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Relativistic all-electron ab initio calculations on the platinum hydride molecule
13May 1994
CHEMICAL
PnYslcs LETTERS ELSEVIER
Chemical Physics Letters 222 (1994) 267-273
Relativistic all-electron ab initio calculations on the platinum hydride molecule Timo Fleig, Christel M. Marian Institutfir Physikalische und Theoretische Chemie der Universitdt Bonn, WegelerstrajJe 12, D-53115 Bonn, Germany
Received 20 January 1994;in final form 17 March 1994
Abstract The results of relativistic self-consistent field, complete active space self-consistent field and multireference configuration interaction calculations for the platinum hydride molecule are presented, using the spin-free no-pair Hamiltonian in a basis set expansion. Spin-orbit interaction effects are taken into account in variational perturbation theory by using a spin-orbit operator derived from no-pair theory for the determination of spindependent integrals. Molecular properties are obtained at multireference configuration interaction and spin-orbit corrected levels of calculation and the results are compared to those of numerical Dirac-Hartree-Fock relativistic configuration interaction, and calculations making use of relativistic effective core potentials.
1. Introduction
The platinum hydride molecule has been the centre of interest in recent studies making use of relativistic effective core potentials [ 1-7 ] and in relativistic allelectron treatments [ 8,9]. PtH being a transition metal compound containing a heavy atom has to be subjected to a relativistic treatment to obtain molecular properties such as bond distances, excitation energies and vibrational frequencies with the desired accuracy. For these purposes spin-free and spin-orbit Hamiltonians derived from no-pair theory [ lo121, respectively modified versions implemented by Samzow and He13 [ 13,141 are employed in the present case. These have already been applied successfully in electronic structure calculations of several transition metal compounds for comparison with experimental data. The spin-free part described in detail below was used in studies of NiH [ 151, CuH [ 161 and AuH [ 17 ] ; in NiH and CuH spin-orbit interaction was considered with the Breit-Pauli Hamiltonian. LiHg [ 181 has been investigated on using the
spin-free as well as the spin-orbit Hamiltonian derived from no-pair theory which is presented below. The purpose of this Letter is to report the results obtained using a one-component spin-averaged formalism as well as those results gained on consideration of spin-orbit coupling. The electronic configuration s’d9 of the platinum atom gives rise to two D terms, the component ‘Da representing the ground state [ 19 1. This configuration is well prepared to form a o bond with atomic hydrogen due to the partially occupied Pt 6s orbital. As expected, the corresponding molecular state multiplet *{Z+, II, A} is the lowest-lying group of states with respect to total energy. This notation is in accordance with Hund’s coupling case (a) but will be replaced later on by the representation of the angular momentum projection w corresponding to Hund’s case (c) because of considerable mixing of spin-orbit split states with equal Sz quantum number. The 4{C+, II, A} states originating from the s1d9 contiguration of atomic platinum are supposed to be repulsive and will not be considered here. At large inter-
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI0009-2614(94)00348-T
T. Fleig, CM. Marian / Chemical PhysicsLetters 222 (1994) 267-273
268
nuclear separations where the recoupling to the atomic states sets in they are expected to mix heavily with the corresponding doublet states. We have, therefore, confined the range of PtH bond distances to values smaller than 5 aO.Comparison with the results obtained from numerical DHF, relativistic CI and RECP calculations confirms the reliability of our computational method especially when dealing with diatomic transition metal compounds.
2. Theory The Hamiltonian used for calculating kinematic relativistic corrections is the spin-free no-pair Hamiltonian with external field projectors correct to second-order in the external potential,
where Ei=J&GGF) V~(i)=-eAi[~~/,,,(i)+Ri~=‘,,(i)Ri]Ai -W;f(i)EiW;f-~{(W~f)2yEi))
&=
pi Ei+mc2’
Wf is an integral operator with the kernel
This Hamiltonian can be derived from QED through several approximations. Additional details of its derivation and implementation in quantum chemical programs can be found in refs. [ 13,141. The zero-field splitting of molecular PtH states is taken into account by variational perturbation theory in the present case. As the spin-orbit Hamiltonian HBP in the Breit-Pauli approximation [ 201 exhibits an rm3 behaviour and, therefore, leads to a variational collapse, we employ the relativistic spinorbit Hamiltonian,
where
Ai=&* It is also derived from no-pair theory by inclusion of the Breit interaction [ 2 1] and is bounded from below which allows a variational treatment of the spinorbit interaction.
3. Methods and technical details
The calculations on the PtH molecule are performed with a finite basis set of Cartesian Gaussian functions. The Pt atom is described by a ( 16s, 14p, 1Od, 5f) primitive set [ 221 augmented by three s, two p, two d and two f functions for consideration of polarization effects. The primitive Gaussians ( 19s, 16p, 12d, 7f) are contracted to [ 9s, 6p, 5d, 3f] according to a generalized scheme. The H atom is represented by a contraction of the 6s primitive set of van Duijneveldt [ 23 ] enlarged by a semi-diffuse s Gaussian along with four p [ 241 and one d function for polarization purposes. The generalized contraction scheme for H then reads (7s, 4p, Id) --t [ 4s, 3p, 1d]. The inclusion of basis functions with low exponents and angular momentum quantum numbers greater than zero, so as to describe an Hg- ion, is reasonable because of an expected slightly negative charge on the hydrogen atom. SCF, CASSCF and CI calculations are performed with the relativistic Hamiltonian H$’ and by use of the SWEDEN program package #‘. SCF molecular orbitals are optimized by calculating an open-shell wavefunction with orbitals of Q, x and 6 symmetry singly occupied in the particular case. The generation of a one-particle basis in common “’ MOLECULE-SWEDEN is an electronic structure program system written by J. Ahnlaf, C.W. Bauschlicher Jr., M.R.A. Blomberg, D.P. Chong, A. Heiberg, S.R. Langhoff, P.-A. Malmqvist, A.P. Rendell, B.O. Roos, P.E.M. Siegbahn and P.R. Taylor.
269
T. Fleig, CM. Marian /Chemical PhysicsLetters 222 (1994) 267-273
for all multiplet states is carried out by including valence correlation effects employing the CASSCF procedure. The use of such an orbital basis is essential if succeeding spin-orbit calculations are to be performed with a zeroth-order CI wavefunction as a starting point in variational perturbation theory. Furthermore, the evaluation of transition density matrices for non-orthogonal sets of orbitals obtained in different symmetries is avoided by these means. The active space contains four active orbitals (CAS 4) of o symmetry which are composed of varying contributions of atomic Pt orbitals 6s, Sd, 5p and the H orbital 1s. Three electrons are distributed among this active space describing the *C+ state. Furthermore, CASSCF calculations with eleven electrons in twelve active orbitals (CA!3 12 ) are carried out for all three symmetries E, II and A but under moderate restriction of the theoretically possible excitations. The active space consists of four o, four R and four 6 orbitals and the symmetry subgroup the calculations are carried out in is CzV. In addition to some minor restrictions for excitations involving electron transfer from one irreducible representation of Czy to another the main constraints can be deduced from Table 1. The number of electrons in orbitals with a given angular momentum quantum number is fmed at three for the symmetry in question whilst four electrons occupy the remaining angular momenta. The following CI calculations are performed with the multireference single and double excitation method (MRD-CI) by Buenker and Peyerimhoff [ 25 ] and molecular orbitals taken from the CA!3 4 optimization. Eleven electrons are correlated and six reference configurations used for *E+, eight for *II and *A. The Pople correction has been added to estimate the effect of unlinked cluster contributions [ 26 1. Table 1 Minimum/maximum number of electrons occupying the active orbitals in CAS 12 calculations depending on angular momentum; total number of generated configurations as due to restrictions u
II:
6
Configurations
313 414 414
414 414 313
414 313 414
13100 15748 16372
The aforementioned Gaussian basis is used for calculating spin-dependent integrals with the operator Hy. The spin-orbit splitting of molecular states is taken into consideration with the CAS 4-CI wavefunction as a zeroth-order approximation.
4. Results and discussion Equilibrium geometries and excitation energies at the various computational levels are presented in Table 2. The extensive consideration of dynamical correlation in the valence region (CAS 4X1) results in a bond length contraction in all molecular fi states compared to the SCF values. This phenomenon is a characteristic feature of such CI treatments in general. It also has to be noted that the CAS 12 calculation making use of a comparatively large active space does not affect bond lengths significantly. The inclusion of electron correlation is responsible for the substantial stabilization of d electron rich states. Table 2 also summarizes the results for excitation energies at the equilibrium geometry at the computational levels used so far. In contrast to the bond lengths an even qualitative difference between SCF and correlation treatments can be made out in this case. Whereas *A is found to be the ground state at SCF level, the d electron rich *Z+ state is stabilized significantly as correlation effects are taken into account. This relative displacement is strong enough to let *Z+ even drop to the ground state at CAS 12 and CA!3 4-CI levels. The energy difference between *A Table 2 Equilibrium bond lengths r, (a,,) and excitation energies Lr, (cm-‘) for the low-lying electronic states rz+
SCF ’ US12 b CAS 4-U =
2l-l
sL\
re
T,
r,
T,
re
r,
2.92 2.91 2.84
392 0 0
2.93 2.92 2.86
0 1548 1873
3.04 3.05 2.97
4281 6252 7092
* Molecular orbitals obtained from an SCF calculation in the particular symmetry. b Molecular orbitals obtained from CASSCF calculation with eleven electrons in twelve active orbitals. c MRD-CI calculations based on CAS 4 optimized MO; selection threshold for CI contigurations: 1.O ).I&,.
210
T. Fleig, C.M. Marian /Chemical Physics Letters 222 (1994) 267-273
Fig. 1. Molecular AE states from MRD-CI calculations including ACPF estimation and CAS 4 x state.
and *II is only slightly enlarged from SCF to CAS 4CI calculations. Fig. 1 displays the potential energy curves for the CAS 4 and corresponding MRD-CI calculations including an ACPF estimation of the energy values. When spin-orbit coupling is taken into consideration the discussed states are subjected to a splitting into five states which can be characterized as o= 5/2(I), 1/2(I), 3/2(I), 3/2(11), l/2(11). 5/2(I) is a pure 2Astate whereas the l/2 and 312 states consist of U-coupled electronic states 2Eie/2-2111,2and respectively. In Table 3 all linear com2A3,2-2b,2, binations of these states including the appropriate mixing coefficients are listed. As o= 512 is a AsI2 state corresponding to the angular momentum projections n = 2 and Z= l/2 it does not mix with any other AZ state. The lower l/2(1) state, however, is affected by 2111,2and can be characterized basically as ‘C h2, but the 2111,2component gains in significance as the nuclei are separated. w = 3/2(I) is mainly of 2As,2 character at short interatomic distances but switches to 211s,2around 2.7 ao. For o= 3/2 (II) the exact opposite is the case whilst o= l/2 (II) is mainly of 2111,2kind, especially at short bond distances. Fig. 2 shows these five states obtained from calculations at seven different geometries. An energy gap between the three lower and the two upper states is distinguishable. The lower corre-
spond to the atomic dS,2 hole states, the upper to atomic dSj2 hole states and this assignment finds its ratification as the dissociation limit is approached. Finally, a collection of the present results is to be compared with appropriate values obtained from Dirac-Hartree-Fock, relativistic CI, and RECP calculations. Table 4 contains equilibrium geometries, the excitation energies at these bond lengths and the harmonic frequencies corresponding to the spin-orbit split states. The most obvious conclusion to be drawn from this comparison is the reliability of the present approach if the four-component RCI and experimental results are taken as guidance. The RCI treatment of ref. [ 9 ] is comparable to our CI expansion. Although Visscher et al. apply four-component methods to the problem, it has to be stated that they make use of the no-pair approximation excluding electron and positron pair processes as well. There is no sense in comparing atomic orbital bases for fourcomponent spinors and one-component molecular orbitals as the description of the former is much more demanding. Furthermore, we have decided to outline the advantages of all-electron calculations in comparison with the RECP-SCF-CI treatment carried out by Wang and Pitzer [2] and the RECPCASSCF-RCI by Balasubramanian and Feng [ 6 1, as these studies are the only ones that hold information at the spin-orbit coupled level. Wang and Pitzer em-
T. Fleig, CM. Marian /Chemical Physics Letters 222 (1994) 267-273
271
Table 3
Spin-orbit split states as linear combinations of AZ states at
severalinternuclear
distances
2.0
1/2W) 3/2(w 3/2(I) 1/2(I)
-0.988 -0.515 0.515 -0.988
IA=l,M,=-0.5)+0.155 IA=O, Ms=OS) IA=2, MS=-0.5)+0.857 iA= 1, h&=0.5) IA= 1, M,=O.5) +0.857 IA=2, MS= -0.5) IA=O, M,=O.5)-0.155 IA=l,Ms=-0.5)
2.3
l/2(11) 3/2(w 3/2(I) 1/2(I)
-0.977 -0.610 0.610 -0.977
IA= 1, MS= -0.5) +0.213 IA=O, M,=O.5) IA=2, MS= -0.5) +0.793 IA=l, M~=0.5) IA= 1, M,=O.5)+0.793 IA=2, MS= -0.5) IA=O, M,=O.5)-0.213 IA=l, MS= -0.5)
2.7
l/2(11) 3/2W) 3/2(I) 1/2(I)
-0.956 -0.708 -0.708 -0.956
IA=l, IA=2, IA=l, IA=O,
2.85
l/2(11) 3/2w 3/2(I) 1/2(I)
-0.946 -0.737 -0.737 -0.946
lA=l,M,= -0.5) +0.325 (A=O, h&=0.5) IA=2, MS=-0.5) +0.676 Id= 1, M,=O.5) IA= 1, h&=0.5) -0.676 IA=2, M,= -0.5) IA=O, h&=0.5) -0.325 IA= 1, MS= -0.5)
3.1
l/2(11) 3/2w) 3/2(I) 1/2(I)
-0.928 -0.779 -0.779 -0.928
IA=l,M,=-0.5)+0.372 lA=O,M,=O.5) IA=2, MS=-0.5) +0.627 IA= l,M,=O.5) IA= 1, M,=O.5) -0.627 IA=2, M,= -0.5) IA=O, M,=O.5) -0.372 IA= 1, MS= -0.5)
4.0
1/2uu 3/2W 3/2(I) 1/2(I)
-0.864 -0.850 -0.850 -0.864
IA= 1, MS= -0.5) +0.504 IA=O, h&=0.5) IA=2, MS= -0.5) +0.527 IA= 1, M,=O.5) IA=l,M,=0.5) -0.527 IA=2, MS= -0.5) (A=O, M,=O.5)-0.504 IA=l, M,= -0.5)
5.0
1/2uu 3/2W 3/2w 1/2(I)
-0.819 -0.879 -0.879 -0.819
IA= 1, MS= -0.5) +0.573 IA=O, M,=O.5) IA=2,MS= -0.5) +0.477 IA=l, M,=0.5) IA= 1, M,=O.5) -0.477 IA=2, MS= -0.5) IA=O,M,=0.5)-0.573 I/l-1,&=-0.5)
ploy small ST0 valence basis sets and limited CI expansion lengths in their investigation. The active space defined in the CASSCF calculation of ref. [ 6 ] is comparable to our CAS 4 space concerning the number of unique configurations in the full-C1 expansion, but our CI treatment is considerably more extensive than the RCI calculation of Balasubramanian et al. Concerning excitation energies excellent agreement is observed between the DHF and RCI values, the results of the present no-pair CI calculations including spin-orbit coupling and experimental values being known so far. As already noticed by Visscher [ 91 the excitation energies of the spin-orbit mixed states are hardly affected by the inclusion of electron correlation, in contrast to our findings for the one component states presented in Table 2. Only a slight stabilization of the l/2(1) state with respect to the
MS= -0.5) +0.294 IA=O, M,=O.5) MS=-0.5) +0.707 IA=l, M,=O.5) M,=O.5)-0.707 (A=2, MS= -0.5) h&=0.5) -0.294 iA= 1, MS= -0.5)
DHF treatment by Dyall [ 8 ] is recognized. On the other hand, the bond length contraction due to the dynamical valence electron correlation in the onecomponent states is carried over to the spin-orbit mixed potentials. The equilibrium bond lengths resulting from the no-pair calculations are slightly shorter than the RCI values, probably due to the more extensive electron correlation treatment in the former. Good agreement with experiment is observed for the 3/2(I) state; the equilibrium distance for w=5/2 turns out to be too short just like the RCI result. The experimentally determined r, values are derived from rotational parameters B,with the latter parameter fitted to a comparatively simple effective Hamiltonian [ 28 1. The considerably different equilibrium bond lengths deduced from the spectra of PtH and PtD on the other hand suggest strong perturbations. The deviation observed for the w= 512 ground
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T. Fleig, C.M. Marian /Chemical PhysicsLetters 222 (1994) 267-273
Fig. 2. Spin-orbit split states composed of mixed AZ states. Table 4 Equilibrium bond lengths r. (a,), excitation energies r, (eV), and harmonic frequencies w, (cm-i)
for spin-orbit split states
W
5/2(I)
WP’ BFb DHF = RCI d present exp. c
3/2(I)
1/2(I)
1/2(II)
re
T,
w.
r,
T,
w.
r,
T,
we
r,
K
w,
r,
T,
w.
3.04 2.93 2.93 2.87 2.84 2.888
0 0 0 0 0 0
2020 2177 2234 2458 2423 2378
3.00 2.91 2.91 2.88 2.86
0.12 0.186 0.34 0.24 0.17
2045 2188 2094 2419 2355
3.19 2.99 2.99 2.91 2.89 2.812
0.34 0.521 0.42 0.44 0.45 0.400
2016 2155 2080 2313 2252 2265
3.00 2.98 2.91 2.88
2.74 1.346 1.43 1.46 1.46 1.45
2179 2162 2365 2331
2.99 3.00 2.95 2.91
1.59 1.348 1.55 1.61 1.54
2021 2097 2217 2253
‘Ref. [2]. bRef. [6]. ‘Ref. [S]. dRef. [9]. e Ref. [27], and revalues obtained for PtD: 2.880for 0=5/2(I)
state
3/2(II)
may thus be ascribed to experimental uncertainties in this case as well. The vibrational frequencies obtained in the present treatment are close to experiment. Finally, the molecular properties obtained by all-electron methods are in much better agreement with experimental results than those derived using the type of spin-dependent relativistic core potentials as used by Wang et al. and Balasubramanian et al. substantiating the general reliability of the former approaches when dealing with such molecules.
and2.878 for w=3/2(1).
Excitation energies from Ref. [28].
5. Conclusion The accomplished study of the PtH molecule provides an analysis of the lowest-lying state multiplet without and with the inclusion of spin-orbit coupling. The overall splitting according to this effect turns out to be up to 17000 cm-’ at short distances and can therefore no longer be considered a perturbation. The importance of kinematic relativistic corrections in the calculations is substantiated for achieving high accuracy of molecular properties. Evidence is given for the competitiveness of the no-pair formalism with four-component all-electron treat-
T. Fleig, CM. Marian / Chemical Physics Letters 222 (1994) 267-273
ments when dealing with heavy transition-metal diatomics.
Acknowledgement
We would like to thank Professor Peyerimhoff for financial support. The computational facilities made available to us by the ‘Deutsche Forschungsgemeinschaft’ through SFB 334 and the WAP-program are greatfully acknowledged.
References [ 1] H. Basch and S. Topiol, J. Chem. Phys. 71 ( 1979) 802. [2]S.W. WangandK.S.Pitzer, J.Chem.Phys. 79 (1983) 3851. [ 3 ] C. McMichael Rohllii P.J. Hay and R.L. Martin, J. Chem. Phys. 85 (1986) 1447. [4] A. Gavezzotti, G.F. Tantardini and M. Simonetta, Chem. Phys. Letters 129 (1986) 577. [ 51 S. Tobisch and G. Rasch, Chem. Phys. Letters 166 (1990) 311. [6] K. Balasubramanian and P.Y. Feng, J. Chem. Phys. 92 (1990) 541. [ 710. Gropen, J. Almlof and U. Wahlgren, J. Chem. Phys. 96 (1992) 8363. [S] K. Dyall, J. Chem. Phys. 98 (1993) 9678. [9] L. Visscher, T. Saue, W.C. Nieuwpoort, K. Faegri and 0. Gropen, J. Chem. Phys. 99 (1993) 6704. [lo] J. Sucher, Phys. Rev. A 22 (1980) 348. [ 111 M. Douglas and N.M. Kroll, Ann. Phys. 82 ( 1974) 89.
273
[ 121 G. Hardekopf and J. Sucher, Phys. Rev. A 31 ( 1985) 2020. [ 131 B.A. He& Phys. Rev. A 33 (1986) 3742. [ 141 R. Samzow and B.A. Hell, Chem. Phys. Letters 184 ( 1991) 491. [ 151 C.M. Marian, J. Chem. Phys. 93 (1990) 1176. [16] C.M. Marian, J. Chem. Phys. 94 (1991) 5574. [ 171 G. Jansen and B.A. He& Z. Physik D 13 (1989) 363. [IS] M.M. Gleichmann and B.A. He& J. Chem. Phys., to be published. [ 191 C.E. Moore, Atomic energy levels, NSRDS-NBS Circular 467 (US GPO, Washington, 1949). [20] S.R. Langhoff and C.W. Kern, in: Modem theoretical chemistry, ed. H.F. Schaefer III (Plenum Press, New York, 1977). [21] G. Breit, Phys. Rev. 34 (1929) 553. [ 2210. Gropen, unpublished work. [23] F.B. van Duijneveldt, IBM Res. Rept. RJ 945. [24] D.P. Chong and S.R. Langhoff, J. Chem. Phys. 84 (1986) 5606. [25] R.J. Buenker and S.D. Peyerimhoff, Theoret. Chim. Acta 35(1974)33;39(1975)217 R.J. Buenker, S.D. Peyerimhoff and W. Butscher, Mol. Phys. 35 (1978) 771; R.J. Buenker, in: Quantum chemistry in the SO’s, ed. P. Burton (University Press, Wollongong, 1980); in: Current aspects of quantum chemistry, ed. R. Carbo (Elsevier, Amsterdam, 1982); R.J. Buenker and R.A. Phillips, J. Mol. Struct. THEGCHEM 123 (1985) 291. [26] J.A. Pople, R. Seeger and R. Krishnan, Intern. J. Quantum Chem. Quantum Chem. Symp. 11 (1977) 149. [ 271 G. Gustafsson and R. Scullman, Mol. Phys. 67 ( 1989) 98 1. [28] M.C. McCarthy, R.W. Field, R. EnglemanandP.F. Bemath, J. Mol. Spectry. 158 ( 1993) 208.
CHEMICAL
PnYslcs LETTERS ELSEVIER
Chemical Physics Letters 222 (1994) 267-273
Relativistic all-electron ab initio calculations on the platinum hydride molecule Timo Fleig, Christel M. Marian Institutfir Physikalische und Theoretische Chemie der Universitdt Bonn, WegelerstrajJe 12, D-53115 Bonn, Germany
Received 20 January 1994;in final form 17 March 1994
Abstract The results of relativistic self-consistent field, complete active space self-consistent field and multireference configuration interaction calculations for the platinum hydride molecule are presented, using the spin-free no-pair Hamiltonian in a basis set expansion. Spin-orbit interaction effects are taken into account in variational perturbation theory by using a spin-orbit operator derived from no-pair theory for the determination of spindependent integrals. Molecular properties are obtained at multireference configuration interaction and spin-orbit corrected levels of calculation and the results are compared to those of numerical Dirac-Hartree-Fock relativistic configuration interaction, and calculations making use of relativistic effective core potentials.
1. Introduction
The platinum hydride molecule has been the centre of interest in recent studies making use of relativistic effective core potentials [ 1-7 ] and in relativistic allelectron treatments [ 8,9]. PtH being a transition metal compound containing a heavy atom has to be subjected to a relativistic treatment to obtain molecular properties such as bond distances, excitation energies and vibrational frequencies with the desired accuracy. For these purposes spin-free and spin-orbit Hamiltonians derived from no-pair theory [ lo121, respectively modified versions implemented by Samzow and He13 [ 13,141 are employed in the present case. These have already been applied successfully in electronic structure calculations of several transition metal compounds for comparison with experimental data. The spin-free part described in detail below was used in studies of NiH [ 151, CuH [ 161 and AuH [ 17 ] ; in NiH and CuH spin-orbit interaction was considered with the Breit-Pauli Hamiltonian. LiHg [ 181 has been investigated on using the
spin-free as well as the spin-orbit Hamiltonian derived from no-pair theory which is presented below. The purpose of this Letter is to report the results obtained using a one-component spin-averaged formalism as well as those results gained on consideration of spin-orbit coupling. The electronic configuration s’d9 of the platinum atom gives rise to two D terms, the component ‘Da representing the ground state [ 19 1. This configuration is well prepared to form a o bond with atomic hydrogen due to the partially occupied Pt 6s orbital. As expected, the corresponding molecular state multiplet *{Z+, II, A} is the lowest-lying group of states with respect to total energy. This notation is in accordance with Hund’s coupling case (a) but will be replaced later on by the representation of the angular momentum projection w corresponding to Hund’s case (c) because of considerable mixing of spin-orbit split states with equal Sz quantum number. The 4{C+, II, A} states originating from the s1d9 contiguration of atomic platinum are supposed to be repulsive and will not be considered here. At large inter-
0009-2614/94/$07.00 0 1994 Elsevier Science B.V. All rights reserved SSDI0009-2614(94)00348-T
T. Fleig, CM. Marian / Chemical PhysicsLetters 222 (1994) 267-273
268
nuclear separations where the recoupling to the atomic states sets in they are expected to mix heavily with the corresponding doublet states. We have, therefore, confined the range of PtH bond distances to values smaller than 5 aO.Comparison with the results obtained from numerical DHF, relativistic CI and RECP calculations confirms the reliability of our computational method especially when dealing with diatomic transition metal compounds.
2. Theory The Hamiltonian used for calculating kinematic relativistic corrections is the spin-free no-pair Hamiltonian with external field projectors correct to second-order in the external potential,
where Ei=J&GGF) V~(i)=-eAi[~~/,,,(i)+Ri~=‘,,(i)Ri]Ai -W;f(i)EiW;f-~{(W~f)2yEi))
&=
pi Ei+mc2’
Wf is an integral operator with the kernel
This Hamiltonian can be derived from QED through several approximations. Additional details of its derivation and implementation in quantum chemical programs can be found in refs. [ 13,141. The zero-field splitting of molecular PtH states is taken into account by variational perturbation theory in the present case. As the spin-orbit Hamiltonian HBP in the Breit-Pauli approximation [ 201 exhibits an rm3 behaviour and, therefore, leads to a variational collapse, we employ the relativistic spinorbit Hamiltonian,
where
Ai=&* It is also derived from no-pair theory by inclusion of the Breit interaction [ 2 1] and is bounded from below which allows a variational treatment of the spinorbit interaction.
3. Methods and technical details
The calculations on the PtH molecule are performed with a finite basis set of Cartesian Gaussian functions. The Pt atom is described by a ( 16s, 14p, 1Od, 5f) primitive set [ 221 augmented by three s, two p, two d and two f functions for consideration of polarization effects. The primitive Gaussians ( 19s, 16p, 12d, 7f) are contracted to [ 9s, 6p, 5d, 3f] according to a generalized scheme. The H atom is represented by a contraction of the 6s primitive set of van Duijneveldt [ 23 ] enlarged by a semi-diffuse s Gaussian along with four p [ 241 and one d function for polarization purposes. The generalized contraction scheme for H then reads (7s, 4p, Id) --t [ 4s, 3p, 1d]. The inclusion of basis functions with low exponents and angular momentum quantum numbers greater than zero, so as to describe an Hg- ion, is reasonable because of an expected slightly negative charge on the hydrogen atom. SCF, CASSCF and CI calculations are performed with the relativistic Hamiltonian H$’ and by use of the SWEDEN program package #‘. SCF molecular orbitals are optimized by calculating an open-shell wavefunction with orbitals of Q, x and 6 symmetry singly occupied in the particular case. The generation of a one-particle basis in common “’ MOLECULE-SWEDEN is an electronic structure program system written by J. Ahnlaf, C.W. Bauschlicher Jr., M.R.A. Blomberg, D.P. Chong, A. Heiberg, S.R. Langhoff, P.-A. Malmqvist, A.P. Rendell, B.O. Roos, P.E.M. Siegbahn and P.R. Taylor.
269
T. Fleig, CM. Marian /Chemical PhysicsLetters 222 (1994) 267-273
for all multiplet states is carried out by including valence correlation effects employing the CASSCF procedure. The use of such an orbital basis is essential if succeeding spin-orbit calculations are to be performed with a zeroth-order CI wavefunction as a starting point in variational perturbation theory. Furthermore, the evaluation of transition density matrices for non-orthogonal sets of orbitals obtained in different symmetries is avoided by these means. The active space contains four active orbitals (CAS 4) of o symmetry which are composed of varying contributions of atomic Pt orbitals 6s, Sd, 5p and the H orbital 1s. Three electrons are distributed among this active space describing the *C+ state. Furthermore, CASSCF calculations with eleven electrons in twelve active orbitals (CA!3 12 ) are carried out for all three symmetries E, II and A but under moderate restriction of the theoretically possible excitations. The active space consists of four o, four R and four 6 orbitals and the symmetry subgroup the calculations are carried out in is CzV. In addition to some minor restrictions for excitations involving electron transfer from one irreducible representation of Czy to another the main constraints can be deduced from Table 1. The number of electrons in orbitals with a given angular momentum quantum number is fmed at three for the symmetry in question whilst four electrons occupy the remaining angular momenta. The following CI calculations are performed with the multireference single and double excitation method (MRD-CI) by Buenker and Peyerimhoff [ 25 ] and molecular orbitals taken from the CA!3 4 optimization. Eleven electrons are correlated and six reference configurations used for *E+, eight for *II and *A. The Pople correction has been added to estimate the effect of unlinked cluster contributions [ 26 1. Table 1 Minimum/maximum number of electrons occupying the active orbitals in CAS 12 calculations depending on angular momentum; total number of generated configurations as due to restrictions u
II:
6
Configurations
313 414 414
414 414 313
414 313 414
13100 15748 16372
The aforementioned Gaussian basis is used for calculating spin-dependent integrals with the operator Hy. The spin-orbit splitting of molecular states is taken into consideration with the CAS 4-CI wavefunction as a zeroth-order approximation.
4. Results and discussion Equilibrium geometries and excitation energies at the various computational levels are presented in Table 2. The extensive consideration of dynamical correlation in the valence region (CAS 4X1) results in a bond length contraction in all molecular fi states compared to the SCF values. This phenomenon is a characteristic feature of such CI treatments in general. It also has to be noted that the CAS 12 calculation making use of a comparatively large active space does not affect bond lengths significantly. The inclusion of electron correlation is responsible for the substantial stabilization of d electron rich states. Table 2 also summarizes the results for excitation energies at the equilibrium geometry at the computational levels used so far. In contrast to the bond lengths an even qualitative difference between SCF and correlation treatments can be made out in this case. Whereas *A is found to be the ground state at SCF level, the d electron rich *Z+ state is stabilized significantly as correlation effects are taken into account. This relative displacement is strong enough to let *Z+ even drop to the ground state at CAS 12 and CA!3 4-CI levels. The energy difference between *A Table 2 Equilibrium bond lengths r, (a,,) and excitation energies Lr, (cm-‘) for the low-lying electronic states rz+
SCF ’ US12 b CAS 4-U =
2l-l
sL\
re
T,
r,
T,
re
r,
2.92 2.91 2.84
392 0 0
2.93 2.92 2.86
0 1548 1873
3.04 3.05 2.97
4281 6252 7092
* Molecular orbitals obtained from an SCF calculation in the particular symmetry. b Molecular orbitals obtained from CASSCF calculation with eleven electrons in twelve active orbitals. c MRD-CI calculations based on CAS 4 optimized MO; selection threshold for CI contigurations: 1.O ).I&,.
210
T. Fleig, C.M. Marian /Chemical Physics Letters 222 (1994) 267-273
Fig. 1. Molecular AE states from MRD-CI calculations including ACPF estimation and CAS 4 x state.
and *II is only slightly enlarged from SCF to CAS 4CI calculations. Fig. 1 displays the potential energy curves for the CAS 4 and corresponding MRD-CI calculations including an ACPF estimation of the energy values. When spin-orbit coupling is taken into consideration the discussed states are subjected to a splitting into five states which can be characterized as o= 5/2(I), 1/2(I), 3/2(I), 3/2(11), l/2(11). 5/2(I) is a pure 2Astate whereas the l/2 and 312 states consist of U-coupled electronic states 2Eie/2-2111,2and respectively. In Table 3 all linear com2A3,2-2b,2, binations of these states including the appropriate mixing coefficients are listed. As o= 512 is a AsI2 state corresponding to the angular momentum projections n = 2 and Z= l/2 it does not mix with any other AZ state. The lower l/2(1) state, however, is affected by 2111,2and can be characterized basically as ‘C h2, but the 2111,2component gains in significance as the nuclei are separated. w = 3/2(I) is mainly of 2As,2 character at short interatomic distances but switches to 211s,2around 2.7 ao. For o= 3/2 (II) the exact opposite is the case whilst o= l/2 (II) is mainly of 2111,2kind, especially at short bond distances. Fig. 2 shows these five states obtained from calculations at seven different geometries. An energy gap between the three lower and the two upper states is distinguishable. The lower corre-
spond to the atomic dS,2 hole states, the upper to atomic dSj2 hole states and this assignment finds its ratification as the dissociation limit is approached. Finally, a collection of the present results is to be compared with appropriate values obtained from Dirac-Hartree-Fock, relativistic CI, and RECP calculations. Table 4 contains equilibrium geometries, the excitation energies at these bond lengths and the harmonic frequencies corresponding to the spin-orbit split states. The most obvious conclusion to be drawn from this comparison is the reliability of the present approach if the four-component RCI and experimental results are taken as guidance. The RCI treatment of ref. [ 9 ] is comparable to our CI expansion. Although Visscher et al. apply four-component methods to the problem, it has to be stated that they make use of the no-pair approximation excluding electron and positron pair processes as well. There is no sense in comparing atomic orbital bases for fourcomponent spinors and one-component molecular orbitals as the description of the former is much more demanding. Furthermore, we have decided to outline the advantages of all-electron calculations in comparison with the RECP-SCF-CI treatment carried out by Wang and Pitzer [2] and the RECPCASSCF-RCI by Balasubramanian and Feng [ 6 1, as these studies are the only ones that hold information at the spin-orbit coupled level. Wang and Pitzer em-
T. Fleig, CM. Marian /Chemical Physics Letters 222 (1994) 267-273
271
Table 3
Spin-orbit split states as linear combinations of AZ states at
severalinternuclear
distances
2.0
1/2W) 3/2(w 3/2(I) 1/2(I)
-0.988 -0.515 0.515 -0.988
IA=l,M,=-0.5)+0.155 IA=O, Ms=OS) IA=2, MS=-0.5)+0.857 iA= 1, h&=0.5) IA= 1, M,=O.5) +0.857 IA=2, MS= -0.5) IA=O, M,=O.5)-0.155 IA=l,Ms=-0.5)
2.3
l/2(11) 3/2(w 3/2(I) 1/2(I)
-0.977 -0.610 0.610 -0.977
IA= 1, MS= -0.5) +0.213 IA=O, M,=O.5) IA=2, MS= -0.5) +0.793 IA=l, M~=0.5) IA= 1, M,=O.5)+0.793 IA=2, MS= -0.5) IA=O, M,=O.5)-0.213 IA=l, MS= -0.5)
2.7
l/2(11) 3/2W) 3/2(I) 1/2(I)
-0.956 -0.708 -0.708 -0.956
IA=l, IA=2, IA=l, IA=O,
2.85
l/2(11) 3/2w 3/2(I) 1/2(I)
-0.946 -0.737 -0.737 -0.946
lA=l,M,= -0.5) +0.325 (A=O, h&=0.5) IA=2, MS=-0.5) +0.676 Id= 1, M,=O.5) IA= 1, h&=0.5) -0.676 IA=2, M,= -0.5) IA=O, h&=0.5) -0.325 IA= 1, MS= -0.5)
3.1
l/2(11) 3/2w) 3/2(I) 1/2(I)
-0.928 -0.779 -0.779 -0.928
IA=l,M,=-0.5)+0.372 lA=O,M,=O.5) IA=2, MS=-0.5) +0.627 IA= l,M,=O.5) IA= 1, M,=O.5) -0.627 IA=2, M,= -0.5) IA=O, M,=O.5) -0.372 IA= 1, MS= -0.5)
4.0
1/2uu 3/2W 3/2(I) 1/2(I)
-0.864 -0.850 -0.850 -0.864
IA= 1, MS= -0.5) +0.504 IA=O, h&=0.5) IA=2, MS= -0.5) +0.527 IA= 1, M,=O.5) IA=l,M,=0.5) -0.527 IA=2, MS= -0.5) (A=O, M,=O.5)-0.504 IA=l, M,= -0.5)
5.0
1/2uu 3/2W 3/2w 1/2(I)
-0.819 -0.879 -0.879 -0.819
IA= 1, MS= -0.5) +0.573 IA=O, M,=O.5) IA=2,MS= -0.5) +0.477 IA=l, M,=0.5) IA= 1, M,=O.5) -0.477 IA=2, MS= -0.5) IA=O,M,=0.5)-0.573 I/l-1,&=-0.5)
ploy small ST0 valence basis sets and limited CI expansion lengths in their investigation. The active space defined in the CASSCF calculation of ref. [ 6 ] is comparable to our CAS 4 space concerning the number of unique configurations in the full-C1 expansion, but our CI treatment is considerably more extensive than the RCI calculation of Balasubramanian et al. Concerning excitation energies excellent agreement is observed between the DHF and RCI values, the results of the present no-pair CI calculations including spin-orbit coupling and experimental values being known so far. As already noticed by Visscher [ 91 the excitation energies of the spin-orbit mixed states are hardly affected by the inclusion of electron correlation, in contrast to our findings for the one component states presented in Table 2. Only a slight stabilization of the l/2(1) state with respect to the
MS= -0.5) +0.294 IA=O, M,=O.5) MS=-0.5) +0.707 IA=l, M,=O.5) M,=O.5)-0.707 (A=2, MS= -0.5) h&=0.5) -0.294 iA= 1, MS= -0.5)
DHF treatment by Dyall [ 8 ] is recognized. On the other hand, the bond length contraction due to the dynamical valence electron correlation in the onecomponent states is carried over to the spin-orbit mixed potentials. The equilibrium bond lengths resulting from the no-pair calculations are slightly shorter than the RCI values, probably due to the more extensive electron correlation treatment in the former. Good agreement with experiment is observed for the 3/2(I) state; the equilibrium distance for w=5/2 turns out to be too short just like the RCI result. The experimentally determined r, values are derived from rotational parameters B,with the latter parameter fitted to a comparatively simple effective Hamiltonian [ 28 1. The considerably different equilibrium bond lengths deduced from the spectra of PtH and PtD on the other hand suggest strong perturbations. The deviation observed for the w= 512 ground
272
T. Fleig, C.M. Marian /Chemical PhysicsLetters 222 (1994) 267-273
Fig. 2. Spin-orbit split states composed of mixed AZ states. Table 4 Equilibrium bond lengths r. (a,), excitation energies r, (eV), and harmonic frequencies w, (cm-i)
for spin-orbit split states
W
5/2(I)
WP’ BFb DHF = RCI d present exp. c
3/2(I)
1/2(I)
1/2(II)
re
T,
w.
r,
T,
w.
r,
T,
we
r,
K
w,
r,
T,
w.
3.04 2.93 2.93 2.87 2.84 2.888
0 0 0 0 0 0
2020 2177 2234 2458 2423 2378
3.00 2.91 2.91 2.88 2.86
0.12 0.186 0.34 0.24 0.17
2045 2188 2094 2419 2355
3.19 2.99 2.99 2.91 2.89 2.812
0.34 0.521 0.42 0.44 0.45 0.400
2016 2155 2080 2313 2252 2265
3.00 2.98 2.91 2.88
2.74 1.346 1.43 1.46 1.46 1.45
2179 2162 2365 2331
2.99 3.00 2.95 2.91
1.59 1.348 1.55 1.61 1.54
2021 2097 2217 2253
‘Ref. [2]. bRef. [6]. ‘Ref. [S]. dRef. [9]. e Ref. [27], and revalues obtained for PtD: 2.880for 0=5/2(I)
state
3/2(II)
may thus be ascribed to experimental uncertainties in this case as well. The vibrational frequencies obtained in the present treatment are close to experiment. Finally, the molecular properties obtained by all-electron methods are in much better agreement with experimental results than those derived using the type of spin-dependent relativistic core potentials as used by Wang et al. and Balasubramanian et al. substantiating the general reliability of the former approaches when dealing with such molecules.
and2.878 for w=3/2(1).
Excitation energies from Ref. [28].
5. Conclusion The accomplished study of the PtH molecule provides an analysis of the lowest-lying state multiplet without and with the inclusion of spin-orbit coupling. The overall splitting according to this effect turns out to be up to 17000 cm-’ at short distances and can therefore no longer be considered a perturbation. The importance of kinematic relativistic corrections in the calculations is substantiated for achieving high accuracy of molecular properties. Evidence is given for the competitiveness of the no-pair formalism with four-component all-electron treat-
T. Fleig, CM. Marian / Chemical Physics Letters 222 (1994) 267-273
ments when dealing with heavy transition-metal diatomics.
Acknowledgement
We would like to thank Professor Peyerimhoff for financial support. The computational facilities made available to us by the ‘Deutsche Forschungsgemeinschaft’ through SFB 334 and the WAP-program are greatfully acknowledged.
References [ 1] H. Basch and S. Topiol, J. Chem. Phys. 71 ( 1979) 802. [2]S.W. WangandK.S.Pitzer, J.Chem.Phys. 79 (1983) 3851. [ 3 ] C. McMichael Rohllii P.J. Hay and R.L. Martin, J. Chem. Phys. 85 (1986) 1447. [4] A. Gavezzotti, G.F. Tantardini and M. Simonetta, Chem. Phys. Letters 129 (1986) 577. [ 51 S. Tobisch and G. Rasch, Chem. Phys. Letters 166 (1990) 311. [6] K. Balasubramanian and P.Y. Feng, J. Chem. Phys. 92 (1990) 541. [ 710. Gropen, J. Almlof and U. Wahlgren, J. Chem. Phys. 96 (1992) 8363. [S] K. Dyall, J. Chem. Phys. 98 (1993) 9678. [9] L. Visscher, T. Saue, W.C. Nieuwpoort, K. Faegri and 0. Gropen, J. Chem. Phys. 99 (1993) 6704. [lo] J. Sucher, Phys. Rev. A 22 (1980) 348. [ 111 M. Douglas and N.M. Kroll, Ann. Phys. 82 ( 1974) 89.
273
[ 121 G. Hardekopf and J. Sucher, Phys. Rev. A 31 ( 1985) 2020. [ 131 B.A. He& Phys. Rev. A 33 (1986) 3742. [ 141 R. Samzow and B.A. Hell, Chem. Phys. Letters 184 ( 1991) 491. [ 151 C.M. Marian, J. Chem. Phys. 93 (1990) 1176. [16] C.M. Marian, J. Chem. Phys. 94 (1991) 5574. [ 171 G. Jansen and B.A. He& Z. Physik D 13 (1989) 363. [IS] M.M. Gleichmann and B.A. He& J. Chem. Phys., to be published. [ 191 C.E. Moore, Atomic energy levels, NSRDS-NBS Circular 467 (US GPO, Washington, 1949). [20] S.R. Langhoff and C.W. Kern, in: Modem theoretical chemistry, ed. H.F. Schaefer III (Plenum Press, New York, 1977). [21] G. Breit, Phys. Rev. 34 (1929) 553. [ 2210. Gropen, unpublished work. [23] F.B. van Duijneveldt, IBM Res. Rept. RJ 945. [24] D.P. Chong and S.R. Langhoff, J. Chem. Phys. 84 (1986) 5606. [25] R.J. Buenker and S.D. Peyerimhoff, Theoret. Chim. Acta 35(1974)33;39(1975)217 R.J. Buenker, S.D. Peyerimhoff and W. Butscher, Mol. Phys. 35 (1978) 771; R.J. Buenker, in: Quantum chemistry in the SO’s, ed. P. Burton (University Press, Wollongong, 1980); in: Current aspects of quantum chemistry, ed. R. Carbo (Elsevier, Amsterdam, 1982); R.J. Buenker and R.A. Phillips, J. Mol. Struct. THEGCHEM 123 (1985) 291. [26] J.A. Pople, R. Seeger and R. Krishnan, Intern. J. Quantum Chem. Quantum Chem. Symp. 11 (1977) 149. [ 271 G. Gustafsson and R. Scullman, Mol. Phys. 67 ( 1989) 98 1. [28] M.C. McCarthy, R.W. Field, R. EnglemanandP.F. Bemath, J. Mol. Spectry. 158 ( 1993) 208.