Correlation energy in the He atom by Fulde's local approach

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Correlation energy in the He atom by Fulde’s local approach

P. Calvini

To cite this version:

P. Calvini. Correlation energy in the He atom by Fulde’s local approach. Journal de Physique, 1986,

47 (10), pp.1775-1783. �10.1051/jphys:0198600470100177500�. �jpa-00210373�

Correlation energy in the He ^atom by Fulde’s local approach (*) (~)

P. Calvini

Dipartimento di Fisica-Università di Genova, G.N.S.M./C.N.R. ^e C.I.S.M./M.P.I., ^{Unità di} Genova, ^via Dodecaneso 33, 16146 Genova, Italy

(Requ ^{le 14 avril} 1986, accept6 ^{le 13} juin 1986)

Résumé.

Dans le cadre du calcul des énergies de corrélation des électrons, ^nous présentons ^des

considérations sur l’approche locale linéarisée (LLA) ^{de Fulde, et nous étudions en détail comment elle se}

compare ^aux techniques d’interactions de configuration (IC). ^Nous analysons ^les règles particulières que ^Fulde

impose ^aux opérateurs ^de fermions pour éliminer les contributions des excitations simples ^{et nous} calculons la fonction d’onde équivalente ^à l’approche ^{IC. Nous} présentons ensuite les résultats de quelques calculs de

l’énergie ^de corrélation dans l’état fondamental de l’atome d’hélium, obtenus dans cette approche ^{locale et}

dans une base d’orbitales de type Slater, ^{et nous} les comparons ^aux résultats correspondants obtenus par Fulde dans une base d’orbitales de type gaussien. Nos résultats sont un peu meilleurs que ^ceux de Fulde _parce que

nous avons essayé d’optimiser ^{les «} regions

^et nous avons aussi souvent pris ^en compte les corrélations entre les densités de régions différentes. On peut ^conclure que ^cette approche ^{locale est une} technique très efficace pour les calculs des énergies de corrélation pourvu que les ^« régions

soient bien optimisées.

Abstract.

In the framework of calculations of electronic correlation energies ^some considerations are made

on Fulde’s Linearized Local Approach ^with emphasis ^{on the} comparison with the usual C.I. techniques. ^The particular ^rules proposed by ^{Fulde in terms of} the second quantization ^{formalism to eliminate} single ^excitation

effects are analyzed ^{and the} equivalent C.I. wavefunction is derived. Subsequently ^one presents ^{the results of}

some computations of the correlation energy in ^the ground ^{state of the He atom} within the Local Approach

and with a S.T.O. basis and one compares these ^{results to} the corresponding calculations performed by ^Fulde

with a G.T.O. basis. Our results are slightly ^{better than} Fulde’s because we have tried to optimize ^the

«

regions

^{and we have also} frequently considered density-density correlations between different regions.

The effectiveness of the Local Approach ^{is confirmed} provided ^{that an} optimized, ^{or at least} reasonable, choice of the

regions ^{» is} performed.

Classification Physics ^Abstracts

31.20T

1. Some comments on Fulde’s local approach.

In a series of papers [1-3] beginning ⁱⁿ 1977, ^Fulde

and coworkers propose the Local Approach (L.A.)

as a simple ^{method to} compute electronic correlation

energies ⁱⁿ atomic and molecular systems. ^This technique has also been extended to the case of

crystals [4].

It is well known that, if 4’HF ^{is the ground state}

of a system ^{in the} independent particle approxima-

tion (Hartree-Fock ground state) ^and EHF ^{is the}

related energy, the number of the excited ^{states to} be included in an expansion ^{of the exact wave}

(*) ^A preliminary report ^{on this work was} presented ^at

the 3rd General Conference of the CMD of EPS

(Lausanne ^{28-30 March} 1983). [Europhysics Conference

Abstracts 7b (1983) P40-101.6].

(t ) ^{Supported in part by a grant} of the Italian Research Council under the French-Italian Scientific Collaboration

Agreement.

function I*..) ^{in order to obtain a reasonably}

accurate value of the exact energy EeX and therefore of the correlation energy [5]

increases [5-a] ^{as fast as} the binomial coefficient

N ^{where M, the} size of the basis set, is typically

one order of magnitude larger than the electron number N.

Therefore the techniques employing ^this strategy of expansion ^for I Oe.), ^{usually referred to as}

Configuration Interaction (C.I.), necessitate a

computational ^effort rapidly increasing ^{with the} complexity ^{of the} system ^under consideration, ^and already become intractable when N is of the order of

a few decades [3].

In order to avoid the shortcomings ^{due to the} large ^{number of} variational parameters ^{to be deter-} mined and to the huge number of matrix elements to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198600470100177500

1776

be computed, Fulde and coworkers propose the Local Approach ^{as a} method based ^on the physical picture of the correlated motion of electrons in an atom or in a molecule. In a Hartree-Fock state the

tendency of electrons to avoid close encounters

owing ^to their mutual Coulomb repulsion ^{is not} fully

taken into account and Ecorr ^is just ^the gain in energy when this tendency ^is fully considered. To compute this energy the Local Approach proposes ^to subdi- vide the atomic or molecular volume into local

subspaces (regions) ^described by localized, ^but

smooth orbitals giu (r) , ^{and to} build up ^a variational wavefunction which allows a reduction in double

occupancies ^{of each} region for electrons of opposite

spins (double occupancies ^{that are} overestimated in

a Hatree-Fock picture) and anyway gives ^the possibi- lity ^{of an} energetically ^better rearrangement ^of electrons.

The problem of the choice of regions requires physical ^and geometrical insight ^to perform ^an

efficient partition ^{of the} system volume. For further details we refer to the papers by ^Fulde [1-3].

For the structure of the variational wavefunction,

we restrict ourselves to the Linearized Local

Approach (L. L. A. ), ^{which can directly be compared}

to C.I. in the pair approximation [6]. ^If niu ^{is the} number operator for electrons of spin a ⁱⁿ region i,

L.L.A. proposes the ^ansatz

where the qui parameters ^can modify the double occupancies ^{in each} region for electrons of opposite spins

whereas the ’TJ ij parameters ^{can allow for} density-density correlations between regions i ^and j. ^For simplicity

we restrict ourselves to a closed shell system. Then in the independent particle approximation ^the

N

2 N electrons completely fill the first N Hartree-Fock space orbitals. A comparison ^{of ansatz} (2) ^{to the} corresponding wavefunction used by ^{C.I. can be} easily performed ^{if we consider} that each region giu ^{can be} expanded ^{in the set} of all the Hartree-Fock orbitals Ok,,., ^both occupied ^and unoccupied

Introducing b ^and blu respectively ^as destruction and creation operators ⁱⁿ region giu’ ^{one can write a}

similar decomposition also for these operators ^{in terms} of creation and destruction operators of Hartree- Fock orbitals [see (A.I) ^and (A.2) ^{in the.} Appendix]. ^{Then a term such as} n r nl r _

I I . ^4HF) turns out to be a mixture of single and double excitation states, all entering (2) ^{with the}

same variational parameter qi. This fact ^can give ^{rise to some troubles} especially ^{in the cases where the}

correlation energy ^can be considered a small quantity ^with respect ^to Eex ^{and then the q} parameters ^are expected ^{to be small. In these cases} perturbative considerations in a scheme where the perturbation ^{is the}

difference between the exact and the Hartree-Fock Hamiltonians [7] point ^{out that} doubly ^excited configurations ^{enter the} perturbative development ^of 14ex) ⁱⁿ first order whereas singly ^excited configurations ^{enter in} higher ^orders only. ^Therefore forcing single excitations and double excitations

together ^{into the same term can} seriously spoil ^{the results.}

In order to avoid these shortcomings, Fulde proposes ^{to subtract} off single excitation contributions from

(2). ^To achieve this goal ^he suggests ^{to utilize} Wick’s theorem in expanding ^the required ^matrix elements. In these expansions ^the contributions coming ^from single excitation terms are easily recognizable ^{and can} consequently ^be rejected. ^{In the} Appendix ^{we show that this} procedure ^is perfectly equivalent ^to performing

double substitutions in I "’HF) with ^{the part ^g,., of the gi,} orbitals which comes from the contributions of the

unoccupied Hartree-Fock orbitals. If the cP ku ^{orbitals are} orthonormalized, ^Schmidt’s orthogonalization procedure gives

The following reasoning ^can explain why restricting ^{oneself to double} excitation terms can be so effective in

lowering the energy of ^a system ^with respect ^{to its} Hartree-Fock value. Aside from constant terms, ^the

expectation value of the Hamiltonian of a system ^with respect ^{to a} given state I qi) ^{can be} expressed by

means of the corresponding ^{one and two} particle density ^{matrices y and F as follows} [5-b]

where xi stands for the position ri plus ^the spin coordinate of the i-th electron and h is the one particle operator representing ^kinetic energy plus interaction energy with nuclei. An inspection ^{of the} expression ^for

the one particle density ^matrix

immediately ^{shows that} the difference ’Y ex ( X, Xl) - Y HF (X" Xl )is ^{of order q2 if in} I "’ex) ^{the terms of}

order _q contain only ^double excitations with respect to I "’HF). ^{On the contrary a similar analysis of the two}

particle density ^matrix

shows that its corresponding variation is of order _q.

Thus, ^{when the} TJ’S ^{start from} zero, ^a certain amount of variation of r is allowed with respect ^{to its} Hartree-Fock expression ^{with a} consequent ^decrease

in interelectronic electrostatic energy in (5) ^before TJ 2 ^order changes ^{in y} play a substantial counterba-

lancing role and reestablish the validity of the virial theorem.

The above reasoning ^also gives, ^{as a} by-product,

some help ^{in common} situations occurring ^for large systems, ^{where the} description ^of 4’HF ^{is not}

accurate, typically because the basis set is far from

being complete. ^{In such cases both terms of the}

difference (1) ^{are affected} by incompleteness ^errors

and some sort of « error compensation ^{» is} expected

in computing ^the correlation energy. This procedure

has some consistency provided ^{that both} EHF ^and Ee.

are really computed ^{with the same basis set. But}

sometimes one computes EeX ^{with a} larger ^{basis set}

than the one used to expand I ’" HF ). ^{Then such a}

lucky ^error compensation ^{is no} longer ^to be expec-

ted, ⁱⁿ particular ^{if no} symmetry reason can « a

priori » exclude any contribution ^to EHF coming

from the added functions. But in such cases including only double excitations can still give reliability ^{to this} procedure provided ^that the q parameters ^{are small.}

In fact the omitted corrections for EHF, ^{which is a}

function of the one particle density ^matrix [5-b], ^are just ^{of order} q 2and ^{can be} neglected ⁱⁿ comparison

to Ecorr’ which shows a linear _q dependence ^for

TJ ^« 1.

2. Application ^{to the} ground ^{state of the helium}

atom.

In references [2] ^Fulde presents, ^{as the first} applica-

tion of L.A. to atomic systems, the results of some

computations ^of correlation energy in the ground

state of the helium atom. The basis set consists of

Gaussian-type ^orbitals (G.T.O.’s). With the purpose of performing comparisons, ^{we have done} equivalent

calculations with Slater-type ^orbitals (S.T.O.’s), exploiting ^the fact that in single ^centre problems

S.T.O.’s exhibit no additional difficulties with res-

pect ^to G.T.O.’s. The calculations were made on a

V AX11/780 VMS.

A S.T.O. has the structure

where the radial factor Rkl ( r ) ^can be written as

The wavefunction for the ground ^{state of He can be} expressed ^{as the} product

of the space factor qi (rll r2) symmetrical ^{in the}

coordinates of the two electrons, by ^the spin ^factor, antisymmetrical ^with respect ^{to electron} interchange.

In order to describe 14,, ^{only one s-type space}

orbital (A HF ^is required, which ^{can be} expanded 9 la

Roothaan as

1778

With the use of 5 S.T.O.’s one can give ^{an accurate} description ^{of the} Hartree-Fock orbital and, ^conse- quently, get ^for EHF the value - 2.8616799947 a.u.

which differs by ^{less than} 10-9 a.u. from the accurate value - 2.861679995613 ^a.u. obtained with 12 S.T.O.’s [8]. ^A comparison between the 5-S.T.O.

expansion for CPHF by ^Clementi [9] ^{and the one}

utilized in this paper is presented in table I.

2.1 IN-OUT CORRELATION ENERGY. - According

to Slater [10], ^the space wavefunction ’" rl, r2 ^is

expanded in the C.I. framework as

Table I. - Expansion of ^the 1 s-type space orbital

l/JHF ^{in S.T.O.’s (*).}

(*) ^{A more accurate value} EHF = - 2.861679995613 a.u.

is obtained with 12 S.T.O.’s [8].

where

The structure of the orbitals ukl,,, (r) ^is given by (7).

If the expansion (11) of the wavefunction is truncated at 1 = ^{0, only a portion} of the total correlation energy ^{can be obtained,} usually ^referred

to as radial correlation energy Errr rad ^or in-out correla- tion energy. In the literature [11] ^{we found the}

accurate value - 2.879028758 ^a.u. for the total energy Eer’d obtained from the s-type orbital space with sophisticated ^C.I. techniques involving ^{the use}

of natural radial orbitals derived from piecewise polynomials. ^{Then from} (1) ^{one can} get - 1.734876

10- 2 a.u. as a good estimate for Ecr rr*

In reference [2] ^Fulde computes Ec’oardr in the L.A.

scheme by partitioning the atomic volume into 4 s-

type regions (in-out regions). The first 3 gi’s ^are simply ^{the first} 3 Gaussian functions entering ^the development of 0 HF ^{with 5 Gaussians} (see ^{Table IX}

of paper [12] by Huzinaga ^{for N}

5) whereas 94 ^{is a}

contraction of the two remaining Gaussians. Fulde gets E,.rr rad ^{= -1.68 x 10-2 a.u.} (96.8 ^{% of the accu-}

rate value) ^with only ^{four q} parameters, ^associated

to the reduction of double occupancies ^{in each} region. ^{We have} performed ^a similar calculation by dividing up the atomic volume into 5 gi’s given exactly by ^{the same S.T.O.’s} entering ^our expansion for 0 HF (see ^{Table I,} Part 1). From these gi’s ^we

derived the corresponding gi’s ^{with the help of}

equations (4). ^{Table II,} part 1 shows the progressive improvement of the estimates for Ecrorr ^{when the}

double occupancies ^{in the first n} gi’s ^{are reduced}

with increasing values of n. Table II, part ^{2 shows}

Table II.

Estimates of E§g£ ^{with the basis set} of

S.T.O.’s of ^{table I,} part ¹ (*).

(*) In the various computations ^the regions ^{from 1 to n}

are included. Percentages ^{are relative to - 1.734876 x} 10-2 a.u., the accurate value for E ad . ^All energies ^{are in}

Hartrees. In part ¹ only ^double occupancies ^{are reduced}

while in part ² density-density correlations between conti- guous regions ^are also included.

the effects of including ^{in the} wavefunction terms such as nitni +11 I ’" HF ) ^{that establish} density-den- sity correlations between contiguous regions. ^{It must}

be pointed ^{out that a} C.I. calculation with the ^same basis set we used to expand 0 HF (Table ^{I, Part} 1) yields ,ad = - 1.7257 x 10-2 a.u. (99.5 %) ^with

15 configurations. ^{The basis set was} optimized ^to

give ^{an accurate} description of 0 HF and thus it

cannot be expected ^to be well suited to give very

good values for E ad either in C.I. calculations, ^{or in}

the L.A. scheme.

Better values for E,,o ff ^can be obtained with ^even fewer _q parameters by extending ^{the overall basis set}

and choosing ^as gk’s different S.T.O.’s from those used to expand OHF. The ako parameters ^associated with these ukw’s undergo ^{an ad-hoc} optimization procedure while the related nkO’s ^{are chosen in a}

reasonable way. Of ^course the S.T.O.’s used to

expand 0 HF undergo ^no change from those given ⁱⁿ

table I, part 1. In table III, part ¹ the results of

decreasing ^double occupancies ⁱⁿ optimized ^in-out

gk’S ^are shown. If we try ^to improve ^{our results} subsequently by including density-density ^correla-

tions between contiguous regions, ^we get ^no signifi-

cant improvement ^{because our} regions ^were optimi-

zed just ^to give the best results in the scheme of

reducing ^double occupancies only. Somewhat better results can be obtained if we try ^to optimize ^the regions ^{while we take into account} both double

occupancies ^and density-density correlations between contiguous regions ^{at the same time. The} corresponding values for Ecr ad rr ^are shown in table III, part ² together with the associated ako’s ^and nkO’S.

Indeed a comparison with table II shows a manifest saturation of our Eoa estimates towards 99 % of the accurate value. We must remember that we are

allowing only for the double excitation contributions and that going beyong ^a 99 % accuracy requires single excitation contributions too.

2.2 CORRELATION ENERGY WITH ANGULAR CONTRIBUTIONS. - Angular contributions to Er,

are included by extending ^{the sum} (11) ^to values of I greater ^{than 0. The accurate value for the exact} energy Eex ^is Frankowsky and Pekeris’ value

-

2.903724377 a.u. [13] from which one can derive

E.,, = - ^{4.2044 x 10-2 a.u. These values are} computed with variational wavefunctions containing

an explicit dependence ^on the interelectronic dis- tance r12. ^This strategy, ^which unfortunately ^cannot

be extented with the same success to systems ^with

more electrons, is very effective because it contains contributions from all values of 1. In C.I. calculations

one usually ^obtains slightly poorer results because the total energy EeX converges [5-c, 14] ^{to the true}

limit rather slowly ^when higher ^and higher ^{values of I}

are included in the calculations.

In the L.A. context we require regions ^{which are}

localized in proper directions. Hybridized ^orbitals give good performances. ^A simple ^case is represen- ted by tetrahedral hybridization, ^which gives ^{a set of}

four hybrids localized in four directions going ^out

from the center of the atom towards the vertices of a

tetrahedron. These hybrids ^are constructed as linear combinations of one s orbital and three p orbitals

and, owing ^to symmetry, ^{one can} reduce double

occupancies in the four hybrid regions belonging ^to

the ^same set with only ^{one q} parameter.

Fulde [2] ^{builds 2} (or 3) ^sets of tetrahedral

sp3 hybrids, all of them with the same s orbital taken

as OHF in its 5-G.T.O. expansion [12] ^{and with} only

Table III.

Improved ^estimates of E§g£ ^{with a basis set} of n ^S.T.O.’s optimized ^{ad hoc} (*).

(*) Percentages ^{are relative to -} 1.734876 ^x 10-2 _a.u., the accurate value for E"’. ^All energies ^are in Hartrees. Part 1 shows the results for Ead and the values of the optimized ako’s ^when only ^double occupancies ^are reduced in n regions. ^{Part 2}

shows the corresponding results when also density-density correlations between contiguous regions ^{are added.}

1780

2 (or 3) q parameters ^{he obtains - 2.20 x} 10-2 a.u.

(or - 2.27 x 10-z ^{a.u.) for} E,,a.1,9,. ^We performed

similar computations ^{with S.T.O.’s} using ^{a number}

of sets of hybrids increasing ^{from 1 to 3 and we}

obtained correspondingly ^the values - 2.042 x

10-2 a.u.,

2.224 x 10-2 a.u. and - 2.275 x

10-2 ^a.u. The values of the optimized akl’s ^{and of}

the chosen nkl’s determining the radial factor of the p part ^{of the} hybrids ^are shown in table IV.

In view of further developments, ^we present ^the

meaning ^of reducing ^double occupancies ⁱⁿ orthogo-

nalized tetrahedral hybrids ^{in terms of C.I.}

wavefunctions. A L.A. ansatz of the following

structure

Table IV.

Estimates of E£2f ^{with n sets} of ^tetra-

hedral hybrids of ^S.T.O.’s optimized ^{ad hoc} (*).

(*) ^{The table} gives ^the optimized ^{values of the} OCk1’S ^and

the chosen values of the nki’s determining ^{the radial factor} of the p part of n tetrahedral hybrids. ^{The values of the corre-}

lation _energy are obtained using CPHF ^{as the s} part ^{for all} the n hybrids. ^All energies ^are in Hartrees.

is equivalent, except ^{for a mere} redefinition of q, ^to

where the four tetrahedral hybrids (r) ^are composed ^{with the} s-type ^orbital U^AOO and the three p-type orbitals ubim (obviously ublm - ublm)’ ^A comparison ^to (11)-(12) shows that wavefunction (13) ^is equivalent

to a C.I. wavefunction where s-type ^and p-type ^{orbitals are} associated to the same variational parameter.

Furthermore one can immediately verify ^{that the L.A. ansatz}

allowing ^{also for} density-density correlations between the different hybrids belonging ^{to the same set is} equivalent ^to

and then closely agrees with the C.I. scheme of (11)- (12).

These considerations suggest ^{to take account of} in-out and angular ^{effects at the same time,} simply by using ^{as the s} part ^{of the} hybrids not OHF, ^{but the} optimized ^in-out regions of table III. So we construc- ted n sets of hybrids (with n ranging ^{from 1 to} 3) ^with

the s-type parts ^{taken from row n} of table III

(Part 1) ^and the radial factors of the p-type parts

taken from ^{row n} of table IV. After subtracting,

with the help ^of equation (4), single ^excitation

contributions from the s parts, ^we simply ^reduced

double occupancies ^{in each} hybrid region (nq para-

meters) ^and subsequently ^we also considered densi-

ty-density correlations _{among the} hybrids belonging

to the ^same set (2 ^nr parameters). ^{The results are}

shown in table V and compared with the correspon-

ding ^{values obtained} in reference [2]. ^{The latter}

were computed ^{in a different} scheme, by using ^the hybrids ^{to include} only angular effects and by retaining the 4 radial regions ^{to take into account in-}

out effects. It must be pointed ^out that, ^as long ^as

tetrahedral hybrids ^are used, ^{the value}

-

3.846 x 10-2 a.u. obtained with 6 q parameters (see ^Table V) ^{is 99 % of - 3.88362 x 10-2} a.u., the limit value for E, ,.,, obtainable with s and p orbitals

only [11].

Finally ^we have also used cubic sp3d3f hybrids, going ^{out of the centre of the atom towards the}

vertices of a cube. As one can see by inspection ^of

Table V.

Estimates of ^total Ecorr (*).

(*) Percentages ^{are relative to - 4.2044 x 10-2 a.u., the accurate value for} Ecorr. ^All energies ^are in Hartrees.

(a) ^Nearest neighbours only.

(’) ^Nearest neighbours plus ^second neighbours ^{in the cubic} hybrids.

(c) All kinds of neighbours within each set.

table V, ^it turns out to be more advantageous ^{to use}

2 sets of tetrahedral hybrids plus ^{one set of cubic} hybrids instead of 2 sets of cubic hybrids. ^{The values}

of the nkl’s and of the optimized au’s determining

the radial factor of the d and f parts entering ^{the set}

of cubic hybrids ^are respectively (3,3.457) ^and (4,4.506). ^{For the} s and p parts ^we have taken

(3,3.952) ^and (4,3.152) respectively (see ^Table III,

Part 1 and Table IV).

3. Conclusions.

For testing purposes ^we have computed the correla- tion energy in the He ^atom within Fulde’s Local

Approach utilizing ^{S.T.O.’s and we have found that}

our results are slightly better than those obtained in reference [2] with G.T.O’s. The reason for the

improvement ^{lies in a} frequent ^{use of} density-density

correlations between different regions and in the

optimization ^{of the} regions themselves. This strategy permits ^some reduction in the number of q parame- ters because in-out correlation effects are already

taken into account by ^{the s} part ^{of the} hybrids. ^We

have extensively applied ^the technique ^of optimizing

each s, p, d ^or f part separately ^{and then} constructing

our hybrid regions with these optimized ^{orbitals. It}

can be verified that optimizing ^all parts ^{with different}

1 values at the same time practically gives ^no

additional improvement and is very costly ^{from the} computational point ^{of view.} According ^{to our} experience, ^the problem ^of determining optimized regions ^{is a crucial one,} especially ^{if one wants to} study ^more complicated systems than the He atom.

Acknowledgments.

The Author is deeply ^{indebted to} Professor P. Fulde and Dr G. Stollhoff for most helpful discussions of their techniques. Stimulating discussions with Profes-

sor J. Friedel ^are gratefully acknowledged. ^The

Author finally ^{wishes to thank} Professors G. Bobel and F. G. Fumi for illuminating discussions and critical readings ^of drafts of the paper.

Appendix.

In the following ^the symbol ( ) ^{means the} expectation

value with respect ^to the Hartree-Fock ground-state.

If ^we indicate with On ^{one of the} operators ni t ni i ^or

L niu nju’ ^{entering (2) and if H is the} Hamiltonian of

aa,

the system, ^the computation of the correlation energy in the L.L.A. scheme requires the matrix

elements (On Om), (On H) ^and On HOm . ^In

1782

order to eliminate single excitation contributions from these elements, ^Fulde [2-3] proposes ^to reject

in their expansions ^{all the terms} containing ^contrac-

tions between the fermion

operators ^{entering the}

same On. ^{If we} indicate with bi., ^and bi. ^{the creation}

and destruction operators ^{relative to the} region gi., ^{this is} equivalent ^to suppose that the expectation

values (biu biu) ^and (biu ^{ba always vanish if bi}

and b,, belong ^{to the same} n. ^,

We can construct 0-type operators automatically obeying ^these prescriptions if, ^{with the} help ^of equations (3), ^we perform ^the decompositions

and

The equations (A.1) show that each bier ^{operator can}

be split ^{into the} part bier coming from the contribu- tion of the occupied Hartree-Fock orbital states plus

the part bier originating from the contributions of the

unoccupied ^{ones. The} equations (A.2) represent ^the corresponding expansions ^{for the} bier operators, ^and in particular they give ^the bier’s ^{as the creation}

operators ^{in the} regions gier ^given by equations (4).

Having ^{in mind} 0-type operators of the kind ni t ni I , ^{one can} easily verify ^{that the} operators

and their Hermitean conjugates

have the required property ^of giving vanishing

contractions between the fermion operators entering

each one of them. In this scheme the required ^matrix

elements must be written as On Ôm), ^{On H),}

(H’on) ^{and On} HOm ^{because the 0’s are not}

Hermitean. The contractions between a b operator coming ^from On and another one belonging ^to Ô m

can be different from zero. Of ^course the above

switching from the 0’s to the O’s ^can also be

performed without any further difficulty for opera- tors of the form 2: nio, ^njo,’*

uu’

In conclusion ^we have introduced the 6-type

operators ^which automatically obey ^{Fulde’s rules.}

The equivalence of these rules to performing ^double

substitutions in I "’HF) ^{with the gU orbitals given by}

(4) ^is proved ^{if we} analyse the way ^an 0-type operator ^works when ^{it is} applied ^to I "’HF). ^{If we}

consider that a b operator ^{absorbs an} electron from the Hartree-Fock occupied ^{orbital states whereas a} t _operator creates an electron in a 9 _orbital, ^{we can}

write oi 14’HF), ^by using (A.1), (A.2) ^and (A.3), ⁱⁿ

the following way

i.e. as a superposition ^{of states} containing ^a pair ^of

electrons with opposite spins ^{in the same « excited »}

space region 4,.- -

References

[1] STOLLHOFF, ^{G. and} FULDE, P., ^Z. Physik ^{B 26} (1977)

257.

[2] STOLLHOFF, ^{G. and} FULDE, P., ^Z. Physik ^{B 29} (1978)

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