electron correlation described by extended geminal models: the exgem7 and exrhf3 models

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, VOL. XXXVII, 585-596 (1990)

Electron Correlation Described by Extended Geminal Models: The EXGEM’I and EXRHF3 Models

I. R0EGGEN Institute of Mathematical and Physical Science, University of Trams(, P.O. Box 953, N-9001 Troms),

Norway

Abstract

Within the framework of the general extended geminal model, two new approximate models, EX GEM^ and EXRHF3, are introduced. Compared with previous models, these new models imply a more sophisti- cated approximation of the four-electron terms {em}. The models are tested by calculations on the beryllium atom and the HF molecule.

1. Introduction

A new approach to the electron correlation problem, denoted extended geminal models, has recently been developed by Rwggen [l-lo]. These models have a hier- archial structure. The basic approximation in all the models is an antisymmetric product of strongly orthogonal geminals. At the next level, single-pair correction terms are added; at the second level, double-pair correction terms, and so on. If all corrections are included, i.e., going to the highest level, we have a model that is equivalent to a full configuration interaction expansion. In practice, we have to truncate the general expansion at the double-pair correction level, and, furthermore, we also have to approximate the double-pair correction terms.

There are two main weaknesses of the extended geminal models developed so far. First, the adopted optimization procedure for the determination of the APSG function is an iterative procedure that converges only slowly. Second, the determination of the double-pair correction term {E,} are partly based on a second-order perturbation-type correction. In some preliminary calculations of the binding energy of the HF dimer, we obtained a value of the binding energy far below the currently accepted experimental value. This failure could essentially be traced back to the second-order perturbation- type correction terms {.$))}.

In this work, we are not concerned with the lack of computational feasibility of the APSG optimization procedure. The purpose of this work is to introduce a set of models where the second-order perturbation-type corrections are replaced by proper eigenvalue corrections; i.e., infinite-order perturbation-type corrections. Moreover, the procedure for the selection of a truncated virtual orbital space used in the determination of the full CI correction terms {E;:} shall also be redefined.

The structure of the paper is as follows: In Section 2, we sketch the essential ele- ments of the general extended geminal model and define the new models EX GEM^ and EXRHF3. In Section 3, we present test calculations on the beryllium atom and the HF

0 1990 John Wiley & Sons, Inc. CCC 0020-7608/90/040585- 12$04.00

586 R0EGGEN

molecule. In the final section, we sketch future work within a research program based on extended geminal models.

2. Truncated Extended Geminal Models

The models considered in this section are derived from the general model by trun- cating at the double-pair correction level. In Subsection 2.A, we give the essential formulas of the general model, which are needed in our approximate treatmeld. In Subsection 2.B, we define the EXGEM7 and the EXRHF3 models. The latter model is a special case of the EX GEM^ model when the antisymmetric product of strongly orthogonal geminals (APSG) root function is replaced by a restricted Hartree-Fock (RHF) function. We comment on the computational aspects of the models in the final subsection.

A . Truncation at the Double-Pair Correction Level

If the general extended geminal model [ S ] is truncated at the double-pair correction level, we have the following ansatz for the electronic wave function of a closed-shell 21-electron system:

N

In Eq. ( l ) , denotes the APSG function, i.e., the antisymmetric product of strongly orthogonal geminals; TK represents a single-pair correction term; and qKL, a double-pair correction term. The energy can formally be evaluated within the framework of the method of moments. Since we, by construction, have

(2 ) (@APSG 1 @APSG) = 1. ,

it follows

N N

where

EXTENDED GEMINAL MODELS 587

EKL = (aAPSG I H q K L )

= ( @;ISG 1 H fiF$ . (6) In Eq. (3, H$ is an effective two-electron Hamiltonian for electron pair K [5 ] , and H $ in Eq. (6) is an effective Hamiltonian for the four-electron cluster defined by electron pairs K and L. Furthermore, A K is the APSG geminal for electron pair K and @iy is the APSG function for the four-electron system ( K , L ) .

The key problem in approximate models within this framework is to estimate the correction term E ~ ~ . It is defined by the solution of a full CI problem for a four-electron system [ 5 ] . With exception for very small basis sets, to obtain the full CI eigenvalue is a heavy computational task. Since there are N(N - 1)/2 four-electron problems to be considered for a 2N-electron system, the need for approximate models to deal with these problems is imperative.

B . The EXGEM7 and EXRHF3 Models

In the recently proposed EX GEM^ model [lo], the double-pair correction term has the following form:

(7) In the EX GEM^ model we have the same ansatz for the double-pair correction term tzKL. The term &: is the same in the two models. The definition of @ is modified in the new models, and the procedure for the selection of the truncated virtual orbital space used in the determination of $1 is also changed.

B.l. Calculation of @: The first step in the modified procedure is to construct a set of dispersion-type natural orbitals (NOS). Let the orbital subsets {(p:; k = 1,. . . , n K } , (9:; 1 = 1 , . . . , nL} and {p:; i = 1,. . . , nc} define, respectively, the geminals AK , AL and the common orbital space or virtual space. From these orbitals, which are mutually orthonormal, we define a set of dispersion-type configuration

E K L = E t j + &?j + &g.

k = 1, . . . , nK

I = 1, . . . , nL

i = I , . . .,nc

j = 1,. . . ,Izc

k = 1, . . . , nK

1 = 1, . . . , nL

i = 1, . . . , nc

j = 1, . . . , nc

k = l r . . . , n K

1 = 1 , . . . ,nL

i = 2, . . . ,nc

j = 1, . . . , i - 1

(9)

588 ROEGGEN

rijikl = det{cpFacpFap#cpFp}, k = 1, . . . , nK

1 = 1, . . . , nL

i = 2, . . . , nc

j = 1, . . . , i - 1 (11) Second-order perturbation theory leads to the following dispersion-type correction to the APSG approximation @y:

k=l / = I i , j = l

k = l / = I t>j

On the basis of @!L, we construct the submatrix involving only virtual orbitals, of the corresponding first-order density matrix:

k = l I=l j = l

Diagonalization of the density matrix (cp::) yields a set of dispersion-type NOS for the electron pairs K and L . We shall denote this set of NOS {$F3xL; i = 1, . . . , nc} and where the numbering of orbitals is in accordance with decreasing occupation numbers of the diagonalized density matrix.

In the next step of the procedure, we define a truncated common orbital space based on the first mKL members of the set {$:KL}; i.e., {$F,"; i = 1, . . . , mKL}. A new set of dispersion-type CFS, {rp}, is then constructed according to Eqs. (8)-(ll), but replacing the orbitals {qF} with the dispersion-type NOS {$: KL; i = 1, . . . , m K L } .

On the basis of the CFS,

*;L"" u {rJ, a matrix representation of the effective four-electron Hamiltonian H $ can be constructed. If Afi is the lowest eigenvalue of this particular matrix representation, then the correction term $1 is simply given by

( 14) EKL KL (@FG I H~;@KL ) .

The purpose of using the dispersion-type NOS in the determination of & was to reduce the dimension of the corresponding eigenvalue problem. However, the dispersion-type NOS shall also be used in the determination of .$L.

The key idea in the determination of .$ in the EXGEM6 model [lo] was the assumption that the main contribution to the coupling between intra- and interpair CFS can be estimated by considering a smaller orbital basis than the full set:

{qf; k = 1,. . . , n K } u {cp;; z = 1, . . . , nL} u {pF; i = I , . . . , nc}.

(2) = - KI M S G

B.2. Calculation of @':


In the present work, we define a truncated common orbital space by selecting the most important NOS in the intrapair correction terms fifl and RF1 [lo] and the most important dispersion-type NOS for the electron pairs K and L . These subsets of orbitals are denoted, respectively, {$J:'~; r = 1, . . . , h,}, ( + F f L ; r = 1, . . . , tEL} and {+: K L ; r = 1, . . . ,mKL}. The three sets of orbitals are merged into one set {I$?; i = 1, . . . , h,}, and orthonormalized such that the following conditions are satisfied:

(Vf Is3 = 0

<d I+?> = 0

(+;KL(+F) = a,,, k = 1 , . . . , n K

1 = I r . . . , n L

i , j = 1, . . . ,hc.

The dimension of the truncated common orbital space, h, , satisfies the relation

h, 5 m, + mL + mKL. ( 16) The inequality sign is due to a possible occurrence of a linear dependence in the set that is subjected to the orthonormalization procedure.

The next step in the determination of ~ $ 1 is formally identical to the one used in the EXGEM6 model. First, the correction terms E,, E L , and E$] are calculated. The super- script - indicates that these calculations are based on the truncated common orbital space, i.e., {+?; i = 1, . . . , h,}. Otherwise, the procedures are the same as the ones used in generating E,, cL , and E$L, respectively. Second, the eigenvalue equation

(17) P H g P %L - KL KL

is solved. In Q. (17), H$ is the effective four-electron Hamiltonian in question and P'" is a projection operator into the space of all singlet-coupled, antisymmetrized four-electron functions that can be generated from the spatial orbitals:

{pf; k = 1 , . . . , n K } u {cpf; I = 1 , . . . , n L } u {@; i = I , . . . ,hC}.

' K L KL'-KL-[4] - @4]

The term $1 is then simply given by

(18)

In the limit h, = n,, this procedure yields the exact value, defined by the adopted basis set, of the double-pair correction term EKL .

A question that has to be answered on the basis of some experience is how to choose the values of mK, hL, and mKL for a fixed &. A priori, we assume that f i K L

should be the largest of these numbers of the following reasons. First, the dispersion- type expansion based on dispersion-type NOS is known to converge more slowly than does the corresponding NO expansion of A, + @ I . Second, compared with the EXGEM6 model [lo], we have neglected a certain class of dispersion-type CFS. This class consists of the Slater determinants generated from the orbitals {&, I&, pf, cp;; k, # k,} and {p:,, pl",, pf, cp;; I, # l , } . These CFS were negelected because they would lead to a very complex program code for the corresponding eigenvalue prob-

APSG KL APSG - E$L = i K L - (@KL I H e f @ K L ) - E, - E L - ~ $ 1 - 5:; .

590 RDEGCEN

lem. However, the most important of these CFS are included in the determination of $1 if mKL is chosen sufficiently large. We shall return to this question in Section 3 .

For computational reasons, we shall not always use a proper APSG function as the basic approximation of our models. In many applications, we might just as well use geminals {AK} with nK > I , for only the bond-pair electrons. The residual geminals are then defined by restricted Hartree-Fock orbitals [4], which we might denote restricted Hartree-Fock (RHF)- geminals. Unfortunately, these RHF-geminals will not be uniquely defined. They might be rotated internally without changing the root function. However, for concep- tual reasons, we strongly favor a localized representation of these RHF-geminals. Ac- cordingly, if not otherwise stated, the RHF-orbitals are mixed such that the Coulomb repulsion between the corresponding electron pairs is minimized. This is essentially Edmiston-Ruedenberg localization of the RHF-orbitals [ 111, but the chosen algorithm is a gradient procedure rather than one based on a sequence of two-by-two rotations of the RHF-orbitals.

In the case nK = n, = 1, the formula defining $2 is slightly different from Eq. (18). Since &! is not defined in this case, the term E!;: has to be dcleted in Eq. (18).

We shall denote the model where there is at least one geminal A, with n, > 1 in the root function, as the EX GEM^ model. When all geminals {A,} are RHF-geminals, the corresponding model is denoted as the EXRHF3 model.

B.3. Definition of the EX GEM^ and E X R H F ~ models:

C . Computational Aspects

For every double-pair correction term eKL that is determined, the approximate two- electron integrals in the selected (molecular) orbital basis have to be calculated. It is therefore of paramount importance to have an efficient procedure for the two-electron integral transformation.

The implemented computer programs of the extended geminal models are based on a nonstandard handling of the two-electron integrals. We have in a previous work [ 121 demonstrated that the Beebe-Linderberg two-electron approximation [ 131 leads to a substantial reduction of the storage requirements associated with the two-electron integrals. However, what is equally important is that the two-electron integral tables generated by the Beebe-Linderberg scheme are extremely well suited for vector computers. A two-electron integral [,uv I ha] , defined by the primary basis set {gP; ,u = 1, . . . , m}, is related to the integral tables .[(L,,;,); t = 1, . . . , r8} by the relation

where r8 is the effective numerical rank of the two-electron matrix [ 121. The calculation of the two-electron integrals in a molecular basis ; i = 1, . . . , n } is then performed in two steps: first, transformation of the two-electron tables:


where

and second, calculation of the transformed two-electron integrals according to the formula

It is obvious that the algorithms defined by Eqs. (20) and (21) can be extremely eff- ciently coded for a vector computer.

3. Test Calculations

In this section, we present test calculations on the beryllium atom and the HF molecule. The purpose of these calculations are as follows: first, to look at the pattern of convergence of @ as the number of dispersion-type NOS used in the calculation is increased; second, to look at the convergence rate of the full CI correction .$; as the terms &, E L , and $? approach their limiting values; and third, to study the effect of different choices of subspaces comprising the truncated common orbital space.

A . The Be Atom

In these calculations, we adopt a set of [8s, 4p, 2 4 contracted GTOS. The s-type functions are identical to the set defined by Harrison and Handy [14]. The exponents of the four sets ofp-type functions are 7.56, 1.28, 0.51, and 0.27, and the two sets of d-type functions have exponents equal to 1.27 and 0.44. The calculations are based on the EXRHF3 model. The total energy obtained for the beryllium atom with this model is then - 14.650118 a.u.

In Table I, we display the double-pair correction term 8:). of Be as a function of the number of dispersion-type NOS used in the calculation. The results suggest that a high accuracy can be obtained by using a considerably smaller dimension of the truncated common orbital space than the maximum dimension n, . Furthermore, the relative occupation number of the last included NO is of the same order of magnitude as the relative error in &if:. If this relation turns out to be generally valid, then we have a practical criterion for evaluating the accuracy of E ~ L without performing a calculation using the full common orbital space.

In Table 11, we consider the double-pair correction term of Be as a function of the dimension and the character of the truncated common orbital space. Pertaining to the results in Table 11, we would like to emphasize the following points. First, for a fixed dimension f i , , the relative error of egL is of the same order of magnitude as is the largest relative error in E K , E L , and Efi. This relation, if generally valid, constitutes a practical procedure for estimating the truncation error in .$?. Second, as for the selection of subspaces comprising the truncated common orbital space, the results are clear. The dimension of the dispersion-type subspace, i.e., i j i K L , should be considerably larger than are the dimensions of the intrapair subspaces. This result is as ex-

592 R0EGGEN

TABLE I. Dispersion-type double-pair correction, @, of Be as a function of increasing number of dispersion-type NOS used in the calculation (relative error in parentheses)a’b,c*d

Relative occupation numbers

6 0.286267 -0.003733 (13.3%) 12 0.033815 -0.004094 (4.9%) 16 0.0 19062 -0.004176 (3.1%) 24 0.005663 -0.004286 (0.5%) 30 0.000041 -0.004309 (0.0%)

‘All energies in atomic units. %asis set: contracted GTOs [8s, 4p, 2 4 . ‘Calculated by the EXRHF3 model, i.e., n, = n2 = 1. ?he magnitude of the largest dispersion-type occupation number: cl = 0.000112.

pected, since the natural orbital expansion of a two-electron function converges much more rapidly than does the dispersion-type expansion. Accordingly, since we should have more or less the same relative error in i& , E L , and ,522, the dimension of the dispersion-type subspace should dominate. Third, the full CI correction E$? can be calculated with a relative error of a few percent using a considerably smaller subspace than the full common orbital space. From a computational point of view, this result is very important since the computation time is proportional to f i ; .

B . The HF Molecule

The basis set adopted in the calculations on HF is a set (lOs, 6p, 2d, lf/5s, 2p, I d ) GTOS contracted to [6s, 4p, 2d, lf/4s, 2p, Id] . The s- and p-type functions on fluorine

TABLE 11. Relative errors of double-pair correction terms of Be as functions of the dimension and the character of the truncated common orbital

4 4 6 14 0.0286 4 4 10 18 0.0190 4 8 6 18 0.0286 4 4 14 22 0.0078 8 8 6 22 0.0050 4 4 18 26 0.0076 4 4 22 30 0.0000

( -0.036655)d

0.0244 0.0583 0.021 1 0.0323 0.0098 0.0583 0.0158 0.0158 0.0056 0.0309 0.0013 0.0012 o.oo00 0.0000

(-0.03301 I)d ( -0.004309)d

0.0658 0.0141 0.0755 0.0107 0.0345 0.0125 o.oO0o

(-0.003 1 90)d

“Footnote b of Table I. bFootnote c of Table I. ‘~$9:’ is the value of &f\ corresponding to f i , = 30. dAbsolute values in parentheses.


are taken from the work of van Duijneveldt [ 151. The contraction scheme for the s- and p-type functions is (5,1,1,1,1,1/3,1,1, l), and the expansion coefficents of the atomic 1s- and 2p-orbitals are used as contraction coefficients. The exponents of the two sets of d-type polarization functions are 0.83043 and 3.00616, and the single set offtype functions has exponents equal to 1.832 [16]. The s-type functions of the hydrogen centered set are also taken from the work of van Duijneveldt [15], and atomic contraction is used as in the fluorine case. The exponents of the p-type functions are chosen as 1.6 and 0.4, respectively, whereas the exponents of the set of d-type functions are chosen equal to 1.5.

In all calculations on HF, we use the EXGEM7 model with nK = 2 for the bond-pair geminal and nK = 1 for the core-pair geminal and the lone-pair geminals. The energy localization yields three equivalent lone-pair geminals.

In Table IT[, we display the double-pair correction term @ involving the bond pair and a lone pair. We notice the same pattern of convergence as in the Be case. By using 32 NOS instead of the complete set consisting of 50 orbitals, the truncation error is only 0.8%. As in the Be case, the relative occupation number of the last included NO is of the same order of magnitude as is the relative truncation error.

Pertaining to Table IV, a full CI calculation for the four-electron system in question (bond pair plus lone pair) using the full orbital space comprising 53 spatial orbitals could not be performed on a CRAY X-MP28 since our code requires all CI vectors in central memory. Our largest full CI calculation is based on 35 orbitals. This calculation is used as one reference for the $1 terms. However, since the .$$ term calculated with this truncated basis is obviously subjected to a truncation error, the relative errors based on this calculation are somewhat misleading. Accordingly, we therefore assume that the relative error in & for A, = 32 (nK + nL + A, = 35) is twice the largest relative error in E K , EL, and i.gL. This assumption leads to the last column of

TABLE 111. Dispersion-type double-pair correction, .A:), of HF as a function of increasing number of dispersion-type NOS used in the calculation (relative error in

parentheses)"' b, c3d,c

Relative occupation numbers

8 0.134030 -0.014032 (20.8%) 16 0.039449 -0.016201 (8.6%) 24 0.016317 -0.017 126 (3.4%) 32 0.005234 -0.017588 (0.8%) 40 0.001361 -0.01 7690 (0.2%) 50 0.0000003 -0.017728 (0.0%)

aAll energies in atomic units. bBasis set: contracted GTOS [6s, 4p, 2d, lf/4s, 2p, I d ] . 'Bond-pair-lone-pair double-pair correction with n , = 2 and n2 = 1 for the bond-pair

dBond length R(H-F) = 1.7329 a.u. 'The magnitude of the largest dispersion-type occupation number: c, = 0.000399.

and lone-pair geminals, respectively.

594 R0EGGEN

TABLE IV. Relative errors of double-pair correction terms of HF as functions of the dimension and the character of the truncated common orbital space.asb

m, mz m,,z A, ( E l - L . , ) / E , (EZ - M E 2 (8111)2 - 4::,/4:: (Ee:' - EyZ)/E(I?:.E

4 4 8 16 0.1155 0.0741 0.1234 0.0596 (0.097)d 4 4 16 24 0.0507 0.0425 0.0460 0.0109 (0.052) 8 8 8 24 0.0426 0.0175 0.0588 0.0118 (0.052) 4 4 20 28 0.0371 0.0294 0.0266 0.0063 (0.045) 8 8 12 28 0.0280 0.0151 0.0397 0.0072 (0.045) 4 4 24 32 0.0205 0.0188 0.0139 0.0000 (0.040)

(-0.011967)' (-0.023861)' (-0.017481)' (-0.014821)'

"Footnote b of Table 111. 'Footnote c of Table 111. E , , ~ IS the value of EC: corresponding to Ac = 32.

dSee the body of the text for explanation of this column.

F (3)' '

'Absolute values in parentheses.

Table IV (the results in parentheses). The estimated relative error in E$; is more or less the same as the maximum relative error in E K , E L , and .$i. This pattern will not change significantly even if we increase our already conservative estimate of the error in E:L for A, = 32. As in the beryllium calculations, the best results for a fixed A, are obtained when the dimension of the subspace of dispersion-type NOS is considerably larger than are the dimensions of the intrapair subspaces.

With the adopted basis set, the complete common orbital space used in the calculation of i$L and subspace dimensions mK = mL = 4 and mKL = 20, the total energy is calculated for four internuclear distances: 1.68 a.u., 1.7329 a.u., 1.78 a.u., and 50.0 a.u. By a parabolic fit, we obtain a binding energy, D, = 6.04 eV, and an equilibrium distance, Re = 0.922 A. The experimental values are respectively, 6.123 eV and 0.917 A 1171. The total energy for the equilibrium distance is -100.386451 a.u. -100.386451 a.u.

There are three sources of errors in these calculations of the spectroscopic parameters: the truncation of the general extended geminal expansion, the approximate calculation of $2, and the incompleteness of the orbital basis set. It is reasonable to assume that the first type of error is the smaller one. Since the purpose of this work was not to calculate highly accurate values for the spectroscopic parameters R, and D, , we have chosen A, = 28 for the truncated common orbital space. According to Table IV, the error in $2 for a bond-pair-lone-pair cluster is then approximately 0.0006 a.u. When the internuclear distance is changed from R = 1.742 a.u. to R = 50.0 a.u., the .$i term in question is reduced in magnitude with roughly 50%. We therefore assume that only half of the error 0.0006 a.u. affect the dissociation energy. Hence, the error in the calculated D, due to approximate values for &:; is approximately 0.001 a.u. = 0.03 eV. The basis set error should then correspond to roughly 0.05 eV. It is demonstrated by Dunning 1181 that by increasing the basis set from [6s, 4p, 2d, If] contracted GTOS to [6s, 4p, 3d, 2f, lg ] the valence shell electron correlation energy of fluorine increases in magnitude by 0.015 a.u. For the HF mole-


cule, this error can be scaled up to 0.018 a.u. The correlation energy (with respect to the APSG energy) changes from -0.2978 to -0.2576 a.u. when the internuclear distance is increased from R = 1.742 to R = 50.0 a.u. The increase in the dissociation energy by this particular basis set expansion should then correspond to (0.018 X

0.04/0.30) a.u. = 0.0024 a.u. = 0.065 eV. This result for the basis set error sug- gests that our previous estimated concerning the errors in the approximate {$i} terms are reasonable, but perhaps slightly too conservative.

4. Concluding Remarks

By the advocated refinement of the extended geminal models, we consider the full CI four-electron problem to be practically solved. Even for strongly correlated electron pairs, the convergence rate of the approximate {E$L} terms is sufficient to guaran- tee a large reduction in computation time compared with a calculation using the full common orbital space.

In our future work on the development of extended geminal models, two problems shall be given priority. The first problem is related to the well-known fact that a one- electron basis yields an extremely slow convergence of the electron correlation energy. Recently, Kutzelnigg [ 191 and Klopper and Kutzelnigg [20] have developed a practical scheme for including the interelectronic distance, r,, , explicitly into the wave function when electron correlation is described by Moller-Plesset perturbation theory to second order. A somewhat similar approach can be incorporated in an extended geminal framework. The second problem is to develop a procedure for including in an approximate way the higher-order terms of the extended geminal expansion.

As for the application of extended geminal models, we have now started on an ex- tensive research program on intermolecular interactions. We are in particular inter- ested in applying a previously advocated energy decomposition scheme [7] in order to look at the origin and character of this type of interactions. A study of the hydrogen- bonded systems (HF),, H,O * * * HF and (H20), is now in press.

Acknowledgments

The calculations were performed on the CRAY X-MP28 supercomputer at RUNIT, Trondheim, Norway. The work was supported by the Norwegian Research Council for Science and the Humanities. The author thanks J. Almlof for giving access to his integral program MOLECULE.

Bibliography

[l] I. R@eggen, Int. J. Quantum Chem. 20, 817 (1981). [2] I. Roeggen, Int. J. Quantum Chem. 22, 149 (1982). [3] I. Roeggen, Chem. Phys. Lett. 92, 398 (1982). [4] 1. Reeggen, J. Chem. Phys. 78, 2485 (1983). [5] I. Roeggen, J. Chem. Phys. 79, 5520 (1983). [6] I. Roeggen, Int. J. Quantum Chem. 31, 951 (1987) [7] I. Roeggen, J. Chem. Phys. 85, 262 (1986). [8] I. Roeggen, J. Chem. Phys. 85, 969 (1986).

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[9] I . Rwggen and E. Wisleff-Nilssen, J. Chem. Phys. 86, 2869 (1987). [lo] I . Reeggen, J. Chem. Phys. 89, 441 (1988). [Ill C. Edmiston and K. Ruedenberg, Rev. Mod. Physics 35, 457 (1963). 1121 I. Reeggen and E. Wisl0ff-Nilssen, Chem. Phys. Lett. W2, 1154 (1986). [13] N. H. F. Beebe and J. Linderberg, Int. J. Quantum Chem. 12, 683 (1977). [14] R. J . Harrison and N. C. Handy, Chem. Phys. Lett. 98, 97 (1983). [15] F.B. van Duijneveldt, IBM Report RJ945 (1971). I161 R. Ahlrichs, F. Driessler, H. Lischka, V. Staemler, J. Chem. Phys. 62, 1235 (1975). [ 171 K. P. Huber and G. Hertzberg, Molecular Spectra and Molecular Structure IV. Constants of Diatomic

[18] T. H. Dunning, Jr., J. Chem. Phys. 90, 1007 (1989). [19] W. Kutzelnigg, Theor. Chim. Acta 68, 445 (1985). [20] W. Klopper and W. Kutzelnigg, Chem. Phys. Lett. W, 17 (1987).

Molecules (D. van Nostrand Reinhold Company, London, 1979).

Received June 27, 1989 Accepted for publication October 9, 1989

electron correlation described by extended geminal models: the exgem7 and exrhf3 models

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