c c ~ ~ l ~ ~ ~ i - ~ c c ~ ~ 2 ~ ~ ~ ~

B (22

where N (12) is the irreducible, basic portion of N(12) a s defined in reference /7/; and s i m i l a r l y

f o r t h e m a t r i x elements o f (U(12) = N' (12). The

B B

l e a d i n g c o n t r i b u t i o n s t o N (12), W (12) a r e t h e i r l o c a l p a r t s ; these are simply e x p r e s s i b l e i n terms of primed and unprimed q u a n t i t i e s a l r e a d y en- countered. I n fact, i t was i n t h e CBF c o n t e x t /7/ t h a t FHNC-prime equations f o r t h e X' were

XY f i r s t d e r i v e d ( a l b e i t f o r t h e CW k i n e t i c energy o p e r a t o r ) . I t i s of course n a t u r a l t h a t t h e i n - g r e d i e n t s o f t h e CBF p e r t u r b a t i o n c o r r e c t i o n s , i n t h e i r d e t a i l e d s t r u c t u r e , a r e r e l a t e d t o t h e i n g r e d i e n t s o f t h e E u l e r equation f o r t h e optimal f (r). A comparable, though much l e s s cornpl i c a t e d , s i tuati'on p r e v a i 1 s w i t h i n t h e paired-phonon a n a l y s i s by Feenberg, Jackson and Campbell /10,25/

f o r t h e o p t i m a l Jastrow t r e a t m e n t o f Bose systems.

I n t h e a c t u a l c a l c u l a t i o n s o f ~ ~ ( 1 2 ) and [crB(l 2 ) we have included, along w i t h rdd(r12) and d i d ( r 1 2 ) , c e r t a i n t r a c t a b l e "elementary" c o n t r i b u t i o n s

(see Fig. 5.4 o f reference /7/), as w e l l as t h e e f f e c t s o f separable three-body c o n t r i b u t i o n s . The subsequent p r e p a r a t i o n o f 6E (' ") f o r numerical computation f o l l o w s an e s s e n t i a l l y standard p a t t e r n /26,27/: p a r t i a l - w a v e a n a l y s i s , angle-averaging, q u a d r a t i c approximation o f h o l e energies E ( k ) , k < k F , e t c . However, i n c o n t r a d i s t i n c t i o n t o t h e procedure o f r e f e r e n c e ,/27/, t h e p a r t i c l e energies

~ ( k ) , k > kF, a r e n o t taken t o be s i m p l y t h e f r e e 2 2

e n e r g i e s

6

k /2m b u t a l s o i n c l u d e t h e constant t e r m Uo. (It i s t o be noted t h a t above t h e Fermi s u r f a c e t h e remaining terms i n t h e ~ ( k ) formula r a p i d l y become n e g l i g i b l e compared t o t h e f r e e k i n e t i c energy. ) A more thorough d i s c u s s i o n o f t h e s e m a t t e r s w i l l be presented i n another a r t i c l e . One q u a l i t a t i v e f a c t emerging from a n a l y s i s o f t h e s t r u c t u r e o f 6 E ( 2 y 2 ) should, nevertheless, be men- t i o n e d here. When t h i s c o r r e c t i o n i s reduced f o r t h e s p e c i a l case o f o p t i m a l f, making use o f t h e E u l e r e q u a t i o n w(k) = 0 and i n t r o d u c i n g and subtrac-

t i n g o u t an i n v e r s e energy d i f f e r e n c e < e - t averaged over holes, one may i s o l a t e a c o n t r i b u t i o n a t t r i b - u t a b l e t o n o n - c e n t r a l (angle-dependent) c o r r e l a t i o n s , a c o n t r i b u t i o n a p p a r e n t l y r e l a t e d t o backflow.

We conclude o u r o u t l i n e o f formal methods by p o i n t i n g o u t t h a t t h e r e s u l t s o f r e f e r e n c e /7/ f o r t h e non-diagonal CBF q u a n t i t i e s Nmn, Hmn-HooNmn and Hm-Hnn n o t o n l y y i e l d t h e i n p u t s necessary f o r e s t i m a t i o n o f t h e CBF p e r t u r b a t i o n c o r r e c t i o n s t o t h e Jastrow energy, b u t a l s o p r o v i d e f u r t h e r valu- a b l e i n f o r m a t i o n on t h e Jastrow model i t s e l f . This has a l r e a d y been seen i n t h e case o f t h e Jastrow e f f e c t i v e mass. To c i t e a more e l a b o r a t e example:

as shown i n r e f e r e n c e /8/, one may t e s t t h e s t a - b i l i t y o f t h e Jastrow ground s t a t e a g a i n s t p a i r condensation i n v a r i o u s p a r t i a l waves, i n terms o f

e f f e c t i v e p a i r i n g m a t r i x elements d e r i v e d from t h e o p e r a t o r ~ ( 1 2 ) . I n a t h i r d a p p l i c a t i o n (which by no means exhausts t h e i n t e r e s t i n g p o s s i b i l i t i e s ) , one may determine t h e magnetic s u s c e p t i b i l i t y

x

o f t h e Jastrow model from CBF m a t r i x elements. I f t h e CW form i s assumed f o r t h e Hmn, t h e simple formula

( a p p l i c a b l e t o u n p o l a r i z e d He) i s obtained, where 3

xF

i s t h e magnetic s u s c e p t i b i l i t y o f t h e f r e e Fermi gas and 6 ( 1 So) i s t h e dimensionless e f f e c t i v e p a i r i n g m a t r i x element i n t h e So channel, d e r i v e d 1 from t h e CBF a n a l y s i s o f r e f e r e n c e /8/. ( I t should be remarked t h a t e q u a t i o n (23) i s j u s t a formal r e - s u l t among CBF q u a n t i t i e s and does n o t i m p l y a c o r - responding p h y s i c a l r e l a t i o n s h i p between magnetic response and p a i r i n g . )

4. A p p l i c a t i o n s t o Helium Systems.- We c o n s i d e r o r d i n a r y , unpol a r i z e d He ( w i t h 3 1 eve1 degeneracy v = 2), t o g e t h e r w i t h f u l l y s p i n - p o l a r i z e d He (v=l). 3 The l a t t e r system w i l l be denoted He+; t h e former, 3 s i m p l y as He. 3 P r o p e r t i e s o f t h e Jastrow model o f

14

C 7 - 2 0 4 JOURNAL DE PHYSIQUE

these systems a r e computed f o r two choices of the safely preserves t h i s property, without e r r i n g too c o r r e l a t i o n function f ( r ) : ( a ) the Schiff-Verlet much in t h e upward d i r e c t i p n (which i s usually the

form /28/ case f o r t h e CW form). To avoid any possible con-

1 b 5

f S V ( r ) =

ex^[-? (T;) 1

³ (24) fusion, we s t r e s s t h a t t h e f opt r e s u l t s i n Table I with b = 2 . 8 8 8

1

independent of density and polar- and Fig. 1 a r e obtained by i n s e r t i o n of the

JF-

i z a t i o n and ( b ) an optimal f ( r ) determined from based optimal f i n t o t h e indicated functionals.

t h e JF version of the Euler equation, f o r

each

v We a l s o note t h a t a l l numerical energies a r e of and

each

density. The usual Lennard-Jones poten- course given i n K per p a r t i c l e .

t i a l i s assumed, v ( r ) = 4 c [ ( a / r ) I 2

-

( a / r ) 6 ] with

0

E = 10.22 K and a = 2.556 A.

Results f o r t h e ground-state energy expecta- t i o n value a r e collected i n Table I and plotted ( f o r the optimal f ) in Fig. 1. The reader should focus h i s a t t e n t i o n on the r e s u l t s f o r EJF = <H> JF.

s i n c e t h e other forms of t h e energy (CW, PB) con- t a i n l a r g e terms involving aspects of t h e three- p a r t i c l e d i s t r i b u t i o n function which FHNC theory may represent poorly /4,6,22/. ~t p=0.0142K3 in t h e case of ordinary He, t h e exact expectation 3 value f o r t h e Schiff-Verlet (SV) correlation func- t i o n specified above i s known by flonte Carlo (MC) c a l c u l a t i o n / 2 9 / t o l i e roughly midway between the JF and PB energy functionals computed i n FHNC/O or FHNC/C approximation /4,22/. However, w i t h i n these approximations t h e PB choice i s t h e "most l i k e l y "

among PB, JF and CW t o v i o l a t e t h e upper-bound

property of t h e exact <H> upon variation of f Fig. 1 : Jastrow ground-state energy versus density f o r unpolarized 3 ~ e and f o r 3 ~ e + , based on around a s e n s i b l e reference function. We expect _{t h e} Lennard-Jones potential ( a = 2.556 A ) .

(Curves and points not marked with

+

r e f e r t h a t in t h e present applications t h e JF choice t o unpolarized 3 ~ e . )

3 3

Table I : Jastrow ground-state energies f o r unpolarized He and f o r He+.

One can immediately draw two i n t e r e s t i n g q u a l i t a t i v e conclusions from t h i s study of the ground-state energetics. F i r s t , the Jastrow model cannot reproduce the experimentally determined energy and density of unpolarized liquid He 3

(Lequil = -2.52 K, pequil = 0.0164 i - 3 ) a t zero temperature and zero external pressure. Important c o r r e l a t i o n e f f e c t s (presumably associated w i t h spin-density f l u c t u a t i o n s and backf low) a r e c l e a r l y missing from t h e Jastrow ansatz. Indeed, t h e margin of f a i l u r e seems especially d i s t u r b i n g u n t i l i t i s realized t h a t t h e t o t a l energy E of t h e system r e s u l t s from near cancellation of r e l a - t i v e l y l a r g e k i n e t i c and potential contributions.

For these individual terms t h e percentage e r r o r of t h e Jastrow model i s only of order 10%.

The second point concerns t h e comparison of t h e two polarization s t a t e s of He. 3 I t i s seen t h a t , beyond a re1 a t i v e l y low density, t h e Jastrow models of He+ a r e e n e r g e t i c a l l y more s t a b l e than 3 t h e corresponding models of the unpol a r i zed system This i s t h e case even f o r t h e SV c o r r e l a t i o n function--the parameter b of which was determined f o r ordinary 3 ~ e . We have, then,another indica- t i o n t h a t t h e Jastrow t r i a l function i s inadequate f o r unpolarized He. 3 On the other hand, t h e Jastrow model appears t o be r a t h e r good f o r He+, a t l e a s t 3 s o f a r a s t h e energy i s concerned: a reasonable i n t e r p o l a t i o n between t h e JF and PB curves f o r t h i s system would put t h e energy minimum somewhere i n t h e range -1.5 K t o -2 K, and the t r u e energy s u r e l y cannot be much lower. (We assume f o r t h e sake of argument t h a t t h e FHNC/C procedure i s no

3 3

l e s s accurate f o r He+ than i t i s f o r ordinary He, a supposition which has y e t t o be f u l l y t e s t e d . In t h i s connection one may observe in Fig. 1 t h a t f o r 3 ~ e + , t h e FHNCIC approximation t o the PB functional c l e a r l y v i o l a t e s the upper-bound property a t

d e n s i t i e s exceeding p = 0.014 i - 3 . )

Inspecting Table I , we notice t h a t f o r ordi- nary He t h e optimal f does not lead t o much im- 3 provement over t h e simple SV choice, EJF being e s p e c i a l l y i n s e n s i t i v e ; even f o r He+ t h e improve- 3 ment i s not dramatic. In some cases i t turns out

t h a t f opt

-

r a i s e s t h e energy s l i g h t l y , compared t o t h e SV energy. While t h i s might a t f i r s t seem con- t r a d i c t o r y , i t must be remembered t h a t s i n c e t h e Euler equation (13) i s derived from an exact energy functional and t h e r e a f t e r approximated, the s o l u t i o n obtained does not necessarily minimize an approxi- mate energy functional (FHNC version of <H>JF, e t c ) .

In Fig. 2 t h e s t r u c t u r e f a c t o r s S(k) of the optimal Jastrow models of unpolarized and polarized systems a r e plotted a t t h e same density (near

3

Pequ i 1 of ordinary He). The r e s u l t s f o r t h e two cases a r e very s i m i l a r , t h e only apparent d i s t i n c - t i o n being t h a t t h e peak is displaced s l i g h t l y in- ward f o r He+ compared t o t h e normal system. 3

Fig. 3 exposes f u r t h e r shortcomings of t h e Jastrow description of unpolarized He. 3 The ef- f e c t i v e masses of optimal and SV models a r e f a r from t h e experimental value, which has recently been s e t /30/ a t (m*/m) = 2.12 ( f o r p = pequil).

exp

An even more s t r i k i n g symptom i s displayed by the s u s c e p t i b i l i t y r a t i o

xF/x

of the SV model, which dives i n t o t h e negative region already a t q u i t e low density. The l a t t e r behavior corresponds t o our previous finding t h a t , within the Jastrow approach, 3 ~ e + i s e n e r g e t i c a l l y favored over i t s unpolarized counterpart, except a t small p. (We should remark t h a t t h e

xF/x

curve drawn i n Fig. 3 is not c o n s i s t e n t through (23) w i t h the values of m*/m and 6 ( 1 so) reported here. This curve i s de- rived from Jastrow pairing matrix elements and e f f e c t i v e masses /8/ based on t h e CW form of t h e Hmn ( f o r which (23) a p p l i e s ) , whereas the other

C 7 - 2 0 6 JOURNAL DE PHYSIQUE

q u a n t i t i e s in Figs. 3 and 4 a r e calculated using a r e purely i l l u s t r a t i v e , having no s i g n i f i c a n c e f o r t h e JF form. A t any r a t e , t h e

xF/x

r e s u l t s given t h e r e a l system.)

Fig. 2 : Jastrow s t a t i c s t r u c t u r e function f o r unpol arized 3 ~ e and f o r 3 ~ e 4 .

F i g . 3 : Effect've mass and magnetic s u s c e p t i b i l i t y

4

Fig. 4 : Dimensionless airing matrix elements of o f unpolarized He, corresponding t o t h e Jastrow unpolarized 3tle, corresponding t o the Jastrow

models. models, f o r various two-body channels.

Fig. 4 supplements t h e CW r e s u l t s of r e f e r - ence /8/ with plots of the (dimensionless) Jastrow pairing matrix elements 6 of normal 3 ~ e in So, 1 3p and 1

D

p a r t i a l waves, a s determined by FHNC/C-

0 2

approximated CBF q u a n t i t i e s of JF form. Negative 6 values signal i n s t a b i l i t y of t h e Jastrow s t a t e with respect t o p a i r condensation in t h e given p a r t i a l wave. Detailed consideration of these r e s u l t s (along t h e 1 ines of'reference /8/) leads once more t o t h e conclusion t h a t t h e Jastrow cor- r e l a t i o n operator i s d e f i c i e n t in important r e s p e c t s - - p a r t i c u l a r l y i n i t s lack of spin (and momentum) dependence.

Table I1 presents some r e s u l t s of an attempt t o c o r r e c t f o r t h e deficiencies of the Jastrow d e s c r i p t i o n by means of CBF perturbation theory.

The required CBF matrix elements a r e evaluated via t h e FHNC/C procedure a s sketched in sections 2,3.

The e n t r i e s f o r bE ( 2 s 2 ) represent our most complete estimate of this quantity, based on formula (22) f o r the N(12) matrix elements and t h e corresponding formula f o r ~ ( 1 2 ) derived by graphical (prime) d i f - f e r e n t i a t i o n . In the l a t t e r fotmul a the denominator D =

r

^[I-Em(., )I[] - i C c ( 0 , ) 1 1 ~ w i l l a l s o appear. As we shall document elsewhere, t h e correction 6 ~ ( (for He systems) i s q u i t e ~ 3 ~ ~ ) s e n s i t i v e t o t h e precise means used t o t r e a t t h i s denominator. Accordingly, the r e s u l t s given i n Table I1 should be regarded as i l l u s t r a t i v e rather than q u a n t i t a t i v e . I t i s seen t h a t t h e c o r r e c t i o n is very l a r g e i n magnitude and c l e a r l y overesti- mates t h e e f f e c t s of t h e non-Jastrow c o r r e l a t i o n s . The other e n t r i e s in t h e t a b l e , labeled 6~,!,:'~), are the r e s u l t s f o r t h e perturbation correction without t h e denominator D, i.e., the f a c t o r

{[I-ice]. ..I-'

i s replaced by unity i n the f i n a l formulas f o r N(12) and W(12). I t may be argued t h a t t h e omission of D from the present treatment

simulates t h e e f f e c t of higher-order, RPA-type con- t r i b u t i o n s t o t h e CBF perturbation expansion. The

( 2 ' 2 ) i s much modified second-order correction 6Ewo

more reasonable in s i z e , though l a r g e r than 6E ( 2 2 ) a s approximated by Woo /13/. Woo used, i n e f f e c t , l e s s highly dressed CBF matrix elements than em- ployed herein; among o t h e r s i m p l i f i c a t i o n s , t h e denominators D do not appear.

Table I1 : Second-order CBF p e r t u r b a t i o n c o r r e c t i o n s t o Jastrow ground-state energies f o r un- polarized 3 ~ e , using Jackson-Feenberg k i n e t i c energy operator.

P

sv

^f

(10-3 i - 3 ) 6E(2Y2) =(232) "wo (292)

We do not r e p o r t any r e s u l t s f o r t h e pertur- bation correction i n the case of He+, f o r t h e 3 following reason, associated with the behavior of

' - i -

Xcc(O ) shown i n Fig. 5. The non-nodal compound- diagrammatic q u a n t i t y Xcc(k) has i t s maximum value

-

a t k=O; i t f a l l s off as k increases through the Fermi s e a , and displays damped o s c i l l a t i o n s about zero f o r k > kF. From Fig. 5 we see t h a t icc(O

+

) r i s e s monotonically with density, reaching unity i n ' ~ e + a t a r e l a t i v e l y low value of p. Beyond t h a t c r i t i c a l d e n s i t y , s i n g u l a r i t i e s appear in t h e expression f o r t h e CBF correction & E ( ~ , ~ ) , because of t h e vanishing, a t some k , of denominators [I-icc(k)]' i n t h e W and N matrix elements and of t h e denominator 1-Xcc(k) i n t h e s i n g l e - p a r t i c l e energies ~ ( k )

.

Consequently, f o r d e n s i t i e s a t which icc(O t ) > 1 , t h e CBF perturbation procedure, i n i t s p r e s e n t computational r e a l i z a t i o n , ceases

--

C 7 - 2 0 8 JOURNAL DE P H Y S I Q U E

t o be meaningful. Formally, i t may be shown t h a t we choose n o t t o quote any m*/m r e s u l t s f o r 3 ~ e + t h e s i n g u l a r i t i e s generated by t h e denominator o f u n t i l t h e o r i g i n o f t h e s i n g u l a r i t y i n t h e s i n g l e - (22) a r e i n f a c t c a n c e l l e d by those o c c u r i n g i n p a r t i c l e energy i s b e t t e r understood.

t h e ~ ( k ) , p r o v i d e d 2 C C ( k ) remains l e s s than u n i t y That t h e values o f i C c ( o t ) i n u n p o l a r i z e d

f o r k > kF. (We a r e reminded o f s i m i l a r compensa- 3

3 ~ e a r e about h a l f those i n He+ may be a t t r i b u t e d t i o n s o f p o s s i b l e [l-iCc]-' s i n g u l a r i t i e s w i t h i n t o t h e presence o f a (rough) o v e r a l l f a c t o r o f v-' t h e FHNC t r e a t m e n t o f t h e Jastrow s p a t i a l and i n t h i s q u a n t i t y . It i s i n t e r e s t i n g t o n o t e t h a t momentum d i s t r i b u t i o n s . ) However, o u r computa-

-

_Xcc(O

+

) f o r o r d i n a r y He e v e n t u a l l y crosses u n i t y , ³ t i o n a l procedure, which i s based on an e f f e c t i v e - a t a v a l u e o f p somewhere beyond t h e c r y s t a l l i z a - mass approximation, has y e t t o b e r e f o r m u l a t e d t i o n d e n s i t y . L a n t t o /31/ has observed t h e t o t a k e advantage o f t h i s c a n c e l l a t i o n . The v a n i s h i n g o f 1-Xcc(O

- ⁺

) i n t h e e l e c t r o n gas a t v e r y [ 1 - i C c ] - l s i n g u l a r i t y i n ~ ( k ) (and hence i n - l o w density--corresponding t o t h e s t r o n g - c o u p l i n g Hm-Hoo) may be merely an a r t i f a c t o f o u r theory, regime o f t h e Coulomb system.

w i t h o u t p h y s i c a l relevance. On t h e o t h e r hand 5. A p p l i c a t i o n s t o Deuterium Systems.- We d e f i n e i t may a c t u a l l y r e f l e c t some i n t e r e s t i n g p h y s i c a l D+ as a system o f deuterium atoms somehow con-

- +

phenomenon; e.g., X (0 ) = 1 may s i g n a l some im-

CC s t r a i n e d so t h a t any p a i r o f atoms i n t e r a c t s ex- minent phase t r a n s i t i o n . F u r t h e r a n a l y s i s ( f o r c l u s i v e l y i n t h e b 3 ~ : s t a t e /32/. S i n c e t h e example, u s i n g t h e p e n e t r a t i n g methods o f CBF l a t t e r i n t e r a c t i o n i s s t r o n g l y r e p u l s i v e a t s h o r t c o u p l e d - c l u s t e r t h e o r y /14/) i s needed t o decide d i s t a n c e s and o n l y v e r y weakly a t t r a c t i v e a t between these two p o s s i b i 1 i t i es. l o n g e r range, we have a system o f f e r m i o n s which

The e f f e c t of a [l-iCc~-' s i n g u l a r i t y on t h e i s expected t o d i s p l a y extreme quanta1 b e h a v i o r - - Jastrow values of mx/m, t h e p a i r i n g m a t r i x elements even more so than He /33/. 3

6 and t h e s u s c e p t i b i l i t y r a t i o

xF/x

w i l l be s l i g h t Two species o f D+ a r e examined here, namely:

u n l e s s t h e s i n g u l a r i t y appears n e a r kF. Even so, ( i - ) D+l, w i t h

one

a l l o w e d n u c l e a r - s p i n s t a t e , and

F i g . 5 : Compound-diagrammatic q u a n t i t y XCc(Ot) f o r

-

u n p o l a r i z e d 3 ~ e and f o r ' ~ e + , as a f u n c t i o n o f d e n s i t y .

( i i ) Df2, w i t h

two

a l l o w e d n u c l e a r - s p i n s t a t e s which a r e assumed t o be e q u a l l y populated /34/.

We n o t e t h a t Dtl, w i t h one Fermi sea, has v = l and corresponds t o ' ~ e t , w h i l e Dt2, w i t h two equal Fermi seas and v=2, corresponds f o r m a l l y t o o r d i n a r y ' ~ e . W i t h such correspondences i n mind, c a l c u l a t i o n s o f t h e t y p e described i n s e c t i o n 4 have been repeated f o r deuterium. It i s found t h a t t h e r e s u l t s pre- s e r v e t h e s t a t e d analogies, t o t h e e x t e n t t h a t many o f t h e q u a l i t a t i v e f e a t u r e s encountered i n s e c t i o n 4 (as w e l l as t h e a s s o c i a t e d judgments) c a r r y o v e r

3 3

w i t h He+ r e p l a c e d by D+l and u n p o l a r i z e d He r e - p l a c e d by De2. I n p a r t i c u l a r , t h e Jastrow model shows an e n e r g e t i c preference f o r

aP,

over D+2

(except a t low d e n s i t y ) ; t h e J a s t r o w t r i a l f u n c t i o n may a g a i n be c o n s i d e r a b l y b e t t e r f o r v = l t h a n f o r v=2.

The c a l c u l a t i o n s a r e based on t h e t h e o r e t i c a l b 3 ~ : p o t e n t i a l o f Kolos and Wolniewicz /35/, as used by M i l l e r and Nosanow /34/. We r e p o r t r e s u l t s o n l y f o r a S c h i f f - V e r l e t c o r r e l a t i o n f u n c t i o n (24).

(Optimal c o r r e l a t i o n f u n c t i o n s f o r t h e deuterium systems w i l l b e generated i n l a t e r work. ) The r e - s u l t s f o r energy e x p e c t a t i o n values, s t r u c t u r e f u n c t i o n s , e f f e c t i v e masses, second-order p e r t u r b a - t i o n c o r r e c t i o n s and 3=(0

+

) values a r e presented i n Tables I 1 1 and I V and Figs. 6-9. Note t h a t i n

T a b l e I 1 1

Jastrow ground-state energies f o r two species o f s p i n - a l i g n e d deuterium. (LO r e f e r s t o lowest c l u s t e r o r d e r approximation.)

these f i g u r e s , a = 3.69

r\,

corresponding t o a Lennard- Jones f i t t o t h e Kolos-Wolniewicz p o t e n t i a l /34/.

T a b l e I V

Second-order CBF p e r t u r b a t i o n c o r r e c t i o n s t o J a s t r o w ground-state e n e r g i e s f o r DT2, u s i n g Clark-Westhaus k i n e t i c energy o p e r a t o r /7/. ( S i x p a r t i a l waves a r e included. )

It i s o f s p e c i a l i n t e r e s t t o compare t h e Jastrow e n e r g i e s o b t a i n e d i n FHNC/C a p p r o x i m a t i o n w i t h t h e e a r l i e r r e s u l t s o f M i 1 l e r and Nosanow (MN).

I n t h e MN work, a h y p e r n e t t e d - c h a i n (HNC) o r BBGKY- KSA procedure /36/ was a p p l i e d t o e v a l u a t e t h e r a d i a1 d i s t r i b u t i o n f u n c t i o n corresponding t o t h e Jastrow f a c t o r IIf ( r i j), and t h e Wu-Feenberg (WF) antisymmetry expansion /37/, c a r r i e d t o t h r e e - i n d e x terms, was used t o c o r r e c t f o r t h e presence o f t h e S l a t e r determinant Oo i n t h e (Fermi) J a s t r o w ansatz

(1 ) - ( 2 ) . T h i s t r e a t m e n t i s based on t h e JF energy f u n c t i o n a l . The c o r r e l a t i o n f u n c t i o n f was taken o f SV form, w i t h b determined f o r each v and p b y m i n i m i z i n g t h e Fermi-system energy i n HNC-WF o r BBGKY-KSA-WF approximation. b!e have adopted t h e b values corresponding t o t h e BBGKY-KSA approxima- t i o n , which were s u p p l i e d by M i l l e r . F o r t h e Dtl system o u r JF energies agree v e r y w e l l w i t h t h e HNC-WF r e s u l t s r e p o r t e d by M i l l e r and Mosanow (see F i g . 6 ) . On t h e o t h e r hand, i n t h e case o f Dt2 t h e JF and MN curves d e p a r t markedly f r o m one

another as t h e d e n s i t y i n c r e a s e s p a s t 0.003

im3

(see f i g . 7). The disagreement o f o u r D.E2 r e s u l t s

C7-210 JOURNAL DE PHYSIQUE

w i t h those o f MN i s even worse f o r t h e i r BBGKY-KSA c a l c u l a t i o n . ( A c a l c u l a t i o n a l s i t u a t i o n analogous t o t h a t f o r Df2 p r e v a i l s w i t h respect t o unpolar- i z e d He: 3 juxtapose t h e f i n d i n g s o f r e f e r e n c e /38/ a g a i n s t those o f references /4,22/.) ^It must be kept i n mind throughout such c o n s i d e r a t i o n s t h a t t h e n e t energies E o f t h e systems under study a r e v e r y small i n magnitude compared t o t h e separate k i n e t i c and p o t e n t i a l energies, so t h a t r e l a t i v e l y small e r r o r s i n t h e e v a l u a t i o n o f these separate p a r t s i s p r o m i n e n t l y r e f l e c t e d on t h e

i s somewhat s u r p r i s i n g f o r t h e D+2 case, where t h e d i f f e r e n c e s between o u r r e s u l t s and those o f MN can exceed 1 K i n t h e d e n s i t y range considered.

s c a l e o f Figs. 6 and 7.

-3

"

-3

p

(10 A

1

F i g . 7 : Jastrow ground-state energy versus d e n s i t y f o r D+ w i t h two a l l o w e d n u c l e a r - s p i n s t a t e s , e q u a l l y populated. ( C a l c u l a - t i o n s based on t h e KO 0s-Wolniewicz p o t e n t i a l ; o = 3.69

B.)

The p r e s e n t c a l c u l a t i o n s t i l l does n o t s e t t l e t h e q u e s t i o n o f whether Dfl ( r e s p e c t i v e l y D f 2 ) i s a l i q u i d o r a gas i n i t s ground s t a t e under zero

F i g . 6 : Jastrow ground-st'ate energy versus d e n s i t y f o r D+ w i t h one allowed n u c l e a r - s p i n s t a t e , based on t h e Kolos-Wolniewicz potential,(corresponding Lennard-Jones a = 3.69 A).

For b o t h deuterium systems we c a r r i e d o u t a search i n t h e v i c i n i t y o f t h e b values p r o v i d e d by M i l l e r . It was found t h a t these parameters are s t i l l v e r y c l o s e t o o p t i m a l f o r t h e JF energy f u n c t i o n a l as approximated here; t h e best parameters produce changes i n t h e JF curves which would h a r d l y be n o t i c e a b l e on t h e s c a l e o f Figs. 6 and 7. T h i s

pressure. ( I n t h e former case one has a Fermi l i q u i d l i k e o r d i n a r y He; i n t h e l a t t e r , one can 3 go from gas t o l i q u i d a t T = 0 under a p p r o p r i a t e pressure. ) However, we may c a l l a t t e n t i o n t o t h e v e r y small n e t e n e r g i e s and s u b s t a n t i a l n e g a t i v e

"

³

p o t e n t i a1 e n e r g i e s (e. g

. , <V> =

-5 K a t p

=

0 . 0 0 4 ~ - ) c h a r a c t e r i z i n g o u r J a s t r o w r e s u l t s . These f e a t u r e s suggest t h a t improvement o f t h e c o r r e l a t i o n operator and/or a r e l i a b l e CBF p e r t u r b a t i o n c a l c u l a t i o n may w e l l depress t h e minimum t h e o r e t i c a l ground-state energies o f t h e deuterium systems t o n e g a t i v e values, i m p l y i n g t h e e x i s t e n c e o f two new Fermi 1 i q u i d s .

Theoretische Physi k , Universi tzt Hamburg. We ex-

Fig. 9 : Compound-diagrammatic q u a n t i t y i C c ( o t ) f o r D+2 and f o r 091, a s a f u n c t i o n o f d e n s i t y . E f f e c t i v e mass of Df2 cor- responding t o J a s t r o w model.

Fig. 8 : J a s t r o w s t a t i c s t r u c t u r e f u n c t i o n s f o r D.r2 and

D q .

1.25

1.00

- ^0.75

Y

Y

V)

0.50

0.25

Acknowledgments.- This r e s e a r c h has been supported i n p a r t by t h e U.S. National Science Foundation

-

^{I I I I}

A 0

- - - -

4"-8@&-4---Q?x-Q4-&-

- - - -

0 A

-

0

-

-

8

-

B - B

&O

-

an

I I 1 I A

under Grant No. DMR78-08552, by t h e U.S. Department

p r e s s o u r a p p r e c i a t i o n t o M. D. M i l l e r f o r informa- t i v e d i s c u s s i o n s and f o r s u p p l y i n g u s w i t h p o t e n t i a l and c o r r e l a t i o n f u n c t i o n d a t a . We a l s o b e n e f i t e d from s t i m u l a t i n g d i s c u s s i o n s w i t h A . D. Jackson, L. J. L a n t t o and R. A. Smith.

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