Revised model core potentials for second-row transition metal atoms from Y to Cd

Chemical Physics Letters 463 (2008) 230–234

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Revised model core potentials for second-row transition metal atoms from Y to Cd You Osanai a, Eiko Soejima b, Takeshi Noro c, Hirotoshi Mori d,e, Ma San Mon b, Mariusz Klobukowski f, 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 060-0810, Japan d Ocha-dai Academic Production, Division of Advanced Sciences, Ochanomizu University, 2-1-1 Ohtsuka, Bunkyo-ku, Tokyo 112-8610, Japan e JST-CREST, Kawaguchi 332-0012, 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 19 June 2008 In final form 16 July 2008 Available online 31 July 2008

a b s t r a c t We have produced new relativistic model core potentials (spdsMCPs) for the second-row transitionmetal atoms from Y to Cd treating explicitly 4s and 4p electrons in addition to 4d and 5s electrons in the same manner as for the first-row transition-metal atoms given in [Y. Osanai, M.S. Mon, T. Noro, H. Mori, H. Nakashima, M. Klobukowski, E. Miyoshi, Chem. Phys. Lett. 452 (2008) 210]. Using suitable correlating functions together with the split valence MCP functions, we demonstrate that the present MCP basis sets show reasonable performance in predicting the electronic structures of atoms and molecules, bringing about accurate excitation energies for atoms and reasonable spectroscopic constants for AgH. Ó 2008 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 excessive singlet–triplet splittings [22,23], for example. On the other hand, the valence orbitals with a nodal structure in the MCP and AIMP methods can describe the valence correlation effects with reasonable accuracy. We have recently shown for 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 and ns electrons. Both of the MCPs can properly describe the

* 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 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.07.091

ground state properties of transition-metal complexes, but not the excited states with equal accuracy. In a previous study [26], we developed new relativistic MCPs, called spdsMCPs, for the first-row transition-metal atoms from Sc to Zn, explicitly treating 3s and 3p electrons as well as 3d and 4s electrons. The restricted Hartree–Fock (RHF) calculations using the spdsMCPs reproduce the energy separation between the lowest LS state of the 3dn4s2 configuration and that of 3dn+14s1 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 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 diatomic molecules, CuH and Cu2, and found to work well for the excited states of molecules. In the present work, we produce new relativistic MCPs for the second-row transition-metal atoms from Y to Cd in the same manner as those for the first-row transition-metal atoms, with explicit treatment of 4s and 4p electrons as well as 4d and 5s electrons. We briefly describe the determination of newly developed MCP parameters for the second-row transition-metal atoms in Section 2. Applications to atoms involving the calculations of their excitation energies at correlated levels are performed and compared with the AE calculations in Section 3, while applications to molecules are discussed in Section 4. All the calculations have been performed using the program packages, ATOMCI [27,28] for atoms and GAMESS [29] for molecules.

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2. Model core potentials of second-row transition metal atoms In the model core potential method, the atomic Hamiltonian HMCP (1, 2, . . ., Nv) of the Nv valence electrons (in atomic units) is chosen as follows:

HMCP ð1; 2; . . . ; Nv Þ ¼

Nv X i¼1

Nv X 1 hMCP ðri Þ þ 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 ðri Þ þ Bc jwc ihwc j 2 c " # 3 3 X X Z  Nc aI r 2 aJ r 2 V MCP ðrÞ ¼  AI e þ AJ re 1þ r I¼1 J¼1

ð2:2Þ ð2:3Þ

Table 1 Relativistic Hartree–Fock energy of the lowest LS state of 4dn+15s1 with respect to that of 4dn5s2 for the second-row transition-metal atoms Atoms

LS states 4dn+15s1/4dn5s2

Y Zr Nb Mo Tc Ru Rh Pd Ag s.d.c

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 4dn+15s1 relative to 4dn5s2 (eV) pdsMCPa

spdsMCPa

QRHFb

0.884 0.129 0.621 2.241 1.047 0.600 1.269 2.102 4.022 0.123

0.780 0.030 0.761 2.340 0.820 0.725 1.381 2.143 3.936 0.042

0.748 0.007 0.797 2.370 0.754 0.743 1.400 2.098 3.872 –

a

where Z is the atomic number and Nc is the number of core electrons replaced by MCPs. {Aj, aj; Aj, aj} and {Bc} (c = 1s, 2s, 2p, 3s, 3p, and 3d) are the MCP parameters and {wc} denotes the core orbital functions, while {Bc} is defined as:

Bc ¼ Fec

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 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 it 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 4dn5s2 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 the Darwin terms of the Pauli Hamiltonian were added to the HF differential equations, which were then solved in a self-consistent fashion. This allowed for incorporation of the major direct and indirect relativistic effects into the radial orbital functions. 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, p-type five, and d-type five. The valence orbital energies and orbital shapes were 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 second-row transition-metal atoms in which the 4s, 4p and 4d electrons were treated explicitly together with the 5s electrons. These new MCPs are called spdsMCPs. The HF energies of the lowest LS states of the 4dn+15s1 configuration with respect to those of 4dn5s2, given by the use of the present spdsMCPs, are listed in Table 1 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 valence functions were uncontracted. The energy separations calculated with the previous pdsMCPs differs from those from the reference QRHF by 0.13–0.18 eV for the atoms except for Tc and Pd; by 0.293 eV for

Valence basis sets [8s5p5d] were used in the uncontracted form. QRHF: numerical quasi-relativistic Hartree–Fock calculations at the level of the Cowan–Griffin approximation. c Standard deviation of the differences in the MCP values from the reference QRHF values. b

Tc and by 0.004 eV for Pd. Those calculated with the present MCPs agree with those from the QRHF for all atoms within the deviation of 0.07 eV. Thus, our new spdsMCPs are found to give a more reliable description for the excited states in the atoms and molecules containing second-row transtion-metal atoms 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 4dn5s2 and 4dn+15s1 for the Y, Tc, and Ag atoms. The restricted HF calculation was followed by a configuration interaction single and double (CISD) calculation considering the correlation effects of the 4d and 5s 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 spdsMCP valence orbitals (8s5p5d) to (55111/5/311) and adding one more diffuse d-type function and correlating set of [2p2f1g] by Osanai et al. [32]. The exponents of the diffuse d-type GTFs were 0.0225882, 0.0524591, and 0.0733166 for Y, Tc, and Ag, respectively. The other smaller set [4s2p4d1f] is composed of decontracted (6611/5/311) valence set, the diffuse d-type function, and correlating set of [1p1f] [32]. In AE calculations, we first carried out restricted HF-CISD calculations with very large GTF sets of [27s23p19d19f19g] for Y and Tc, and [28s24p20d20f20g] for Ag. These sets were an extension of Huzinaga’s well-tempered sets [33], [27s20p17d] for Y and Tc, and [28s20p17d] for Ag, augmented with the p- and d-type diffuse functions having the same exponents as s-type diffuse functions and the f- and g-type GTFs having the same exponents as d-type GTFs. Then, for comparison with MCP calculations, CISD calculations were carried out with each of the [8s5p5d2f1g] and [7s4p5d1f] sets which was generated by adding each of the correlating [3s2p3d2f1g] and [2s1p3d1f] atomic natural orbital (ANO) sets to the occupied [5s3p2d] HF orbitals. The sets of [8s5p5d2f1g] and [7s4p5d1f] in AE calculations had the same number of correlating orbitals as the sets of [5s3p4d2f1g] and [4s2p4d1f] in MCP calculations, respectively. In the AE calculations, the relativistic effects were taken into account through the third-order Douglas–Kroll (DK3) approximation [34]. Furthermore, we followed similar procedures including the correlation effects of the 4s and 4p electrons in addition to the 4d and 5s electrons with the larger orbital sets. In these calculations, however, 4p orbital was split to (41), except for the Ag atom. We did not split the 4p orbital of Ag because of its linear dependency.

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Table 2 Energy of the lowest LS state of 4dn+15s1 with respect to that of 4dn5s2 for the Y, Tc, and Ag atoms including the correlation effects of the 4d and 5s electrons Atoms

Y

Method/basis set

spdsMCP [5s3p4d2f1g]

[4s2p4d1f]

AE [8s5p5d2f1g]

[7s4p5d1f]

Calculation level

Term energy (a.u.)

Relative energy (eV)a

4dn5s2

4dn+15s1

2

4

D

F

HF CISD CISD + Q HF CISD CISD + Q

37.96686 38.00690 38.01054 37.96662 38.00431 38.00775

37.93848 37.95599 37.95639 37.93827 37.95367 37.95402

0.772 1.385 1.474 0.771 1.378 1.462

HF CISD CISD + Q CISD CISD + Q

3382.95286 3382.99769 3383.00225 3382.99588 3383.00035

3382.92527 3382.94420 3382.94469 3382.94241 3382.94285

0.751 1.456 1.566 1.455 1.565 1.398

6

6

Exptl.b Tc

spdsMCP [5s3p4d2f1g]

[4s2p4d1f]

AE [8s5p5d2f1g]

[7s4p5d1f]

S

spdsMCP [5s3p4d2f1g]

[4s2p4d1f]

AE [8s5p5d2f1g]

[7s4p5d1f]

80.29998 80.42279 80.43118 80.29974 80.40347 80.40921

80.27096 80.40322 80.41012 80.27080 80.38538 80.39107

0.790 0.533 0.573 0.787 0.492 0.494

HF CISD CISD + Q CISD CISD + Q

4283.09272 4283.22642 4283.23690 4283.20990 4283.21873

4283.06484 4283.20588 4283.21404 4283.18796 4283.19480

0.759 0.559 0.622 0.597 0.651 0.406

2

2

HF CISD CISD + Q HF CISD CISD + Q

D 145.40375 145.72152 145.74356 145.40356 145.67060 145.68614

HF CISD CISD + Q CISD CISD + Q

5312.74586 5313.08409 5313.11008 5313.03231 5313.05254

Exptl.b a b

(0.607) (0.691)

(0.705) (0.815) (0.704) (0.814)

D

HF CISD CISD + Q HF CISD CISD + Q

Exptl.b Ag

(0.613) (0.702)

S 145.54882 145.88119 145.90125 145.54845 145.83726 145.85357

5312.88779 5313.23843 5313.26142 5313.19200 5313.21063

3.948 4.345 4.291 3.943 4.535 4.556 3.862 4.200 4.118 4.346 4.302 3.971

(0.257) (0.217) (0.295) (0.293)

(0.200) (0.137) (0.162) (0.108)

(0.397) (0.343) (0.592) (0.613)

(0.338) (0.256) (0.484) (0.440)

Values in parentheses are correlation effects on the relative energy. Ref. [41].

Table 2 gives the energy of the lowest LS state of the 4dn+15s1 configuration with respect to that of 4dn5s2, calculated with the correlation effects of the 4d and 5s electrons, together with the observed value. The correlation contribution to the relative energy is also given in parentheses in the last column. Table 3 shows a similar result that was given by the calculations including the correlation effects of the 4s, 4p, 4d, and 5s electrons. The relative energies given by HF in both the AE and MCP calculations agree closely with those achieved by the QRHF calculations. Including only the correlation effects of the 4d and 5s electrons, the relative energies given by the MCP calculations differ from those by the AE calculations by 0.05–0.17 and 0.10–0.25 eV with the larger and smaller sets, respectively. Including the correlation effects of the 4s, 4p, 4d, and 5s electrons, the differences between the MCP and AE calculations were considerably reduced: the difference in the relative energies is 0.08 eV at most. We also see that inclusion of the correlation effects of the 4s and 4p electrons is important to obtain accurate relative energies. Especially for the Ag atom, deviation in the relative energy from the observed value by the MCP calculations was 0.32 eV when only the correlation effects of the 4d and 5s electrons were included, but it was reduced to 0.05 eV by inclusion of the correlation effects of the 4s, 4p, 4d, and 5s electrons.

4. Molecular applications The present spdsMCP have also been applied in the calculations of molecular spectroscopic constants in the ground X 1R+ state and the excited A1R+ state of the AgH molecule. The basis sets for Ag were [4s2p4d1f] and [5s3p4d2f1g], which were used in the atomic applications, and the basis set of hydrogen was a triple-zeta type [3s2p1d] from Ref. [35]. To obtain the spectroscopic constants of the three low-lying 1R+ states, we performed CASSCF calculations on the average of three low-lying states in the Ag symmetry of D2h with 12 electrons in 7 orbitals, followed by second-order multi-configuration quasi-degenerate perturbation theory (MCQDPT2) calculations [36] to take into account dynamical correlation. In the correlated calculations, the Ag 4d and 5s and H 1s (twelve) electrons were correlated for both basis sets, while the Ag 4s and 4p electrons (total twenty electrons) were also correlated for the largest set. We used the intruder state avoidance (ISA) method [37] with the ISA shift of 0.02. The energies of the dissociation limit for these single reference calculations were obtained by supermolecular calculations at an internuclear distance of 100 a.u. The results of the spectroscopic constants for the low-lying 1R+ states of AgH obtained by a third-degree polynomial fitting are

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Y. Osanai et al. / Chemical Physics Letters 463 (2008) 230–234 Table 3 Energy of the lowest LS state of 4dn+15s1 with respect to that of 4dn5s2 for the Y, Tc, and Ag atoms including the correlation effects of the 4s, 4p, 4d, and 5s electrons Atoms

Y

Method/basis set

spdsMCP [5s4p4d2f1g]

AE [8s6p5d2f1g]

Calculation level

Relative energy (eV)a

Term energy (a.u.) 4dn5s2

4dn+15s1

2

4

D

F

HF CISD CISD + Q

37.96692 38.17980 38.19798

37.93851 38.13856 38.15088

0.773 1.122 (0.349) 1.282 (0.509)

HF CISD CISD + Q

3382.95286 3383.24084 3383.26597

3382.92527 3383.20192 3383.22070

0.751 1.059 (0.308) 1.232 (0.481) 1.398

6

6

80.30029 80.65771 80.68686 4283.09272 4283.51822 4283.55412

80.27097 80.64075 80.66875 4283.06484 4283.50276 4283.53694

2

D 145.40375 145.93851 145.98096 5312.74586 5313.36677 5313.41953

2

Re (Å)

xe (cm1)

Te (eV)

De (eV)

CASSCF MCQDPT2/12ea CASSCF MCQDPT2/12ea MCQDPT2/20eb MCQDPT2/12ea MCQDPT2/12ea

1.763 1.623 1.742 1.608 1.589 1.564 1.564 1.618

1223 1721 1322 1774 1837 2024 2073 1760

0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

1.45 2.00 1.46 2.04 2.13 – 2.53 2.39

CASSCF MCQDPT2/12ea CASSCF MCQDPT2/12ea MCQDPT2/20eb MCQDPT2/12ea MCQDPT2/12ea

1.686 1.658 1.703 1.670 1.654 1.750 1.717 1.665

1620 1531 1521 1429 1575 1166 1422 1490

3.615 4.105 3.413 4.019 3.839 4.03 3.993 3.714

Exptl.b Tc

spdsMCP [5s4p4d2f1g]

AE [8s6p5d2f1g]

S

HF CISD CISD + Q HF CISD CISD + Q

D 0.798 0.461 0.493 0.759 0.421 0.468 0.406

b

Exptl. Ag

spdsMCP [5s3p4d2f1g]

AE [8s5p5d2f1g]

HF CISD CISD + Q HF CISD CISD + Q

S 145.54882 146.08911 146.12881 5312.88779 5313.51516 5313.56453

3.948 4.098 4.023 3.862 4.038 3.946 3.971

b

Exptl. a b

(0.351) (0.545) (0.280) (0.484)

(0.150) (0.075) (0.176) (0.084)

Values in parentheses are correlation effects on the relative energy. Ref. [41].

Table 4 Spectroscopic constants of the ground X1R+ and excited states A1R+ of AgH State X1R+

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

AE/[13s10p8d4f]c AE/[13s10p8d4f3g]d Exptl.e A1R+

MCP/[4s2p4d1f] MCP/[5s3p4d2f1g]

AE/[13s10p8d4f]c AE/[13s10p8d4f3g]d Exptl.e a b c d e

Calculation level

Ag 4d and 5s and H 1s electrons were correlated. Ag 4s and 4p electrons were correlated, as well the Ag 4d and 5s and H 1s electrons. Ref. [39]. Ref. [40]. Ref. [38].

listed in Table 4 together with the corresponding experimental values [38]. For the ground state, CASSCF/MCP calculations with both basis sets gave a somewhat longer equilibrium distance, Re, and a much smaller vibrational frequency xe and binding energy De. These results were significantly improved by considering the electron correlations within the 12 (Ag 4d and 5s and H 1s) electrons in MCQDPT2/MCP. However, the inclusion of the dynamical correlation of Ag 4s and 4p electrons gave slightly worse spectroscopic constants, indicating that the agreement of the results of the calculation with those including electron correlations of twelve electrons was accidental. Relativistic MCQDPT2 calculations with very large all-electron (AE) basis sets [13s10p8d4f] and

[13s10p8d4f3g] have been performed by Lo and Klobukowski [39] and Witek et al. [40], respectively, using state averaged CASSCF orbitals. The twelve electrons were correlated in the MCQDPT2/ AE calculations. It is notable that the present MCQDPT2/MCP provided spectroscopic constants that are comparable with the experimental values and the results of the AE calculations in spite of the use of much smaller basis sets; this indicates that the present MCP set provides sufficiently high performance for description of electronic structures in highly correlated calculations. As for the A1R+ state, CASSCF and MCQDPT2 calculations with MCPs gave nearly identical results for Re and xe. For the term energy, Te, however, the MCQDPT2 calculations with twelve corre-

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lated electrons increased the deviation between the calculated and experimental values worse, whereas CASSCF gave fairly good calculated values. The inclusion of dynamical correlation including 4s and 4p electrons largely improved Te; the calculations MCQDPT2 with [5s3p4d2f1g] including the correlation of Ag 4s and 4p electrons reproduced the observed Te values within 0.1 eV. The calculated molecular constants for the A state are also comparable with the experimental and AE values. Thus, the present spdsMCPs are capable of describing electronic structures of excited states as well as those of the ground state of molecules at highly correlated levels. The accuracy in calculated Te values will still be improved if the correlations of Ag 4s and 4p electrons are taken into account in the AE calculations [39,40]. 5. Conclusions New model core potentials, spdsMCPs, have been proposed for the second-row transition-metal atoms from Y to Cd, in which the 4s, 4p, and 4d electrons are treated explicitly together with the 5s electrons. The present spdsMCP basis sets have been tested in the calculation of the energy differences between the 4dn5s2 and 4dn+15s1 states for Y, Tc, and Ag atoms, and the spectroscopic constants of the X 1R+ and A1R+ states of the AgH molecule. It has been shown that the proposed spdsMCP basis sets describe well these physical properties. The MCP sets are available in the homepage of Ref. [35]. Acknowledgments The present study has been supported partly by a Grant-in-Aid for Scientific Research from the Japan Society for 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.

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