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)



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.



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




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


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


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

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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



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


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



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.


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].


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.


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.



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


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.


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.


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


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



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


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,’*


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,.- -


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