Universal Gaussian basis functions in relativistic quantum chemistry: atomic Dirac-Fock-Coulomb and Dirac-Fock-Breit calculations

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Universal Gaussian basis functions in relativistic quantum chemistry: atomic Dirac-Fock-Coulomb and Dirac-Fock-Breit calculations G. L. MALLI' AND A. B. F.

1 Universal Gaussian basis functions in relativistic quantum chemistry: atomic Dirac-Fock-Coulomb and Dirac-Fock-Breit calculations G. L. MALLI' AND A. B. F. DA SILVA' Department of Chemistry, Simon Fraser University, Burnaby, BC, Canada V5A IS6 AND YASUYUKI ISHIKAWA Department of Chemistry and the Chemical Physics Program, Universiry of Puerto Rico, Rio Piedras, Puerto Rico 00931, U.S.A. Received January 2, 1992 G. L. MALLI, A. B. F. DA SILVA, and YASUYUKI ISHIKAWA. Can. J. Chem, 70, 1822 (1992). Matrix Dirac-Fock-Coulomb and Dirac-Fock-Breit self-consistent field calculations are performed for a number of neutral atoms, He (Z = 2) through Xe (Z = 54), using the universal Gaussian basis set (l8s, 12p, 1 ld) reported recently by Da Silva et al. The total Dirac-Fock-Coulomb, the Dirac-Fock-Breit, and the Breit interaction energies calculated with this universal Gaussian basis set are in good agreement with the corresponding values obtained by using an extensive well-tempered Gaussian basis set for the He through Ca (Z = 20) atoms. Although this universal Gaussian basis set is inadequate for the calculation of total Dirac-Fock-Coulomb and Dirac-Fock-Breit energies for the Kr, Sr, and Xe atoms, the Breit interaction energies calculated with this basis for these three atoms are in very good agreement with the corresponding Breit interaction energies obtained by using the extensive well-tempered Gaussian basis sets. Work is in progress to generate a more extensive and energetically better universal Gaussian basis set for He through Xe for its use in non-relativistic Hartree-Fock as well as Dirac-Fock self-consistent field calculations on polyatomics involving heavy atoms. G. L. MALLI, A. B. F. DA SILVA et YASUYUKI ISHIKAWA. Can. J. Chem. 70, 1822 (1992). Utilisant la base gaussienne universelle (18s, 12p, 1 Id) proposce rccemment par Da Silva et al., on a effectuc des calculs en champ auto-coherent des matrices de Dirac-Fock-Coulomb et de Dirac-Fock-Breit d'un certain nombre d'atomes neutres, He (Z = 2) a Xe (Z = 54). Les Cnergies totales d'interaction de Dirac-Fock-Coulomb, et les Cnergies d'interaction de Dirac-Fock-Breit et de Breit sont en bon accord avec les valeurs correspondantes obtenues par l'utilisation d'une base gaussienne extensive bien tempcrce pour les atomes de He a Ca (Z = 20). MCme si cette base gaussienne universelle est inadequate pour calculer les Cnergies totales de Dirac-Fock-Coulomb et de Dirac-Fock-Breit des atomes de Kr, Sr et Xe, les Cnergies d'interaction de Breit calculces pour ces trois atomes ti l'aide de cette base sont en bon accord avec les Cnergies correspondantes d'interaction de Breit obtenues en utilisant la base gaussienne extensive bien tempcree. Des travaux sont en cours dans le but de gcncrer une base gaussienne universelle plus Ctendue et meilleure d'un point de vue de 1'Cnergie que l'on pourrait appliquer du He au Xe et qui pourrait &tre utilisce dans des calculs en champ auto-coherent non-relativiste de Hartree-Fock ainsi que de Dirac-Fock pour des ensembles polyatomiques impliquant des atomes lourds. [Traduit par la rcdaction] Introduction Ab initio electronic structure calculations for atoms and especially for molecules in general are mostly camed out within the finite basis set expansion method of Roothaan (1). There is a considerable degree of freedom in choosing the basis functions for atomic and molecular calculations, since any complete set of functions can be employed. Although Slater-type basis functions (STF) have been widely used for atoms and diatomics, Gaussian-type basis functions (GTF) are invariably the popular choice for polyatomic calculations. The prime reason for the preference of the GTF as basis for polyatomics lies in the fact that all the multicenter integrals can be evaluated exactly by closed analytical formulas. However, a much larger basis set of GTF is needed than the basis set of STF because the GTF's behave incorrectly both in the region near the nucleus (if approximated as a point nucleus) and in the long range. These deficiencies are, however, relevant to the non-relativistic quantum chemistry based upon the Schrodinger '~uthor to whom correspondence may be addressed. 'permanent address: Departamento de Fisica e Quimica Molecular, Instituto de Fisica e Quimica de S5o Carlos, Universidade de Slo Paulo, C.P. 369, SHo Carlos, SP, Brazil. equation where the point nucleus approximation is usually employed. Relativistic quantum chemistry, however, is based on the Dirac equation whose solutions for an electron in a finite nucleus have been shown to be Gaussians (2, 3). Thus the GTF basis set is the natural choice for ab initio relativistic quantum chemistry, and spherical Gaussians were introduced as basis sets in relativistic quantum chemistry more than a decade ago (4). Since the computational cost of finite basis set atomic and molecular calculations increases as -p (where N is the number of basis functions used), various attempts have been made to economize the cost as much as possible by adopting various strategies. One such approach was the introduction of the universal STF basis set (5) for atoms following the earlier work on the even-tempered STF and GTF basis sets (6, 7). As a result of extensive studies, it was observed that the optimum exponents of the rigorously optimized GTF for ab initio non-relativistic Hartree-Fock (HF) atomic calculations could be related by Gaussian rule (8). The existence of such a rule suggested a high degree of universality in the optimum GTF representation of different atoms, and it is obvious that considerable computational savings would accrue if a universal Gaussian basis set were employed for each atom in a molecular calculation. The transferability

2 MALLI ET AL alone of all the one- and two-electron integrals over the primitive universal GTF from system to system (with trivial multiplicative scale factors due to change in nuclear charges) would lead to remarkable computational savings. Therefore, there has been a considerable effort involved in the design of the so-called "universal Gaussian basis set" (8-10). A universal basis set is a single, sufficiently flexible basis set that can be used for any atomic or molecular environment without much loss of accuracy (9). Needless to reiterate, it is always possible to construct a basis set for an atom which is smaller and still as accurate as the universal basis set; however, the design of such basis sets involves nonlinear exponent optimization for each atom, which is computationally very expensive and time consuming. Recently, well-tempered GTF (1 I), geometrical GTF, and various other GTF basis sets have been used for Dirac-Fock- Coulomb (DFC) and Dirac-Fock-Breit (DFB) (1 2-15) calculations without exponent reoptimization along with the "kinetic balance" and the finite model of the atomic nucleus. The results have been very encouraging not only for DFC and DFB calculations, but also for relativistic manybody perturbation theory calculations on atoms (1 6). In this paper, we report the results of the DFC and DFB calculations for a number of atoms, He (Z = 2) through Xe (Z = 54), using the universal Gaussian type functions (UGTF) (18s, 12p, 1 ld) basis set reported recently by Da Silva et al. (10). The purpose of the present study is to assess the performance of the UGTF in atomic Dirac-Fock calculations. The calculations are performed using the matrix Dirac-Fock- Breit SCF methodology detailed by us previously (15). Gaussian basis set for relativistic calculations The basis set expansion method for relativistic calculations was pioneered by Kim (17), who introduced the STF basis set with a non-integer exponent of r and developed the Dirac-Fock-Roothaan SCF method for closed-shell atoms. Kagawa (18) extended Kim's method to open-shell atoms and results were reported for a large number of transition elements (18) and Rn (19). In these studies, the same STF basis set was used for the large and small radial components, and often the problem of so-called variational failure was encountered. The spherical Gaussian basis set was introduced in relativistic atomic structure calculations by Malli (4), and in Dirac-Fock-Roothaan molecular calculations by Matsuoka et al. (21). Recently, there has been great interest in the use of GTF as a basis set for Dirac-Fock-Coulomb, Dirac-Fock-Breit, and relativistic many-body perturbation calculations (12-16). This is mainly due to the work of Ishikawa et al. (2, 3, 12, 15, 16), who have emphasized that the imposition of finite nuclear boundary conditions for solutions of the Dirac-Fock equation results in a solution that is Gaussian at the origin and, therefore, the GTF of integer power of r are appropriate basis functions for the finite nuclear model. Moreover, it has been shown that the failure to satisfy proper boundary conditions near the origin can lead to a spurious solution. The Gaussian-type functions that satisfy the boundary conditions for the finite nucleus automatically satisfy the condition of the so-called "kinetic balance" for a finite speed of light. This is due to the fact that the exponent of r for the GTF does not depend on the speed of light, c, in contrast to the non-integer STF whose exponent depends on the speed of light and, thus, does not satisfy the kinetic balance conditions for a finite value of c (3). The ki- TABLE 1. Orbital exponents of the universal Gaussian basis set (18s, 12p, 1 ld) taken from ref. 10 netic balance simply guarantees that the solutions of the matrix DFC and DFB equations approach the correct nonrelativistic limit as c approaches infinity (3). Choice of the universal Gaussian basis set There are a great variety of GTF's to be used in relativistic atomic and molecular calculations; however, if the GTF basis is sufficiently large and flexible, the particular choice of the GTF basis is not expected to be reflected in the calculated properties. One is forced, however, to use a moderately large basis set, since its flexibility in general increases as it is extended. It is also well established that it is generally more profitable to increase the basis set size rather than optimize the individual basis function exponent since the exponent optimization is computationally very costly. The concept of a universal basis set arose from these considerations and basis set exponent non-optimization has been so far almost the rule in relativistic quantum chemistry. We chose to adopt the UGTF (18s, 12p, 116) basis set recently reported by Da Silva et al. (lo), which is based on the earlier work of Mohallem et al. (22). The significance of the UGTF basis set (10, 22) lies in the fact that no a priori information is forced on an atom, which is characterized in the SCF algorithm only by the atomic number in the Fock operator, i.e., an atom is characterized only by its Harniltonian. This method of designing a UGTF basis introduces a new algorithm for basis set exponent selection. The basis set exponents are not adjustable parameters to optimize a property (e.g., energy), but values generated by a certain criterion for the integration of the SCF equations for the atom, i.e., the exponent values are generated by discretization of an integral equation (with the best numerical integration as the goal). The number of basis functions, N, is related to the number of discretization points, and as the number N is increased, better numerical integration is achieved. We believe that this procedure of generating UGTF basis has sound theoretical background and deserves further investigation both in non-relativistic and relativistic quantum chemistry.

3 CAN. J. CHEM. VOL. 70, 1992 TABLE 2. Total DFC SCF, DFB SCF, and variational Breit interaction energies of the rare-gas and alkaline-earth atoms (in au) Atom UGTF GTP ~umerical~ "Calculated by using well-tempered GTF basis (15). bnurnerical finite-difference calculations (15). 'Calculated by using even-tempered GTF basis (15). Results and discussion The UGTF (I&, 12p, 1 Id) basis set was used for the atoms He, Be, Ne, Mg, Ar, Ca, Kr, Sr, and Xe. The exponents of this basis set are derived from the non-relativistic generator coordinate version of the Hartree-Fock equations (22) and are collected in Table 1. The exponents were used in our DFC and DFB SCF calculations without further optimization. The radial functions with different K quantum number but of the same C quantum number are expanded in terms of the same set of basis functions, e.g., the radial functions of p,/, and p3i2 symmetries are expanded in the same set of p-type GTF's listed in Table 1. The speed of light used in our calculations is au (atomic units). The nuclei are modeled as spheres as uniform proton charge distribution and the atomic masses used for the rare gas atoms He, Ne, Ar, Kr, and Xe are, respectively, , 20.18, , 83.80, and The atomic masses used for the alkaline earth atoms Be, Mg, Ca, and Sr are, respectively, , , 40.08, and The results of the calculated DFC, DFB, and Breit interaction energies (designated as EDFc, EDFB, and EB, respectively) using the UGTF (18s, 12p, 1 Id) basis are given in Table 2, where we have also included the results obtained using large well-tempered (1 1) GTF basis sets (6). In the last column of Table 2, the DFC energies obtained by using the numerical finite-difference Dirac-Fock program (20) are also tabulated. Here, E, denotes the variational Breit interaction energy computed as the difference EDFB - EDFC The variational Breit energy is the level shift in the total SCF energy due to the inclusion of the Breit term in the SCF process. The results clearly demonstrate that the total DFC and DFB energies computed with the UGTF basis set are in very good agreement with those obtained by using large well-tempered GTF for He, Be, Ne, Mg, Ar, and Ca atoms. However, for Kr, Sr, and Xe atoms, the calculated DFC and DFB energies are relatively poor in comparison with those obtained from the well-tempered GTF basis set calculations (15); the differences in EDFc and EDFB for the case of the Sr atom are and au, respectively. For the Xe atom, these differences are an order of magnitude larger than in the case of the Sr atom, viz., and au, respectively. Moreover, it turns out that both ED, and ED, are in error by almost the same amount for each atom and, therefore, the variational Breit interaction energies predicted by using both the UGTF and well-tempered GTF basis sets are in excellent agreement for all the atoms, He through Xe, treated in this paper. Although the UGTF ( 18s, 12p, 1 ld) basis is inadequate for the calculations of EDFc and ED,, for the Kr, Sr and Xe

4 MALL1 ET AL. TABLE 3. DFC and DFB orbital energies of Xe (in au) DFC SCF Orbital energies DFB SCF Orbital UGTP GTF~ UGTP G T ~ "Computed by using UGTF basis set (l8s, 12p, 1 Id) (10) bcomputed by using well-tempered 23s21p14d GTF basis set (15) atoms, EB calculated with this UGTF basis set is in very good agreement with the corresponding EB obtained from using an extensive GTF basis set. We also performed calculations for these atomic systems using the UGTF (18s, 18p, 184 basis set. Although the basis set is much larger for the p and d symmetries, only a marginal improvement in EDFc and ED, over the results of UGTF (18s, 12p, 114 was obtained for the heavier atoms Kr, Sr, and Xe. The calculated orbital energies for the Xe atom obtained by using the UGTF and well-tempered GTF (15) are collected in Table 3. A comparison of the orbital energies calculated by using these two basis sets indicates significant differences for the innermost shells. An improvement of the UGTF basis is therefore expected to lead to better total DFC and DFB energies and also the orbital energies of the innermost shells. Conclusion The universal Gaussian basis set employed in our calculations is fairly good for the DFC and DFB calculations for the atoms He through Ca. However, this basis set needs much improvement for the Kr, Sr, and Xe atoms, since the DFC and DFB energies calculated with this basis set for these three atoms are rather poor. The errors in the calculated DFC and DFB energies are, however, of the same magnitude and, therefore, the calculated Breit interaction energies even for these three atoms are in perfect agreement with those obtained by using the well-tempered GTF basis set (1 1, 15). It is clear that there is an urgent need for an improved UGTF basis set for He through Xe atoms. Work is in progress in the design of the improved UGTF basis set that can consistently provide more accurate results for the calculations of ED, and ED, for many-electron atoms. The improved UGTF basis set should lead to remarkable savings in computations for non-relativistic as well as relativistic calculations on molecular systems. Acknowledgments This work was supported in part by the Natural Sciences and Engineering Research Council of Canada (grant no. A3598 to G. L. Malli), by the Brazilian CNPQ (to A. B. F. Da Silva), and by the National Science Foundation (PHY to Y. Ishikawa), which are gratefully acknowledged. All calculations were performed on an IBM RISC/ 6000 workstation at Simon Fraser University. The authors thank Dr. Steve Kloster of the Academic Computing Services at Simon Fraser University for continuous help with the calculations on the IBM RISC/ C. C. J. Roothaan. Rev. Mod. Phys. 23, 69 (1951). 2. Y. Ishikawa, R. Baretty, and R. C. Binning, Jr. Chem. Phys. Lett. 121, 130 (1985). 3. Y. Ishikawa and H. M. Quiney. Int. J. Quantum Chem. S21, 523 (1987). 4. G. Malli. Chem. Phys. Lett. 68, 529 (1979). 5. D. M. Silver and W. C. Nieuwpoort. Chem. Phys. Lett. 57, 421 (1978). 6. K. Ruedenberg, R. C. Raffenetti, and R. D. Bardo. Energy, structure, and reactivity. Proceedings of the 1972 Boulder Seminar Research Conference on Theoretical Chemistry. Edited by D. W. Smith. Wiley, New York p R. C. Raffenetti. J. Chem. Phys. 59, 5936 (1973). 8. P. G. Mezey. Theor. Chim. Acta, 53, 183 (1979). 9. S. Wilson. Adv. Chem. Phys. 67, 439 (1987). 10. A. B. F. Da Silva, H. F. M. Da Costa, and M. Trsic. Mol. Phys. 68, 433 (1989). 11. S. Huzinaga and M. Klobukowski. J. Mol. Struct. Theochem. 167, 1 (1988); S. Huzinaga, M. Klobukowski, and H. Tatewaki. Can. J. Chem. 63, 1812 (1985). 12. Y. Ishikawa, H. Sekino, and R. C. Binning, Jr. Chem. Phys. Lett. 160, 206 (1989). 13. S. Okada and 0. Matsuoka. J. Chem. Phys. 91,4193 (1989). 14. A. K. Mohanty and E. Clementi. J. Chem. Phys. 93, 1829 (1990).

5 1826 CAN. J. CHEM. VOL. 70, Y. Ishikawa, H. M. Quiney, and G. L. Malli. Phys. Rev. A, 20. J. P. Desclaux. Comput. Phys. Commun. 9, 31 (1975). 43, 3270 (1991) Matsuoka, N. Suzuki, T. Aoyama, and G. L. Malli. J. 16. Y. Ishikawa. Phys. Rev. A, 42, 1142 (1990). Chem. Phys. 73, 1320 (1980). 17. Y.-K. Kim. Phys. Rev. 154, 17 (1967). 22. J. R. Mohallem and M. Trsic. J. Chem. Phys. 86, 5043 (1987); 18. T. Kagawa. Phys. Rev. A, 12, 2245 (1975). H. F. M. Da Costa, M. Trsic, and J. R. Mohallem. Mol. Phys. 19. T. Kagawa and G. L. Malli. Can. J. Chem. 63, 1550 (1985). 62, 91 (1987).

Highly accurate Gaussian basis sets for low-lying excited states of some positive and negative ions

Indian Journal of Chemistry Vol. 46A, September 2007, pp. 1383-1387 Papers Highly accurate Gaussian basis sets for low-lying excited states of some positive and negative ions P J P de Oliveira & F E Jorge*