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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.17751783. �10.1051/jphys:0198600470100177500�. �jpa00210373�
Correlation energy in the He ^{atom } by Fulde’s local approach (*) (~)
P. Calvini
Dipartimento di FisicaUniversità 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.
^{2014 }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.
^{2014 }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 densitydensity 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 [13] 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 (HartreeFock 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 ^{2830 } ^{March } 1983). [Europhysics Conference
Abstracts 7b (1983) P40101.6].
(t ) ^{Supported } ^{in } ^{part } ^{by } ^{a } ^{grant } of the Italian Research Council under the FrenchItalian 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 [5a] ^{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 HartreeFock 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 HatreeFock 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 } [13].
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 } densitydensity 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 HartreeFock 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 HartreeFock 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 HartreeFock 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 HartreeFock 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 } HartreeFock 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 } [5b]
where xi stands for the position ri plus ^{the } spin coordinate of the ith 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} HartreeFock 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 } byproduct,
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 } [5b], ^{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
Gaussiantype ^{orbitals } (G.T.O.’s). With the purpose of performing comparisons, ^{we } ^{have done } equivalent
calculations with Slatertype ^{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 } ^{stype } ^{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 } HartreeFock orbital and, ^{conse} quently, get ^{for } EHF the value  2.8616799947 a.u.
which differs by ^{less than } 109 a.u. from the accurate value  2.861679995613 ^{a.u. } obtained with 12 S.T.O.’s [8]. ^{A } comparison between the 5S.T.O.
expansion for CPHF by ^{Clementi } [9] ^{and the } ^{one}
utilized in this paper is presented in table I.
2.1 INOUT 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 stype 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 } inout 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 stype 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 (inout 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 } ^{102 } ^{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} 102 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 } densitydensity correlations between conti guous regions ^{are } also included.
the effects of including ^{in the } wavefunction terms such as nitni +11 I ’" HF ) ^{that } ^{establish } densityden 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 102 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 } ^{adhoc } 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 ^{inout}
gk’S ^{are } shown. If we try ^{to } improve ^{our } ^{results} subsequently by including densitydensity ^{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 } densitydensity 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 } ^{102 } ^{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 [5c, 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 5G.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 } 102 _{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 densitydensity correlations between contiguous regions ^{are } ^{added.}
1780
2 (or 3) q parameters ^{he } ^{obtains  } ^{2.20 } ^{x } 102 a.u.
(or  2.27 x 10z ^{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
102 a.u.,
^{ }2.224 x 102 a.u. and  2.275 x
102 ^{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 } stype ^{orbital } U^AOO and the three ptype orbitals ubim (obviously ublm  ublm)’ ^{A } comparison ^{to } (11)(12) shows that wavefunction (13) ^{is } equivalent
to a C.I. wavefunction where stype ^{and } ptype ^{orbitals } ^{are } associated to the same variational parameter.
Furthermore one can immediately verify ^{that } ^{the L.A. } ^{ansatz}
allowing ^{also for } densitydensity 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} inout and angular ^{effects } ^{at } ^{the } ^{same } ^{time, } simply by using ^{as } ^{the } ^{s } part ^{of the } hybrids not OHF, ^{but the} optimized ^{inout } regions of table III. So we construc ted n sets of hybrids (with n ranging ^{from 1 } ^{to } 3) ^{with}
the stype parts ^{taken } ^{from } ^{row n } of table III
(Part 1) ^{and } the radial factors of the ptype 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
tydensity 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}
