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Revised model core potentials for first-row transition-metal atoms from Sc to Zn
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Chemical Physics Letters 452 (2008) 210–214 www.elsevier.com/locate/cplett
Revised model core potentials for first-row transition-metal atoms from Sc to Zn You Osanai a, Ma San Mon b, Takeshi Noro c, Hirotoshi Mori b,1, Hisaki Nakashima b, Mariusz Klobukowski d, Eisaku Miyoshi b,e,* a Faculty of Pharmaceutical Sciences, Aomori University, Aomori 030-0943, Japan Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga Park, Fukuoka 816-8580, Japan c Division of Chemistry, Graduate School of Science, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan d Department of Chemistry, University of Alberta Edmonton, Alberta, Canada T6G 2G2 e JST-CREST, Kawaguchi 332-0012, Japan b
Received 5 November 2007; in final form 7 December 2007 Available online 10 January 2008
Abstract We have developed new relativistic model core potentials (MCPs) for the first-row transition-metal atoms from Sc to Zn, in which 3s and 3p electrons are treated explicitly together with the 3d and 4s electrons. By adding suitable correlating functions, we demonstrated that the present MCP basis sets show reasonable performance in describing the electronic structures of atoms and molecules, bringing about accurate excitation energies for atoms and good molecular spectroscopic constants. Ó 2007 Elsevier B.V. All rights reserved.
1. Introduction During the past two decades, we have developed nonrelativistic and relativistic model core potentials (MCPs) together with valence functions for atoms up to Rn [1–5], while Seijo et al. developed ab initio model potentials (AIMPs) [6–9]. The MCP and AIMP methods are both based on the theory originally proposed by Huzinaga et al. [10–21]; the MCP and AIMP methods are unique among various effective core potential (ECP) methods in that they are naturally capable of producing valence orbitals with a nodal structure. The nodeless pseudo-orbitals in ECP approaches may yield too large exchange integrals, resulting in exaggerated correlation energies and too large singlet–triplet splittings [22,23]. Thus, due to the accurate * Corresponding author. Address: Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga Park, Fukuoka 816-8580, Japan. Fax: +81 92 583 7677. E-mail address: [email protected] (E. Miyoshi). 1 Present address: Department of Chemistry, Iowa State University, Ames, IA 50011, USA.
0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.12.019
nodal structure of the valence orbitals, the MCP and AIMP methods are well suited to accurately describe the correlation effects of valence electrons. Recently, we developed effective valence function sets (MCPdzp, MCPtzp, and MCPqzp) by a combination of split MCP valence orbitals and correlating contracted GTFs [24,25] to describe the correlation effects for main group elements. For transition-metal elements, we have developed two types of MCPs, dsMCP and pdsMCP. In the former type, only (n 1)d and ns electrons are explicitly treated, while in the latter, (n 1)p electrons are also treated as well as (n 1)d and ns electrons. Both MCPs have described well the ground state properties of transition-metal complexes [26]. However, the MCPs have a weak point for describing the excited states: for example, the relative energies of the 3dn+14s1 state with respect to the 3dn4s2 state at the Hartree–Fock level using pdsMCPs differ significantly (0.7 eV) from those given by the reference numerical Hartree–Fock calculations (see Table 1). In this Letter, we report the newly developed MCPs that reproduce the Hartree–Fock relative energies within 0.1 eV.
Y. Osanai et al. / Chemical Physics Letters 452 (2008) 210–214 Table 1 Energy of the lowest LS state with 3dn+14s1 relative to that with 3dn4s2 for the first transition-metal atoms at the relativistic Hartree–Fock level Atoms
LS states 3dn+14s1/3dn4s2
Sc Ti V Cr Mn Fe Co Ni Cu
4
2
F/ D F/3F 6 G/4F 7 5 S/ D 6 D/6S 5 F/5D 4 F/4F 3 D/3F 2 2 S/ D 5
Energy of 3dn+14s1 relative to 3dn4s2 (eV) pdsMCPa
spdsMCPa
QRHFb
0.915 0.329 0.192 1.564 2.869 1.395 1.227 1.076 0.566
1.128 0.673 0.273 1.111 3.440 2.001 1.832 1.711 0.118
1.122 0.681 0.290 1.066 3.532 2.053 1.831 1.631 0.051
a
Valence basis sets [6s4p5d] were used in the uncontracted form. QRHF: Numerical quasi-relativistic Hartree–Fock calculations at the level of the Cowan–Griffin approximation. b
In Section 2, we briefly describe the determination of newly developed MCP parameters for the first-row transition-metal atoms. The reference state for the determination of the MCP parameters was taken to be 3dn4s2. Atomic applications, involving the calculations of the excitation energies of atoms at correlated levels are performed and compared with all-electron (AE) calculations in Section 3, while the molecular applications are discussed in comparison with AE calculations in Section 4. All calculations of the atomic and molecular applications were performed using the program packages, ATOMCI [27,28] and MOLCAS6 [29], respectively. 2. Model core potentials of first-row transition-metal atoms In the model core potential method, the atomic Hamil^ MCP ð1; 2; . . . ; N v Þ of the Nv valence electrons (in tonian H atomic units) is chosen as follows: ^ MCP ð1; 2; . . . ; N v Þ ¼ H
Nv X i¼1
^ hMCP ðri Þ þ
Nv X 1 rij i>j
ð2:1Þ
with the one-electron Hamiltonian term defined as: X 1 ^ hMCP ðri Þ ¼ Di þ V MCP ðri Þ þ Bc jwc ihwc j ð2:2Þ 2 c " # 3 3 X X Z Nc 2 2 1þ V MCP ðrÞ ¼ AI eaI r þ AJ reaJ r ; r I¼1 J ¼1 ð2:3Þ where Z is the atomic number of the atom and Nc is the number of core electrons replaced by MCPs. {Aj, aj; Aj, aj} and {Bc} (c = 1s, 2s, 2p) are the MCP parameters and {wc} denotes the core orbital functions, while {Bc} is defined as: Bc ¼ F ec
ð2:4Þ
where ec is the orbital energy of the core orbital and F is usually taken to be 2 [30]. The MCP, VMCP in Eq. (2.3), given as a simple spherically-symmetric local potential, approximates the local core Coulomb potential and the
211
non-local core exchange potential as well as the nuclear attraction potential term. The projection operator of the third term in Eq. (2.2) prevents the valence orbitals from collapsing into the core region, and may also be regarded as an energy shift operator. By the use of this operator, the energy levels of core electrons are shifted far above zero energy, thus allowing for finding, in each symmetry, the lowest valence orbitals with appropriate nodal structure. Prior to the determination of the MCP parameters, we performed atomic quasi-relativistic (QR) restricted Hartree–Fock (HF) calculations for the 3dn4s2 configuration at the level of the Cowan and Griffin approximation [31] to prepare reference data (radial functions and orbital energies). In the QRHF calculations, the mass–velocity and Darwin terms of the Pauli Hamiltonian are added to the HF differential equations, which are then solved in a selfconsistent fashion. This allows for incorporation of the major direct and indirect relativistic effects into the radial orbital functions. Although only the valence orbitals are treated explicitly in the MCP method, the core orbitals are required in order to construct the projection operator of Eq. (2.2). The core orbitals are expanded in terms of GTFs, whose exponents and expansion coefficients are determined by a leastsquares fit to the reference QRHF core orbitals. The valence orbitals are expanded in terms of a smaller number of GTFs. The exponents of the valence GTFs are also determined by a least-squares fit to the reference QRHF valence orbitals: s-type functions were expanded with six GTFs, p-type four, and d-type five. The valence orbital energies and orbital shapes are chosen as the criteria for the determination of the MCP parameters and the coefficients of the valence orbitals. Following this approach, we optimized the MCP parameters and valence orbitals for the first-row transition-metal atoms in which the (n 1)s, (n 1)p and (n 1)d electrons were treated explicitly together with the ns electrons. These new MCPs are called spdsMCPs. The relative energies of the 3dn+14s1 state with respect to the 3dn4s2 state at the Hartree–Fock level given by the use of the present spdsMCPs, are listed and compared with those given by the use of the previous pdsMCPs and the numerical results of the reference QRHF. In both MCP calculations, the basis sets were uncontracted. The present relative energies agree with those of the reference QRHF for all atoms, the differences being within 0.1 eV. Thus, our new spdsMCPs provide a more reliable description for the excited states in the atoms and molecules containing firstrow transition-metal atoms than the previous pdsMCPs, which sometimes gave relative energies that differed by 0.7 eV from those obtained by the reference QRHF. 3. Atomic applications The quality of the present spdsMCP sets was examined through the correlated calculations of the relative energies of the 3dn+14s1 state with respect to the 3dn4s2 state for Sc,
212
Y. Osanai et al. / Chemical Physics Letters 452 (2008) 210–214
Mn, and Cu atoms. We first performed the restricted Hartree–Fock (RHF) calculations followed by a configuration interaction single and double (CISD) and its Davidson correction (+Q), in order to include the electron correlation effects among the 3s, 3p, 3d, and 4s electrons. The correlating functions [2p2f1g] from the sets given by Noro et al. [32] were utilized with our valence orbitals in the spdsMCP sets (6s4p5d), which were decontracted to (33111/31/311). One more diffuse d-type GTF was augmented (0.0376626 for
Sc, 0.0761019 for Mn, and 0.0969874 for Cu). Thus, the resulting basis set was [5s4p4d2f1g]. A smaller set [4s3p4d1f] with correlating functions [1p1f] [32], contraction (4411/31/311), and diffuse d-function, was also examined. For comparison, we performed all-electron (AE) calculations with very large basis sets that were formed from Huzinaga’s well-tempered sets [33], [26s17p13d] for Sc and Mn, and [26s17p14d] for Cu, augmented by p-type and d-type diffuse functions with the same exponents of s-type diffuse
Table 2 Energy of the lowest LS state with 3dn+14s1 relative to that with 3dn4s2 for Sc, Mn, and Cu atoms at the HF and correlated calculations Atoms
Method/basis set
Calculation level
Term energy (a.u.) n
Sc
spdsMCP [5s4p4d2f1g]
[4s3p4d1f]
AE [7s5p4d2f1g]
[6s4p4d1f]
2
n+1
3d 4s
3d
2
4
4s
1
HF CISD CISD+Q
D 46.06216 46.31554 46.33576
F 46.02342 46.26918 46.28423
1.054 1.262 (0.206) 1.402 (0.348)
HF CISD CISD+Q
46.06179 46.27749 46.29453
46.02330 46.22637 46.23787
1.048 1.391 (0.343) 1.542 (0.494)
HF CISD CISD+Q
763.29120 763.62548 763.65211
763.24988 763.58020 763.60139
1.125 1.232 (0.107) 1.380 (0.255)
CISD CISD+Q
763.58817 763.61138
763.53893 763.55639
1.340 (0.215) 1.496 (0.371) 1.427
HF CISD CISD+Q
6 S 102.72621 103.11814 103.14458
6 D 102.60625 103.03562 103.06437
3.264 2.246 (1.018) 2.183 (1.081)
HF CISD CISD+Q
102.72448 103.03336 103.05400
102.60581 102.94637 102.96818
3.229 2.367 (0.862) 2.335 (0.894)
HF CISD CISD+Q
1157.40179 1157.88456 1157.91743
1157.27178 1157.79486 1157.82976
3.538 2.441 (1.097) 2.385 (1.153)
CISD CISD+Q
1157.80590 1157.83248
1157.70894 1157.73650
2.638 (0.900) 2.612 (0.926) 2.145
2
2
Experimentalb Mn
spdsMCP [5s4p4d2f1g]
[4s3p4d1f]
AE [7s5p4d2f1g]
[6s4p4d1f] Experimentalb Cu
spdsMCP [5s4p4d2f1g]
[4s3p4d1f]
AE [7s5p4d2f1g]
[6s4p4d1f]
HF CISD CISD+Q
D 195.87751 196.48705 196.52643
S 195.87970 196.53486 196.57595
0.060 1.301 (1.241) 1.348 (1.288)
HF CISD CISD+Q
195.87492 196.35261 196.38220
195.87876 196.39865 196.42899
0.104 1.253 (1.149) 1.273 (1.169)
HF CISD CISD+Q
1653.19053 1653.92042 1653.96670
1653.18836 1653.96310 1654.01005
0.059 1.161 (1.220) 1.180 (1.239)
CISD CISD+Q
1653.75690 1653.78925
1653.79955 1653.83211
1.161 (1.220) 1.166 (1.225) 1.490
Experimentalb a b
Relative energy (eV)a
Values in parentheses are correlation effects on the relative energy. Ref. [34].
Y. Osanai et al. / Chemical Physics Letters 452 (2008) 210–214
functions. The f-type and g-type GTFs having the same exponents as the d-type GTFs were also considered. Thus, the uncontracted GTFs in the AE calculations were [26s20p15d15f15g] for Sc and Mn, and [26s21p16d16f16g] for Cu. Using these basis functions we performed CISD calculations for each atom, followed by CISD calculations with a [7s5p4d2f1g] basis set composed of occupied [4s2p1d] Hartree–Fock orbitals and correlating [3s3p3d2f1g] atomic natural orbitals (ANO) for the sake of comparison. The AE [7s5p4d2f1g] basis set has the same number of correlating orbitals as the MCP/[5s4p4d2f1g] basis set. A small AE [6s4p4d1f] basis set with the occupied orbitals and correlating [2s2p3d1f] orbitals was also used for the comparison with the MCP/[4s3p4d1f] basis set. The relativistic effects were taken into account in the AE calculations through the third-order Douglas–Kroll (DK3) approximation [35]. The resulting relative energies are listed in Table 2 together with the observed values. Correlation effects on the relative energies are also given in parentheses in the last column. The spdsMCP calculations reasonably well reproduce the relative energies given by the AE calculations with the same size of the ANO sets. The HF relative energies given in AE almost exactly coincide with those achieved by the QRHF. The deviations of HF relative energies by spdsMCP from those by AE are somewhat larger than the deviations of spdsMCP HF from QRHF shown in Table 1. This deterioration of HF energies in spdsMCP is caused by the contraction of the basis sets. It is notable that the correlation effects on the relative energies in MCP calculations are almost the same as those in AE calculations: the differences between the correlation effects by MCP [5s4p4d2f1g] and AE [7s5p4d2f1g] calculations are within 0.1 eV and those by MCP [4s3p4d1f] and AE [6s4p4d1f] are within 0.13 eV.
mial fitting are listed in Table 3 together with the corresponding experimental values [37]. For CuH, CASSCF calculations with the [4s3p4d1f] basis set gave somewhat longer equilibrium distance, Re, and much smaller binding energy, De. These results were significantly improved by considering electron correlations within the 12 (Cu 3d and 4s and H 1s) electrons in all of MRCI, MP2, and CCSD(T). It is notable that MCP CCSD(T) calculations with [4s3p4d1f] give nearly equal spectroscopic constants as all-electron CCSD(T) calculations with [8s6p4d1f], where 12 electrons were also correlated [38]. Both expansion of the basis set and inclusion of the correlation of Cu 3s and 3p electrons bring about little change to the calculated spectroscopic constants. The most striking change is 0.19 eV in De, which is brought about by basis set expansion in CCSD(T) calculation. For Cu2, on the other hand, both expansion of the basis set and inclusion of the correlation of Cu 3s and 3p electrons bring about more significant change to the calculated spectroscopic constants. The former improves Re by 0.05– ˚ , xe by 15–17 cm1, and De by 0.3–0.4 eV. The latter 0.06 A ˚ , xe by 13–14 cm1, and additionally improves Re by 0.02 A De by 0.1–0.2 eV. As in CuH, MCP CCSD(T) calculations
Table 3 Spectroscopic constants of the ground states X1R+ of CuH and Cu2 Molecules Methods
CuH
spdsMCP
4. Molecular applications The present spdsMCP have also been applied in the calculations of molecular spectroscopic constants in the ground states of CuH and Cu2 molecules. The basis sets used for Cu were [4s3p4d1f] and [5s4p4d2f1g], and the basis set of hydrogen was a triple-zeta type [3s2p1d] from Ref. [36]. The computational methods employed here were MP2 and CCSD(T). The energies of the dissociation limit for these single reference calculations were obtained by supermolecular calculations for a triplet open shell state at an internuclear distance of 100 a.u. We also performed three-states averaged CASSCF calculations with 12 electrons in 7 orbitals for CuH, which were followed by multi-reference singly and doubly excited configuration interaction (MRCI) calculations. In the correlated calculations, the Cu 3d and 4s and H 1s (12 and 22 electrons for CuH and Cu2, respectively) electrons were correlated for both basis sets, while the Cu 3s and 3p electrons were correlated only for the largest set. Results of the spectroscopic constants for the ground state of CuH and Cu2 obtained by a third-degree polyno-
213
AEc Experimentald Cu2
spdsMCP
Basis set
Calculation Spectroscopic level constants Re ˚) (A
xe De (cm1) (eV)c
CASSCF MRCIa MP2a CCSD(T)a
1.520 1.466 1.451 1.482
1798.7 1746.7 1972.3 1836.4
1.230 2.428 2.560 2.556
[5s4p4d2f1g] CASSCF MRCIa MP2a CCSD(T)a MRCIb MP2b CCSD(T)b CCSD(T)a
1.520 1.465 1.443 1.473 1.453 1.418 1.457 1.490 1.463
1810.7 1768.6 2014.7 1877.6 1794.8 1922.0 1766.7 1839.8 1941.3
1.232 2.424 2.655 2.741 2.465 2.736 2.769 2.55 2.85
[4s3p4d1f]
[4s3p4d1f]
MP2a 2.279 CCSD(T)a 2.307
[5s4p4d2f1g] MP2a CCSD(T)a MP2b CCSD(T)b AE [5s3p3d1f]e Experimentald a
2.226 2.249 2.205 2.229
CCSD(T)a 2.278 2.220
265.3 1.513 243.0 1.536 279.6 259.7 295.3 272.6
1.852 1.830 2.012 1.930
246.8 1.576 264.6 2.05
Cu 3d and 4s and H 1s electrons were correlated. Cu 3s and 3p electrons were correlated, as well as the Cu 3d and 4s and H 1s electrons. c Ref. [38]. d Ref. [37]. e Using a contracted basis set from Ref. [36]. b
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Y. Osanai et al. / Chemical Physics Letters 452 (2008) 210–214
Table 4 Spectroscopic constants of an excited state A1R+ of CuH Methods
spdsMCP
Basis set
[4s3p4d1f] [5s4p4d2f1g]
Experimentalc
Calculation level
Spectroscopic constants Re ˚) (A
xe (cm1)
Te (eV)
CASSCF MRCIa
1.652 1.623
1585.8 1548.4
1.521 3.006
CASSCF MRCIa MRCIb
1.645 1.622 1.607 1.572
1600.0 1569.8 1654.5 1698.4
1.555 3.001 2.858 2.905
a
Cu 3d and 4s and H 1s electrons were correlated. Cu 3s and 3p electrons were correlated, as well as Cu 3d and 4s and H 1s electrons. c Ref. [37]. b
with the [4s3p4d1f] set yield nearly equal spectroscopic constants as all-electron CCSD(T) calculations with the [5s3p3d1f] set from Ref. [36], indicating that the present MCP set provides sufficiently high performance for description of electronic structures in highly correlated calculations. For CuH, the excited state A1R+ was also studied at the three-states averaged CASSCF and MRCI calculations. Calculated spectroscopic constants are listed in Table 4. The CASSCF calculation gave the excitation energy, Te, of only a half of the observed value, but inclusion of electron correlation through MRCI improved the Te enormously. The spectroscopic constants given by MRCI with [5s4p4d2f1g] including correlation of Cu 3s and 3p electrons agree well with observed values, again demonstrating the superior capabilities of the present spdsMCP. 5. Conclusions We proposed new model core potentials, spdsMCPs, for the first-row transition-metal atoms from Sc to Zn, in which the (n 1)s and (n 1)p electrons are treated explicitly in addition to the (n 1)d and ns electrons. The present MCP basis sets were tested in the calculation of the energy differences between the 3dn4s2 and 3d(n+1)4s1 states for Sc, Mn, and Cu atoms and spectroscopic constants of the ground and excited states of CuH and the ground state of Cu2 molecules. It has been shown that the newly developed MCP basis sets describe well these physical properties. The MCP sets are available in the homepage of Ref. [36]. Acknowledgements The present study has been supported partly by a Grantin-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS). A part of the calculations
reported here were performed using computing resources in the Research Center for Computational Science, Okazaki, Japan. References [1] Y. Sakai, E. Miyoshi, M. Klobukowski, S. Huzinaga, J. Comput. Chem. 8 (1987) 226. [2] Y. Sakai, E. Miyoshi, M. Klobukowski, S. Huzinaga, J. Comput. Chem. 8 (1987) 264. [3] Y. Sakai, E. Miyoshi, M. Klobukowski, S. Huzinaga, J. Chem. Phys. 106 (1997) 8084. [4] E. Miyoshi, Y. Sakai, K. Tanaka, M. Masamura, J. Mol. Struct.: Theochem 451 (1998) 73. [5] Y. Sakai, E. Miyoshi, H. Tatewaki, J. Mol. Struct.: Theochem 451 (1998) 143. [6] S. Huzinaga, L. Seijo, Z. Barandiara´n, M. Klobukowski, J. Chem. Phys. 86 (1987) 2132. [7] L. Seijo, Z. Barandiara´n, S. Huzinaga, J. Chem. Phys. 91 (1989) 7011. [8] Z. Barandiara´n, L. Seijo, S. Huzinaga, J. Chem. Phys. 93 (1990) 5843. [9] Z. Barandiara´n, L. Seijo, Can. J. Chem. 70 (1992) 409. [10] V. Bonifacic, S. Huzinaga, J. Chem. Phys. 60 (1974) 2779. [11] Y. Sakai, J. Chem. Phys. 75 (1981) 1303. [12] Y. Sakai, S. Huzinaga, J. Chem. Phys. 76 (1982) 2537. [13] Y. Sakai, S. Huzinaga, J. Chem. Phys. 76 (1982) 2552. [14] J. Andzelm, S. Huzinaga, M. Klobukowski, E. Radzio-Andzelm, Mol. Phys. 52 (1981) 1495. [15] M. Klobukowski, J. Comput. Chem. 4 (1983) 350. [16] L.G.M. Pettersson, U. Wahlgren, O. Gropen, J. Chem. Phys. 86 (1987) 2176. [17] S. Katsuki, S. Huzinaga, Chem. Phys. Lett. 152 (1988) 203. [18] S. Katsuki, Can. J. Chem. 70 (1992) 285. [19] S. Katsuki, J. Chem. Phys. 98 (1993) 496. [20] H. Tatewaki, S. Katsuki, Y. Sakai, E. Miyoshi, J. Comput. Chem. 17 (1996) 1056. [21] S. Huzinaga, Can. J. Chem. 73 (1995) 619 and references cited therein. [22] B. Pittel, W.H.E. Schwartz, Chem. Phys. Lett. 46 (1977) 121. [23] C. Teichteil, J.P. Malrieu, J.C. Barthelat, Mol. Phys. 33 (1977) 181. [24] Y. Osanai, T. Noro, E. Miyoshi, J. Chem. Phys. 117 (2002) 9623. [25] E. Miyoshi, H. Mori, R. Hirayama, Y. Osanai, T. Noro, H. Honda, M. Klobukowski, J. Chem. Phys. 122 (2005) 074104. [26] E. Miyoshi, Y. Sakai, J. Comput. Chem. 9 (1988) 719. [27] F. Sasaki, Int. J. Quantum Chem. 8 (1974) 605. [28] F. Sasaki, M. Sekiya, T. Noro, K. Ohtsuki, Y. Osanai, MOTECC-94, in: E. Clementi (Ed.), STEF, Calgari, 1993. [29] G. Karlstro¨m et al., Computat. Mater. Sci. 28 (2003) 222. [30] G. Ho¨jer, J. Chung, Int. J. Quantum Chem. 14 (1978) 623. [31] R.D. Cowan, D.C. Griffin, J. Opt. Soc. Am. 66 (1976) 1010. [32] T. Noro, M. Sekiya, T. Koga, H. Matsuyama, Theor. Chem. Acc. 104 (2000) 146. [33] S. Huzinaga, B. Miguel, Chem. Phys. Lett. 212 (1990) 289. [34] J.E. Sansonetti, W.C. Martin, S.L. Young, Handbook of Basic Atomic Spectroscopic Data. Available from:. [35] T. Nakajima, K. Hirao, J. Chem. Phys. 133 (2000) 7786. [36] These basis sets are available at the following home pages: . [37] K.P. Huber, G. Herzberg, Constants of Diatomic Molecules, Van Nostrand Reinhold, New York, 1979. [38] J. Hrusak, S. Ten-no, S. Iwata, J. Chem. Phys. 106 (1997) 7185.
Chemical Physics Letters 452 (2008) 210–214 www.elsevier.com/locate/cplett
Revised model core potentials for first-row transition-metal atoms from Sc to Zn You Osanai a, Ma San Mon b, Takeshi Noro c, Hirotoshi Mori b,1, Hisaki Nakashima b, Mariusz Klobukowski d, Eisaku Miyoshi b,e,* a Faculty of Pharmaceutical Sciences, Aomori University, Aomori 030-0943, Japan Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga Park, Fukuoka 816-8580, Japan c Division of Chemistry, Graduate School of Science, Hokkaido University, Sapporo, Hokkaido 060-0810, Japan d Department of Chemistry, University of Alberta Edmonton, Alberta, Canada T6G 2G2 e JST-CREST, Kawaguchi 332-0012, Japan b
Received 5 November 2007; in final form 7 December 2007 Available online 10 January 2008
Abstract We have developed new relativistic model core potentials (MCPs) for the first-row transition-metal atoms from Sc to Zn, in which 3s and 3p electrons are treated explicitly together with the 3d and 4s electrons. By adding suitable correlating functions, we demonstrated that the present MCP basis sets show reasonable performance in describing the electronic structures of atoms and molecules, bringing about accurate excitation energies for atoms and good molecular spectroscopic constants. Ó 2007 Elsevier B.V. All rights reserved.
1. Introduction During the past two decades, we have developed nonrelativistic and relativistic model core potentials (MCPs) together with valence functions for atoms up to Rn [1–5], while Seijo et al. developed ab initio model potentials (AIMPs) [6–9]. The MCP and AIMP methods are both based on the theory originally proposed by Huzinaga et al. [10–21]; the MCP and AIMP methods are unique among various effective core potential (ECP) methods in that they are naturally capable of producing valence orbitals with a nodal structure. The nodeless pseudo-orbitals in ECP approaches may yield too large exchange integrals, resulting in exaggerated correlation energies and too large singlet–triplet splittings [22,23]. Thus, due to the accurate * Corresponding author. Address: Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga Park, Fukuoka 816-8580, Japan. Fax: +81 92 583 7677. E-mail address: [email protected] (E. Miyoshi). 1 Present address: Department of Chemistry, Iowa State University, Ames, IA 50011, USA.
0009-2614/$ - see front matter Ó 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2007.12.019
nodal structure of the valence orbitals, the MCP and AIMP methods are well suited to accurately describe the correlation effects of valence electrons. Recently, we developed effective valence function sets (MCPdzp, MCPtzp, and MCPqzp) by a combination of split MCP valence orbitals and correlating contracted GTFs [24,25] to describe the correlation effects for main group elements. For transition-metal elements, we have developed two types of MCPs, dsMCP and pdsMCP. In the former type, only (n 1)d and ns electrons are explicitly treated, while in the latter, (n 1)p electrons are also treated as well as (n 1)d and ns electrons. Both MCPs have described well the ground state properties of transition-metal complexes [26]. However, the MCPs have a weak point for describing the excited states: for example, the relative energies of the 3dn+14s1 state with respect to the 3dn4s2 state at the Hartree–Fock level using pdsMCPs differ significantly (0.7 eV) from those given by the reference numerical Hartree–Fock calculations (see Table 1). In this Letter, we report the newly developed MCPs that reproduce the Hartree–Fock relative energies within 0.1 eV.
Y. Osanai et al. / Chemical Physics Letters 452 (2008) 210–214 Table 1 Energy of the lowest LS state with 3dn+14s1 relative to that with 3dn4s2 for the first transition-metal atoms at the relativistic Hartree–Fock level Atoms
LS states 3dn+14s1/3dn4s2
Sc Ti V Cr Mn Fe Co Ni Cu
4
2
F/ D F/3F 6 G/4F 7 5 S/ D 6 D/6S 5 F/5D 4 F/4F 3 D/3F 2 2 S/ D 5
Energy of 3dn+14s1 relative to 3dn4s2 (eV) pdsMCPa
spdsMCPa
QRHFb
0.915 0.329 0.192 1.564 2.869 1.395 1.227 1.076 0.566
1.128 0.673 0.273 1.111 3.440 2.001 1.832 1.711 0.118
1.122 0.681 0.290 1.066 3.532 2.053 1.831 1.631 0.051
a
Valence basis sets [6s4p5d] were used in the uncontracted form. QRHF: Numerical quasi-relativistic Hartree–Fock calculations at the level of the Cowan–Griffin approximation. b
In Section 2, we briefly describe the determination of newly developed MCP parameters for the first-row transition-metal atoms. The reference state for the determination of the MCP parameters was taken to be 3dn4s2. Atomic applications, involving the calculations of the excitation energies of atoms at correlated levels are performed and compared with all-electron (AE) calculations in Section 3, while the molecular applications are discussed in comparison with AE calculations in Section 4. All calculations of the atomic and molecular applications were performed using the program packages, ATOMCI [27,28] and MOLCAS6 [29], respectively. 2. Model core potentials of first-row transition-metal atoms In the model core potential method, the atomic Hamil^ MCP ð1; 2; . . . ; N v Þ of the Nv valence electrons (in tonian H atomic units) is chosen as follows: ^ MCP ð1; 2; . . . ; N v Þ ¼ H
Nv X i¼1
^ hMCP ðri Þ þ
Nv X 1 rij i>j
ð2:1Þ
with the one-electron Hamiltonian term defined as: X 1 ^ hMCP ðri Þ ¼ Di þ V MCP ðri Þ þ Bc jwc ihwc j ð2:2Þ 2 c " # 3 3 X X Z Nc 2 2 1þ V MCP ðrÞ ¼ AI eaI r þ AJ reaJ r ; r I¼1 J ¼1 ð2:3Þ where Z is the atomic number of the atom and Nc is the number of core electrons replaced by MCPs. {Aj, aj; Aj, aj} and {Bc} (c = 1s, 2s, 2p) are the MCP parameters and {wc} denotes the core orbital functions, while {Bc} is defined as: Bc ¼ F ec
ð2:4Þ
where ec is the orbital energy of the core orbital and F is usually taken to be 2 [30]. The MCP, VMCP in Eq. (2.3), given as a simple spherically-symmetric local potential, approximates the local core Coulomb potential and the
211
non-local core exchange potential as well as the nuclear attraction potential term. The projection operator of the third term in Eq. (2.2) prevents the valence orbitals from collapsing into the core region, and may also be regarded as an energy shift operator. By the use of this operator, the energy levels of core electrons are shifted far above zero energy, thus allowing for finding, in each symmetry, the lowest valence orbitals with appropriate nodal structure. Prior to the determination of the MCP parameters, we performed atomic quasi-relativistic (QR) restricted Hartree–Fock (HF) calculations for the 3dn4s2 configuration at the level of the Cowan and Griffin approximation [31] to prepare reference data (radial functions and orbital energies). In the QRHF calculations, the mass–velocity and Darwin terms of the Pauli Hamiltonian are added to the HF differential equations, which are then solved in a selfconsistent fashion. This allows for incorporation of the major direct and indirect relativistic effects into the radial orbital functions. Although only the valence orbitals are treated explicitly in the MCP method, the core orbitals are required in order to construct the projection operator of Eq. (2.2). The core orbitals are expanded in terms of GTFs, whose exponents and expansion coefficients are determined by a leastsquares fit to the reference QRHF core orbitals. The valence orbitals are expanded in terms of a smaller number of GTFs. The exponents of the valence GTFs are also determined by a least-squares fit to the reference QRHF valence orbitals: s-type functions were expanded with six GTFs, p-type four, and d-type five. The valence orbital energies and orbital shapes are chosen as the criteria for the determination of the MCP parameters and the coefficients of the valence orbitals. Following this approach, we optimized the MCP parameters and valence orbitals for the first-row transition-metal atoms in which the (n 1)s, (n 1)p and (n 1)d electrons were treated explicitly together with the ns electrons. These new MCPs are called spdsMCPs. The relative energies of the 3dn+14s1 state with respect to the 3dn4s2 state at the Hartree–Fock level given by the use of the present spdsMCPs, are listed and compared with those given by the use of the previous pdsMCPs and the numerical results of the reference QRHF. In both MCP calculations, the basis sets were uncontracted. The present relative energies agree with those of the reference QRHF for all atoms, the differences being within 0.1 eV. Thus, our new spdsMCPs provide a more reliable description for the excited states in the atoms and molecules containing firstrow transition-metal atoms than the previous pdsMCPs, which sometimes gave relative energies that differed by 0.7 eV from those obtained by the reference QRHF. 3. Atomic applications The quality of the present spdsMCP sets was examined through the correlated calculations of the relative energies of the 3dn+14s1 state with respect to the 3dn4s2 state for Sc,
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Mn, and Cu atoms. We first performed the restricted Hartree–Fock (RHF) calculations followed by a configuration interaction single and double (CISD) and its Davidson correction (+Q), in order to include the electron correlation effects among the 3s, 3p, 3d, and 4s electrons. The correlating functions [2p2f1g] from the sets given by Noro et al. [32] were utilized with our valence orbitals in the spdsMCP sets (6s4p5d), which were decontracted to (33111/31/311). One more diffuse d-type GTF was augmented (0.0376626 for
Sc, 0.0761019 for Mn, and 0.0969874 for Cu). Thus, the resulting basis set was [5s4p4d2f1g]. A smaller set [4s3p4d1f] with correlating functions [1p1f] [32], contraction (4411/31/311), and diffuse d-function, was also examined. For comparison, we performed all-electron (AE) calculations with very large basis sets that were formed from Huzinaga’s well-tempered sets [33], [26s17p13d] for Sc and Mn, and [26s17p14d] for Cu, augmented by p-type and d-type diffuse functions with the same exponents of s-type diffuse
Table 2 Energy of the lowest LS state with 3dn+14s1 relative to that with 3dn4s2 for Sc, Mn, and Cu atoms at the HF and correlated calculations Atoms
Method/basis set
Calculation level
Term energy (a.u.) n
Sc
spdsMCP [5s4p4d2f1g]
[4s3p4d1f]
AE [7s5p4d2f1g]
[6s4p4d1f]
2
n+1
3d 4s
3d
2
4
4s
1
HF CISD CISD+Q
D 46.06216 46.31554 46.33576
F 46.02342 46.26918 46.28423
1.054 1.262 (0.206) 1.402 (0.348)
HF CISD CISD+Q
46.06179 46.27749 46.29453
46.02330 46.22637 46.23787
1.048 1.391 (0.343) 1.542 (0.494)
HF CISD CISD+Q
763.29120 763.62548 763.65211
763.24988 763.58020 763.60139
1.125 1.232 (0.107) 1.380 (0.255)
CISD CISD+Q
763.58817 763.61138
763.53893 763.55639
1.340 (0.215) 1.496 (0.371) 1.427
HF CISD CISD+Q
6 S 102.72621 103.11814 103.14458
6 D 102.60625 103.03562 103.06437
3.264 2.246 (1.018) 2.183 (1.081)
HF CISD CISD+Q
102.72448 103.03336 103.05400
102.60581 102.94637 102.96818
3.229 2.367 (0.862) 2.335 (0.894)
HF CISD CISD+Q
1157.40179 1157.88456 1157.91743
1157.27178 1157.79486 1157.82976
3.538 2.441 (1.097) 2.385 (1.153)
CISD CISD+Q
1157.80590 1157.83248
1157.70894 1157.73650
2.638 (0.900) 2.612 (0.926) 2.145
2
2
Experimentalb Mn
spdsMCP [5s4p4d2f1g]
[4s3p4d1f]
AE [7s5p4d2f1g]
[6s4p4d1f] Experimentalb Cu
spdsMCP [5s4p4d2f1g]
[4s3p4d1f]
AE [7s5p4d2f1g]
[6s4p4d1f]
HF CISD CISD+Q
D 195.87751 196.48705 196.52643
S 195.87970 196.53486 196.57595
0.060 1.301 (1.241) 1.348 (1.288)
HF CISD CISD+Q
195.87492 196.35261 196.38220
195.87876 196.39865 196.42899
0.104 1.253 (1.149) 1.273 (1.169)
HF CISD CISD+Q
1653.19053 1653.92042 1653.96670
1653.18836 1653.96310 1654.01005
0.059 1.161 (1.220) 1.180 (1.239)
CISD CISD+Q
1653.75690 1653.78925
1653.79955 1653.83211
1.161 (1.220) 1.166 (1.225) 1.490
Experimentalb a b
Relative energy (eV)a
Values in parentheses are correlation effects on the relative energy. Ref. [34].
Y. Osanai et al. / Chemical Physics Letters 452 (2008) 210–214
functions. The f-type and g-type GTFs having the same exponents as the d-type GTFs were also considered. Thus, the uncontracted GTFs in the AE calculations were [26s20p15d15f15g] for Sc and Mn, and [26s21p16d16f16g] for Cu. Using these basis functions we performed CISD calculations for each atom, followed by CISD calculations with a [7s5p4d2f1g] basis set composed of occupied [4s2p1d] Hartree–Fock orbitals and correlating [3s3p3d2f1g] atomic natural orbitals (ANO) for the sake of comparison. The AE [7s5p4d2f1g] basis set has the same number of correlating orbitals as the MCP/[5s4p4d2f1g] basis set. A small AE [6s4p4d1f] basis set with the occupied orbitals and correlating [2s2p3d1f] orbitals was also used for the comparison with the MCP/[4s3p4d1f] basis set. The relativistic effects were taken into account in the AE calculations through the third-order Douglas–Kroll (DK3) approximation [35]. The resulting relative energies are listed in Table 2 together with the observed values. Correlation effects on the relative energies are also given in parentheses in the last column. The spdsMCP calculations reasonably well reproduce the relative energies given by the AE calculations with the same size of the ANO sets. The HF relative energies given in AE almost exactly coincide with those achieved by the QRHF. The deviations of HF relative energies by spdsMCP from those by AE are somewhat larger than the deviations of spdsMCP HF from QRHF shown in Table 1. This deterioration of HF energies in spdsMCP is caused by the contraction of the basis sets. It is notable that the correlation effects on the relative energies in MCP calculations are almost the same as those in AE calculations: the differences between the correlation effects by MCP [5s4p4d2f1g] and AE [7s5p4d2f1g] calculations are within 0.1 eV and those by MCP [4s3p4d1f] and AE [6s4p4d1f] are within 0.13 eV.
mial fitting are listed in Table 3 together with the corresponding experimental values [37]. For CuH, CASSCF calculations with the [4s3p4d1f] basis set gave somewhat longer equilibrium distance, Re, and much smaller binding energy, De. These results were significantly improved by considering electron correlations within the 12 (Cu 3d and 4s and H 1s) electrons in all of MRCI, MP2, and CCSD(T). It is notable that MCP CCSD(T) calculations with [4s3p4d1f] give nearly equal spectroscopic constants as all-electron CCSD(T) calculations with [8s6p4d1f], where 12 electrons were also correlated [38]. Both expansion of the basis set and inclusion of the correlation of Cu 3s and 3p electrons bring about little change to the calculated spectroscopic constants. The most striking change is 0.19 eV in De, which is brought about by basis set expansion in CCSD(T) calculation. For Cu2, on the other hand, both expansion of the basis set and inclusion of the correlation of Cu 3s and 3p electrons bring about more significant change to the calculated spectroscopic constants. The former improves Re by 0.05– ˚ , xe by 15–17 cm1, and De by 0.3–0.4 eV. The latter 0.06 A ˚ , xe by 13–14 cm1, and additionally improves Re by 0.02 A De by 0.1–0.2 eV. As in CuH, MCP CCSD(T) calculations
Table 3 Spectroscopic constants of the ground states X1R+ of CuH and Cu2 Molecules Methods
CuH
spdsMCP
4. Molecular applications The present spdsMCP have also been applied in the calculations of molecular spectroscopic constants in the ground states of CuH and Cu2 molecules. The basis sets used for Cu were [4s3p4d1f] and [5s4p4d2f1g], and the basis set of hydrogen was a triple-zeta type [3s2p1d] from Ref. [36]. The computational methods employed here were MP2 and CCSD(T). The energies of the dissociation limit for these single reference calculations were obtained by supermolecular calculations for a triplet open shell state at an internuclear distance of 100 a.u. We also performed three-states averaged CASSCF calculations with 12 electrons in 7 orbitals for CuH, which were followed by multi-reference singly and doubly excited configuration interaction (MRCI) calculations. In the correlated calculations, the Cu 3d and 4s and H 1s (12 and 22 electrons for CuH and Cu2, respectively) electrons were correlated for both basis sets, while the Cu 3s and 3p electrons were correlated only for the largest set. Results of the spectroscopic constants for the ground state of CuH and Cu2 obtained by a third-degree polyno-
213
AEc Experimentald Cu2
spdsMCP
Basis set
Calculation Spectroscopic level constants Re ˚) (A
xe De (cm1) (eV)c
CASSCF MRCIa MP2a CCSD(T)a
1.520 1.466 1.451 1.482
1798.7 1746.7 1972.3 1836.4
1.230 2.428 2.560 2.556
[5s4p4d2f1g] CASSCF MRCIa MP2a CCSD(T)a MRCIb MP2b CCSD(T)b CCSD(T)a
1.520 1.465 1.443 1.473 1.453 1.418 1.457 1.490 1.463
1810.7 1768.6 2014.7 1877.6 1794.8 1922.0 1766.7 1839.8 1941.3
1.232 2.424 2.655 2.741 2.465 2.736 2.769 2.55 2.85
[4s3p4d1f]
[4s3p4d1f]
MP2a 2.279 CCSD(T)a 2.307
[5s4p4d2f1g] MP2a CCSD(T)a MP2b CCSD(T)b AE [5s3p3d1f]e Experimentald a
2.226 2.249 2.205 2.229
CCSD(T)a 2.278 2.220
265.3 1.513 243.0 1.536 279.6 259.7 295.3 272.6
1.852 1.830 2.012 1.930
246.8 1.576 264.6 2.05
Cu 3d and 4s and H 1s electrons were correlated. Cu 3s and 3p electrons were correlated, as well as the Cu 3d and 4s and H 1s electrons. c Ref. [38]. d Ref. [37]. e Using a contracted basis set from Ref. [36]. b
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Y. Osanai et al. / Chemical Physics Letters 452 (2008) 210–214
Table 4 Spectroscopic constants of an excited state A1R+ of CuH Methods
spdsMCP
Basis set
[4s3p4d1f] [5s4p4d2f1g]
Experimentalc
Calculation level
Spectroscopic constants Re ˚) (A
xe (cm1)
Te (eV)
CASSCF MRCIa
1.652 1.623
1585.8 1548.4
1.521 3.006
CASSCF MRCIa MRCIb
1.645 1.622 1.607 1.572
1600.0 1569.8 1654.5 1698.4
1.555 3.001 2.858 2.905
a
Cu 3d and 4s and H 1s electrons were correlated. Cu 3s and 3p electrons were correlated, as well as Cu 3d and 4s and H 1s electrons. c Ref. [37]. b
with the [4s3p4d1f] set yield nearly equal spectroscopic constants as all-electron CCSD(T) calculations with the [5s3p3d1f] set from Ref. [36], indicating that the present MCP set provides sufficiently high performance for description of electronic structures in highly correlated calculations. For CuH, the excited state A1R+ was also studied at the three-states averaged CASSCF and MRCI calculations. Calculated spectroscopic constants are listed in Table 4. The CASSCF calculation gave the excitation energy, Te, of only a half of the observed value, but inclusion of electron correlation through MRCI improved the Te enormously. The spectroscopic constants given by MRCI with [5s4p4d2f1g] including correlation of Cu 3s and 3p electrons agree well with observed values, again demonstrating the superior capabilities of the present spdsMCP. 5. Conclusions We proposed new model core potentials, spdsMCPs, for the first-row transition-metal atoms from Sc to Zn, in which the (n 1)s and (n 1)p electrons are treated explicitly in addition to the (n 1)d and ns electrons. The present MCP basis sets were tested in the calculation of the energy differences between the 3dn4s2 and 3d(n+1)4s1 states for Sc, Mn, and Cu atoms and spectroscopic constants of the ground and excited states of CuH and the ground state of Cu2 molecules. It has been shown that the newly developed MCP basis sets describe well these physical properties. The MCP sets are available in the homepage of Ref. [36]. Acknowledgements The present study has been supported partly by a Grantin-Aid for Scientific Research from the Japan Society for the Promotion of Science (JSPS). A part of the calculations
reported here were performed using computing resources in the Research Center for Computational Science, Okazaki, Japan. References [1] Y. Sakai, E. Miyoshi, M. Klobukowski, S. Huzinaga, J. Comput. Chem. 8 (1987) 226. [2] Y. Sakai, E. Miyoshi, M. Klobukowski, S. Huzinaga, J. Comput. Chem. 8 (1987) 264. [3] Y. Sakai, E. Miyoshi, M. Klobukowski, S. Huzinaga, J. Chem. Phys. 106 (1997) 8084. [4] E. Miyoshi, Y. Sakai, K. Tanaka, M. Masamura, J. Mol. Struct.: Theochem 451 (1998) 73. [5] Y. Sakai, E. Miyoshi, H. Tatewaki, J. Mol. Struct.: Theochem 451 (1998) 143. [6] S. Huzinaga, L. Seijo, Z. Barandiara´n, M. Klobukowski, J. Chem. Phys. 86 (1987) 2132. [7] L. Seijo, Z. Barandiara´n, S. Huzinaga, J. Chem. Phys. 91 (1989) 7011. [8] Z. Barandiara´n, L. Seijo, S. Huzinaga, J. Chem. Phys. 93 (1990) 5843. [9] Z. Barandiara´n, L. Seijo, Can. J. Chem. 70 (1992) 409. [10] V. Bonifacic, S. Huzinaga, J. Chem. Phys. 60 (1974) 2779. [11] Y. Sakai, J. Chem. Phys. 75 (1981) 1303. [12] Y. Sakai, S. Huzinaga, J. Chem. Phys. 76 (1982) 2537. [13] Y. Sakai, S. Huzinaga, J. Chem. Phys. 76 (1982) 2552. [14] J. Andzelm, S. Huzinaga, M. Klobukowski, E. Radzio-Andzelm, Mol. Phys. 52 (1981) 1495. [15] M. Klobukowski, J. Comput. Chem. 4 (1983) 350. [16] L.G.M. Pettersson, U. Wahlgren, O. Gropen, J. Chem. Phys. 86 (1987) 2176. [17] S. Katsuki, S. Huzinaga, Chem. Phys. Lett. 152 (1988) 203. [18] S. Katsuki, Can. J. Chem. 70 (1992) 285. [19] S. Katsuki, J. Chem. Phys. 98 (1993) 496. [20] H. Tatewaki, S. Katsuki, Y. Sakai, E. Miyoshi, J. Comput. Chem. 17 (1996) 1056. [21] S. Huzinaga, Can. J. Chem. 73 (1995) 619 and references cited therein. [22] B. Pittel, W.H.E. Schwartz, Chem. Phys. Lett. 46 (1977) 121. [23] C. Teichteil, J.P. Malrieu, J.C. Barthelat, Mol. Phys. 33 (1977) 181. [24] Y. Osanai, T. Noro, E. Miyoshi, J. Chem. Phys. 117 (2002) 9623. [25] E. Miyoshi, H. Mori, R. Hirayama, Y. Osanai, T. Noro, H. Honda, M. Klobukowski, J. Chem. Phys. 122 (2005) 074104. [26] E. Miyoshi, Y. Sakai, J. Comput. Chem. 9 (1988) 719. [27] F. Sasaki, Int. J. Quantum Chem. 8 (1974) 605. [28] F. Sasaki, M. Sekiya, T. Noro, K. Ohtsuki, Y. Osanai, MOTECC-94, in: E. Clementi (Ed.), STEF, Calgari, 1993. [29] G. Karlstro¨m et al., Computat. Mater. Sci. 28 (2003) 222. [30] G. Ho¨jer, J. Chung, Int. J. Quantum Chem. 14 (1978) 623. [31] R.D. Cowan, D.C. Griffin, J. Opt. Soc. Am. 66 (1976) 1010. [32] T. Noro, M. Sekiya, T. Koga, H. Matsuyama, Theor. Chem. Acc. 104 (2000) 146. [33] S. Huzinaga, B. Miguel, Chem. Phys. Lett. 212 (1990) 289. [34] J.E. Sansonetti, W.C. Martin, S.L. Young, Handbook of Basic Atomic Spectroscopic Data. Available from: