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Submitted on 1 Jan 1986
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Correlation energy in the He atom by Fulde’s local approach
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 (*) (~)
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
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 
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 .
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
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 . 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  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  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
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 . A comparison between the 5-S.T.O.
expansion for CPHF by Clementi  and the one
utilized in this paper is presented in table I.
2.1 IN-OUT CORRELATION ENERGY. - According
to Slater , 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 .
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  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  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  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
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.  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  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  and with only
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
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
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 . 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