On the performance of molecular model core potential orbitals in spin-orbit and electron correlation studies

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1 On the performance of molecular model core potential orbitals in spin-orbit and electron correlation studies Dietmar Krause and Mariusz Klobukowski ~bstract: The role of improved parametrization and accurate basis sets in model core potentials was studied in calculations of the spin-orbit coupling constants (in PH, ASH, and SbH) and of the electron correlation effects (in P2, Asz, and Sb2). An effective method of identifying and removing the intruder quasi-core orbitals from the virtual orbital space was proposed in connection with the post-hartree-fock calculations. The results demonstrated that (a) the flexible valence basis sets allow evaluation of the spin-orbit effects without resorting to any scaling techniques and (b) the intruder quasicore orbitals, even if left imbedded in the virtual space, have negligible effect on the values of the electron correlation energy. Key words: effective core potentials, basis sets, spin-orbit effects, electron correlation. Resume : On a CtudiC le r6le d'une paramctrisation amcliorce et d'ensembles de base prccis dans les modkles des potentiels des noyaux dans les calculs des constantes de couplage spin-orbitale (dans le PH, ASH et le SbH) et les effets de corrclation des Clectrons (dans P2, AS? et Sb2). En relation avec les calculs post-hartree-fock, on propose une mcthode efficace d'identifier et d'enlever les orbitales quasi-nucliaires inopportunes de I'espace orbitalaire virtuel. Les rcsultats ont dcmontrc que (a) les ensembles de base i valence flexible permettent d'ivaluer les effets spin-orbitale sans faire appel i des techniques scalaires et (b) les orbitales quasi-nucltaires inopportunes, mcme si on les laisse dans I'espace virtuel, n'ont qu'un effet nkgligeable sur les valeurs de 1'Cnergie de corrclation Clectronique. Mots clps : potentiels efficace des noyaux, ensembles de base, effets spin-orbitale, corrclation Clectronique. [Traduit par la redaction] I. Introduction basis set calculations they become mixed with the virtual orbitals. Both the model core potential (MCP) method (1, 2) and the To probe the region near the nucleus, we studied the spinwidely used effective core potential (ECP) method (3) treat orbit coupling of the lowest 3 ~ - state of PH, ASH, and SbH. only the valence electrons of an atomic or molecular system explicitly. Two distinctive features of the MCP method make it different from the ECP family of methods: (a) the capability of reproducing the full nodal structure of the valence orbitals and (b) the use of projection operators. The model potential orbitals not only reproduce the all-electron (AE) valence orbitals in the region around the outermost maximum but they also match the inner part of the AE valence orbitals, resulting in an improved description of the region near nuclei. (Depending on the basis set, the full nodal structure may be maintained in model potential orbitals.) The projection operator technique used in the MCP method shifts the core orbitals into the energy continuum, where in the finite Received November 6, This paper is dedicated to Professor Richard F. W. Bader on the occasion of his 65th birthday. D. Krausel and M. ~lobukowski.~ Department of Chemistry, University of Alberta, Edmonton, AB T6G 2G2, Canada. Hany Emmett Gunning Graduate Fellow Author to whom correspondence may be addressed. Telephone: (403) Fax: (403) mariusz.klobukowski@ualberta.ca Can. J. Chem. 74: (1996). Printed in Canada I Imprimt au Canada These systems were the subject of a comparative study in the past (4), showing rather uneven performance of the model core potential method. In our present study, the performance of the model potential method is significantly improved by choosing a better parametrization scheme and employing a high-quality basis set. Furthermore, in previous work (5, 6) we compared the performance of the model core potential method and the ECP method in studies of valence-electron correlation energies in atoms and molecules. Both methods were shown to yield essentially equivalent results. The question arose, however, whether the core orbitals, shifted into the virtual orbital (VO) space by the projection operators, might lead to overestimated correlation energies. Excitations of valence electrons into these quasi-core orbitals have no counterpart in all-electron CI calculations, and it was hoped that the corresponding correlation energies are negligible compared to the overall valence-electron correlation energies. To study the effect of the quasi-core orbitals, we performed CI-SD calculations of P2, As2, and Sb2 with an active virtual orbital space containing only quasi-core orbitals, without quasi-core orbitals, and in full virtual orbital space. To observe the effect of higher excitations, Moiler-Plesset calculations up t~ fourth order were done. For the model core potential calculations, an all-electron basis was employed that enables both

2 Krause and Klobukowski Table 1. Size and contraction of the all-electron well-tempered basis for P, As, and Sb. Atom Size Contraction an unambiguous assignment of the quasi-core orbitals and model potentials and their performance in molecular calcua direct comparison with the correlation energies obtained lations will be reported in a forthcoming paper. from all-electron SCF orbitals. B. Basis sets 11. Details of calculations Basis sets for all atoms were contracted to give the flexibility of five primitive Gaussian-type functions in the outermost A. Model core potential method valence region. For hydrogen, the well-balanced (12s) basis (7) was contracted to (711111) and augmented with a single The detailed derivation of the model core potential method p-type poiarisation function (, = has been published (1, 2). In the following, only the terms The basis sets for phosphorus, arsenic, and antimony were that are relevant to the present work are defined. taken from the compilation of the well-tempered Gaussian The one-electron Hamiltonian hi in the model core potenbasis sets (8-10). The size of the basis and the contractial formalism contains the potential Via and the projection tion of the all-electron basis are shown in Table For the operator Pq at the atomic center a: valence-electron basis, the s and p valence shells were uncontracted in the outermost reg& to yield one contracted shell plus five primitive Gaussian functions, e.g., for phosphorus (18,11111/11,11111). All basis sets, except for hydrogen, were augmented with two d-type polarization funcwhere za is the core charge of atom a (charge of the bare tions (11). For the electron correlation studies, we implenucleus minus the number of core electrons). mented the model core potential method into the HONDO (12) The spherically symmetric local potential Via, which ap- program package in order to take advantage of the ability proximates the exact atomic nonlocal core potential, has the of the HONDO system to remove the spherically symmetric form: contribution of Cartesian d-type Gaussian functions. where I and J refer to the sets of parameters {ai,al) and {~J~AJ). In the projection operator the parameter Bc is taken as twice the absolute value of the atomic orbital energy of the core shell c on atom a. An initial set of the model potential parameters {a,a) for a given atom was determined by fitting the potential Via to a numerical core potential consisting of the core electron nuclear attraction, the core Coulomb and core exchange potentials acting on the valence p-type radial function. Further optimization of the parameters was done to maximize the overlap between the atomic model potential orbitals and the all-electron valence functions. The expectation values (rn), with n = (-3, -2, -1, 1,2), of the model potential valence functions are within 1% of those obtained in atomic allelectron calculations using the same well-tempered basis set. The performance of the model potentials was assessed in SCF calculations of P2, As2, and Sb2. The equilibrium distances are within 2% and the harmonic vibrational frequencies are within 10% of the corresponding all-electron results. A detailed description of the parametrization scheme of the Ill. Results of calculations A. Spin-orbit coupling For the radicals PH, ASH, and SbH, the ground electronic configuration 02n2 gives rise to the states 3~-, 'A, and 'c+. The zero-field splitting of the 3 ~ - state is dominated by the matrix element ( 3~- I Hsol lc+) of the one-electron spin-orbit Hamiltonian Hso The spin-orbit matrix elements of PH, ASH, and SbH at their experimental bond distances (1.42, 1.53, and 1.72 A, respectively) were computed using the program package GAMESS (13), enhanced by the addition of the MCP one-electron integral modules. We used a minimal description of the CI wave functions, in which two determinants span the states 'A and 'c+, and one determinant represents the 3 ~ state. - The determinants were generated by the ground state SCF orbitals in a well-tempered basis for the all-electron calculations and in a valence basis for the model core potential calculations. In Table 2, the spin-orbit matrix elements, the energy gaps between the 'c+ and 3 ~ - states, and the second-order spin-orbit couplings of the 3 ~ - states, based on the dominant spin-orbit matrix element, are reported. The model potential results agree within 1% with the respective all-electron values and it should be stressed that this excellent agreement was achieved without employing effective core charges different from the actual bare nucleus charges, and without using any other scaling factors. The purpose of the present work was to establish degree

3 Can. J. Chem. Vol. 74, 1996 Table 2. Spin-orbit matrix elements, energy differences, and spin-orbit splitting of 0' and 1 states in PH, ASH, and SbH calculated using all-electron (AE) and model core potential (MCP) wave functions (all values are in cm-'). XH Method (3C~IHs,I'C') AE(3C-,1C') Spin-orbit (0' - 1) splitting PH AE MCP ASH AE MCP CAS SCFISOCI (14) Experiment (16) SbH AE MCP Relativistic CI (15) Experiment (17) of agreement between all-electron and MCP results. It is, nevertheless, interesting to compare the present results with both experimental values and results from earlier calculations. The two heavier systems were studied by Balasubramanian and Nannegari, who used CAS SCF followed by the second-order CI method for ASH (14) and relativistic CI for SbH (15). Their results are reported in Table 2, together with experimental values of the spin-orbit splitting between the states 0+ and 1. The main difference between the present results and those of Balasubramanian is the larger separation between the 3 ~ - and 'c+ states; this difference is expected because of the extremely limited form of the wave function used in the present work. In consequence, the present values of spin-orbit splitting are smaller than the experimental ones. B. Electron correlation energy A model core potential and an all-electron SCF calculation of P2 was performed at its equilibrium distance (1.89 A) using the same all-electron well-tempered basis in both cases. The model potential valence orbitals reproduce almost exactly the corresponding all-electron orbitals; the deviation of the overlap from 1 is smaller than 4 x lop4. As expected, due to the flexibility of the all-electron basis set, all 10 core MOs could be found in the model potential virtual orbital space of P2. These quasi-core orbitals could be assigned unambiguously, as the overlap with the corresponding all-electron occupied core MOs was larger than After reordering the quasi-core VOs, a valence CI-SD calculation with an active virtual orbital space spanned only by the quasi-core orbitals was done. The correlation energy is au, which is less than 0.4% of the all-electron CI-SD correlation energy ( au) obtained with excitations from the valence space into the full virtual orbital space. The selection of the quasi-core orbitals based on overlap with the all-electron core molecular orbitals requires that the all-electron calculations be performed. A simpler and more general technique of identifying and shifting the quasi-core orbitals was accomplished by a transformation within the virtual orbital space, using a procedure similar to the method of modified virtual orbitals (18): after the last SCF step, the canonical eigenvector matrix is saved and a new Fock matrix is constructed using model potential one-electron integrals based on scaled B, parameters (see eq. [3]). The Fock matrix is then transformed into the virtual orbital space, diagonalized, and the resulting eigenvector matrix is transformed back into the A0 (basis set) space. This procedure generates transformed virtual orbitals while leaving the occupied orbitals unchanged. Choosing a scaling factor of -100 for all B, parameters results in a virtual orbital space containing the quasi-core orbitals as the energetically lowest virtual orbitals. The overlap between the canonical model potential SCF VOs and the transformed VOs deviates less than 6 x lop4 from 1, so that we used this transformation method as a simple substitute for reordering VOs based on their overlaps. It must be stressed that this technique is more general than the one based on calculation of overlaps between the AE and MCP MOs as it does not require the all-electron calculations. Applying the above transformation procedure, the correlation effects of quasi-core VOs were studied in P2, As2, and Sb2 at their respective equilibrium distances (1.89, 2.10, and 2.34 A) using model core potential CI-SD wave functions generated by single and double excitations from the valence space into the space spanned by the quasi-core VOs. The resulting correlation energies (Table 3) are smaller than au, being less than 1% of the all-electron CI-SD correlation energies. The all-electron CI-SD wave functions were generated by excitations from the valence space of the all-electron SCF wave functions into the full virtual orbital space. Using the same all-electron well-tempered basis sets, model potential CI-SD calculations were done employing the full virtual orbital space without the quasi-core VOs. This was achieved by using a large positive scaling factor for B (lo4 for P2 and As2, lo7 for Sb2) in the transformation of the VOs, resulting in the quasi-core VOs as the energetically highest orbitals. These truncated model potential VO spaces have the same size as the corresponding all-electron full virtual orbital spaces. Consequently, the number of configuration state functions (CSFs) generated in the model potential CI-SD calculations is the same as the number of CSFs in the all-electron calculations. A direct comparison of the model potential correlation energies with the all-electron correlation energies is

4 A - Krause and Klobukowski Table 3. CI-SD correlation energies of P,, As,, and Sb2 (all values in atomic units). - Method" Active spaceb p? AS? Sbz MCP-AE Core AE Full MCP-AE Trunc MCP-AE Full MCP Full "All-electron calculation (AE); model core potential calculation using an all-electron basis (MCP-AE) or a valence basis (MCP). *Active space spanned by the quasi-core VOs (core), by all other VOs (trunc.), and the full virtual orbital space (full). Table 4. Second-, third-, and fourth-order Meller-Plesset correlation energies of P,, As,, and Sb, in full virtual orbital space (all values in atomic units). PZ As2 Sb, Method" MP2 MP3 MP4 MP2 MP3 MP4 MP2 MP3 MP4 AE MCP-AE MCP "All-electron calculation (AE); model core potential calculation using an all-electron basis (MCP-AE) or a valence basis (MCP). therefore iustified and gives a measure of the quality of the molecul& model potential valence and virtual orbitals. The respective correlation energies (Table 3) agree within au (a relative error of less than 1%). In the next set of model potential CI-SD calculations the full virtual orbital space was employed, expanded in the same all-electron basis sets. The correlation energies obtained (Table 3) agree very well with those of the previous calculations, in which the quasi-core VOs were discarded. The deviations, smaller than 5% of the total AE CI-SD correlation energies (0.003 au for P2, au for AsZ, and au for Sb2) are larger than the correlation energies resulting from excitations only into the quasi-core orbital space. The difference may be attributed to contributions from CSFs generated via mixed excitations connecting quasi-core orbitals with the remaining virtual orbitals. Furthermore, it is reassuring to see the good agreement between the all-electron and model potential correlation energies obtained in M~ller-Plesset calculations up to fourth order (Table 4). Finally, model potential CI-SD calculations were performed using the valence basis sets and employing the full virtual orbital space. A direct comparison with the allelectron correlation energies is not justified due to the different active spaces. Nevertheless, it can be seen from Table 3 that model potential orbitals, spanned by valence basis functions, are very well suited to recover the valence-electron correlation energy -- based on AE SCF orbitals. Furthermore, the MOller-Plesset correlation results (Table 4) based on model potential orbitals closely match the corresponding allelectron correlation energies. IV. CO~C~US~O~S By reproducing the AE spin-orbit couplings of PH, ASH, and SbH within I%, we demonstrated the usefulness of molecular model potential orbitals in the computation of properties that depend on the core region of the wave function. The model potential correlation energies, calculated employing an all-electron basis, agree very well with the AE correlation energies, and the inclusion of quasi-core orbitals in the active virtual orbital space leads to only slightly (less than 5%) overestimated correlation energies. It should be noted that the all-electron basis sets are capable of generating all core orbitals, whereas the basis sets usually chosen for valence-electron calculations are, at most, flexible enough only to reproduce the core orbitals of the outermost core shell. Contamination of the virtual orbital space by the remaining quasi-core orbitals is expected to have a negligible effect on CI-SD calculations. The occupation numbers of the natural orbitals in the space of the quasi-core orbitals (MCP-AElcore results in Table 3) are smaller than 2 x Furthermore, our results from Moller-Plesset calculations up to fourth order do not indicate any problems in the evaluation of higher terms in the correlation energy expansion. We conclude that the occurrence of quasi-core orbitals in the canonical SCF virtual orbital space does not seem to be a drawback of the model core potential method in the study of valence-electron correlation energies. Acknowledgments The calculations were done on SUN and RSl6000

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