Download - Formulation ofN- and ?-representable density functional theory. V. Exchange-only self-consistent field

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. 43,769-782 (1992)

Formulation of N- and v-Representable Density Functional Theory. V. Exchange-Only

Sel f-Consis ten t Field EUGENE S. KRYACHKO*

Quantum Chemistry Group, University of Uppsala, Box 518, 9751 20 Uppsala, Sweden, and CRSI, Centre for Advanced Studies, Research and Development in Sardinia, Via Nazario Sauro 10, I-09123

Cagliari, Italy

EDUARDO v. LUDE~~A Centro de Quimica, Insiituto Venezolano de Investigaciones Cientificas, I.W.C., Apartado 21827,

Caracas I020-A, Venezuela

Abstract

The concept of a self-consistent field is developed within the version of density functional theory based on local-scaling transformations. It is shown that in this context there arise two types of consistency: one relating to the charge-consistency within an orbit and another to “orbit jumping.” The latter is analyzed in terms of one-particle equations. The connection with other methods is discussed. Q 1992 John Wiley & Sons, Inc.

1. Introduction

Recently, the rigorous formulation of density functional theory has been advanced [l-31. It is based on local-scaling transformations, namely, on such transformations that locally distort %’ by means of “swelling” or “contractions” of the given coordinate frame. Of course, the rationale underlying this formulation is that it provides the possibility of mimicking the behavior of the one-electron density in real molecular systems, whereupon the formation of a molecule atoms be- come distorted so that, literally, the one-electron density presents zones where it swells and others where it contracts.

In laying the groundwork for the local-scaling version of the density functional theory, the main interest in the previous papers [l-31 of this series has had to do with arriving at a coherent and well mathematically polished presentation of its fundamental ideas; at that time, in 1986-1988, this had led inevitably to a some- what abstract construct that has not made immediately apparent its relationship with other approaches or its possibilities for carrying out practical applications. The situation now remarkably changes. The publication of the series [3] is accom- panied by the works [4-12a] where the local-scaling version is directly computa-

*On leave from The Institute for Theoretical Physics, Kiev, Ukraine 252143.

Q 1992 John Wiley & Sons, Inc. CCC 0020-7608/92/060769-14$04.00

770 KRYACHKO AND LUDEGA

tionally realized. Also, in the forthcoming paper [12b], we have made the bridge of our formulation of density functional with the so-called density-driven approach developed in [lk].

The energy density functional theory is faced with two basic problems, namely, what an energy density functional will look like, and how to reformulate the self- consistent field concept in terms of the one-electron density. The up-to-date progress in the local-scaling version has already been answered formally [l-31 as well as practically [4-12b] on the first question. In particular, the atoms He, Li, Be, and the heliumlike ions within the Hartree-Fock approach and beyond it were calculated. The application of local scaling to the direct variational determi- nation of the electron-pair density has also been successful [8]. The numerical scheme has also been extended to the spin-decomposed electron densities [9].

The present study aimed to resolve the second problem. Needless to say, our interest in the self-consistent field concept is intimately related to the fact it has served as the basis for the development of powerful and useful techniques in many branches of physics. Moreover, the considerable success and popularity of the different versions of density functional theory in quantum chemistry and con- densed matter physics are undoubtedly related to the self-consistent field concept. To place the present discussion in a proper perspective, let us mention that Slater [13-151, Ghspir [16], and Kohn and Sham [17] have been the first to incorporate the self-consistent field concept into their versions of density functional theory. These approaches have been extended by several authors (see, e.g., [17-25a] and references therein). In general, all these formulations share the following two basic features:

(i) The energy density functional is represented by some approximate expres-

(ii) The one-electron density is written as the sum of the squares of N single- sion, particularly, non-N- and v-representable.

particle functions or orbitals.

In (i) we follow exactly the same line of definitions of N- and v-representabilities commonly used in this area (see, e.g., [25b] and also Part IV in our paper [3a]). In the density functional theory, the one-electron density is by definition the unique variable so that it is in terms of this variable that the theory should be expounded. However, the use of approximate energy density functionals [assump- tion (i)] leads to non-N-representable formulations that do not satisfy the variational principle that comes from the routine many-body quantum theory [l-31. In other words, the Euler-Lagrange equations for the one-electron density that one obtains when these nonrepresentable functionals are varied do not provide de- scriptions of a many-body problem that are equivalent to that given by the Schrodinger equation (remember self-free corrections). However, the above- mentioned cannot be considered as the definition of N-representability (for definition, see [3a, 25a]), but just the corollary of that. Furthermore, since assump- tion (ii) implies the existence of a set of individual one-electron states described by orbitals from which the one-electron density is constructed, the question arises as to the status of these N one-particle equations variationally derived from these

N- AND PREPRESENTABLE DENSITY FUNCTIONAL THEORY. VI 771

approximate functionals. This question has been already posed in the literature [19,20,23] (see also [2b], Parts 8.3-4).

In contrast to this situation, it has been shown in Paper I [3a] that in the context of the local-scaling version of density functional theory within each orbit one can construct an N-representable energy density functional. Variation of this functional leads to an orbit-dependent Euler-Lagrange equation for the density. Since as a complementary step an “orbit jumping” procedure may be formulated, we may transform the search for the energy minimum as a successive application of these two steps. Thus, in the present work, we discuss how this rigorous two- step procedure can be expressed in terms of a self-consistent field formulation. Since we deal with the exchange-only density functional approach, we develop here the density-functional-theory equivalent to the Hartree-Fock method.

2. Self-Consistency in Exchange-Only Density Functional Theory

Following the basic ideas developed in Paper I, the exchange-only approach re- quires the treatment of wave functions belonging to the domain SN C LN of Slater determinants. This domain can be partitioned into single Slater-determinant orbits {S!]} with respect to the group B of local-scaling transformations. Clearly, it is unlikely that SN contains only a single orbit, which, by definition, should be just the Hartree-Fock one. If this were the case, the Hartree-Fock theory could be formulated only in terms of p(r) and, thus, we could have a unique Euler-La- grange equation of motion for the one-electron p(r) in place of the well-known N one-electron eigenvalue problems, a fact that would considerably simplify the search for the Hartree-Fock state. But, certainly, for systems with more than two electrons, viz., N > 2, this is not the case.

So, assume that the domain SN contains a rather large, if not infinite, number of single Slater-determinant orbits where one of them is the Hartree-Fock orbit. In Paper I, we showed that the energy functional E[@] on the given orbit 0l.I C SN takes the form

E[@]JoEap = E[’][p(r)] . (1) This orbit 0l.I is generated by an arbitrary reference “configuration” described by the wave function E SN. The energy density functional given by the r.h.s. of Eq. (1) has an im licit dependence on the generating wave function @jl E SN of the given orbit O! 7 , namely,

~[‘l[p(r)] = E[p(r); @{I]. (2) Equation (2) is, in fact, a prescription for constructing the energy density functional E[’I[p(r)]. Applying the variational principle to E[’][p(r)], we obtain the Euler-Lagrange equation for p(r); its solution, naturally, gives us the optimal one-electron density ptit. It is important to keep in mind that this density is the optimal one within the orbit SS1 generated by the wave function @$I, but it is not, in general, the optimal one for other orbits in SN.

In the language of density functional theory, a self-consistency is defined in terms of an agreement between the absolute optimal density PSI, which, by defi-

772 KRYACHKO AND LUDESA

nition, must necessarily occur at the Hartree-Fock orbit, and the optimal N-electron wave function @$I, which again must also be realized in this orbit. The search for the optimal density and the optimal wave function is carried out along two different paths. In the first, which we denote as the “charge- consistency” step, the Euler-Lagrange equation of motion for p(r) is solved in- side the given orbit Okl. In this manner, we obtain the optimal density p:it (r), relative to the energy functional

E[p(r) ; @/I] with @jl E S N , where @:I is fixed, (3)

which is consistent with the chosen and fixed generating wave function @/I . Of course, it is clear that from this “fixed generating wave function that we may obtain another wave function in the orbit simply by replacing the vector r by the vector f(r); hence, in this fashion, all the wave functions belonging to a given orbit can be generated by spanning over all permissible f(r)’s.

In the second step, which involves an “orbit consistency,” this optimal density is used to calculate-the deformation function

g, opt (r) = fg,opt([p!]; pgt1; r) 7

which, in turn, allows us to construct the new Hamiltonian:

and, moreover, to define the energy functional

Ep; fLLpt(r)] = E[@; pa@) = p/](r)] for fixed p j ’ and p:jt;

= (@lap~;tl@), @ E S N , pa = p p , pgt E XC.

Here, p/](r) is the one-electron density coming from the orbit generating the wave function @!I used in the first step.

The possibility of attaining an “orbit consistency” that is dealt with in the second step stems from the fact that the energy density functional on the r.h.s. of Eq. (2) implicitly depends on a generating wave function. Applying the variational principle to Eq. (6) and varying the function @ subject to the density constraint pa = p!], we obtain, in general, a new wave function, which, of course, must be an optimal wave function for this ^variation. Since this wave function is optimal with respect to the Hamiltonian Hp$,, we shall denote it by Qopt (p:jt>. But in view of the density constraint, this new wave function has to have the same density p / ] as the generating wave function of orbit 0:’. Because there exists a one- to-one correspondence between wave functions and one-electron densities in a given orbit [3a], if the new optimal wave function @opt(p&!t) differs from @/I , but yields the one-electron density pi1 E Xs(= pa), then it cannot belong to the orbit 0:’. This means that QOpt (p$J must belong to another orbit 0:’ C YN withj # i. Thus, we may regard this wave function as the generating one of the orbit St]:

(7) @ ( PI ) @!I E 0 []I* opt Popt S


Notice that in view of the one-to-one correspondence between wave functions in a given orbit 0:’ and one-electron densities in X,”, the search for a global minimum for a fixed one-electron density implies that we must compare the energies of those wave functions belonging to different orbits in SN but that have the same density p!]. Therefore, there occurs in this second step what we may call “orbit jumping.” This part of the self-consistent process that takes us from one orbit to another is what we denote as “orbit-consistency.” To clarify the ideas, we consider below these two steps in detail.

3. Charge-Consistency

Consider the energy functional corresponding to tbe orbit 0:’ generated by the wave function O!] E SN and having a Hamiltonian H,

E[@!~I = E[p!l(r); 6$11

where the wave function is

@!l(r1, sl; . . . ; rN, sN) = dN[@$(rl, s1) . . . +LIN(rN, sN)]. (9)

For a fixed single Slater determinant given by Eq. (9), the energy E[@jl] appearing in Eq. (8) is just a number. However, according to Eq. (2), it follows that from Eq. (8) we can obtain a bona fide functional of the one-particle density p(r) within the given and fixed orbit. This functional is given by the following expression:

The difference between Eqs. (8) and (10) is that in the latter pfl(r) has been replaced by p(r); notice, furthermore, that +il(r; r’), the nonlocal part of the 1-matrix andfF,(r; r‘), the exchange factor of the 2-matrix,

774 KRYACHKO AND L U D E ~ A

have been fixed and replaced by their p(r)-th transformed analogs,

-$([p(r)]; r; r’) = +!‘(fj$(r); fkb(r’))

f!&([p(r)]; r; r‘) = f!&(f:\(r); fi’,,(r’)),

(12)

(13)

and

respectively. The transformation function fL\(r) appearing in Eqs. (12) and (13) takes the initial density pj](r) and changes it into p(r) .

The variation of p(r) subject to the normalization condition, introduced by means of the Lagrange multiplier pF1, yields the Euler-Lagrange equation (for its explicit form, see Paper I; for other “density” equations, see [26-291):

Instead of dealing with the density p(r ) as the basic variable in the variational problem, we may introduce the “shape-wave function” [30] u(r) and its complex conjugate u*(r) which are related to the density through

p ( r ) = u*(r)u(r) . (15)

u(r) = G@ e x p [ ~ ) l (16)

Notice that it follows from Eq. (15) that

where B(r) is, in general, an arbitrary phase factor. In terms of t_he shape-wave functions, the energy functional may be written as E[u*(r), u(r); @!I]. Since the normalization condition becomes

jd’ru*(r)u(r) = N , (17)

we may introduce the auxiliary functional,

Hence, the Euler-Lagrange equations equivalent to Eq. (14) are

[plus the corresponding complex conjugate equation for u*(r)] where

v‘([u(r)]; r) = [VrVr+F1([u*(r), u(r)]; r; r’)]r-r,

(20) s

6U + u*(r) T([VrVr.?i’([u*(r), u(r)l; r; r’)]r-rf)

is the contribution to the potential arising from the nonlocal part of the kinetic energy; v(r) is the external potential; v”([u(r)]; r) is the ordinary Hartree

N- AND PREPRESENTABLE DENSITY FUNCTIONAL THEORY. VI

potential ,

775

(21)

and vx([u(r)]; r) is the exchange potential,

Let us call p!jt(r) the optimum one-particle density obtained by solving either Eq. (14) or, the equivalent one, Eq. (19). The lowest energy that results from this "charge-consistency" procedure is, of course, E[p!jt(r);

Let us look now at the transformation filOpt(r) = f([&(r); p!jt(r)]; r) generated in the above charge-consistency step. Certainly, within each orbit, we have two one-particle densities, #(r) and p!jt(r). These two densities define uniquely the transformation f!bPt (r) via the first-order nonlinear differential equation

for a given direction r/r E R. Let us assume that we have solved this equation and that spanning over all directions we have obtained fjbpt(r). Now, using this function we can transform the orbitals appearing in Eq. (9) into the optimal ones:

This generates, in turn, the optimal wave function

Clearly, the optimal density is given by

j -1 . Notice that the following inequality holds between the optimal "charge-consistent" energy E[@!it] and that corresponding to the generating wave function, E[@/]]:

E[@&] s E[@ji]]. (27)

4. Orbit Consistency

Consider now the energy functional i[@; p*(r) = p/](r)] given by EAq. (6). This functional is the expectation value of the transformed Hamiltonian H,%:

776

where

KRYACHKO AND L U D E ~ ~ A

and

kptj,(rk; r,) = (.f!!opt(rk>.f!~oPt(tj))-’lrk - r,1 -1 fg,opt(rk).f!!opt(r,) 7 1 1 . (31)

in the state with the wave function @ E SN. The energy functional E[@;p*(r) = p/](r)] is well defined only on those Slater determinants @ E SN whose one-electron densities are just p/](r). It is clear, bearing in mind the one- to-one correspondence between one-electron densities and wave functions in a given orbit, that all these wave functions-belong to different orbits in SN and, therefore, the constrained variation of E[@; pa(r) = p/](r)] implies the orbit jumps. Hence, the extremum of this constrained variation is attained at the wave function Qopt (p!Jt) E SN whose one-electron density is &r). This wave function defines a new orbit O t l C SN (i # j), and, in fact, it may be considered as the generating function aOpt (p! j t ) = %$‘I of this orbit.

If we apply the transformation F$;I,, (which is the densitywise optimal transformation to be found in the i-th orbit; see Paper I) to the generating wave function @!I, we obtain another N-particle wave function @,$!-opt belonging to the orbit Okl but which is the i-th optimal,

and has the same density p!Jt as the wave function This new wave function possesses the corresponding energy E[@!&opt], which, in view of the fact that the wave fynction @$‘I was obtained through a minimization of the energy functional E[@; p*(r) = p/](r)], obeys the following variational-type inequality with respect to the energy E[@&!,]:

E[@,th-opt] Q E[@!Jt]. (33) Applying now the charge-consistency step to the energy functional E[p(r); @$‘I], we obtain the optimized one-electron density p$(r), which is realized within the orbit Okl at the point @kit E SN. Hence, the following chain of the variational- type inequalities holds:

E[@&It] Q E[@//l!.opt] Q E[@$’]] Q E[@p], i # J . (34) The identities take place at that particular orbit OF’, which is reached upon variation of the energy functional E[@; pQ(r) = p!](r)]. A graphical representation of a “charge-consistency” and of an “orbit-jumping” is given below. Let us remember that the global minimum in terms of the energy is reached at the Hartree-Fock orbit. The global self-consistency between the density and the wave function is realized at the lower loop of the Hartree-Fock orbit, as is shown in the following diagram (Fig. 1). Charge-consistency results from the fact that in the variation of


1

-c

t

t

t

r

t

,t

1

t pf:? c -r

Figure 1 . Schematic representation of charge- and orbit-consistency.

E[p(r); @$!I one obtains again p$(r). Further, the orbit-consistency via the functional E[@; pe(r) = p p l ( r ) ] gives us again @ $ I . Furthermore, when one ap- plies charge-consistency to E[p(r); @%I], one obtains once more p!,$](r). This, in turn, means that the following equation

fE:it(r) = r (35)

is satisfied and, hence, that neither the wave function @$I = eHF with the one- electron density pAzl(r) = pHF(r) nor the energy E[@Azl] may change upon the

778 KRYACHKO AND LUDEGA

further variation. The above just shows that in the final iteration there is no “orbit jumping” and that the two-step self-consistent procedure is completed. In [12], the numerical scheme for the orbit jumping optimization has been recently proposed.

5. Orb&-Cunsistency and Siagle-Particle Equations

Assume that for a given orbit O:’, the optimal density p!jt has been calculated. Equation (23) gives us the optimal local-scaling transformation flkpt(r), which, together with Eqs. (24) and (25), permits us to obtain the optimal set of orbitals {#&r); j = 1,2, . . . , N} and the optimal wave function Q,& E SN. Taking Eqs. (6) and (25) into account, one writes

E[~,jl] = E[Q; = p!’(r)l lo-.pi = ( ~ ~ I I $ ~ ~ I ~ ~ I ) (36) in the form

E[@j]] = (*!jt1$p!jt) = E [ W opt ] . (37) The equivalence between Eqs. (36) and (37) indicates that we may interpret the orbit-consistency step in either one of the following two ways:

(i) As a variation of the energy function$ I?[@; p*(r) = ptl(r)] corresponding to the transformed Hamiltonian HPgt over those Slater determinants Q, E SN whose one-electron densities p&) are identical to pil(r).

(ii) As a variation of the energy fuytional E[@; p&) = p!A(r)] corresponding to the initial Hamiltonian H over those Slater determinants Q, E S N

whose one-electron densities p*(r) are identical to p!jt(r).

Both of the aforementiuned procedures are similar to the Hartree-Fock variational problem subject to the imposed density constraints. In what follows, we shall describe the first one in &tail in order to show that the orbit-jumping condition may be cast in terms of single-particle equations.

Let a set of N orbitals @kl(r) E L l ; k = 1,. . . , N} be given. Construct the Slater determinant,

@(rl, . . . , rN) = (N!)-”’ det[#!’(rl) a . . $kl(rN)] @I1. (38) Assume that it minimizes the energy functional E[@; pa(r) = pj1(r)], where

N

pa(r) = C. I+yl(r)lz 3 pyl(r). k-1

If the normalization condition on the density is

(39)

(40)

the auxiliary functional takes the form N

n[{1+h1]>f-11 E[{I&’};-I; po(r) = #(r)] - % ( I d : c I@(r)l2 - N) . (41) k-1


Consider now the modified k-th orbital $F1(r) for 1 S k S N. It is defined through the original orbital plus an arbitrary variation:

$&r> = t,@(r> + g&r) E L ~ .

Substituting $kl(r) for +F1(r) in the auxiliary functional (41) results in

a[+pl(rl), . . . , $F1<rk>, . . . , &l(rN)l = a[{+&f=ll + 8 1 d3rrpr1*(r) ( & & I - %)&I(r) + C.C. + O(’18’). (42)

In Eq. (42) the one-electron Hamiltonian, or Fockian is

with

and

(44)

the integral kernel of the exchange operator F x. Here, ?([p!jt(r)]; r; r‘) is the nonlocal part of the 1-matrix [see Eq. (12)]. From Eq. (43) one obtains the one- body equation for the extremizing, or optimal orbitals {+rl}f-l:

F[ptjt]&](r) = @I+kI(r) . (46)

6. Discussion

Consider in detail the Fockian &&!,I and the related Eq. (46). Comparing with the exchange-only Kohn-Sham theory, the following points are noteworthy. In the first place, within the approach presented, the kinetic energy operator is the p!&(r)-th transform of the usual one, and, hence, it differs from the one used in the Kohn-Sham approach. In the second place [Eq. (43)], the external potential operator is CPkJt(r), namely, a transformed operator; its presence emphasizes the important role played by the local-scaling transformations in the procedure leading to the self-consistency and illustrates the one-to-one correspondence es- tablished in the present approach between “exter\al” potentials and densities.

Let us notice that the only term in the Fockian F[p$ of Eq. (46) that depends upon the functions {I,@} (k = 1 , . . . , N) is the exchange potential since if the orbitals obey Eq. (46), the related 1-matrix becomes

N P([p!jt(r)]; r; r’) = Z [D{fklopt(r); r}D{fkLpt(r’); rr>ll’’

k-1

. x ~ ~ ’ ( f ~ b p t ( r ) ) ~ ~ l * ( f ~ b p t ( r ’ ) ) . (47)

780 KRYACHKO AND L U D E ~ ~ A

For this reason, Eq. (46) must be solved self-consistently. However, Eq. (46) lends itself to a number of approximations. We can deal, for example, with the following simplified equation:

&p!$t(r)~+iL(r) = %d’](r), (48) where in the Fockian the orbitals {+&r)} (k = 1,. . . , N ) appearing in the 1- matrix defined by Eq. (47) and, hence, entering in the construction of the exchange potential have been replaced by the orbitals {+i],(r)} (k = 1,. . . , N ) . Since, clearly this is no longer a self-consistent equation, it does not yield an extremum for the energy functional. For this reason, we denote the orbital in the r.h.s. by to differentiate it from the true minimizing orbital &’I. In Eq. (48), the Ek’s are just Lagrange multipliers introduced to provide the appropriate normalization to the orbital set.

An additional simplification of the self-consistent one-particle Eq. (48) can be attained by writing

fi[p&!t(r)~c$’(r) = gkc$l(r) . (49) This is a traditional one-body Schrodinger equation whose solution presents no difficulties in a computational sense. Notice that both Eqs. (48) and (49) also lead to “orbit jumping.”

Return to the rigorous self-consistent Eq. (46). Its first N solutions {+F1(r); k = 1,. . . , N } allow us to create the new wave function

@kl(r,, . . . , rN) = (N!)-’” det[+[’l(r,) . . . +kl(rN)] . (50) ‘,t generates the new orbit 0:’ C SN. It is clear that the wave function @~:h]-opt =

F$: I~, (@[’I) yields an energy value that is lower than E $ . Using these procedures, we finally are led to the Hartree-Fock orbit at which Eq. (46) is equivalent to the Hartree-Fock problem:

P[p$~+/~](r) = ~ ~ b I + ~ F l ( r ) . (51)

Our present way of regarding this problem as being of the intermediate kind is very close to the Slater’s interpretation of the Slater-Kohn-Sham Ansatz, according to which the eigenvalue problem for orbitals yields a reference or trial Slater determinant. If we focus attention on the self-consistent procedure based on the single-particle equations of the type given by Eq. (46), we see that here again we have two complementary paths: One involves a charge-consistency, and the other, an orbit-consistency. The existence of an intermediate Fockian leads to an orbit jumping that eventually ends up at the Hartree-Fock orbit. As we have seen above, in this orbit, Eq. (46) holds and, thus, the loop is completed.

We would like to stress here the fact that the approach developed differs meth- odologically from the Slater-Kohn-Sham theory, although, evidently, they do in- tersect in a nontrivial way. In the present approach, the eigenvalue problem (46) must be consid5red as a simple Schrodinger-type one-electron problem for the linear Fockian F[p!Jt], which is well defined only in the orbit Oil. For this reason, the corresponding eigenvalues {%J appearing in Eq. (46) for the occupied orbitals


have no physical meaning. Only when SF’ coincides with the Hartree-Fock orbit may we give these quantities the approximate physical interpretation embodied by the Koopmans’ theorem.

The approach we have developed has the very important quality of maintaining at all steps its N-representability condition, which, of course, also implies that it remains self-interaction free. In a way, it may be regarded as an adequate procedure for improving local-density approaches, as one could generate the initial orbit by means of the single Slater determinants of the Slater, Ghpir-Kohn-Sham, or, generally, Xa-th forms (in the latter, the reasonable value of the exchange pa- rameter a has to be chosen). The last comment that is worth mentioning con- cerns what this approach is computationally about. That is not the aim of the present work, though. However, few computational examples [lo-121 (see also [3a], Part V) clearly demonstrate that in this scheme only the first few iterations are sufficient to get a result close to the optimal one. For instance, in making the interorbit optimization in [ll], the energy was improved at every iteration and is saturated to the eighth decimal place at the sixth one. In the other case (see also [ll]), the results of the zero and first iteration steps appeared to be crucial.

Acknowledgments

One of the authors (E. S. K.) thanks Jean-Louis Calais and Per-Olov Lowdin for their warm hospitality and fruitful discussion and C.-0. Almbladh, U. von Barth, B. M. Deb, L. Hedin, T. Koga, J. K. Percus, and S. Trickey for the useful discussion. E. Clementi and P. Zannella are greatly acknowledged for offering the position in Cagliari, Italy. The authors thank the referee for the useful comments.

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Received September 10, 1991 Revised manuscript received March 23, 1992 Accepted for publication March 24, 1992