' Revision received April 15, A.B.F. da Silva and M. Trsic. 1. Introduction

Transcription

1 Gaussian- and Slater-type bases for ground and certain low-lying excited states of positive and negative ions of the atoms H through Xe based on the generator coordinate Hartree-Fock method 1. Introduction A.B.F. da Silva and M. Trsic Abstract: We applied a discretized version of the generator coordinate Hartree-Fock (GCHF) method to generate Gaussian- and Slater-type functions for mono positive and mono negative ions of the atoms H through Xe. The basis sizes for Slater-type functions are (12s, lop, 106) for positive ions, and (13s, 1 lp, 106) for negative ions. In the case of Gaussian-type functions the bases are (18s, 12p, 116) for both positive and negative species. Ground and excited state Hartree-Fock energies are calculated with these bases and the results compared with the best atom-optimized calculations and numerical HF results available. A discussion on the role of weight functions in the evaluation of electronic energies emphasizes the integral character of the GCHF method. Key words: Slater-type bases, Gaussian-type bases, generator coordinate Hartree-Fock, atomic ions. RCsumC : On a appliqut une version numcrique de la mcthode de la coordonnke gineratrice Hartree-Fock (GCHF) pour gcncrer des fonctions des types de Gauss et de Slater des mono-ions tant positifs que ncgatifs des atomes allant de H?i Xe. Les tailles des ensembles de base pour les fonctions de type Slater sont (12s, lop, 106) pour les ions positifs et (13s, llp, 106) pour les ions ncgatifs. Dans le cas des fonctions de type de Gauss, les bases sont (l8s, 12p, 116) tant pour les espkces positives que ncgatives. Utilisant ces bases, on a calcult les Cnergies Hartree-Fock de Ctats tant fondamentaux qu'excitts et on a compare les rcsultats avec les meilleurs rtsultats disponibles de calculs optimiscs des atomes et de rcsultats numtriques de HF. Une discussion sur le r6le des fonctions de pois dans 1'Cvaluation des Cnergies Clectroniques met en relief le caractkre inttgral de la mcthode du GCHF. Mots clis : fonctions du type de Slater, fonctions du type de Gauss, coordonnce generatrice Hartree-Fock, ions atomiques. [Traduit par la rcdaction] The generator coordinate (GC) method for the nuclear bound state was introduced in 1957 by Griffin and Wheeler (1). In this method the variational trial function is an integral transform over a nucleonic wave function and a weight function depending on a parameter (generator coordinate), i.e., f(a). In such a manner, the common variational principle, SElSa = 0, was, a priori, made more powerful with the requirement SElSfla) = 0, leading to an integral equation. Most applications in nuclear physics (for a review see, for instance, Wong (2)) relied on the Gaussian overlap approximation (GOA), Received June 27, 1995,' A.B.F. da SilvaZ and M. Trsic. Departamento de Quimica e Fisica Molecular, Instituto de Quimica de SHo Carlos, Universidade de SHo Paulo, P.O. Box 780, , SHo Carlos, SP, Brazil. ' Revision received April 15, Author to whom correspondence may be addressed. Telephone: (55)(16) Fax: (55)(16) ALBERICO@IQSC.SC.USP.BR although there were some attempts at numerical solution (3), and later Lathouwers discussed the GC method against the background of the Fredholm theory of linear integral equations (4). Perhaps the earliest application of GC to electrons in a molecular (hydrogen mclecule) system was due to Laskowski (5). In 1978 Chattopadhyay et al. (6) applied the method to several model problems, including the He atom, with special emphasis on various aspects of the discretization technique. Further developments in discretization techniques were made by Broeckhove, Deumens and co-workers (7). On the other hand, as early as 1968, Somorjai (8) had introduced the integral transform method, closely related to GC, for atomic and molecular systems. In the applications, Somorjai chose explicit forms forfla) (such as the delta function), which led to a variational treatment for the integration limits. An extensive bibliography on the method may be found in ref. 9. In the context of this method, accurate correlated functions for two- and three-electron atomic systems were obtained by Thakkar et al. (10). The generator coordinate Hartree-Fock (GCHF) method was introduced in 1986 (1 l), and one of the first applications was in the generation of Gaussian- and Slater-type orbital Can. J. Chem. 74: (1996). Printed in Canada 1 Imprimt au Canada

2 da Silva and Trsic (GTF and STF) universal basis sets for the first and second rows of the periodic table (12, 13). Later, da Silva and Trsic presented GTF and STF universal basis sets encompassing the atoms from H through Xe for atomic and molecular nonrelativistic calculations (14, 15). Also, the GCHF method was recently tested successfully in the generation of universal Gaussian basis sets for relativistic calculations (16-19). Here it is important to mention that the concept of the universal basis set was first introduced by Silver, Wilson, and Nieuwpoort (20). In this work we are extending the bases of ref. 4 to embrace the positive and negative ions of the atoms H through Xe. The total HF energies of the ground and some excited states are calculated and compared with the best HF values available. We also present a further discussion on the importance of the weight functions in evaluating atomic energies. 2. Results and discussion 2.1. Total energies The total energies reported in this section were calculated by using Slater-type functions (STF) and Gaussian-type functions (GTF) generated with the GCHF method. In this methodology, the one-electron functions are chosen as integral transforms, i.e., where the +i are the generator functions (they may be STF, GTF, or other types of functions), thefi are the weight functions, and a is the generator coordinate. Using eq. [I] to build a Slater determinant for the multi-electronic wave function, and minimizing the total energy E with respect to fi, one obtains the Griffin-Wheeler-HF (GWHF) equations or integral HF equations (11, 14). The GWHF equations are solved by discretization through a technique called integral discretization (21), which preserves the continuous representation (integral character) of the Generator Coordinate Method (GCM) (1, 22). This technique is implemented through a relabelling of the generator coordinate space, i.e., Here, an equally spaced N-point mesh {a,] is selected for each s, p, and d symmetry. The integration range is then characterized by an initial point, a,,,, an increment, AR, and N (discretization point number). The scaling factor A is the same (A = 6.0) for all calculations. We also note from eq. [2] that the numerical integration of the GWHF equations provides a set {ai] and simultaneously an exponent set {ai] for the basis functions. Since eq. [I ] represents a continuous and infinite superposition, we do not need, in the GWHF equations, to specify the principal quantum numbers, n, for the various symmetries of the generator functions, +i, as the weight functions,f,, distinguish the n states. Thus, our bases in this work consist of the simplest Is, 2p, and 3d STF or GTF. The basis size for STF positive ions is (12s, lop, 106) with the following discretization parameters (i.e., orbital exponents): Symmetry Qmin AQ N The GTF (18s, 12p, 1 16) basis for positive ions is generated with: Symmetry Qmin AQ N The basis size for STF negative ions is (13s, 1 lp, 106) with discretization parameters as follows: Symmetry Qmin AQ N The GTF (1 8s, 12p, 1 16) basis for negative ions is generated with: Symmetry Qm,, AQ N The basis sets (discretization parameters) for STF and GTF positive ions are the same as those we previously employed for neutral atoms (14). This coincidence was to be expected, as the ionization produces orbital contraction, and thus the bases for the neutral atoms were adequate for describing the total energies for ground and low-lying excited states of positive ions. For negative ions, therefore, it was not possible to describe the total energies by employing the same basis sets as for neutral atoms. This is a consequence of the entrance of one electron in the frontier atomic orbital, making it more diffuse; thus, lower values for a,,, are required for STF and GTF bases. In Tables 1 4 we present the total STF and GTF HF energies for ground and low-lying excited states of positive and negative ions. Our results are compared with the numerical results of Koga et al. (23) and, only when numerical results are not available, with the atom-optimized values of Clementi and Roetti (24). One can see from Tables 1 4 that our STF energy values (ground and low-lying excited states) compare very favorably with the numerical results and with the atom-optimized results of Clementi and Roetti, often being lower than

3

4 da Silva and Trsic Table 2. Low-lying excited state energies (au) of the STF and GTF bases for positive ions. STF energy Virial GTF energy Virial Clementi- (12% lop, theorem 12p, theorem Roetti energy Ion Configuration State 106)" STF~ 116)" GTP (24) (CR) Sc' [Ar] 4s0 3d2 3~ 1.7 (-4) 3.8 (-6) 1.O (-2) 4.1 (-6) Sc' [Ar] 4s' IS 1.1 (-4) 2.7 (-6) 1.1 (-2) 3.9 (-6) Ti+ [Ar] 4s0 3d3 4F 3.6 (-4)' 1.0 (-6) 1.1 (-2) 7.6 (-6) Ti' [Ar] 4s2 3d' 2~ 5.0 (-5) 3.6 (-7) 1.2 (-2) 8.6 (-6) V' [Ar] 4s2 38 F 1.8 (-4) 3.5 (-6) 1.3 (-2) 6.2 (-6) Cr' [Ar] 4s' 3d' 6~ 2.0 (-4)' 5.2 (-6) 1.4 (-2) 2.6 (-6) Cr+ [Ar] 4s2 3d3 4F (-6) 1.5 (-2) 2.4 (-6) Mn' [Ar] 4s0 3d 5~ 1.0 (-4)' 5.1 (-6) 1.6 (-2) 1.3 (-5) Mn+ [Ar] 4s2 3d' 'D 3.0 (-4) 5.1 (-6) 1.7 (-2) 1.3 (-5) Fe' [Ar] 4s' 3d5 4.0 (-4) 3.8 (-6) 2.0 (-2) 2.1 (-5) Co+ [Ar] 4s' 366 5D 9.0 (-4) 1.9 (-6) 2.4 (-2) 2.4 (-5) Ni+ [Ar] 4s23d7 4F 7.0 (-4) 2.2 (-7) 2.8 (-2) 2.2 (-5) Cu' [Ar] 4s23d F 1.1 (-3) 1.1 (-6) 3.2 (-2) 1.6 (-5) As+ [Ar] 4s23di04p2 I D 1.5 (-3) 1.2 (-6) 5.0 (-2) 1.5 (-6) As' [Ar] 4s' 3d1 4p2 'S 1.5 (-3) 9.9 (-7) 5.0 (-2) 2.1 (-6) Se+ [Ar] 4s' 3dI0 4p3 'D 1.8 (-3) 2.4 (-7) 5.5 (-2) 2.4 (-6) Se+ [Ar] 4s23dI0 4p3 'P 1.9 (-3) 8.4 (-8) 5.5 (-2) 2.5 (-6) Br+ [Ar] 4s23d1 4p4 'D 2.2 (-3) 2.3 (-6) 6.1 (-2) 9.9 (-6) Br' [Ar] 4s23d1 4p4 'S 2.1 (-3) 2.4 (-6) 6.1 (-2) 1.O (-5) Zr' [Kr] 5s04d3 F 3.5 (-2)' 2.5 (-5) 7.6 (-2) 6.4 (-5) Zr+ [Kr] 5s24d'?D 4.6 (-3)' 2.5 (-5) 1.1 (-1) 6.5 (-5) Nb' [Kr] 5s' 4d3 5~ 9.0 (-4) 2.8 (-5) 1.3 (-1) 6.7 (-5) Nb' [Kr] 5s24d2 3~ 1.9 (-3)' 2.8 (-5) 1.3 (-1) 6.7 (-5) Mo' [Kr] 5s' 4d' 6~ 1.7 (-3) 2.8 (-5) 1.5 (-1) 6.5 (-5) Mo' [Kr] 5s24d3 4F 1.9 (-3) 2.8 (-5) 1.5 (-1) 6.4 (-5) Tc' [Kr] 5s04d 5~ 2.1 (-2)' 2.5 (-5) 1.4 (-1) 6.1 (-5) Tc+ [Kr] 5s24d' 5D 9.1 (-3) 2.5 (-5) 1.7 (-1) 6.0 (-5) Ru+ [Kr] 5s' 4d 6~ 1.3 (-2) 1.8 (5) 1.9 (-1) 5.5 (-5) Ru' [Kr] 5s24d5 1.4 (-2) 1.7 (-5) 2.0 (-1) 5.4 (-5) Rh+ [Kr] 5s' 4d7 5F 1.5 (-2) 6.6 (-6) 2.2 (-1) 5.0 (-5) Rh' [Kr] 5s24d 5~ 1.5 (-2) 6.6 (-6) 2.2 (-1) 5.0 (-5) Pd' [Kr] 5s' 4d 4F 3.1 (-2) 7.0 (-6) 2.5 (-1) 4.7 (-5) Pd' [Kr]5s24d7 4F 1.6 (-2) 6.9 (-6) 2.4 (-1) 4.7 (-5) Ag' [Kr] 5s' 48 3~ 1.8 (-2) 2.1 (-5) 2.7 (-1) 4.7 (-5) Ag' [Kr] 5s24 d F 1.8 (-2) 2.1 (-5) 2.7 (-1) 4.8 (-5) Cd' [Kr] 5s24$ 2~ 1.6 (-2) 3.3 (-5) 2.9 (-1) 5.4 (-5) "This column shows AE = IE(NHF) - E(calcu1ated) 1. qhis column shows the values of A = 12 + virial ratiol. 'Calculated energy is lower than CR energy. those obtained with the latter. Our GTF energies are only slightly above the STF values in spite of a rather modest increase in the number of basis functions. For the case of positive ions (Tables 1 and 2), we notice a better performance for both STF and GTF bases for excited states than for ground states when compared with the Clementi and Roetti results. Also for positive and negative ions, the STF energy values are close to the numerical results and competitive with the excited state values of Clementi and Roetti. This capacity to describe excited states with the GCM was commented on in previous papers (6, 14, 21). We would also like to make a comparison between the STF energy results obtained in this work for the ground state of cations and anions and the STF energy results obtained recently by Koga et al. (25) using doubly even-tempered basis sets for the same ionic species. Here it is important to pay attention to the fact that we are working with universal STF basis sets, and thus the sizes of our basis sets are always the same, namely, (12s, lop, lod) for the cations and (13s, 1 lp, lod) for the anions. In his work, Koga is working with atom-adapted basis sets, and thus his STF basis set size increases with increase in Z (atomic nuclear charge). For the cations with s and p orbitals, Koga's basis set size varies from (7s, 5p) to (1 ls, 7p) for C+ through Ca+. For the cations with s, p, and d orbitals, Koga's basis size varies from (1 ls, 7p, 5 4 to (l3s, 12p, 84 for Sc+ through Xe+. In general, for the cations, our STF energy results are on average one decimal figure more accurate than Koga's results but, although

5 1530 Can. J. Chem. Vol. 74, 1996 Table 3. Total ground state energies (au) of the STF and GTF bases for negative ions. STF energy Virial GTF energy Virial Numerical-HF lip, theorem (1% Up, theorem energy (23) Ion Configuration State 1 Od)" STF~ I id)" GTF~ (NHF) H- Li- B- c- N- 0- F Na- Al- S i- P- s- c1- K- sc- Tiv- Cr- Mn- Feco- Ni- Cu - Ga- Ge- As- Se- B r- Rb- Y- Zr- Nb- Mo- Tc- Ru - Rh- Pd- Ag- In- Sn- Sb- Te- 1-1 s2 [He] 2s2 [He] 2s2 2p2 [He] 2s2 2p3 [He] 2s2 2p4 [He] 2s2 2pS [He] 2s2 2p6 [Ne] 3s2 [Ne] 3s' 3p2 [Ne] 3s2 3p3 [Ne] 3s2 3p4 [Ne] 3s2 3pS [Ne] 3s2 3p6 [Ar] 4s2 [Ar] 4s2 3 8 [Ar] 4s2 3d3 [Ar] 4s2 3 d [Ar] 4s2 3dS [Ar] 4s2 3d6 [Ar] 4s' 3d7 [Ar] 4sZ 3d8 [Ar] 4s2 3d) [Ar] 4s2 3d1 [Ar] 4s2 3dI0 4p2 [Ar] 4s2 3d1 4p3 [Ar] 4s2 3dI0 4p4 [Ar] 4s' 3dI0 4pS [Ar] 4s2 3dI0 4p6 [Kr] 5s2 [Kr] 5s2 4dZ [Kr] 5s2 4d3 [Kr] 5s2 4 d [Kr] 5s2 4dS [Kr] 5s2 4d6 [Kr] 5s2 4 8 [Kr] 5s2 4d8 [Kr] 5s2 4d) [Kr] 5s2 4d1 [Kr] 5s' 4dI0 5pZ [Kr] 5s' 4dI0 5p3 [Kr] 5s2 4d1 5p4 [Kr] 5s2 4dI0 5pS [Kr] 5s2 4dI0 5p6 "This column shows AE = I E(NHF) - E(calcu1ated) I. This CO~UINI shows the values of A = (2 + virial ratiol. our results are better for the cations lighter than Zr', one has to recognize that Koga used smaller basis sets. From Zr+ on, Koga's basis set size becomes more similar to our universal basis size (l2s, lop, 106). The average errors in our STF energies with respect to the numerical HF energies for the cations are, respectively, , , , and millihartrees for the first, second, third, and fourth rows. The average errors in the energies found by Koga are , , 0.061, and millihartrees for the respective first-, second-, third-, and fourth-row cations. For the anions, Koga's basis set size varies from (8s, 6p) to (1 ls, 9p) for B- through C1-, and for K- it is (12s, 8p). For SCthrough I-, it varies from (12s, 8p, 66) to (14s, 13p, 96). When we compare our STF energy results with Koga's results for the anions, we notice that our results are on average one decimal

6 da Silva and Trsic 1531 Table 4. Low-lying excited state energies (au) of the STF and GTF bases for negative ions. STF energy Virial GTF energy Virial Clementi- (13s, 1 lp, theorem (1% 12p, theorem Roette energy Ion Configuration State 106)" STF~ 1 Id)'' GTF~ (24)(CR) B- B- c- c- N- N- Al- Al- S i- S i- P- P- sc- Tiv- Cr- Ni- Ge- As- As- Y- Nb- Mo- Te- [He] 2s' 2p2 [He] 2s2 2p2 [He] 2s2 2p3 [He] 2s' 2p3 [He] 2s' 2p4 [He] 2s' 2p4 [Ne] 3s' 3pZ [Ne] 3s' 3pZ [Ne] 3s' 3p3 [Ne] 3s' 3p3 [Ne] 3s' 3p4 [Ne] 3s' 3p4 [Ar] 4s' 3d3 [Ar] 4s' 3dl [Ar] 4s' 38 [Ar] 4s' 3d6 [Ar] 4s' 3d'" [Ar] 4s' 3d'" 4p3 [Ar] 4s' 3d1" 4p4 [Ar] 4s' 3d1" 4p4 [Kr] 5s' 4d3 [Kr] 5s' 48 [Kr] 5s' 4d6 [Kr] 5s' 4d7 "This column shows AE = IE(NHF) - E(calcu1ated) 1. qhis column shows the values of A = 12 + virial ratio 1. 'Calculated energy is lower than CR energy. figure more accurate than Koga's results from H- to Se-. From Br to Ag-, both STF basis set sizes become more similar and, in general, our results have the same accuracy as Koga's results. From In- on, Koga's results become more accurate than our results, but he is working with a slightly larger basis set (l4s, 13p, 96) than our universal basis set (13s, 1 lp, 106). The average errors in our STF energies with respect to the numerical HF limits for the anions are, respectively, , ,0.0026, and millihartrees for the first, second, third, and fourth rows. Koga found, for the respective first-, second-, third-, and fourth-row anions, the following average errors in the energies: , , 0.087, and millihartrees. In conclusion, we would like to say that comparison of our results with Koga's results brings to attention the fact that when we are developing a universal basis set instead of a fully optiniized basis set, we face the penalty of using, mainly for lighter atomic systems (from H through Ca), a larger number of basis functions than a fully optimized basis set needs to obtain the same degree of accuracy. Indeed, our experience in developing universal basis sets always showed that this penalty is reduced when we work with atomic systems from the third row on (14, 16-19) The role of the weight functions in the evaluation of the total electronic energies Since the GWHF equations are obtained from the minirnization of the functional E with respect to the weight functions,j, the description off, governs the quest for the total energies for any atomic system. Achieving the best HF energy for an atomic system means obtaining the best description of J through the numerical approximation outlined in Sect The quest for the best weight function associated with any atomic orbital is implemented by the integral discretization described previously, and the discretization parameters amin, An, and N are responsible for attaining the best weight function. When the GCHF method is employed to generate basis sets, the number of points N determines the size of the basis and, certainly, the larger the value of N, the lower is the ground state energy obtained. But the chosen value of N is a compromise between accuracy and size of the basis set. In ref. 14 we presented a series of weight functions generated by the integration mesh of the neutral atoms, H through Xe, and discussed some practical aspects and properties of the weight functions. All the observations of that paper are valid with respect to the weight functions of the positive and negative ions studied here but, at this time, we add a few relevant

7 Fig. 1. The 2s Gaussian weight functions for Be', Be, Li, and Li-. The point nmi, = for s functions is indicated. Can. J. Chern. Vol remarks especially related to the ionic species, and we will focus our remarks on the Gaussian weight functions. In Fig. 1 we have plotted the Gaussian 2s weight functions for the isoelectronic species Be' and Li and Be and Li-. It appears that the integration interval with R, = is satisfactory for the Be+ and Be species, but it is not adequate for the alkaline atom, and it is particularly insufficient for the negative ion of Li. This is a feature that also appears for the highest occupied orbital of the other alkaline atoms considered, i.e., Na through Cs. For this reason, it was necessary to shift R, for the GTF basis set of the negative ions to R, = for s symmetry (R,, for s symmetry STF weight functions was shifted to -0.21). Thep and d symmetry weight functions also required a revision of the integration limits for the negative ions (see discretization parameters in Sect. 2.1). In Fig. 2 we show how the Gaussian 2s weight functions obtained in this work for the species Li and Li- compare with the corresponding fully integrated weight functions. Both for Li (this work) and Li- (this work) there appears the need for higher values of the weight functions to compensate the truncation of the integration range. In view of our goal of generating "universal" (a unique set of exponents to be used for all atoms under consideration) bases of tractable size, we opted to retain R, values that, in a few cases, did not accomplish a complete numerical integration. It is relevant to point out that this limitation of the R, values has little effect on the total HF energy, even for the alkaline atoms and negative ions. Of course, the present selection of lower limits fo;the numerical integration range should be reconsidered with caution if properties demanding very diffuse orbitals were of interest (26). Otherwise, for most of the atoms and ions considered, the numerical integration ranges were adequate. This is illustrated in Fig. 3 with the example of the Gaussian 2p weight functions for neutral and charged fluorine atoms. 3. Concluding remarks We have obtained STF and GTF basis sets for mono-charged positive and negative atomic species from H through Xe. Our bases stand comparison with numerical results and the best atom-optimized bases available, both for ground and lowlying excited states. The bases for the positive ions are the same as those that we generated in ref. 14 for neutral atoms. In the case of the negative ions, the numerical integration range, which defines a basis set, was shifted to lower values of the generator coordinate to accommodate the need for very diffuse orbitals. The weight functions play a very important role in evaluating the total electronic energies. In fact, plots of the weight functions will provide adequate ranges for the numerical integration of the GWHF equations, and consequently we will have accurate basis sets for atomic species. A penalty is faced when we are developing universal instead of fully optimized basis sets, mainly when we are working with atoms of the first and second rows. The penalty is characterized by the number of extra basis functions needed in the universal basis set to obtain the same degree of accuracy as in a fully optimized basis set. ~cknow~edgements We would like to acknowledge the financial support of CNPq, FAPESP, and FINEP (Brazilian Agencies).

8 da Silva and Trsic Fig. 2. Comparison of the Gaussian 2s weight function for Li and Li- for the universal bases (this work) and the complete integration range. Fig. 3. Plot of the 2p Gaussian weight functions for F, and F References 1. J.J. Griffin and J.A. Wheeler. Phys. Rev. 108, 311 (1957). 2. C.W. Wong. Phys. Rev. 15,283 (1975). 3. J.D. Justin, M.V. Mihailovic, and M. Rosina. Nucl. Phys. A, 182,54 (1971). 4. L. Lathouwers. Ann. Phys. 102, 347 (1976); L. Lathouwers, P. Van Leuven, and M. Bouten. Chem. Phys. Lett 52,439 (1977). 5. B. Laskowski. In Quantum science methods and structure, a tribute to Per-Olov Lowdin. Edited by J.L. Calais, 0. Goscinski, J. Linderberg, and Y. ohm. Plenum, New York p P. Chattopadhyay, R.M. Dreizler, M. Trsic, and M. Fink. Z. Phys. A: At. Nucl. 285,7 (1978). 7. J. Broeckhove and E. Deumens. Z. Phys. A: At. Nucl. 292,243 (1979); F. Arickx, J. Broeckhove, E. Deumens, and P. Van Leuven. J. Comput. Phys. 39,272 (1981). 8. R.L. Somorjai. Chem. Phys. Lett 2, 399 (1968); Phys. Rev. Lett. 23,329 (1969); J. Math. Phys. 12,206 (1971). 9. D.M. Bishop and B.E. Schneider. Int. J. Quantum Chem. 9, 67 (1975). 10. A.J. Thakkar and V.H. Smith, Jr. Phys. Rev. A: Gen. Phys. 15, 1 (1977); 15, 16 (1977); 15, 2143 (1977); A.J. Thakkar. J. Chem. Phys. 75,4496 (1981). 11. J.R. Mohallem, R.M. Dreizler, and M. Trsic. Int. J. Quantum. Chem. Symp. 20,45 (1986). 12. J. R. Mohallem and M. Trsic. J. Chem. Phys. 86,5043 (1986). 13. H.F.M. da Costa, M. Trsic, and J.R. Mohallem. Mol. Phys. 62, 91 (1987). 14. A.B.F. da Silva and M. Trsic. Mol. Phys. 68,433 (1989). 15. A.B.F. da Silva and M. Trsic. Mol. phis. 78, 1301 (1993). 16. G.L. Malli, A.B.F. da Silva, and Y. Ishikawa. Chem. Phys. Lett 201,37 (1993). 17. A.B.F. da Silva, G.L. Malli, and Y. Ishikawa. Chem. Phys. Lett 203,201 (1993). 18. G.L. Malli, A.B.F. da Silva, and Y. Ishikawa. Phys. Rev. A: At. Mol. Opt. Phys. 47, 143 (1993). 19. A.B.F. da Silva, G.L. Malli, and Y. Ishikawa. Can J. Chem. 71, 1713 (1993). 20. D.M. Silver and W.G. Nieuwpoort. Chem. Phys. Lett 57,421 (1978); D.M. Silver, S. ~ilson, and W.G. Nieuwpoort. Int. J. Quantum Chem. 14, 635 (1978); S. Wilson and D.M. Silver. Chem. Phys. Lett. 63,367 (1979). 21. J.R. Mohallem. Z. Phys. D: At. Mol. Clusters, 3,339 (1986).

9 1534 Can. J. Chern. Vol D.L. Hill and J.A. Wheeler. Phys. Rev. 89, 1102 (1953). 25. T. Koga, E. Shibata, and A.J. Thakkar. Theor. Chirn. Acta, 91, 23. T. Koga, H. Tatewaki, and A.J. Thakkar. J. Chem. Phys. 100, 47 (1995) (1994). 26. J.R. Mohallem and M. Trsic. Int. J. Quantum Chem. 33, E. Clementi and C. Roetti. At. Data Nucl. Data Tables, 14, 177 (1988). (1974).