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

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1 Indian Journal of Chemistry Vol. 46A, September 2007, pp 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* Departamento de Física, Universidade Federal do Espírito Santo, Vitória, ES, Brazil jorge@ cce.ufes.br Received 18 January 2007; revised 7 August 2007 The improved generator coordinate Hartree-Fock (HF) method has been used to generate highly accurate Gaussian basis sets for low-lying excited states of some mono-positive and -negative ions. From these basis sets, total HF energies have been calculated and compared with results reported in the literature. The accuracy of HF energy achieved herein is the best so far obtained with finite basis set expansion of Gaussian-type functions (GTFs). Using GTFs, difference between corresponding energies calculated with our sets and those reported in the literature is achieved up to 13.8 mhartree. Hartree-Fock (HF) wave functions for atoms and atomic ions can be computed numerically 1. HF- Roothaan (HFR) 2,3 wave functions are algebraic approximations to the HF wave functions in which the radial orbitals are expanded in a set of basis functions, such as Gaussian-type functions (GTFs). The greater convenience of HFR wave functions has led to a great demand for them in atomic and molecular calculations. In 1986, the generator coordinate HF (GCHF) method 4 was introduced as a new technique for the design of GTFs and Slater-type functions (STFs). This methodology was used to generate universal basis sets (UBSs) 5-7 and adapted Gaussian basis sets (AGBSs) Jorge and de Castro 12 developed the improved GCHF (IGCHF) method, which was applied with success to construct GBSs for atomic and molecular systems The main objective of the present investigations is to test for the first time the performance of the IGCHF method in the generation of GBSs for excited states of ions. Basis sets for low-lying excited states of the cations and anions have been generated and, from these sets, total HF energies evaluated and compared with the results obtained by other basis sets reported in literature 5,6,18. Theoretical (Improved Generator Coordinate Hartree-Fock Method) The GCHF approach 4 is based in choosing the oneelectron functions as a continuous superposition, i.e., ( ) f ( ) Ψ =, i n i (1) φi 1, α i α dα = 1,...,, (1) where the φ i are the generator functions (GTFs in our case), the f i are the weight functions, and α is the generator coordinate. For closed-shell systems, using Eq. (1) to build a Slater determinant for multielectronic wave functions, and minimizing the total energy, E, with respect to the f i, one obtains the Griffin-Hill-Wheeler-HF (GHWHF) equations 4 [ F( α, β ) ε is( α, β )] f i ( β ) d β = 0, i = 1,..., n, (2) where the ε i are the HF eigenvalues, and the Fock, F( α, β ), and overlap, S( α, β ), kernels are already defined 4. The GHWHF equations are integrated numerically through a technique called integral discretization (ID) 19. This technique is implemented through a relabelling of the generator coordinate space, i.e., Ω = ln α / A, A > 1, (3) where A is a scaling parameter determined numerically. The new generator coordinate space, Ω, is discretized for each s, p, d and f symmetry in an equally spaced mesh { Ω i } so that: Ωk = Ωmin + ( k 1 ) Ω, k = 1,..., N (4)

2 1384 INDIAN J CHEM, SEC A, SEPTEMBER 2007 N is the number of discretization points of each atomic symmetry, Ω min is the initial point, and Ω an increment. Thus, the original GCHF method 4 corresponds to use only one arithmetic sequence [Eq. (4)] of equally spaced points {Ω i } to generate basis sets. In 1999, Jorge and de Castro 12 found a simple modification in the GCHF method that produces improvements in the HF atomic energy values without adding more functions; this methodology has been named IGCHF method. In this new approach 12, the new generator coordinate space,ω, is discretized for each s, p, d and f symmetry in three independent arithmetic sequences, Ω ( 1), 1,..., min + k Ω k = j Ω = ' ( k 1) ', k j 1,..., M k Ω + Ω = + min '' Ω min + ( k 1) Ω '', k = M + 1,..., N (5) For a given value of N, the number of parameters to be optimized for each symmetry when one uses Eq. (5) is three times that of the original GCHF method 4. As reported already 12, the basis sets generated through Eqs (3) and (5) were called triple-optimized Gaussian basis sets (TOGBSs). We call attention to the fact that when Eq. (5) is used, there is no longer an equally spaced mesh {Ω i }, because now there are three independent arithmetic sequences to generate the basis function exponents for each atomic symmetry. Results and Discussion Using the IGCHF method 12, we have performed HF self-consistent field calculations for low-lying excited states of the positive and negative ions as presented in Tables 1 and 2, respectively. The scaling parameter A [see Eq. (3)] is the same (6.0) for all calculations. The TOGBSs generated in this work are available from the author upon request through the e- mail address (jorge@cce.ufes.br). Tables 1 and 2 summarize the total HF energies for low-lying excited states of some cations and anions calculated with our TOGBSs, with the UBSs of GTFs and STFs of Jorge and Barros 6, and with the fullyoptimized basis sets of STFs generated by Clementi and Roetti 18. It is important to say that the two GBSs presented in these tables have the same size and they are larger than the other ones. In addition, we recall that when comparing total HF energies calculated with different basis sets, the lowest energy is always obtained with the most accurate basis set, e.g. for Sc + ( 3 F) the best energy ( hartree) is obtained with the UBS of STFs 6 (see the first-row of Table 1). Table 1 shows that the total energies for low-lying excited states of the positive ions obtained with our TOGBSs are always better than those evaluated with the universal Gaussian basis set (UGBS) 6 and that the difference between corresponding energies computed with these two approaches arrives to ~13.8 mhartree for Ni + ( 4 F). On the other hand, the energies computed with the UBS of STFs (ref. 6) improve when compared with the UGBS ones and, in this case, our results are more stable only for the heaviest cations [Tc + ( 5 D)-Cd + ( 2 D)]. When compared with the widely used basis sets of STFs reported by Clementi and Roetti 18, the TOGBS HF energies are worse for the cations from Sc + ( 3 F) to Br + ( 1 S), whereas for Zr + ( 4 F)- Cd + ( 2 D) the opposite occurs. Besides, our TOGBSs are more accurate than the UGBS presented 5. For the case of the low-lying excited states of the negative ions presented in Table 2, one can again verify that our total HF energies are always lower than the corresponding ones evaluated with the reported UGBSs 5,6, and that except for the anions Si - ( 2 D), P - ( 1 D), P - ( 1 S), Cr - ( 6 D), As - ( 1 D), and As - ( 1 S), they are also better than the results of Clementi and Roetti 18 (see the last column of Table 2). The largest difference between corresponding energies obtained with the TOGBSs and the UGBS 6 is ~12.2 mhartree for Nr - ( 2 S). The results presented in this work, when compared with the energies calculated with the UBS of STFs reported by Jorge and Barros 6, with rare exceptions [Al - ( 1 D), Al - ( 1 S), Ti - ( 6 D), Cr - ( 6 D), Ni - ( 2 S), Mo - ( 6 D), Tc - ( 5 F)] are worse. The good performance of the TOGBSs for lowlying excited states of the ions studied here is attributed mainly to the IGCHF method 12, since this method has three independent arithmetic sequences to describe each s, p, d, atomic symmetry [see Eq. (5)], whereas the GCHF method has only one. Thus, when the IGCHF method is used, one has for each atomic symmetry independent descriptions of the inner, intermediate, and outer electrons. When a UBS (a unique set of exponents that describe a sample of atomic systems) is developed instead of adapted basis

3 DE OLIVEIRA & JORGE: GAUSSIAN BASIS SETS FOR LOW-LYING EXCITED STATES OF IONS 1385 Table 1 Total HF energies (in hartree) for low-lying excited states of some positive ions Z Ion Configuration GBS size E (TOGBS) E (UGBS) a E (UBS) b E (STFs) c 21 Sc + ( 3 F) [Ar]4s 0 3d 2 20s13p10d Sc + ( 1 S) [Ar]4s 2 20s13p Ti + ( 4 F) [Ar]4s 0 3d 3 20s13p10d Ti + ( 2 D) [Ar]4s 2 3d 1 20s13p10d V + ( 3 F) [Ar]4s 2 3d 2 20s13p10d Cr + ( 6 D) [Ar]4s 1 3d 4 20s13p10d Cr + ( 4 F) [Ar]4s 2 3d 3 20s13p10d Mn + ( 5 D) [Ar]4s 0 3d 6 21s13p10d Mn + ( 5 D) [Ar]4s 2 3d 4 21s13p10d Fe + ( 6 S) [Ar]4s 2 3d 5 20s13p10d Co + ( 5 D) [Ar]4s 2 3d 6 20s13p10d Ni + ( 4 F) [Ar]4s 2 3d 7 20s13p10d Cu + ( 3 F) [Ar]4s 2 3d 8 20s13p10d As + ( 1 D) [Ar]4s 2 3d 10 4p 2 22s15p11d As + ( 1 S) [Ar]4s 2 3d 10 4p 2 22s15p11d Se + ( 2 D) [Ar]4s 2 3d 10 4p 3 21s16p11d Se + ( 2 P) [Ar]4s 2 3d 10 4p 3 21s16p11d Br + ( 1 D) [Ar]4s 2 3d 10 4p 4 22s16p11d Br + ( 1 S) [Ar]4s 2 3d 10 4p 4 22s16p11d Zr + ( 4 F) [Kr]5s 0 4d 3 23s16p12d Zr + ( 2 D) [Kr]5s 2 4d 1 23s16p12d Nb + ( 5 F) [Kr]5s 1 4d 3 22s16p13d Nb + ( 3 F) [Kr]5s 2 4d 2 22s16p13d Mo + ( 6 D) [Kr]5s 1 4d 4 23s16p13d Mo + ( 4 F) [Kr]5s 2 4d 3 23s16p13d Tc + ( 5 D) [Kr]5s 0 4d 6 26s16p13d Tc + ( 5 D) [Kr]5s 2 4d 4 26s16p13d Ru + ( 6 D) [Kr]5s 1 4d 6 25s16p13d Ru + ( 6 S) [Kr]5s 2 4d 5 25s16p13d Rh + ( 5 F) [Kr]5s 1 4d 7 25s17p14d Rh + ( 5 D) [Kr]5s 2 4d 6 25s17p14d Pd + ( 4 F) [Kr]5s 1 4d 8 24s17p13d Pd + ( 4 F) [Kr]5s 2 4d 7 24s17p13d Ag + ( 3 D) [Kr]5s 1 4d 9 24s17p13d Ag + ( 3 F) [Kr]5s 2 4d 8 24s17p13d Cd + ( 2 D) [Kr]5s 2 4d 9 25s16p14d a Total HF energies obtained from the universal Gaussian basis set (UGBS) generated by Jorge and Barros 6. b Total HF energies obtained from the universal basis set (UBS) of STFs generated by Jorge and Barros 6. c Total HF energies obtained by Clementi and Roetti 18 using fully-optimized basis sets of STFs.

4 1386 INDIAN J CHEM, SEC A, SEPTEMBER 2007 Table 2 Total HF energies (in hartree) for low-lying excited states of some negative ions Z Ion Configuration GBS size E (TOGBS) E (UGBS) a E (UBS) b E (STFs) c 5 B - ( 1 D) [He]2s 2 2p 2 20s11p B - ( 1 S) [He]2s 2 2p 2 20s11p C - ( 2 D) [He]2s 2 2p 3 20s11p C - ( 2 P) [He]2s 2 2p 3 20s11p N - ( 1 D) [He]2s 2 2p 4 20s11p N - ( 1 S) [He]2s 2 2p 4 20s11p Al - ( 1 D) [Ne]3s 2 3p 2 26s18p Al - ( 1 S) [Ne]3s 2 3p 2 26s18p Si - ( 2 D) [Ne]3s 2 3p 3 19s14p Si - ( 2 P) [Ne]3s 2 3p 3 19s14p P - ( 1 D) [Ne]3s 2 3p 4 19s14p P - ( 1 S) [Ne]3s 2 3p 4 19s14p Sc - ( 5 F) [Ar]4s 1 3d 3 23s14p10d Ti - ( 6 D) [Ar]4s 1 3d 4 23s24p10d V - ( 7 S) [Ar]4s 1 3d 5 24s14p10d Cr - ( 6 D) [Ar]4s 1 3d 6 24s14p10d Ni - ( 2 S) [Ar]4s 1 3d 10 24s15p10d Ge - ( 2 D) [Ar]4s 2 3d 10 4p 3 24s17p10d As - ( 1 D) [Ar]4s 2 3d 10 4p 4 24s17p10d As - ( 1 S) [Ar]4s 2 3d 10 4p 4 24s17p10d Y - ( 5 F) [Kr]5s 1 4d 3 25s16p13d Nb - ( 7 S) [Kr]5s 1 4d 5 25s16p14d Mo - ( 6 D) [Kr]5s 1 4d 6 26s15p14d Tc - ( 5 F) [Kr]5s 1 4d 7 26s16p14d a Total HF energies obtained from the universal Gaussian basis set (UGBS) generated by Jorge and Barros 6. b Total HF energies obtained from the universal basis set (UBS) of STFs generated by Jorge and Barros 6. c Total HF energies obtained by Clementi and Roetti 18 using fully-optimized basis sets of STFs. sets (one different set of GTF exponents for each atomic system, e.g. the TOGBSs), one faces the penalty to lose a little accuracy in calculating total HF energy. This loss of accuracy is due to the fact that when a UBS is being developed with the GCHF method 4, one has to find a unique set of discretization points (basis function exponents) that could be able to describe well the energy of a large number of atomic systems, in summary, a UBS is not the best basis set for a specific atom. Thus, with exception of the UBS of STFs constructed by Jorge and Barros 6, which will be discussed bellow, one can affirm that this is another reason for TOGBSs to be more accurate than the UBSs generated already 5,6. It is known that STFs describe better the regions close and far from the nucleus than GTFs; thus, it is necessary about 3 times more GTFs than STFs to obtain comparable accuracies. As in this work, the number of GTFs is only 1.5 larger than the STFs one, this explains why the STFs UBS of Jorge and Barros 6 in general give better total HF energies than the TOGBSs. Conclusions This work presents TOGBSs for low-lying excited states of some mono-positive and -negative ions generated with the IGCHF method 12. The success of

5 DE OLIVEIRA & JORGE: GAUSSIAN BASIS SETS FOR LOW-LYING EXCITED STATES OF IONS 1387 TOGBS comes from its increased variational flexibility in comparison to the basis set constructed with the GCHF method 4. The use of three independent arithmetic sequences permits a better distribution of small, intermediate, and large GTF exponents. Except for the total HF energies calculated with the UBS of STFs of Jorge and Barros 6, our results in general are better than those reported already 5,6,18. In addition, it is important to say that the accuracy of the wave functions presented in this work represents an achievement that has not been obtained so far with GBSs. Thus, the IGCHF method is a powerful tool to generate basis sets to be used in atomic and molecular calculations. Acknowledgement We acknowledge the financial support of CNPq and CAPES (Research Brazilian Agencies). References 1 Froese Fischer C, The Hartree-Fock Method for Atoms (Wiley, New York), Roothaan J C C, Rev Mod Phys, 23 (1951) Roothaan J C C, Rev Mod Phys, 32 (1960) Mohallem J R, Dreizler R M & Trsic M, Int J Quantum Chem Symp, 20 (1986) Da Silva A B F & Trsic M, Can J Chem, 74 (1996) Jorge F E & Barros C L, Comput Chem, 26 (2002) De Castro E V R & Jorge F E, J Chem Phys, 108 (1998) Treu O, Pinheiro J C & Kondo R T, J Mol Structure (Theochem), 671 (2004) Treu O, Pinheiro J C & Kondo R T, J Mol Structure (Theochem), 716 (2005) Morgon N H, Custodio R & Mohallem J R, J Mol Structure (Theochem), 394 (1997) Custodio R, Giordan M, Morgon N H & Goddard J D, Int J Quantum Chem, 42 (1992) Jorge F E & de Castro E V R, Chem Phys Lett, 302 (1999) Centoducatte R, Jorge F E & de Castro E V R, Int J Quantum Chem, 82 (2001) P R Librelon & F E Jorge, Int J Quantum Chem, 95 (2003) Pinheiro J C, Jorge F E & de Castro E V R, Int J Quantum Chem, 78 (2000) Pinheiro J C, Jorge F E & de Castro E V R, J Mol Structure (Theochem), 491 (1999) Pires J M & Jorge F E, Indian J Chem, 44A (2005) Clementi E & Roetti C, At Data Nucl Data Tables, 14 (1974) Mohallem J R, Z Phys D, 3 (1986) 339.

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

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