Revised model core potentials for third-row transition–metal atoms from Lu to Hg

Chemical Physics Letters 476 (2009) 317–322

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Revised model core potentials for third-row transition–metal atoms from Lu to Hg Hirotoshi Mori a,b, Kaori Ueno-Noto a, You Osanai c, Takeshi Noro d, Takayuki Fujiwara e, Mariusz Klobukowski f, Eisaku Miyoshi b,e,* a

Ocha-dai Academic Production, Division of Advanced Sciences, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan JST-CREST, Kawaguchi 332-0012, Japan c Faculty of Pharmaceutical Sciences, Aomori University, Aomori 030-0943, Japan d Division of Chemistry, Graduate School of Science, Hokkaido University, Sapporo 060-0810, Japan e Graduate School of Engineering Sciences, Kyushu University, 6-1 Kasuga Park, Fukuoka 816-8580, Japan f Department of Chemistry, University of Alberta Edmonton, Alberta, Canada T6G 2G2 b

a r t i c l e

i n f o

Article history: Received 13 April 2009 In final form 5 June 2009 Available online 9 June 2009

a b s t r a c t We have produced new relativistic model core potentials (spdsMCPs) for the third-row transition–metal atoms from Lu to Hg explicitly treating explicitly 5s and 5p electrons in addition to 5d and 6s electrons in the same manner for the first- and second-row transition–metal atoms given in the previous Letters [Y. Osanai, M.S. Mon, T. Noro, H. Mori, H. Nakashima, M. Klobukowski, E. Miyoshi, Chem. Phys. Lett. 452 (2008) 210; Y. Osanai, E. Soejima, T. Noro, H. Mori, M.S. Mon, M. Klobukowski, E. Miyoshi, Chem. Phys. Lett. 463 (2008) 230]. Using suitable correlating functions with the split-valence MCP functions, we demonstrate 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 proper spectroscopic constants for Au2, Hg2, and AuH. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction In the last two decades, we have developed non-relativistic and relativistic model core potentials (MCPs) accompanied by valence functions for atoms up to Rn [1–5]. The MCP method as well as the ab initio model potential (AIMP) method by Seijo et al. [6–9] is based on the theory proposed by Huzinaga et al. [10–21], and has the advantage of producing valence orbitals with a nodal structure over various other effective core potential (ECP) methods. The nodeless pseudo-orbitals in the usual ECP approaches may produce a large exchange integral and thus overestimate the correlation energies, resulting in singlet–triplet splittings that are too large [22,23], for example. On the other hand, the valence orbitals with a nodal structure in the MCP and AIMP methods can accurately describe the valence correlation effects. We have recently shown for the main group elements that the valence correlation effects can be described adequately by a combination of split MCP valence orbitals and correlating contracted Gaussian-type functions (GTFs) [24,25]. For transition–metal atoms, we have already developed two types of MCPs; one is dsMCP, in which only (n  1)d and ns electrons are treated explicitly, and the other is pdsMCP, in which (n  1)p electrons are treated explicitly in addition to (n  1)d

* 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). 0009-2614/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2009.06.019

and ns electrons. Both of the MCPs can properly describe the ground state properties of transition–metal complexes but are less competent to describe the excited states accurately. In previous studies [26,27], we have developed new relativistic MCPs, called spdsMCPs, for the first- and second-row transition–metal atoms, explicitly treating (n  1)s and (n  1)p electrons as well as (n  1)d and ns electrons. The restricted Hartree–Fock (RHF) calculations using the spdsMCPs reproduce the energy separation between the lowest LS state of the (n  1)dmns2 configuration and that of (n  1)dm+1ns1 given by numerical quasi-relativistic Hartree–Fock (QRHF) calculations within the deviation of 0.1 eV. Comparison with the results of all-electron (AE) configuration interaction (CI) calculations shows that the CI calculations using the new MCPs with a combination of decontracted MCP valence functions and suitable correlation functions yield reasonable correlation energies for both of the atomic states. The new MCPs have been applied to the calculation of spectroscopic constants for some diatomic molecules and found to work well for the excited states of molecules. In the present work, we present new relativistic MCPs for the third-row transition–metal atoms from Lu to Hg in the same manner as those applied for the first-row transition–metal atoms by explicitly treating 5s and 5p electrons as well as 5d and 6s electrons. In Section 2, we briefly describe the determination of newly developed MCP parameters for the third-row transition–metal atoms. Atomic applications involving calculations of the excitation energies of atoms at correlated levels are performed and compared

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with the 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 [28,29] and GAMESS [30], respectively. 2. Model core potentials of third-row transition–metal atoms In the model core potential method, the atomic Hamiltonian HMCP ð1; 2; . . . ; N v Þ of the Nv valence electrons (in atomic units) is chosen as follows:

HMCP ð1; 2; . . . ; N v Þ ¼

Nv X i¼1

hMCP ðr i Þ þ

Nv X 1 r ij i>j

ð2:1Þ

with the one-electron Hamiltonian term defined as

X 1 hMCP ðr i Þ ¼  Di þ V MCP ðr i Þ þ Bc jwc ihwc j 2 c

ð2:2Þ

$ % 3 3 X X Z  Nc aI r 2 aJ r2 ; V MCP ðrÞ ¼  AI e þ AJ re 1þ r I¼1 J¼1

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, 3s, 3p, 3d, 4s, 4p, 4d, and 4f) 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 [31]. The MCP, VMCP in Eq. (2.3), given as a simple spherically-symmetric local potential, approximates the local core Coulomb potential and the 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 HF calculations for the 5dm6s2 configuration at the level of the Cowan and Griffin (CG) approximation [32] 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 self-consistent fashion. This allows for incorporation of the major direct and indirect relativistic effects into the radial orbital functions. Although the CG approximation does not work for 1s core electrons when the atomic number increases, it works well for valence electrons: we verified [25] that the CG calculations reproduce valence orbitals and their energies were almost the same as the results given by the third-order Douglas–Kroll (DK3) approximation proposed by Nakajima and Hirao [33] up to the atomic number Z = 102. Although only the valence orbitals were treated explicitly in the MCP method, the core orbitals were required in order to construct the projection operator of Eq. (2.2). The core orbitals were expanded in terms of GTFs, whose exponents and expansion coefficients were determined by a least-squares fit to the reference QRHF core orbitals. The valence orbitals were expanded in terms of a smaller number of GTFs. The exponents of the valence GTFs were also determined by a least-squares fit to the reference QRHF valence orbitals: s-type functions were expanded with eight GTFs,

Table 1 Energies of the lowest LS state with 5dm+16s1 relative to that with 5dm6s2 for the third-row transition–metal atoms at the relativistic Hartree–Fock level. Atoms

LS states 5dm+16s1/5dm6s2

Lu Hf Ta W Re Os Ir Pt Au rmsc

4

2

F/ D F/3F 6 G/4F 7 5 S/ D 6 D/6S 5 5 F/ D 4 4 F/ F 3 D/3F 2 2 S/ D 5

Energy of 5dm+16s1 relative to 5dm6s2 (eV) pdsMCPa

spdsMCPa

QRHFb

1.691 (+0.050) 0.989 (+0.036) 0.256 (+0.040) 1.122 (+0.155) 2.025 (+0.263) 0.716 (+0.175) 0.218 (+0.133) 0.376 (+0.028) 1.822 (+0.046) 0.129

1.756 (+0.115) 1.064 (+0.111) 0.316 (+0.100) 1.182 (+0.095) 1.893 (+0.131) 0.633 (+0.092) 0.170 (+0.085) 0.355 (+0.049) 1.827 (+0.041) 0.095

1.641 0.953 0.216 1.277 1.762 0.541 0.085 0.404 1.868 –

a Valence basis sets [8s6p6d] were used in the decontracted form. The differences between the MCP and the reference QRHF values are given in parentheses. b Numerical quasi-relativistic Hartree–Fock calculations at the level of the Cowan–Griffin approximation. c Root mean square differences in the MCP values from the reference QRHF values.

p-type functions were expanded with six GTFs, and d-type functions were expanded with six GTFs. 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 third-row transition–metal atoms in which the 5s, 5p, and 5d electrons were treated explicitly together with the 6s electrons. These new MCPs are called spdsMCPs. In Table 1, the HF energies of the lowest LS states of the 5dm+16s1 configuration with respect to those of 5dm6s2 given by the use of the present spdsMCPs, are listed and compared with those given by the use of the previous pdsMCPs and the reference data obtained by QRHF. In both MCP calculations, the valence functions were decontracted. The differences of the MCPs values from the reference QRHF values are given in parentheses in each column. For all the atoms, the present spdsMCPs gave the difference of 0.1 eV. For the W, Re, and Os atoms, the differences of the previous pdsMCPs, as large as 0.16, 0.26, and 0.18 eV, respectively, are considerably improved in the new spdsMCP calculations. On the other hand, the differences of the previous pdsMCPs are very small (<0.05 eV) for the Lu, Hf, and Ta atoms, while the differences in the spdsMCPs are 0.1 eV. It may arise from that the criteria for the determination of our MCP parameters are not the total energies but the shapes and orbital energies of the valence orbitals. Although the energy separations are not monotonically improved, the new spdsMCPs are expected to give a more reliable description for both the ground and excited states in the third-row transition– metal atoms and molecules containing them than the previous pdsMCPs.

3. Atomic applications In order to examine the quality of the present spdsMCPs, MCP and AE calculations were carried out for the lowest LS state of each configuration of 5dm6s2 and 5dm+16s1 for the Lu, Re, and Au atoms. A restricted HF calculation was followed by a configuration interaction single and double (CISD) calculation considering the correlation effects of the 5d and 6s electrons, and its Davidson’s correction (+Q) was also calculated. Two MCP orbital sets were tested. The larger set [5s3p4d2f1g] was generated by decontracting our (8s6p6d) spdsMCP valence orbitals to (55111/6/411) and adding one more diffuse d-type GTF and a correlating set of [2p2f1g] by Osanai et al. [34]. The exponents

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of the diffuse d-type GTFs were 0.02550870, 0.05125471, and 0.06885332 for Lu, Re, and Au, respectively. The other smaller set [4s2p4d1f] is composed of the decontracted (6611/6/411) valence set, the diffuse d-type function, and the correlating set of [1p1f] [34]. In the AE calculations, we first carried out a restricted HF-CISD calculation with a very large GTF set of [28s24p20d17f17g], which was an expansion of Huzinaga’s [28s21p18d12f] well-tempered set [35] augmented with the p-, d-, and f-type diffuse GTFs having the same exponents as the s-type diffuse GTFs and the g-type GTFs having the same exponents as the f-type GTFs. Then, for comparison with the MCP calculations, CISD calculations were carried out with each of the [9s6p6d3f1g] and [8s5p6d2f] sets, which were generated by adding each of the correlating [3s2p3d2f1g] and [2s1p3d1f] atomic natural orbital (ANO) sets to the occupied [6s4p3d1f] HF orbital set. The sets of [9s6p6d3f1g] and [8s5p6d2f] in the AE calculations have the same number of correlating orbitals as the sets of [5s3p4d2f1g] and [4s2p4d1f] in the MCP calculations, respectively. In the AE calculations, the relativistic effects were taken into account through the DK3 approximation [33] with a Gaussian nucleus model by Visscher and Dyall [36].

Furthermore, we performed similar procedures including the correlation effects of the 5s and 5p electrons in addition to the 5d and 6s electrons. We used almost the same orbital sets as the above larger sets. The difference between them was only in the following point: the 5p orbital was split into (51) in the MCP calculations and [3p] correlating ANO set was employed in the AE calculations. Consequently, the basis sets employed were [5s4p4d2f1g] and [9s7p6d3f1g] in the MCP and AE calculations, respectively. Table 2 gives the energy of the lowest LS state of the 5dm+16s1 configuration with respect to that of 5dm6s2, calculated with the correlation effects of the 5d and 6s electrons, together with the observed value [37]. The correlation contribution to the relative energy is also given in parentheses in the last column. Table 3 shows the corresponding values yielded by including the correlation effects of the 5s, 5p, 5d, and 6s electrons. The relative energies given by the AE-HF calculations (Tables 2 and 3) are close to those achieved by the QRHF calculations (Table 1). At the HF level, differences between the relative energies given by the MCP and AE calculations were 0.03–0.12 eV. Inclusion of the correlation effects of the 5d and 6s electrons considerably reduced

Table 2 Energies of the lowest LS state of 5dm+16s1 with respect to that of 5dm6s2 for the Lu, Re, and Au atoms including the correlation effects of the 5d and 6s electrons. Atoms

Method/basis set

Lu

spdsMCP [5s3p4d2f1g]

[5s3p4d2f1g]

AE [5s3p4d2f1g]

[5s3p4d2f1g]

Calculation level

Term energy (a.u.)

Relative energy (eV)a

5dm6s2

5dm+16s1

2

4

HF CISD CISD+Q HF CISD CISD+Q

D 40.96149 41.00800 41.01239 40.96138 41.00569 41.01000

HF CISD CISD+Q CISD CISD+Q

14558.99804 14559.04469 14559.04903 14559.04304 14559.04733

14558.93769 14558.95638 14558.95687 14558.95474 14558.95520

6

F 40.89745 40.91678 40.91733 40.89737 40.91477 40.91529

Exptl.b Re

spdsMCP [5s3p4d2f1g]

[5s3p4d2f1g]

AE [5s3p4d2f1g]

[5s3p4d2f1g]

S 79.00496 79.14151 79.15270 79.00459 79.12165 79.13126

6

HF CISD CISD+Q HF CISD CISD+Q HF CISD CISD+Q CISD CISD+Q

16689.18472 16689.31728 16689.32783 16689.30108 16689.31013

16689.11986 16689.25292 16689.26143 16689.23525 16689.24248

2

D 78.93567 79.07382 79.08316 78.93563 79.05320 79.06138

Exptl.b Au

spdsMCP [5s3p4d2f1g]

[5s3p4d2f1g]

AE [5s3p4d2f1g]

[5s3p4d2f1g]

D 136.39524 136.71796 136.74456 136.39466 136.65939 136.68023

2

HF CISD CISD+Q HF CISD CISD+Q HF CISD CISD+Q CISD CISD+Q

19010.06969 19010.39061 19010.41679 19010.33706 19010.35735

19010.13807 19010.46124 19010.48438 19010.41183 19010.43020

Exptl.b a b

Values in parentheses are correlation effects on the relative energy. Ref. [37].

S 136.46203 136.78771 136.81156 136.46185 136.73364 136.75305

1.743 2.482 2.587 1.742 2.474 2.577 1.642 2.403 2.508 2.403 2.507 2.359 1.885 1.842 1.892 1.877 1.863 1.902 1.765 1.751 1.807 1.791 1.841 1.759

(0.739) (0.844) (0.732) (0.835)

(0.761) (0.866) (0.761) (0.865)

(0.043) (0.007) (0.014) (0.025)

(0.014) (0.042) (0.026) (0.076)

1.817 1.898 1.823 1.828 2.021 1.982 1.861 1.922 1.839 2.034 1.982 1.745

(0.081) (0.006) (0.193) (0.154)

(0.061) (0.022) (0.173) (0.121)

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Table 3 Energy of the lowest LS state of 5dm+16s1 with respect to that of 5dm6s2 for the Lu, Re, and Au atoms including the correlation effects of the 5s, 5p, 5d, and 6s electrons. Atoms

Lu

Method/basis set

spdsMCP [5s4p4d2f1g]

AE [9s7p6d3f1g]

Calculation level

Term energy (a.u.)

Relative energy (eV)a

5dm6s2

5dm+16s1

2

D 40.96151 41.14682 41.16227

4

HF CISD CISD+Q HF CISD CISD+Q

14558.99804 14559.24963 14559.26956

14558.93769 14559.17192 14559.18492

6

F 40.89746 41.06382 41.07285

Exptl.b Re

spdsMCP [5s4p4d2f1g]

AE [9s7p6d3f1g]

S 79.00496 79.32611 79.35409

6

HF CISD CISD+Q HF CISD CISD+Q

16689.18472 16689.55837 16689.58907

16689.11986 16689.49697 16689.52526

2

D 78.93567 79.26158 79.28781

Exptl.b Au

spdsMCP [5s4p4d2f1g]

AE [9s7p6d3f1g]

D 136.39525 136.89847 136.94398

2

HF CISD CISD+Q HF CISD CISD+Q

19010.06969 19010.62376 19010.67238

19010.13807 19010.68933 19010.73425

Exptl.b a b

S 136.46204 136.96259 137.00470

1.743 2.259 (0.516) 2.433 (0.690) 1.642 2.115 (0.473) 2.313 (0.671) 2.359 1.885 1.756 (0.129) 1.803 (0.082) 1.764 1.671 (0.093) 1.736 (0.028) 1.759 1.818 1.745 (0.073) 1.652 (0.166) 1.861 1.784 (0.077) 1.694 (0.167) 1.745

Values in parentheses are correlation effects on the relative energy. Ref. [37].

the differences, but the calculated relative energies deviated noticeably from the observed values. Especially in the calculations with a small basis set for the Au atom, the deviation amounted to less than 0.3 eV. Considering also the correlation effects of the 5s and 5p electrons, both the relative energies calculated by the MCP and AE calculations well reproduced the observed values within 0.1 eV. However, the AE-CISD calculations without Davidson’s correction gave a relative energy that is 0.24 eV smaller than the observed value for the Lu atom. 4. Molecular applications The present spdsMCPs have also been applied in calculations for molecular spectroscopic constants of the ground states of Au2 and AuH and those of a van der Waals molecule, Hg2. The spectroscopic constants of the first excited states for AuH were also calculated. The basis sets for Au and Hg were twofold, [4s2p4d1f] and [5s3p4d2f1g], which correspond to those used in the atomic applications, and the basis set of hydrogen was a triple-zeta polarization type [3s2p1d] from Ref. [38]. To obtain the spectroscopic constants of Au2 and AuH, we performed complete active space self-consistent field (CASSCF) calculations followed by second-order multi-configuration quasidegenerate perturbation theory (MCQDPT2) calculations [39] to take into account the dynamical correlation. For Hg2, we performed single reference MP2 and coupled-cluster CCSD and CCSD(T) calculations. The active orbitals in the above multi reference calculations were the 5d and 6s orbitals in Au, and the 1s orbital of H. Thus, 22 electrons in 12 orbitals and 12 electrons in 7 orbitals were treated as active electrons for Au2 and AuH, respectively. The CASSCF wave function of Au2 was optimized for the ground state. For AuH, three low-lying 1A1 states in C2v were equally averaged in the CASSCF calculation to obtain the potential energy curves in

the ground and excited states. In the MCQDPT2 calculations, the Au 5d and 6s and H 1s electrons (22 electrons in Au2 and 12 electrons in AuH) were correlated, while the Au 5s and 5p electrons were also correlated (38 electrons in Au2 and 20 electrons in AuH) only in the MCQDPT2 calculation with the larger set. For comparison, we also performed also all-electron (AE) CASSCF and MCQDPT2 calculations for the three low-lying 1A1 states of AuH with the general contracted AE [9s7p6d3f1g] basis of Au, where the relativistic effects were considered through the DK3 approximation [33] with a point nucleus. We used the intruder state avoidance (ISA) method [40] with the ISA shift of 0.02 in the MCQDPT2 calculations. Single reference post Hartree–Fock calculations, MP2, CCSD, and CCSD(T), were carried out for Hg2, where twenty four 5d and 6s electrons (24 in total) were correlated. 5s and 5p electrons as well as 5d and 6s electrons (40 electrons) were also correlated only in calculations with the larger set. The energies of the dissociation limit for all calculations were obtained by supermolecular calculations at an internuclear distance of 100 a.u. Calculated results of the spectroscopic constants for each molecule obtained by a third-degree polynomial fitting are listed in Tables 4 and 5, together with the corresponding experimental values [41] and those obtained by all-electron (AE) calculations [42–44]. For Au2 and AuH in their ground states, CASSCF/MCP calculations with both basis sets gave longer equilibrium distances Re and much smaller vibrational frequencies xe and binding energies De. These results were significantly improved by considering the electron correlations within Au 5d and 6s electrons (including H 1s electrons for AuH) in MCQDPT2/MCP. For Hg2 in the ground state, MP2/MCP calculations with both basis sets gave shorter equilibrium distances Re, and much larger vibrational frequencies xe and binding energies De. The inclusion of correlation effects with highly correlated methods, CCSD or CCSD(T), significantly improved the calculated results. In all cases, the inclusion of the

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H. Mori et al. / Chemical Physics Letters 476 (2009) 317–322 Table 4 Spectroscopic constants of the ground X1R+ states of Au2 and Hg2. Molecule

Method/basis set MCP/[4s2p4d1f]

Au2

MCP/[5s4p4d2f1g]

AE/[11s12p6d4f2g]d Exptl.e Hg2

MCP/[4s2p4d1f]

MCP/[5s4p4d2f1g]

Re (Å)

xe (cm1)

De (eV)

CASSCF MCQDPT2/22ea CCSD/22ea CCSD(T)/22ea CASSCF MCQDPT2/22ea MCQDPT2/38eb CCSD/22ea CCSD/38eb CCSD(T)/22ea CCSD(T)/38eb CCSD(T)/34ec

2.671 2.532 2.559 2.559 2.630 2.472 2.448 2.509 2.497 2.504 2.490 2.484 2.472

135.0 182.9 174.1 172.6 140.6 194.1 202.4 181.5 185.6 182.2 186.9 192.2 190.9

1.05 1.97 1.66 1.81 1.16 2.23 2.44 1.87 1.97 2.06 2.18 2.25 2.31

MP2/24ea CCSD/24ea CCSD(T)/24ea MP2/24ea MP2/40eb CCSD/24ea CCSD/40eb CCSD(T)/24ea CCSD(T)/40eb

3.535 4.087 3.965 3.455 3.337 4.185 4.005 3.942 3.769 3.63

27.0 14.9 16.3 29.7 35.2 11.0 12.7 14.5 17.9 19.6

0.09 0.03 0.04 0.09 0.12 0.02 0.03 0.03 0.04 0.05

Calculation level

Exptl.e a b c d e

5d and 6s electrons were correlated. 5s, 5p, 5d, and 6s electrons were correlated. 5p, 5d, and 6s electrons were correlated. Ref. [44]. Ref. [41].

and relativistic CCSD(T) calculations have been obtained with very large AE basis sets (29s26p15d12f2g)/[11s12p6d4f2g] for Au2 and AuH [44]. The Au 5d, 6s, and H 1s electrons were correlated in the MCQDPT2/AE calculations [42,43] and the 5p, 5d, 6s, and H 1s electrons were correlated in the CCSD(T)/AE calculations [44]. It is seen that the present MCP calculations provided spectroscopic constants similar to the results of the AE calculations and were comparable with the experimental values, even though the present basis sets were small, indicating that the present MCP sets provide sufficiently high performance for the description of electronic structures in highly correlated calculations. The spectroscopic constants of AuH in the first excited state, A1R+, were also obtained with the state averaged CASSCF/MCP and MCQDPT2/MCP calculations. Both the CASSCF and MCQDPT2 calculations gave almost the same results for Re. However, the excitation energy Te was largely improved by including dynamic correlation. The Te value given by the MCQDPT2 calculations with the [5s4p4d2f1g] set including the dynamic correlation of 20 electrons (including Au 5s and 5p), 3.52 eV, agrees well with the observed value of 3.43 eV within 0.1 eV. The calculated spectroscopic constants of the excited state are also comparable with the corresponding experimental and AE values. The present AE-DK3 calculations with [9s7p6d3f1g] gave almost the same results as those given by the above MCP calculations. Thus, the present spdsMCPs are regarded as capable of describing the electronic structures of the excited states as well as that of the ground state of molecules at a highly correlated level. 5. Conclusions

dynamical correlation of Au/Hg 5s and 5p electrons gave little effect on the spectroscopic constants. Relativistic MCQDPT2 calculations have been obtained with AE basis sets [13s11p7d4f] [42] and [20s17p12d6f] [43] using state averaged CASSCF orbitals for AuH,

We have proposed new model core potentials, spdsMCPs, for the third-row transition–metal atoms from Lu to Hg, in which the 5s, 5p, and 5d electrons are treated explicitly together with the 6s electrons. The present MCP basis sets have been tested

Table 5 Spectroscopic constants of the ground X1R+ and the first excited A1R+ states of AuH. State X1R+

Method/Basis set MCP/[4s2p4d1f] MCP/[5s3p4d2f1g]

AE/[9s7p6d3f1g]

A1R+

AE/[13s11p7d4f]d AE/[20s17p12d6f]e AE/[11s12p6d4f2g]f Exptl.g MCP/[4s2p4d1f] MCP/[5s4p4d2f1g]

AE/[9s7p6d3f1g]

AE/[13s11p7d4f]d AE/[20s17p12d6f]e Exptl.g a b c d e f g

Calculation level CASSCF MCQDPT2/12ea CASSCF MCQDPT2/12ea MCQDPT2/20eb CASSCF MCQDPT2/12ea MCQDPT2/20eb MCQDPT2/12ea MRMP/12ea CCSD(T)/18ec CASSCF MCQDPT2/12ea CASSCF MCQDPT2/12ea MCQDPT2/20eb CASSCF MCQDPT2/12ea MCQDPT2/20eb MCQDPT2/12ea MRMP/12ea

Au 5d and 6s and H 1s electrons were correlated. Au 5s, 5p, 5d, and 6s, and H 1s electrons were correlated. Au 5p, 5d, and 6s, and H 1s electrons were correlated. Ref. [42]. Ref. [43]. Ref. [44]. Ref. [41].

Re (Å)

xe (cm1)

Te (eV)

De (eV)

1.574 1.513 1.569 1.506 1.499 1.573 1.510 1.502 1.494 1.493 1.520 1.524 1.636 1.638 1.638 1.626 1.630 1.650 1.639 1.650 1.586 1.572 1.673

1919.8 2321.5 2004.5 2390.3 2451.2 2003.6 2400.4 2454.7 2480 2414 2328 2305 1794.7 1525.7 1990.5 1821.4 1805.3 1942.1 1754.0 1741.4 2118 2198 1669.6

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 3.02 3.86 2.84 3.82 3.52 2.96 3.96 3.66 3.71 3.27 3.43

1.89 3.01 1.96 3.08 3.25 1.99 3.10 3.24 – – 3.19 3.36

322

H. Mori et al. / Chemical Physics Letters 476 (2009) 317–322

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