Ab initio group model potentials including electron correlation effects

J. Chem. Phys. 113, 6082 (2000); https://doi.org/10.1063/1.1290003 113, 6082 © 2000 American Institute of Physics. Ab initio group model potentials including electron correlation effects Cite as: J. Chem. Phys. 113, 6082 (2000); https://doi.org/10.1063/1.1290003 Submitted: 03 February 2000 . Accepted: 07 July 2000 . Published Online: 02 October 2000 Norge Cruz Hernández, and Javier Fdez. Sanz Ab initio group model potentials including electron correlation effects Norge Cruz Herna ´ndez and Javier Fdez. Sanza) Departamento de Quı ´mica Fı ´sica, Facultad de Quı ´mica, E-41012, Sevilla, Spain 共Received 3 February 2000; accepted 7 July 2000兲 A method for determination of ab initio group model potentials, with the intention of describing the effects of a whole molecule or a chemical group within a density functional theory framework, is reported. The one-electron part of the Kohn–Sham equations is modified by incorporation of a Coulomb operator, which accounts for the classical electron interaction arising from the group. Exchange and correlation effects are introduced by a suitable modification of the exchange-correlation functionals. The strong orthogonality condition, usually required by the theory of separability of many electron systems, is written in terms of first order reduced density matrices. In order to check the method a group model potential for H2O共environment兲was obtained and employed in the calculation of HF¯H2O and H2O¯H2O complexes using several functionals. Equilibrium intergroup distances and binding energies are compared with all-electron calculations. ©2000 American Institute of Physics. 关S0021-9606共00兲30337-3兴 I. INTRODUCTION A winning approximation in quantum chemistry, especially helpful in reducing the huge computational effort usually involved, is the classification of electrons of a given system as active or inactive.1–4 The inactive electrons may be assumed to behave as spectators and therefore can be frozen. This idea has been successfully exploited to describe the behavior of atoms or ions to model environment effects in cluster embedded calculations,5,6,7 as well as for the determination of effective potentials for fragments that could be employed as saturator groups. In this sense, effective potentials for NH3,8SiH3,9COOH,10 and cyclopentadienyl ligand11 have been reported. On the other hand, some efforts have been devoted to replace a complete molecule, or chemical group, in calculations in which the environment is described through an embedding type technique. Along this line, in 1993 Mejı ´as and Sanz12 reported on the determination of ab initio group model potentials 共GMPs兲for the HF molecule, together with some applications. Later, Frank et al.13 reported on the feasibility of representing spectator molecules through effective potentials generated by a semiempirical NDDO calculation. Quite recently, Day et al.14 came up with a method to obtain what they called an ‘‘effective fragment potential’’ which makes use of a distributed multipole expansion, and which allowed them to replace water molecules by an effective potential, with excellent results.15 More recently we have developed polycenter compact model potentials in which the short-range contribution was expanded as a spectral representation. Also, we reported an algorithm to solve the problem associated with rotation of the model potential.16 A straightforward way to improve group model potentials would involve the incorporation of the electron correlation effects between active and frozen electrons. On a general basis, this contribution could in principle be incorporated into the density functional theory 共DFT兲共Refs. 17, 18兲framework through a modification of the exchangecorrelation term. This topic has been considered for atomic cores in several works19,20,21 and, in this context, one of the most widely pseudopotentials used in solid-state calculations is the so-called ‘‘ultrasoft pseudopotential’’ 共UPP兲of Vanderbilt.22 As to the representation beyond atomic cores, Abarenkov et al.23 reported a method in which the wave function of a small cluster of Li2Mg was computed by incorporating the exchange-correlation interaction with a neighboring group in an iterative way. More recently Carter et al.24 presented a new embedding technique that combines periodic-DFT methods and explicit electron-correlation procedures. Ground-state properties of chemical groups have also been studied by means of the ‘‘freeze-and-thaw’’ cycle of Kohn–Sham equations with constrained electron density 共KSCED兲as proposed by Wesolowski and Weber.25 In this approach, the electron density of each fragment is kept frozen while the interaction energy is calculated using terms derived from DFT, and, in addition to the exchange correlation functional, the analytical form of the approximate kinetic energy functional is needed. Examples like the water dimer, HF–HF, HCl–HCl or HCN–HF may be found as applications to the study of the hydrogen bond.26 The main goal of this paper is to model the effects of a chemical group, or the environment in cluster embedded calculation, but now within the one-particle Kohn–Sham selfconsistent field 共SCF-KS兲framework. The starting point is the electronic separability principle proposed by McWeeny1 and Huzinaga2–4 but involving, in this case, the first-order reduced density matrices of the separable systems. Using the Kohn–Sham orbitals such conditions lead to an operator similar to the Vanderbilt22 and Phillips–Kleinman27 pseudopotentials. The Coulomb interaction between electrons of a given group was handled by fitting the electronic density to a set of auxiliary Gaussian functions,28 according to one of the a兲Author to whom correspondence should be addressed. Electronic mail: [email protected] JOURNAL OF CHEMICAL PHYSICS VOLUME 113, NUMBER 15 15 OCTOBER 2000 60820021-9606/2000/113(15)/6082/6/$17.00 © 2000 American Institute of Physics strategies of Dunlap et al.29 The article is arranged as follows. The theoretical frame is briefly overviewed in Sec. II. A new separability condition in terms of group density matrices is proposed and the way to incorporate the exchangecorrelation effect coming from the environment is presented. In Sec. III some test calculations concerning the H2O model potential are reported. Finally the main conclusions are summarized in Sec. IV. II. THEORETICAL ASPECTS In the density functional theory, the SCF Kohn–Sham 共SCF-KS兲equations17 have to be solved, 兵 ⫺1 2ⵜ2⫹Vc共r兲⫹ ␮ xc共 ␳ 兲 其 ␺ i KS共r兲⫽␧i ␺ I KS共r兲.共1兲 The term ⫺1 2ⵜ2is the kinetic energy operator, Vc(r) represents the electrostatic 共or Coulomb兲potential, and ␮ xc( ␳ ) introduces the exchange and correlation contribution, which, in general, may be written as ␮ xc共 ␳ 兲⫽ ⳵ Exc共 ␳ 兲 ⳵␳ ,共2兲 where Exc( ␳ ) is the so-called exchange and correlation functional. In local density approximation 共LDA兲or in the generalized gradient approximation 共GGA兲, it is generally written as Exc共 ␳ 兲⫽ 冕 fxc共 ␳ ,ⵜ ␳ 兲d3r.共3兲 The total electron density ␳ in Eq. 共1兲is related to the first-order density matrix for an Nparticles system in the form ␳ 共r兲⫽ 冕 ⌫共x,x⬘兲d ␴ ,共4兲 where the spin–orbital notation x⫽(r ␴ ) is used and the first-order reduced density matrix is defined as30 ⌫共x,x⬘兲⫽N 冕冕 ¯ 冕 ⌽*共x,x1,x2,...,xN⫺1兲 ⫻⌽共x⬘,x1,x2,...,xN⫺1兲dx1dx2¯dxN⫺1, 共5兲 ⌽(x1,x2,..,xN) being the electronic wave function. The necessary and sufficient condition for the reduced density matrix to be N-representable18 can be expressed as ⌫ ˆ⫽兺 ini 兩 ␸ i 典具 ␸ i 兩 共6兲 with ␸ ibeing the spin–orbitals and 0⭐ni⭐1. Choosing ni ⫽0,1 in Eq. 共5兲, the total electron density may be spanned on a set of spinless single-particle orbitals, each one occupied with two electrons, ␳ 共r兲⫽兺 occ 兩 ␺ i共r兲 兩 2.共7兲 Let us now assume a system constituted by two molecules or chemical groups, Aand B, weakly interacting, with ⌽Aand ⌽Bbeing the wave functions. According to the building block principle of McWeeny and Huzinaga, the total electronic wave function ⌽can be written as ⌽⫽MA ˆ共⌽A⌽B兲,共8兲 where Mis a normalization factor and A ˆis the intergroup antisymetrizer operator. For mathematical convenience in the derivation of ab initio model potentials, the group wave functions are selected so as to be strongly orthogonal,1i.e., 冕 ⌽A *共x,i,j,...兲⌽B共x,k,l,...兲dx⫽0. 共9兲 In order to keep the same condition in terms of electron density. Eq. 共9兲is multiplied by ⌽A共2,i,j,...兲⌽B *共3,k,l,...兲 and integrated with respect to i,j,..,k,l,.... Bearing in mind the form of the reduced density matrix, Eq. 共9兲becomes 冕 dx⌫B共3,x兲⌫A共x,2兲⫽0. 共10兲 Since in Eq. 共9兲, it is possible to write Ain place of B and Bplace of A, then the strong orthogonality condition 共9兲 in terms of reduced density matrices may be written as ⌫ ˆA⌫ ˆB⫽⌫ ˆB⌫ ˆA⫽0. 共11兲 If this condition is fulfilled, the problem of finding the local properties of the group of interest 共cluster, A or B兲,inthe presence of a spectator group 共environment, B or A兲can be set up using the DFT formalism. According to the definition of the density matrix given in Eq. 共6兲, the strong orthogonality condition 共11兲is satisfied if 具 ␺ i,clus KS 兩 ␺ j,env KS 典 ⫽0. 共12兲 The total reduced density matrix in a separable system may be written as ⌫ ˆall⫽⌫ ˆclus⫹⌫ ˆenv ,共13兲 and the electron density, ␳ 共r兲⫽ ␳ clus共r兲⫹ ␳ env共r兲,共14兲 where the electron densities ␳ clus(r) and ␳ env(r) are normalized to the number of electrons of the cluster (Nclus) and the environment (Nenv), respectively. Under these conditions, the cluster energy in the presence of the environment 共the effective energy Eclus eff )is Eclus eff 共 ␳ clus兲 ⫽E0共 ␳ clus ,nclus兲 ⫹Eint共 ␳ clus⇔ ␳ env , ␳ clus⇔nenv ,nclus⇔ ␳ env,nclus⇔nenv兲. 共15兲 The first term on the right-hand side of Eq. 共15兲is the isolated cluster energy, excluding all components depending on the environment. The second term is the interaction of electron density in cluster ␳ clus , with electrons and nuclei of the environment ␳ env and nenv , respectively. This last term also includes the interaction between the cluster nuclei, nclus , and the environment negative and positive charges, which can be 6083J. Chem. Phys., Vol. 113, No. 15, 15 October 2000 Group model potentials added at the end of the electronic energy calculation. Notice that following Eq. 共7兲the cluster electron density ␳ clus can be written as ␳ clus共r兲⫽兺 occ 兩 ⌿i,clus KS 共r兲 兩 2.共16兲 Application of the variational principle to Eq. 共15兲, together with the condition 共12兲leads to the set of SCF-KSlike equations 兵 ⫺1 2ⵜ2⫹Vc clus⫹Veff共r兲⫹ ␮ xc eff共 ␳ clus , ␳ env兲 其 ␺ t,clus KS 共r兲 ⫽␧i ␺ i,clus KS 共r兲.共17兲 In this equation the third and fourth terms represent the interaction between the cluster and the environment, represented through a group model potential 共KS-GMP兲. Analyzing these contributions, one of the central points in the present work, the third term expands as Veff共r兲⫽⫺ 兺 i nucl. env. zi 兩 r⫺Ri 兩 ⫹ 冕 ␳ env共r⬘兲 兩 r⫺r⬘ 兩 d3r⬘⫺P ˆenv ,共18兲 where the summation represents the interaction between the cluster electrons and the environment nuclei, while the integral accounts for the electron–electron interaction between the two groups. In the present work, such an interaction has been simulated by fitting the charge density of the spectator group to a set of coefficient and Gaussian functions. The algorithm used has been one of those proposed by Dunlap et al.29 whose main point is to find the set of coefficients that minimizes the functional D⫽ 冕 ⌬ ␳ 共1兲⌬ ␳ 共2兲 r12 dr1dr2共19a兲 with ⌬ ␳ 共r兲⫽ ␳ env exact共r兲⫺ ␳ env共r兲.共19b兲 The set of Gaussian functions used here was taken from Ref. 28. Finally P ˆenv is a weighted projection operator over the occupied orbitals of the spectator group that allows for restriction of the variational space to the cluster space reflecting the condition given by Eq. 共11兲, P ˆenv⫽ ␣ 兺 occ 兩 ␺ occ,env KS 典 ␧occ,env KS 具 ␺ occ,env KS 兩 ,共20兲 where the sum runs over the occupied molecular orbitals of the environment with eigenvalues ␧occ,env KS . The projector factor ␣ is set to 2.31 Concerning the fourth term of Eq. 共17兲, the operator ␮ xc eff( ␳ clus , ␳ env) arises from the exchange-correlation component of the interaction between the two groups. It can be written as ␮ xc eff⫽ ⳵ Exc,clus eff 共 ␳ clus兲 ⳵␳ clus ⫽ ␮ xc共 ␳ clus⫹ ␳ env兲,共21兲 where Exc,clus eff ( ␳ clus) is the effective exchange-correlation energy for the cluster Exc,clus eff 共 ␳ clus兲⫽Exc共 ␳ clus⫹ ␳ env兲⫺Exc共 ␳ env兲共22兲 If a fraction of the exact Hartree–Fock exchange is used, or in a simple Hartree–Fock calculation, the exchange operator can be simulated using a nondiagonal spectral representation.5 Equations 共21兲and 共22兲introduce the exchange and electron correlation contribution between cluster and the surrounding system in a DFT framework. These two equations are similar to those proposed by Vanderbilt22 in the UPP formulation in order to include the correlation between core and valence electrons, although in our case the determination of the Veff(r) does not need the use of a cutoff radius. To build Veff(r) we only need to know an approximate density and the occupied orbitals of the surrounding system, whatever the functional used in the cluster group is. Moreover, in contrast with the KSCED,25,26 in our method it is not necessary to use an approximated kinetic energy functional, and the inclusion of the weighted projection operator P ˆenv confers a nonlocal component to Veff(r) as in the UPP approach. The evaluation of Eq. 共22兲and the part of the SCF-KS equation related to Eq. 共21兲was solved using the grid of points generated by the cluster system as if it were isolated.32 These algorithms were implemented in the HONDO program.33 The integrals in the atomic basis were evaluated using the King, Dupuis, and Rys34,35 and Gauss–Hermite quadratures. For integrals concerning Eqs. 共19兲and 共20兲, the recursive formulas of Obara and Saika were used.36 III. NUMERICAL EXAMPLES We report in this section a few numerical applications of the procedure developed above. Among other aspects, the suitability of the KS-GMPs to simulate a water molecule in the computation of the 共H2O兲2dimer and the HF–H2O complex is explored. More extensive examples in which the method is tested in surface chemistry calculations will be reported in a forthcoming work. For the present we have centered our attention on the use of a wide variety of functionals commonly used in DFT calculations. For the exchange we have selected that of Slater,37 共SLT兲as well as that proposed by Becke,38 共B88兲. For the correlation part we employed the functional of Lee–Yang–Parr39 共LYP兲, which added to SLT or B88 gives two different exchangecorrelation functionals: SLYP and BLYP. The widely used hybrid three parameter functional B3LYP has also been considered.40 To set up the procedure, a first single calculation of the isolated water molecule was carried out using each of these functionals and from the KS molecular orbitals the KS-GMP was determined. These KS-GMPs were then incorporated into the DFT calculations of the active group (H2OorHF兲 and the interaction energy was estimated at several distances. The resulting profiles were compared with those arising from reference DFT all electron 共AE兲calculations carried out on the supermolecules HF¯H2O and H2O¯H2O. All the calculations were performed using the standard TZP basis set. Results for the HF¯H2O and H2O¯H2O complexes are reported in Tables I and II, respectively. Starting with the HF¯H2O system we can see that the equilibrium intermolecular distance estimated with the KS-GMP agrees reason6084 J. Chem. Phys., Vol. 113, No. 15, 15 October 2000 N. C. Herna ´ndez and J. F. Sanz ably with those obtained from AE calculations. With the exception of SLT based functionals, the KS-GMP distances are found to be larger than the AE ones, the errors ranging between 0.9% and 6.9%. The KS-GMP binding energies computed at the optimized intermolecular distances appear to be significantly lower than the AE ones; however, it should be noted that the AE binding energies are overestimated because of the basis set superposition error 共BSSE兲. Since the KS-GMP are free of BSSE, we corrected the AE energies using the well-known counterpoise method of Boys and Bernaldi.41 These corrected energies are also reported in the table under Ecor entry. As can be seen, the agreement is clearly improved, with deviations between 3% and 20%. The observed general trend is an underestimation of the binding energies reflecting the fact that the model potential is frozen and, therefore, no repolarization of the electron distribution is allowed. Also, because no basis set is present in the GMP, the electron density and, therefore, the intergroup region are not properly described. An additional approximation introduced in the KS-GMP calculations arises from the way of computing the energy and integrals related to Eq. 共21, 22兲, where only the grid generated by the cluster is used. Finally, in Fig. 1, the profile energies obtained from KS-GMP and AE calculations are plotted. It shows that beyond the numerical reasonable agreement, the shape of these curves appears to be quite satisfactory, with smooth behavior and a suitable description of the equilibrium region. Results obtained for the H2O¯H2O dimer are reported in Table II and Fig. 2. Both intergroup and binding energies are found to follow the trends already mentioned above. The equilibrium distances appear to be overestimated 共10%– 14%兲again with the exception of the SLT functionals, while the binding energies are underestimated with respect to the BSSE corrected AE ones. The energy profiles also show a suitable behavior, and as in the HF¯H2O case, the curvatures are found to be lower according to the underestimation of the group–group interaction. Finally, we will comment on one of the differences between our KS-GMP and the KSCED method reported in Ref. 25. This difference concerns the use of the projection operator defined in Eq. 共20兲, which is not present in the KSCED formalism. In order to show the importance of this term, a set of calculations for the HF¯H2O system was carried out but now setting the projection factor at ␣ ⫽0 in Eq. 共20兲. The profile energies thus obtained are reported in Fig. 3. As can be seen, the absence of the projector gives rise to a completely attractive system whatever the functional is. This result shows the necessity of the presence of the projection operator in our representation. The presence of such a projector comes directly from the strong orthogonality condition as expressed in Eq. 共11兲in terms of the first-order density matrices, and taking into account the N-representability condition leads to Eq. 共12兲. IV. CONCLUSIONS In this work a method to obtain polycenter group model potentials within a DFT framework is reported. The main idea underlying this approach is to carry out the calculation of an active subsystem 共cluster兲taking into account the influence of the surrounding groups 共environment兲through an ab initio model potential, including electron correlation effects in the cluster–environment interaction. With this purFIG. 1. Binding energy profiles for the H2O–HF complex from all-electron and KS-GMP calculations. TABLE I. Optimized intergroup distances 共Å兲and binding energies 共kcal/ mol兲obtained for the H2O–HF system from all electron and 共KS-GMP兲 calculations using SLT, B88, SLYP, BLYP, and B3LYP functionals. Deviations in percent are between parentheses. DFT functional dBinding energy All electron KS-GMP All electron KS-GMP Ecor SLT 1.690 1.573 共6.9兲⫺13.2 ⫺11.2 共3.4兲⫺11.6 B88 1.936 1.954 共0.9兲⫺6.6 ⫺4.7 共14.5兲⫺5.5 SLYP 1.594 1.540 共3.4兲⫺17.5 ⫺15.1 共3.8兲⫺15.7 BLYP 1.786 1.822 共2兲⫺9.6 ⫺6.8 共18兲⫺8.3 B3LYP 1.763 1.849 共4.9兲⫺10.1 ⫺7.2 共20兲⫺9.0 TABLE II. Optimized intergroup distances 共Å兲and binding energies 共kcal/mol兲obtained for the H2O–H2O system from all electron and 共KS-GMP兲calculations using SLT, B88, SLYP, BLYP, and B3LYP functionals. Deviations in percent are between parentheses. DFT functional dBinding energy All electron KS-GMP All electron KS-GMP Ecor SLT 1.839 1.785 共2.9兲⫺8.4 ⫺6.5 共6.1兲⫺6.9 B88 2.200 2.576 共14.6兲⫺3.2 ⫺1.9 共24.9兲⫺2.5 SLYP 1.719 1.708 共0.6兲⫺11.8 ⫺8.9 共15.3兲⫺10.5 BLYP 1.976 2.192 共10.9兲⫺5.3 ⫺3.5 共22.2兲⫺4.5 B3LYP 1.953 2.182 共11.7兲⫺5.7 ⫺3.8 共22.4兲⫺4.9 6085J. Chem. Phys., Vol. 113, No. 15, 15 October 2000 Group model potentials pose in mind, two main modifications are introduced into the KS equations. The first one accounts for the classical electron–electron interaction between the cluster and the environment. Such an interaction involves a new term in the one-electron part and is represented through a Coulomb operator built up by fitting the electron density of the environment group to a set of Gaussian functions. The second modification involves an adequate correction to the exchangecorrelation operator, depending on the functional used. The exchange-correlation effects are introduced into the KS equations by means of a grid of points defined in the cluster system. In order to preserve the strong orthogonality condition between group wave functions, as required by the building block principle, an equivalent restriction is proposed using the density matrix formalism. This allows for the definition of a projection operator similar to that used in the group model potential implementation at the Hartree–Fock level.16 The whole procedure was implemented in an all-purpose program 共HONDO兲,33 permitting us to compute the cluster electron density with several exchange-correlation functionals 共SLT, B88, SLYP, BLYP, B3LYP兲within the DFT framework. In order to test the method a group model potential for H2O was determined and used in the calculations of HF¯H2O and H2O¯H2O complexes. Compared to allelectron DFT calculations, our results show a satisfactory behavior of the water model potentials. ACKNOWLEDGMENTS This work was financially supported by the DGES and by the Junta de Andalucı ´a共Spain, Projects Nos. PB98-1125 and FQM132兲. We are grateful to Dr. A. Ma ´rquez for his help with the HONDO implementation. 1R. McWeeny, Methods of Molecular Quantum Mechanics 共Academic, New York, 1989兲. 2S. Huzinaga and A. A. Cantu, J. Chem. Phys. 55, 5543 共1971兲. 3S. Huzinaga, D. McWilliams, and A. A. Cantu, Adv. Quantum Chem. 7, 187 共1973兲. 4V. Bonifacic and S. Huzinaga, J. Chem. Phys. 60, 2770 共1974兲. 5Z. Barandiara ´n and L. Seijo, J. Chem. Phys. 89,5739共1988兲. 6J. A. Mejı ´as and J. F. Sanz, J. Chem. Phys. 102, 327 共1995兲. 7J. A. Mejı ´as and J. F. Sanz, J. Chem. Phys. 102, 850 共1995兲. 8K. Otha, Y. Yoshioka, K. Morokuma, and K. Kitaura, Chem. Phys. Lett. 101,12共1983兲. 9Ph. Durand and J. P. Malrieu, in Advances in Chemical Physics: Ab initio Methods in Quantum Chemistry, edited by K. P. Lawley 共Wiley, Chichester, 1987兲, Part I, p. 321. 10S. Katsuki, J. Chem. Phys. 98, 496 共1993兲. 11F. Alary, R. Poteau, J. Barthelat, H. Abou el Makarim, and J. Daudey, XXV Congresso Internazionale dei Chimici Teorici di Espressione Latina, Napoli, 13–18 Settembre, Comunicazioni orali 27, 1999. 12J. A. Mejı ´as and J. F. Sanz, J. Chem. Phys. 99, 1255 共1993兲. 13I. Frank, S. Grimme, M. von Arnim, and S. D. Peyerimhoff, Chem. Phys. 199, 145 共1995兲. 14P. N. Day, H. H. Jensen, M. S. Gordon, S. P. Webb, W. J. Stevens, M. Krauss, D. Garmer, H. Basch, and D. Cohen, J. Chem. Phys. 105,1968 共1996兲. 15W. Chen and M. S. Gordon, J. Chem. Phys. 105, 11081 共1996兲. 16N. C. Herna ´ndez and J. F. Sanz, J. Comput. Chem. 20,1145共1999兲. 17W. Kohn and L. J. Sham, Phys. Rev. A 140,1133共1965兲. 18R. G. Parr and W. Yang, Density Functional Theory in Atoms and Molecules 共Oxford University Press, New York, 1989兲. 19D. R. Hamann, M. Schlu ¨ter, and C. Chang, Phys. Rev. Lett. 43,1494 共1979兲. 20L. Kleinman and D. M. Bylander, Phys. Rev. Lett. 48, 1425 共1982兲. 21D. R. Hamann, Phys. Rev. B 40, 2980 共1989兲. 22D. Vanderbilt, Phys. Rev. B 41, 7892 共1990兲. 23Y. V. Abarenkov, V. L. Bulatov, R. Godby, V. Heine, M. C. Payne, P. V. Souchko, A. V. Titov, and I. I. Tupitsyn, Phys. Rev. B 56, 1743 共1997兲. 24N. Govind, Y. A. Wang, A. J. R. da Silva, and E. A. Carter, Chem. Phys. Lett. 295, 129 共1998兲. 25T. A. Wesolowski and J. Weber, Chem. Phys. Lett. 248,71共1996兲. 26T. A. Wesolowski, J. Chem. Phys. 106,8516共1997兲. 27J. C. Phillips and L. Kleiman, Phys. Rev. 116, 287 共1959兲. 28N. Godbout, D. R. Salahub, J. Andzelm, and E. Wimmer, Can. J. Chem. 70,560共1992兲. 29B. I. Dunlap, J. W. D. Connolly, and J. R. Sabin, J. Chem. Phys. 71, 3396 共1979兲. FIG. 2. Binding energy profiles for the H2O–H2O dimer from all electron and KS-GMP calculations. FIG. 3. Binding energy profiles for the H2O–HF complex computed without the projector operator in the KS-GMP. 6086 J. Chem. Phys., Vol. 113, No. 15, 15 October 2000 N. C. Herna ´ndez and J. F. Sanz 30P. O. Lo ¨wdin, Phys. Rev. 97, 1474 共1955兲;97, 1490 共1955兲. 31B. G. Ho ¨er and Chung, Int. J. Quantum Chem. 14, 623 共1978兲. 32C. W. Murray, N. C. Handy, and G. J. Laming, Mol. Phys. 78, 997 共1993兲. 33M. Dupuis, S. Chin, and A. Ma ´rquez, in Relativistic and Electron Correlation Effects in Molecules and Solids, NATO ASI Series, edited by G. L. Malli 共Plenum, New York, 1994兲. 34H. F. King and M. Dupuis, J. Comput. Phys. 21,144共1976兲. 35M. Dupuis, J. Rys, and H. F. King, J. Chem. Phys. 65,111共1976兲. 36S. Obara and A. Saika, J. Chem. Phys. 84, 3963 共1986兲. 37J. C. Slater, Quantum Theory of Molecular and Solids,Vol.4inThe Self-Consistent Field for Molecular and Solids 共McGraw–Hill, New York, 1974兲. 38A. D. Becke, Phys. Rev. A 38, 3098 共1988兲. 39C. L. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 共1988兲. 40A. D. Becke, J. Chem. Phys. 98,5648共1993兲. 41S. F. Boys and F. Bernaldi, Mol. Phys. 19,553共1970兲. 6087J. Chem. Phys., Vol. 113, No. 15, 15 October 2000 Group model potentials