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An approximate method for treating spin-orbit effects in platinum
3 November 1995
CHEMICAL PHYSICS LETTERS
iZj,:! ELSEVIER
Chemical Physics Letters 245 (1995) 509-518
An approximate method for treating spin-orbit effects in platinum Christoph Heinemann, Wolfram Koch, Helmut Schwarz Institut fdr Organische Chemie der Technischen Universitiit Berlin, Strafle des 17. Juni 135, D-10623 Berlin. Germany
Received 2 July 1995; in final form 24 August 1995
Abstract
Spin-orbit coupling in platinum-containing species can be treated via a one-electron spin-orbit operator and a single scaling parameter Zcff(Pt) in conjunction with an effective core potential for the description of scalar relativistic effects. Our calibration calculations cover the five low-lying electronic states of platinum hydride PtH and the lowest fourteen levels in the atomic spectrum of the platinum atom Pt. Here, qualitative and semi-quantitative agreement between the presented semi-empirical approach and four-component Dirac-Fock calculations is found if Zeff(Pt) is chosen between 950 and 1200. Further applications concern the low-lying levels of the platinum cation Pt +, the theoretical determination of ground states for the diatomic oxides PtO and PtO + as well as spin-orbit effects in the cationic carbene complex PtCH~-.
1. Introduction
The major influences of relativity on the electronic structure of heavy-atom containing molecules are well understood at least from a pragmatic point of view and it has become feasible to perform qualitatively as well as quantitatively predictive ab initio calculations on molecular systems incorporating heavy third-row transition-metals, actinides and trans-actinides [1]. While different approaches have been devised for the incorporation of scalar relativistic effects in the established one-component formalism of molecular quantum chemistry [2], the treatment of spin-orbit effects [3] is far less straightforward and requires additional, sometimes prohibitively expensive computational effort [4,5]. Thus, in spectroscopy semi-empirical schemes have been developed to estimate fine-structure splittings and off-diagonal spin-orbit matrix elements such that perturbations in observed spectra can be traced back
tO atomic s p i n - o r b i t or s p i n - s p i n coupling constants 1. This approach becomes, however, impractical for nonlinear polyatomic molecules or even diatomics with considerable multi-configurational character where wavefunctions composed of a single configuration state function built from one-electron orbitals with well defined /-quantum numbers cannot be formulated in a straightforward manner. With regard to the interplay between computational quantum chemistry and spectroscopic experiments there is a need for a practical procedure for incorporating spin-orbit effects into theoretical calculations. High-resolution spectroscopy is able to provide detailed information about spin-orbit coupling, for example, in clusters of metal and rare-gas atoms [7]. From our particular point of view, we would like to point out that spin-orbit coupling is
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-26 14(95)01042-4
i An overview of these methods is given in Ref. [6].
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C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
also of paramount importance for an understanding of the reactions of small unsaturated organometallic complexes with simple inorganic substrates such as dihydrogen or unfunctionalized hydrocarbons [8]. The spin-orbit interaction represents one of the major perturbations of the potential energy surfaces of such reactions. Electronically excited states may influence the ground state chemistry of 'bare' and ligated transition-metal cations via spin-orbit mediated curve-crossing mechanisms, such that the reactivity of an intrinsically unreactive ground state is enhanced via avoided crossings with more reactive excited states. Recently, Koseki et al. took up an earlier idea [9-13] of calculating spin-orbit effects via the singly parametrized one-electron operator Hso n s•o =
1 2 E[zeff(A)/r?a]LiA.Si, -~Ol i,A
where a denotes the fine-structure constant, LiA and Si angular momentum and spin operators, ria an electron-nucleus distance and i and A run over all electrons and nuclei, respectively [14,15]. Zeff(A) is an effective nuclear charge, which was adjusted for each element within a given basis set description such that experimental fine-structure splittings of 2II and 317 states of simple diatomic hydrides were in agreement with those calculated by the help of /-/so operating on complete active-space self-consistent field (CAS-SCF) wavefunctions. The usefulness of this operator in spin-orbit calculations is based on an approximate cancellation between neglected terms of the full Breit-Pauli Hamiltonian. Zeff should thus be regarded as a genuine semi-empirical parameter rather than a physical 'charge' since it adjusts the errors in the approximation, in particular, when effective core potentials with the corresponding nodeless valence pseudo-orbitals are employed for heavy atoms [ 16]. In this Letter, we investigate the performance of this model for the treatment of spin-orbit effects in platinum. Using earlier fully relativistic four-component Dirac-Fock calculations (where available) and experimental data as a reference, we derive a useful value for the parameter Zeff(Pt). to be used in conjunction with valence CAS-SCF wavefunctions, a relativistic pseudo-potential and a valence basis of triple-zeta plus two polarization functions quality.
The calibration of the method is carried out for the diatomic platinum hydride molecule PtH and atomic platinum Pt, for which reliable reference data are available. In the second part, we apply this scheme to systems such as the platinum cation Pt ÷, the diatomic neutral and cationic platinum oxides PtO and PtO ÷ as well as the cationic platinum carbene complex PtCH~-, for which there is still a significant need for accurate calculations. Particular emphasis is given to spin-orbit effects on spectroscopic constants, excitation energies between low-lying electronic states and electronic ground-state stabilization.
2. Computational details The inner 60 electrons of platinum ([Kr] + 4d m + 4f 14) were treated by the relativistic effective core potential given by Stevens et al. [17] thus treating the scalar relativistic effects from the mass-velocity and Darwin terms en gros. The associated Gaussian-type (7s7p5d)/[4s4p3d] (4111141111311) basis set was augmented by two f-type polarization functions (exponents: 0.2, 0.08) to generate a valence basis of triple-zeta plus two polarization functions (TZ2P) quality. For carbon and oxygen, Dunning's (lls6p)/[5s, 3p] (621111411) sets [18] were augmented by two d-type polarization functions (C-exponents of 1.00, 0.40; O-exponents of 2.56, 0.64). For hydrogen, two p-type polarization functions (exponents: 1.25, 0.45) were added to the same author's (5s)/[3s] (311) get. All Cartesian components of the d- and f-type basis functions were retained in the calculations. These one-particle sets served to construct CAS-SCF wavefunctions [19], which will be explicitly described for each case in the course of the discussion. The natural orbitals of these wavefunctions were employed to calculate matrix elements of /-/so and this matrix was diagonalized in order to obtain spin-orbit coupled states. For the diatomic molecules 10-20 single point energy calculations in the vicinity of the minima were performed to generate a potential curve in which subsequently the nuclear Schr/Adinger equation for the most abundant isotopes was solved using Numerov's method to obtain spectroscopic constants. All calculations were carried out without symmetry restrictions (technical reasons) using the latest version (March 10, 1995) of the GAMESS program system [20] on an I B M / R S
C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
6000 workstation. Since our treatment of dynamic electron correlation is almost none, comparison of the theoretical results to experimental data should be carried out with caution. However, some discussion of dynamic correlation effects will be given in Section 3.
Table 2 Spectroscopic constants (equilibrium distances Re in A, harmonic frequencies ~oe in cm-J and adiabatic excitation energies Te in eV) for the five low-lying states of PtH from CAS-SCF calculations as a function of Zeff(Pt). The four-component Dirac-Fock results are from Ref. [4] 0=5/2
12=1/2 (1)
g2=3/2 (1)
12=3/2 (II)
12=1/2 (II)
1.590 2061 0.29
1.581 2050 0.33
1.599 2103 0.87
1.611 2017 1.03
1.595 2045 0.33
1.589 2026 0.37
1.590 2121 I. 10
1.607 2034 1.26
Zeff(Pt) = 950 R c 1.558 ~oe 2190 Te 0.00
1.597 2040 0.35
1.593 2027 0.38
1.588 2125 1.20
1.604 2040 1.35
Zerr(Pt) = 1200 R e 1.558 coe 2190 T~ 0.00
1.601 2028 0.38
1.598 2018 0.40
1.583 2128 1.45
1.600 2054 1.59
Dirac-Fock R e 1.551 o)e 2234 Te 0.00
1.573 2094 0,34
1.584 2080 0.42
1.577 2162 1.43
1.590 2097 1.55
Zeff(Pt) = 600 R e 1.559 we 2189 Te 0.00
3. Results and discussion
3.1. Platinum hydride PtH
511
Z~ff(Pt)= 850 A small active space of six orbitals and eleven electrons was chosen in the CAS-SCF treatment (5s and 5p orbitals of Pt always doubly occupied). While close to the equilibrium geometry near-degeneracy effects (nondynamic electron correlation) are absent in PtH, dynamic electron correlation has been shown to shorten the P t - H bond lengths of the low-lying states by = 0.04 ,~ and to increase their frequencies by = 200 cm-1 [5]. The results of a state-avera§ed AS-coupled calculation for the three low-lying A, 2~ +, and 2II states (treated with equal weights in the orbital optimization) are given in Table 1. Using these natural orbitals for the spin-orbit treatment one obtains five low-lying electronic states with 12 = 5 / 2 , 3 / 2 and 1/2, whose spectroscopic constants as a function of the scaling parameter Zeff(Pt) a r e given in Table 2. In accord with earlier results [4,5], spin-orbit coupling in the ground state of PtH is a pure finestructure effect in the sense that the /2 = 5 / 2 state can be identified as the only weakly t~erturbed g2 = 5 / 2 component of the AS-coupled =A state. Conversely, the four low-lying excited states are mixtures of the ,(-2= 1 / 2 and ,(2 = 3 / 2 components of the {2A, 2 ~ +, 2 11} manifold. The spectroscopic constants obtained for the five low-lying electronic states Table I Spectroscopic constants (equilibrium distances R e in/~, harmonic frequencies o9¢ in cm I and adiabatic excitation energies Te in eV) for the three low-lying AS-coupled states of PtH from CAS-SCF calculations
Re w~
2A
2E+
2H
1.559 2235 0.00
1.579 2130 0.13
1.624 2002 0.54
Re coe Te
1.559 2190 0.00
are in remarkably good agreement with two recently reported uncorrelated four-component Dirac-Fock calculations which include spin-orbit coupling at the level of the self-consistent procedure [4,5]. Here, we will compare our results to the more comprehensive data of Dyall [4]. Further theoretical analyses of PtH have been discussed elsewhere in the literature [4,5,21]. First, we find the correct qualitative state ordering with relatively small excitation energies from the [2 = 5 / 2 ground state to the lowest ,(2 = 1 / 2 (I) and 12 = 3 / 2 (I) excited states and the larger energetic gap to the two higher excited states. The overall energetic splitting of the considered state manifold increases with larger Z~ff(Pt). Increasing the scaling parameter also shortens the bond lengths and raises the frequencies of the second pair of excited states (12 = 1/2(II), ,(2 = 3/2(II)), while the opposite applies to their lower counterparts. Trends in the calcu-
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c. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
lated frequencies and bond distances parallel those from Dyall's four-component study [4], except for the bond length o f the 1-2 = 1 / 2 (I) state, which is a bit overestimated in our calculation. Quantitatively, our frequencies are slightly lower and our bond distances somewhat larger than the respective Dirac-Fock (DF) values and the absolute average deviations (50 cm -1 and 0.013 ,~) are almost independent of Zeff(Pt) in the considered range. These differences are partly due to our use of state-averaged natural CAS orbitals while the DF treatment includes individual orbital optimization for each state. As far as the excitation energies are concerned, the best agreement between the method employed here and the D i r a c - F o c k calculations is found for Zeff(Pt) = 1200 (absolute average deviation: 0.03 eV). For smaller scaling parameters, the excitation energies as derived from the present study are smaller than the four-component results. Independent of Zeff(Pt), the absolute splitting between the lowest two excited states is always somewhat underestimated. Finally, we note that with respect to the experimental parameters of PtH ( g 2 = 5 / 2 : R e = 1 . 5 2 8 A, we = 2 3 7 8 cm -1", O = 3 / 2 (I): R e = 1.519 A, o~e = 2265 cm -I [22]) our calculations show the typical behavior of dynamically uncorrelated wavefunctions for transition-metal systems (overestimation of bond lengths by = 0.05 A, underestimation of frequencies by = 250 c m - ~ for Zeff(Pt) = 1200). On the other hand, the calculated electronic excitation energies exactly match the experimentally observed ones (g2 = 5 / 2 ---, g2 = 3/2(1): 0.40 eV; /2 - 5 / 2 --*/2 = 3/2(11): 1.45 eV [23]) for Zeef(Pt)= 1200. The quantitative side of this agreement is probably due to a fortuitous cancellation of errors. 3.2. P l a t i n u m a t o m P t
Recent studies [4,5,24] have shown that the relative energies of the lowest 14 atomic energy levels of the platinum atom depend strongly on the relative weights of the three electron configurations 5dl°6s °, 5d96s 1 and 5da6s 2 in the optimization of the orbitals from which these levels are constructed. In order to approach agreement with the experimental spectrum [25], both electron correlation and quantum electrodynamic corrections were found to be important. In view of these difficulties, we restrict ourselves primarily to a comparison to an available numerical
Table 3 Relative energies of the low-lying states of the Pt atom (in cm- i ) from CAS-SCF calculations. The four-component Dirac-Fock results are from Ref. [4] J
zat(Pt) = 950
Zeff(Pt)= 1200 Dirac-Fock
Exp.[25]
3 4 2 2 1 3 2 2 0 1 0 4 2 0
0 1010 1217 7021 8334 8141 12752 16357 16662 20736 21471 25151 26812 53579
0 342 1201 7856 10528 9457 14833 17092 17052 22054 22556 26259 29345 55614
0 824 776 6567 10132 10116 15502 13496 18567 6140 21967 26639 -
0 941 1312 6910 9227 10287 15055 17473 17673 22418 24350 25054 29210 54687
four-component Dirac-Fock calculation [4] without additional corrections (the basis set error with a one-particle space of approximately valence TZ2P quality was found to be small [5]). Similar to this study, singlet and triplet orbitals were optimized separately for CAS-SCF wavefunctions (10 electrons in 6 active orbitals; 5s, 5p always doubly occupied) with equal weights for the 5d96s I and 5dS6s 2 configurations and a relative weight of 10% for the 5d1°6s ° configuration. After a corresponding orbital transformation [26], the respective natural orbitals were used in a spin-orbit calculation which yielded the atomic spectrum of platinum as listed in Table 3. The so-obtained level structure agrees well with the four-component result. Qualitatively, only the first J = 1 and the second J = 3 levels have the incorrect ordering for Ze,(Pt) = 950. Using a scaling parameter Z e f f ( P t ) = 1200, also the fourth J = 2 and the first J = 0 level are interchanged but their splitting is small (40 c m - J ) and they are also close in the Dirac-Fock treatment (200 c m - l ) . The absolute average deviation in the excitation energies between the present results and the Dirac-Fock study is 1224 cm-1 for Zeff(Pt)= 950 and 726 cm -1 for Zeff(Pt) = 1200. However, the energy difference between the J = 3 ground state and the J = 4 first excited state is closer to the Dirac-Fock result for Z a e ( P t ) = 950. The experimentally observed atomic energy levels for platinum are also compiled in Table 3. Absolute average deviations between calculated and observed
C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
excitation energies amount to 2847 c m - l (Z~ff(Pt) = 950) and 3129 cm - l (Z~ff(Pt)= 1200), compared to 3110 cm -1 for the D i r a c - F o c k calculation. The errors are smaller (1599 cm -I for Zeff(Pt)=950, 1800 cm -1 for Z~ff(Pt)= 1200, 1601 cm - l for Dirac-Fock) if the first J = 0 level, which depends strongly on the choice of the one-particle basis [4], is omitted. The operator Hso used with a scaling parameter Zeff(Pt) between 950 and 1200 reproduces more rigorous treatments of spin-orbit effects on spectroscopic constants and low-lying excited state energies in platinum-containing species in a semi-quantitative manner. The large magnitude of the optimum parameter arises from the missing nodal structure of the valence pseudo-orbitals in the region near the nucleus from which spin-orbit effects predominantly arise [13,15,16]. The main purpose of the present investigation is not a rigorous first-principles treatment of spin-orbit coupling in platinum, but to establish the applicability of this simple semi-empirical approach. Thus, after the successful calibration of the theoretical four-component Dirac-Fock results for PtH and Pt, we will now continue with applications to cases in which spin-orbit effects are expected to be important but where, to the best of our knowledge, no rigorous theoretical reference data are available. It is important to reiterate that in spite of being called an effective nuclear charge, no physical relevance should be attributed to the scaling parameter Zeff(Pt). Its sole purpose is to correct for errors in the approximation. 3.3. P l a t i n u m c a t i o n P t +
The platinum cation was chosen as the first test case because comparison of the calculated energy levels to the low-lying part of the experimentally observed spectrum should be less difficult than for the neutral platinum atom: only two LS terms (2D, 4F) arising from two electronic configurations (5d96s °, 5d86s t ) dominate the spectrum of Pt + up to excitation energies of 2 eV [25] and, thus, the individual J-levels are expected to be less mixed than in the neutral platinum atom. For Pt +, CAS-SCF wavefunctions were obtained separately for the 2D and 4F states with 9 active electrons in 6 active orbitals (6s, 5d) and the 5s and 5p orbitals always doubly occupied. Neglecting spin-orbit coupling, the ZD-~ 4F
513
Table 4 Relative energies of the low-lying states of the Pt + cation (in cm -~ ) from CAS-SCF calculations. The experimental values are from Ref. [25] J 5/2 9/2 3/2 7/2 5/2 3/2
Zeff(Pt) Zeff(Pt) = 950 = 950 a
Zeff(Pt)
Zcff(Pt)
= 1200
= 1200 a
0 2897 7671 8164 12320 15405
0 2357 9653 9010 14267 18200
0 3516 9653 10169 15426 19359
0 4056 7671 9323 13479 16564
Exp. 0 4787 8420 9356 13329 15791
a The calculated levels deriving from the 4F term are shifted upwards by 1159 cm-~ to correct for the disagreement between the experimental and calculated ~D -~ 4F excitation energies. excitation energy is calculated to be 4968 cm - I , 1159 cm -l lower than the experimental value (obtained as the weighted average of the corresponding J-levels [25]). Part of this disagreement reflects the larger dynamic correlation energy in the low-spin 5d96s ° configuration. After a corresponding orbital transformation [26] //so was diagonalized in the basis of the CAS-SCF wavefunctions and the obtained level spectrum of Pt ÷ is listed in Table 4, along with the experimental data. To approximately correct for the neglect of dynamic correlation in the CAS-SCF wavefunctions, the calculated spin-orbit levels deriving from the 4F state have been shifted upwards by 1159 cm - l . After this manipulation, the absolute average deviation between the calculated and the observed excitation energies amounts to 487 cm -1 for Zeff(Pt)= 950 and 1796 cm -1 for Zeff(Pt) = 1200. Larger than average errors occur in particular for the two highest J = 5 / 2 and J = 3 / 2 levels which are expected to interact with higher-lying terms that are not included in the present calculation. The final deviation between computed and observed excitation energies for Z~ff(Pt) = 950 are of the order of magnitude of quantum electrodynamic corrections in the spectrum of neutral atomic platinum [4,5]. Thus, we can conclude that for similar charges Z~ff(Pt) the chosen semi-empirical approach yields consistent results for spin-orbit interaction in both the neutral and the cationic platinum atom. 3.4. P l a t i n u m o x i d e P t O
The ground state of PtO has been under some discussion in recent years [27]. The latest spectro-
514
C. Heinemann et aL / Chemical Physics Letters 245 (1995) 509-518 4or E
T
2¢J
Fig. 1. Valence molecular orbital diagram for PtO and PtO +. The electron configuration corresponds to the 3E- ground state of PtO. Removal of the spin-down electron from the 3s orbital affords the 4 ~ - ground state of PtO +. Orbitals inactive in the CAS-SCF calculations (Is(O), 5s(Pt) and 5p(Pt)) are not shown.
scopic experiments [28] suggest a Hund's case (c) [6,29] description with an O = 0 ground state arising from a 3 E state (valence configuration: 1842"rr23tr2, see Fig. 1) via spin-orbit coupling with electronically excited states, in particular the ~E+ state (valence configurations: 18427r43tr°/ 1842 xr 23tr 2). The excitation energy from the ground state to the first excited state ( O = 1) was found to be 0.12 eV [28]. Density functional calculations [30] have investigated the lowest state arising from the 1842 ~ 23tr 2 configuration but a definitive theoretical answer to the question of the ground state symmetry could not be given due to the multi-configurational character of the problem. Thus, we have calculated CAS-SCF wavefunctions for the lowest AS-coupled singlet and triplet states of PtO distributing 16 electrons in 10 active orbitals as shown in Fig. 1 (5s, 5p orbitals of platinum and is orbital of oxygen doubly occupied in all CSFs). The lowest CAS-SCF energies were always obtained for the 3 ~ - state and the lowest singlet states were of ~A and ~E + symmetry. Thus, the electronic situation in PtO bears some similarity to the dioxygen molecule 02 [31], which also has a 3E- ground state and two low-lying excited IA and i E + states [29]. The diagonalization of /-/so was performed using the natural orbitals for
the 3~- state. The reason for using the ground state's natural orbitals in the spin-orbit treatment was that the corresponding orbital transformation [26], which is necessary prior to the evaluation of matrix elements with two different sets of orbitals, failed for PtO. However, reoptimization of the excited states' natural orbitals lowers their total energies by only 2-3 mhartree and using different sets of singlet or triplet orbitals affects the computed spinorbit matrix elements by less than 10%. These effects are considerably smaller as compared to other approximations made in this study. Spectroscopic constants for the three lowest AScoupled states of PrO and their four spin-orbit levels are given in Table 5. In the AS-coupling scheme, the 3E - ground state of PtO has a shorter P t - O distance and higher vibrational frequency as compared to the two excited singlet states. Within the {3~ - , 1A, l~ +} manifold, a nonvanishing spin-orbit matrix element occurs only between the M s = 0 component of the 3 ~ - state and the 1E+ state, giving rise to an /2 = 0 ground state, a low-lying O = 1 excited state (unperturbed with respect to 3E-), the unperturbed IA state ( O = 2) and an /2 = 0 state deriving mainly from 1E +. Similar to the PtH case (see above), spin-orbit coupling increases the overall energetic manifold splitting and mixes the states such that the bond length of the lower O = 0 state is slightly increased and its frequency decreased while the opposite applies for the higher /2 = 0 state. The calculated adiabatic excitation energies between the 12 = 0 ground state and the /-2 = 1 first excited state (0.06 eV with Zeff(Pt)=950, 0.10 eV with Zeff(Pt)= 1200) compare reasonably well with the experimental result of 0.12 eV [28] thus confirming the interpretation of the PtO ground state as derived from the spectroscopic experiments [28]. Since the CAS-SCF method accounts for near-degeneracy effects in the valence region we expect that the inclusion of dynamic correlation will not alter the qualitative ordering of states as obtained here. However, these effects are expected to shorten the P t - O bond length and to enhance the stretching frequency relative to the CAS-SCF results, which deviate from the experimental data (R~ = 1.727 c m - l , toc = 851 cm-1 for the /-2 = 0 ground state [28]) within the expected range. In addition, the calculated trends for the bond lengths and frequencies of the 12 = 0 ground state and the
C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518 Table 5 Spectroscopic constants (equilibrium distances R e in ,~, harmonic frequencies oJ~ in c m - t and adiabatic excitation energies Te in eV) for low-lying states of PtO from CAS-SCF calculations. The experimental values are from Ref. [28] R,~
¢oe
Te
a]£ IA i y, +
1.791 1.819 1.849
723 648 588
0.00 0.59 0.94
Zc,(Pt) = 950 ~,h O = 0 ~(2 = I -(2 = 2 .(2 = 0
1.792 1.791 1.819 1.848
722 723 648 595
0.00 0.06 0.65 1.06
Zcrf(Pt) = 1200 a.h ~f2 = 0 1.792 .(2 = 1 1.791 .(2 = 2 1.819 .(2 = 0 1.848
719 723 648 597
0.00 0.10 0.69 1.14
Zcfr(Pt) = 1200 a,c f2 = 0 u 1.797 -(2 = I 1.801 .(2 = 2 1.843 g2 = 0 1.880
705 658 635 484
0.00 0.03 0.52 0.92
expt. ,(2 = 0 O = 1
851 832
0.00 0.12
1.727 1.729
Z~tr(O) = 5.6, as recommended in Ref. [14]. S p i n - o r b i t operator diagonalized in the space of the states {35~ - ,
IA, 1~+}.
c S p i n - o r b i t operator diagonalized in the space of the states { 3 ~ -
iA, 1~;+, 3rl ' 117, 34,}. a This state is stabilized by 0.12 eV with respect to the 3 ~ - state.
12 = 1 first excited state are opposite to the experimentally observed ones [28]. Besides dynamic correlation effects, spin-orbit interaction with higher excited states might constitute the reason for this disagreement. To investigate the influence of the next excited states, a second diagonalization of H~o including two additional triplet states (3II and 3 ~ ) and one additional singlet state (l I I ) was performed 2. This enlargement of the basis for the representation of H~o increases the P t - O distances and lowers the frequencies of all four low-lying states. Moreover, the 12 = 1 first excited state is stabilized with respect to the
515
12 = 0 ground state and the calculated excitation energy (0.03 eV) is reduced to only 20% of the experimental value. On the other hand, inclusion of the higher excited states affords the correct trend for the bond lengths and frequencies of these two states. These effects indicate a drawback of the employed method in treating spin-orbit effects: Since spiqorbit coupling is neglected during the self-consistent field procedure, it might be neccesary to include a large number of low-lying electronically excited states in the spin-orbit treatment to obtain meaningful results for the properties under study. Thus, we regard the presented methodology as a semi-quantitative treatment, which allows one to derive the magnitude of spin-orbit effects within the correct ballpark but should be used with caution when quantitative agreement with spectroscopic data is intended. 3.5. Platinum oxide cation PtO +
This diatomic molecule is of interest in organometallic gas-phase ion chemistry since it constitutes an important intermediate in the Pt+-cata lyzed gas-phase oxidation of methane by molecular oxygen [32,33]. Based on the electronic spectrum of neutral PtO (see above) one can expect low-lyhlg states of the AS symmetries 4 ~ - (l~42,rr23o.J configuration, see Fig. I), 21] (l~42"rr33cr°), 2A (l~42q'r23cr 1) or 2~+ (l~42,rr23crl). There should also be a large number of additional low-lying states with a hole in the l~-shell. Using CAS-SCF wavefunctions with 15 active electrons in 10 active orbitals (similar to the treatment of neutral PtO, see above), we find a high-spin 4E_ ground state for PtO +, similar to the isoelectronic NiO + molecule [8]. The natural orbitals for this state were also used to calculate the potential energy curves for the lowest low-spin doublet state ( 2 E - ) and the three lowest spin-orbit coupled states (12 = 3 / 2 , 12 = 1/2(I) and 12= 1/2(II)), which arise from the 2/4~- couple
2 These states have vertical excitation energies of 1.02 (3I/), 1.16 ( l i d and 1.42 eV (3alP) from the ground state at a P t - O distance of 1.80 A. The next excited state has a vertical excitation energy of 1.71 eV and is of I I-I symmetry.
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C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
Table 6 Spectroscopic constants (equilibrium distances R e in ~,, harmonic frequencies toe in c m - t and adiabatic excitation energies Te in eV) for low-lying states of PtO + from CAS-SCF calculations
4~2y..,Zetr(Pt) = 1200 a,b ~O = 3 / 2 c /'2 = 1/2(1) O = 1/2(1I)
Re
toe
:re
1.805 1.815
675 671
0.00 0.51
1.815 1.807 1.815
683 728 610
0.00 0.05 0.55
a Zeff(O) = 5.6, as recommended in Ref. [14]. b Spin-orbit operator diagonalized in the space of the states {4~ - , 2 Z - ' 21-i, 2A, 4A, 2~.+, 2[i ' 2A 41-i}.
4, E / eV ~
Pt
l.,OOX/H 1.084A \H
1
2b 1 (-5.6)
4a 1 (-10.8)
T
3al (-18.9) 1~2 (-1;'.1) 2b2 (-18.2) lb, (-18.5)
TJ. Tl T,I. TI
2a I (-21.1)
T J,
lb 2 (-23,9)
T,L
la I (-31,6)
T,I,
c This state is stabilized by 0.20 eV with respect to the 4 ~ - state.
via spin-orbit interaction with excited states 3 (see Table 6). Since the 4 E - state has the angular momentum quantum number A = 0, it does not undergo a finestructure splitting [6]. However, the splitting into the two /2 = 3 / 2 and O = 1 / 2 components occurs by inclusion of the excited states via nonvanishing offdiagonal elements of the spin-orbit operator. The /2 = 3 / 2 state is found to be the ground state of PtO +, as evident from the two matrix elements for the 4E-/Ell interaction (R = 1.80 .~): ( 4 E - , M s = 3 / 2 , /2=3/21/-/,ol21-I, Ms=l~2, / 2 = 3 / 2 ) = 1823+488i cm -1 and ( 4 E - , M , = l / 2 , /2 M, = - 1 / 2 , /2 = 1 / 2 ) = 1053-281i cm -1 The excitation energy to the first excited O = 1 / 2 state (0.05 eV) is of a similar order of magnitude as the /2 = 0 / / 2 = 1 state splitting in neutral PtO (see above). The next excited state deriving from the AS-coupled 2 E - state has also /2 = 1 / 2 and the respective excitation energy from the ground state is only marginally affected by spin-orbit interaction. Thus, we conclude that the interesting gas-phase chemistry of PtO ÷ [32,33] is dominated by the high-spin quartet ground state. An experimen-
1/21nsol2rl,
3 Individual orbital optimization for the 2 ~ - state lowers its total energies by typically 1 mhartree. In the spin-orbit treatment, the following states were included (symmetry, vertical excitation energy at 1.80 .~): 4 ~ - , 0.00 eV; 2 ~ - , 0.51 eV, 21I, 0.83 eV; 2A, 0.92 eV; 4A, 1.08 eV; 2~+, 1.22 eV; 21I, 1.24 eV; 2A, 1.60 eV; 41-[, 1.80 eV.
Fig. 2. Geometry employed in the calculations of PtCH~ and valence molecular orbital diagram obtained from a restricted open-shell Harttee-Fock (ROHF) calculation for the 2A I ground state. Orbitals inactive in the CAS-SCF calculations (1 s(C), 5s(Pt) and 5p(Pt)) are not shown.
tal challenge would be to try to differentiate between the two O = 3 / 2 and O = 1 / 2 components of the 4]~- ground state as monitored by different chemical behavior.
3.6. Cationic platinum carbene complex PtCH2+ This organometallic fragment constitutes the key intermediate in the Pt÷-mediated activation [33] and subsequent oxidation of methane [32] in the gas phase. It has been shown that scalar relativistic effects (mass-velocity and Darwin terms) strengthen the Pt+-CH2 bond by as much as 55 kcal/mol [34], and it, therefore, appears interesting to investigate spin-orbit effects in this species with the method proposed in this study. Here, we report on the energetic stabilization of the 2A 1 ground state by spinorbit effects at its minimum geometry [33,34], which is shown in Fig. 2 together with a valence molecular orbital diagram. The CAS-SCF active space comprised 9 orbitals for 15 active electrons (5s, 5p orbitals of Pt and ls orbital of C always doubly occupied) and three sets of natural orbitals were optimized, one for the ground state (CAS-1), one for an average over the lowest five doublet states with
C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
517
Table 7 Energetic stabilization AEso of the 2A l ground state of PtCH~ by interaction with excited doublet and quartet states CAS treatment a
States considered in spin-orbit calculation b
A E~o (eV)
CAS- 1
five lowest doublets five lowest doublet and two lowest quartets
0.10 0.13
CAS-2
five lowest doublets five lowest doublet and two lowest quartets
0.18 0.18
CAS-3
five lowest doublets five lowest doublet and two lowest quartets
0.14 0.18
a See
text.
"ZL~,.(Pt) = 1200; ZeIT(C) = 3.6, as recommended in Ref. [14].
equal weights for each state (CAS-2) and one for the lowest quartet state (CAS-3). Contrary to the isoelectronic PtO + cation with its 4 ~ - ground state, a perfect-pairing situation prevails in the PtCH~ molecule: If CAS-SCF calculations are performed with individual orbital optimization for each state, the four lowest excited states have doublet multiplicity (vertical excitation energies: 2A l 2A2:0.70 eV; 2A 1 ~ 2A l (II): 0.87 eV; 2A 1 ~ 2 B2:1.36 eV; 2A 1 --* 2A2:1.55 eV) and the vertical excitation energies to the accidentally degenerate lowest two quartet s t a t e s ( 4 B 1, 2 8 2, arising from the one-electron excitations 3a I ~ 2b I and la 2 ~ 2b 1, respectively) amount to 1.81 eV. Nonvanishing spin-orbit matrix elements with the ground state are found lor all excited states except for 2A1(II). As evident from the results shown in Table 7, the absolute spin-orbit stabilization of the ground state depends somewhat on the sets of natural orbitals and the choice of excited states in the spin-orbit treatment, but its order of magnitude can be recognized as 0.14 eV. Further refinement by optimizing the orbital basis would be possible; however, the uncertainty of the result is within the expected overall error of the employed theoretical model for the treatment of spin-orbit effects. More important is our finding that the spin-orbit stabilization of the 'bare' Pt ÷ cation (0.48 eV 4) is considerably larger
4 According to Table 4, the J = 5 / 2 ground state of Pt + lies 0.48 eV below the 2D state (weighted average of the first J = 5 / 2 and J = 3 / 2 states) for Zen.(Pt)= 1200.
than for the cationic carbene complex PtCH~-. As a consequence, earlier theoretical values for the binding energy of CH 2 towards Pt ÷, which included only the mass-velocity and Darwin terms for the relativistic treatment, would have to be lowered by = 6130.34 eV (8 kcal/mol). Dirac-Fock- and spin-free Douglass-Kroll calculations [35] corroborate this conclusion from both its qualitative and quantitative side and we expect similar effects also for the binding energies of other unsaturated open-shell platinum species. Overall, spin-orbit effects weaken the interaction between Pt + and CH 2 to a much smaller extent than the scalar relativistic terms strengthen it. Thus, our earlier conclusion [34] that large relativistic effects stabilize the PtCH~- molecule to an extent that exothermic methane activation by Pt ÷ cations in the gas-phase becomes thermochemically feasible, remains valid also upon inclusion of spin-orbit effects.
4. Summary Spin-orbit effects in platinum species can be treated in a semi-empirical manner by use of the operator /-/so with a scaling parameter Zeff(Pt) between 950 and 1200. In this manner the qualitative ordering of excited states is usually well reproduced. Using a pseudo-potential approximation for the treatment of the 60 core electrons of platinum and CASSCF wavefunctions expanded in basis sets of TZ2P quality, errors in calculated excitation energies between spin-orbit coupled energy levels are kept to
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the order o f 0.1 e V w h e n f o u r - c o m p o n e n t D i r a c F o c k calculations are taken as a reference. C o m p a r i son to e x p e r i m e n t a l data yields s e m i - q u a n t i t a t i v e a g r e e m e n t d e p e n d i n g on the d e g r e e o f d y n a m i c electron correlation, a point w h i c h should be considered further. W e h o p e this p r a g m a t i c way o f handling valence-shell s p i n - o r b i t effects in platinum c o m pounds can be applied to m o r e c o m p l i c a t e d p o l y atomic potential e n e r g y surfaces in the near future. The ultimate goal is to understand the influence o f s p i n - o r b i t c o u p l i n g on the activity o f p l a t i n u m - c o n taining catalysts on a m o l e c u l a r basis.
Acknowledgements W e thank P r o f e s s o r M a r k S. G o r d o n , Professor Shiro Koseki and Dr. M i c h a e l W. S c h m i d t for a copy o f the latest v e r s i o n o f G A M E S S , for a preprint o f their recent w o r k on s p i n - o r b i t c o u p l i n g [15] and for useful discussions. F i n a n c i a l support by the Deutsche Forschungsgemeinschaft (Schwerpunktprogramm " R e l a t i v i s t i s c h e E f f e k t e " ) , the F o n d s der C h e m i s c h e n Industrie ( K 6 k u l 6 - f e l l o w s h i p for CH), the V o l k s w a g e n - S t i f t u n g and the E u r o p e a n S c i e n c e F o u n d a t i o n ( R E H E p r o g r a m ) are gratefully acknowledged.
References [1] P. PyykkiS, Relativistic theory of atoms and molecules, Vols. 1, II (Spinger, Berlin, 1986, 1993). [2] R.J. Martin and P.J. Hay, J. Chem. Phys. 75 (1981) 4539, and references therein. [3] W.G. Richards, H.P. Trivedi and D.L. Cooper, Spin-orbit coupling in molecules (Clarendon Press, Oxford, 1981). [4] K.G. Dyall, J. Chem. Phys. 98 (1993) 9678. [5] L. Visscher, T. Saue, W.C. Niuewpoort, K. Faegri and O. Gropen, J. Chem. Phys. 99 (1993) 6704. [6] H. Lef'ebvre-Brion, R.W. Field, Perturbations in the spectra of diatomic molecules (Academic Press, New York, 1986). [7] J.S. Pilgrim, C.S. Yeh, K.R. Berry and M.A. Duncan, J. Chem. Phys. 100 (1994) 7945. [8] A. Fiedler, D. SchriSder, S. Shaik and H. Schwarz, J. Am. Chem. Soc. 116 (1994) 10734.
[9] T.E.H. Walker and W.G. Richards, J. Chem. Phys. 52 (1970) 1311. [10] S.R. Langhoff, J. Chem. Phys. 73 (1980) 2379. [11] P.J. Hay, W.R. Wadt, L.R. Kahn, R.C. Raffenetti and D.H. Phillips, J. Chem. Phys. 71 (1979) 1767. [12] J.S. Cohen, W.R. Wadt and P.J. Hay, J. Chem. Phys. 71 (1971) 2955. [13] W.R. Wadt, Chem. Phys. Letters 89 (1982) 245. [14] S. Koseki, M.W. Schmidt and M.S. Gordon, J. Phys. Chem. 96 (1992) 10768. [15] S. Koseki, M.S. Gordon, M.W. Schmidt and N. Matsunaga, J. Phys. Chem., in press. [16] M. Klobukowski, Chem. Phys. Letters 183 (1991) 417. [17] W.J. Stevens, M. Krauss, H. Basch and P.G. Jasien, Can. J. Chem. 70 (1992) 612. [18] T.H. Dunning, J. Chem. Phys. 55 (1971) 716. [19] K. Ruedenberg, M.W. Schmidt, M.M. Gilbert and S.T. Elbert, Chem. Phys. 71 (1982) 41. [20] M.W. Schmidt, K.K. Baldridge, J.A. Boatz, S.T. Elbert, M.S. Gordon, J.H. Jensen, S. Koseki, N. Matsunaga, K.A. Nguyen, S. Su, T.L. Windus, M. Dupuis and J.A. Montgomery, J. Comput. Chem. 14 (1993) 1347. [21] T. Fleig and C.M. Marian, Chem. Phys. Letters 222 (1994) 267. [22] G. Gustafsson and R. Scullman, Mol. Phys. 67 (1989) 981. [23] M.C. McCarthy, R.W. Field, R. Engleman and P.F. Bernath, J. Mol. Spectry. 158 (1993) 208. [24] M. Casarrubios and L. Seijo, Chem. Phys. Letters 236 (1995) 510. [25] C.E. Moore, Atomic energy levels, NSRDS-NBS, Circular No. 467 (US GPO, Washington, 1949). [26] B.H. Lengsfield, J.A. Jafri, D.H. Phillips and C.W. Bauschlicher Jr., J. Chem. Phys. 74 (1981) 6849. [27] V.I. Srdanov and D.O. Harris, J. Chem. Phys. 89 (1988) 2748, and references therein. [28] U. Sassenberg and R. Scullman, Physica Scripta 28 (1983) 139. [29] G. Herzberg, Molecular spectra and molecular structure, Vol. 1, Reprint Edition (Krieger, Malabar, 1989). [30] S.-C. Chang, S. Kriiger, G. Pacchioni and N. R~Ssch, J. Chem. Phys. 102 (1995) 3695. [31] K. Kayama and J.C. Baird, J. Chem. Phys. 46 (1967) 2604. [32] R. Wesendrup, D. SchriSder and H. Schwarz, Angew. Chem. 195 (1994) 1232; Angew. Chem. Intern. Ed. Engl. 33 (1994)
1174. [33] C. Heinemann, R. Wesendrup and H. Schwarz, Chem. Phys. Letters 239 (1995) 75 [34] C. Heinemann, R.H. Hertwig, R. Wesendrup, W. Koch and H. Schwarz, J. Am. Chem. Soc. 117 (1995) 495. [35] C. Heinemann, H. Schwarz, W. Koch and K.G. Dyall, to be published.
CHEMICAL PHYSICS LETTERS
iZj,:! ELSEVIER
Chemical Physics Letters 245 (1995) 509-518
An approximate method for treating spin-orbit effects in platinum Christoph Heinemann, Wolfram Koch, Helmut Schwarz Institut fdr Organische Chemie der Technischen Universitiit Berlin, Strafle des 17. Juni 135, D-10623 Berlin. Germany
Received 2 July 1995; in final form 24 August 1995
Abstract
Spin-orbit coupling in platinum-containing species can be treated via a one-electron spin-orbit operator and a single scaling parameter Zcff(Pt) in conjunction with an effective core potential for the description of scalar relativistic effects. Our calibration calculations cover the five low-lying electronic states of platinum hydride PtH and the lowest fourteen levels in the atomic spectrum of the platinum atom Pt. Here, qualitative and semi-quantitative agreement between the presented semi-empirical approach and four-component Dirac-Fock calculations is found if Zeff(Pt) is chosen between 950 and 1200. Further applications concern the low-lying levels of the platinum cation Pt +, the theoretical determination of ground states for the diatomic oxides PtO and PtO + as well as spin-orbit effects in the cationic carbene complex PtCH~-.
1. Introduction
The major influences of relativity on the electronic structure of heavy-atom containing molecules are well understood at least from a pragmatic point of view and it has become feasible to perform qualitatively as well as quantitatively predictive ab initio calculations on molecular systems incorporating heavy third-row transition-metals, actinides and trans-actinides [1]. While different approaches have been devised for the incorporation of scalar relativistic effects in the established one-component formalism of molecular quantum chemistry [2], the treatment of spin-orbit effects [3] is far less straightforward and requires additional, sometimes prohibitively expensive computational effort [4,5]. Thus, in spectroscopy semi-empirical schemes have been developed to estimate fine-structure splittings and off-diagonal spin-orbit matrix elements such that perturbations in observed spectra can be traced back
tO atomic s p i n - o r b i t or s p i n - s p i n coupling constants 1. This approach becomes, however, impractical for nonlinear polyatomic molecules or even diatomics with considerable multi-configurational character where wavefunctions composed of a single configuration state function built from one-electron orbitals with well defined /-quantum numbers cannot be formulated in a straightforward manner. With regard to the interplay between computational quantum chemistry and spectroscopic experiments there is a need for a practical procedure for incorporating spin-orbit effects into theoretical calculations. High-resolution spectroscopy is able to provide detailed information about spin-orbit coupling, for example, in clusters of metal and rare-gas atoms [7]. From our particular point of view, we would like to point out that spin-orbit coupling is
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-26 14(95)01042-4
i An overview of these methods is given in Ref. [6].
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also of paramount importance for an understanding of the reactions of small unsaturated organometallic complexes with simple inorganic substrates such as dihydrogen or unfunctionalized hydrocarbons [8]. The spin-orbit interaction represents one of the major perturbations of the potential energy surfaces of such reactions. Electronically excited states may influence the ground state chemistry of 'bare' and ligated transition-metal cations via spin-orbit mediated curve-crossing mechanisms, such that the reactivity of an intrinsically unreactive ground state is enhanced via avoided crossings with more reactive excited states. Recently, Koseki et al. took up an earlier idea [9-13] of calculating spin-orbit effects via the singly parametrized one-electron operator Hso n s•o =
1 2 E[zeff(A)/r?a]LiA.Si, -~Ol i,A
where a denotes the fine-structure constant, LiA and Si angular momentum and spin operators, ria an electron-nucleus distance and i and A run over all electrons and nuclei, respectively [14,15]. Zeff(A) is an effective nuclear charge, which was adjusted for each element within a given basis set description such that experimental fine-structure splittings of 2II and 317 states of simple diatomic hydrides were in agreement with those calculated by the help of /-/so operating on complete active-space self-consistent field (CAS-SCF) wavefunctions. The usefulness of this operator in spin-orbit calculations is based on an approximate cancellation between neglected terms of the full Breit-Pauli Hamiltonian. Zeff should thus be regarded as a genuine semi-empirical parameter rather than a physical 'charge' since it adjusts the errors in the approximation, in particular, when effective core potentials with the corresponding nodeless valence pseudo-orbitals are employed for heavy atoms [ 16]. In this Letter, we investigate the performance of this model for the treatment of spin-orbit effects in platinum. Using earlier fully relativistic four-component Dirac-Fock calculations (where available) and experimental data as a reference, we derive a useful value for the parameter Zeff(Pt). to be used in conjunction with valence CAS-SCF wavefunctions, a relativistic pseudo-potential and a valence basis of triple-zeta plus two polarization functions quality.
The calibration of the method is carried out for the diatomic platinum hydride molecule PtH and atomic platinum Pt, for which reliable reference data are available. In the second part, we apply this scheme to systems such as the platinum cation Pt ÷, the diatomic neutral and cationic platinum oxides PtO and PtO ÷ as well as the cationic platinum carbene complex PtCH~-, for which there is still a significant need for accurate calculations. Particular emphasis is given to spin-orbit effects on spectroscopic constants, excitation energies between low-lying electronic states and electronic ground-state stabilization.
2. Computational details The inner 60 electrons of platinum ([Kr] + 4d m + 4f 14) were treated by the relativistic effective core potential given by Stevens et al. [17] thus treating the scalar relativistic effects from the mass-velocity and Darwin terms en gros. The associated Gaussian-type (7s7p5d)/[4s4p3d] (4111141111311) basis set was augmented by two f-type polarization functions (exponents: 0.2, 0.08) to generate a valence basis of triple-zeta plus two polarization functions (TZ2P) quality. For carbon and oxygen, Dunning's (lls6p)/[5s, 3p] (621111411) sets [18] were augmented by two d-type polarization functions (C-exponents of 1.00, 0.40; O-exponents of 2.56, 0.64). For hydrogen, two p-type polarization functions (exponents: 1.25, 0.45) were added to the same author's (5s)/[3s] (311) get. All Cartesian components of the d- and f-type basis functions were retained in the calculations. These one-particle sets served to construct CAS-SCF wavefunctions [19], which will be explicitly described for each case in the course of the discussion. The natural orbitals of these wavefunctions were employed to calculate matrix elements of /-/so and this matrix was diagonalized in order to obtain spin-orbit coupled states. For the diatomic molecules 10-20 single point energy calculations in the vicinity of the minima were performed to generate a potential curve in which subsequently the nuclear Schr/Adinger equation for the most abundant isotopes was solved using Numerov's method to obtain spectroscopic constants. All calculations were carried out without symmetry restrictions (technical reasons) using the latest version (March 10, 1995) of the GAMESS program system [20] on an I B M / R S
C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
6000 workstation. Since our treatment of dynamic electron correlation is almost none, comparison of the theoretical results to experimental data should be carried out with caution. However, some discussion of dynamic correlation effects will be given in Section 3.
Table 2 Spectroscopic constants (equilibrium distances Re in A, harmonic frequencies ~oe in cm-J and adiabatic excitation energies Te in eV) for the five low-lying states of PtH from CAS-SCF calculations as a function of Zeff(Pt). The four-component Dirac-Fock results are from Ref. [4] 0=5/2
12=1/2 (1)
g2=3/2 (1)
12=3/2 (II)
12=1/2 (II)
1.590 2061 0.29
1.581 2050 0.33
1.599 2103 0.87
1.611 2017 1.03
1.595 2045 0.33
1.589 2026 0.37
1.590 2121 I. 10
1.607 2034 1.26
Zeff(Pt) = 950 R c 1.558 ~oe 2190 Te 0.00
1.597 2040 0.35
1.593 2027 0.38
1.588 2125 1.20
1.604 2040 1.35
Zerr(Pt) = 1200 R e 1.558 coe 2190 T~ 0.00
1.601 2028 0.38
1.598 2018 0.40
1.583 2128 1.45
1.600 2054 1.59
Dirac-Fock R e 1.551 o)e 2234 Te 0.00
1.573 2094 0,34
1.584 2080 0.42
1.577 2162 1.43
1.590 2097 1.55
Zeff(Pt) = 600 R e 1.559 we 2189 Te 0.00
3. Results and discussion
3.1. Platinum hydride PtH
511
Z~ff(Pt)= 850 A small active space of six orbitals and eleven electrons was chosen in the CAS-SCF treatment (5s and 5p orbitals of Pt always doubly occupied). While close to the equilibrium geometry near-degeneracy effects (nondynamic electron correlation) are absent in PtH, dynamic electron correlation has been shown to shorten the P t - H bond lengths of the low-lying states by = 0.04 ,~ and to increase their frequencies by = 200 cm-1 [5]. The results of a state-avera§ed AS-coupled calculation for the three low-lying A, 2~ +, and 2II states (treated with equal weights in the orbital optimization) are given in Table 1. Using these natural orbitals for the spin-orbit treatment one obtains five low-lying electronic states with 12 = 5 / 2 , 3 / 2 and 1/2, whose spectroscopic constants as a function of the scaling parameter Zeff(Pt) a r e given in Table 2. In accord with earlier results [4,5], spin-orbit coupling in the ground state of PtH is a pure finestructure effect in the sense that the /2 = 5 / 2 state can be identified as the only weakly t~erturbed g2 = 5 / 2 component of the AS-coupled =A state. Conversely, the four low-lying excited states are mixtures of the ,(-2= 1 / 2 and ,(2 = 3 / 2 components of the {2A, 2 ~ +, 2 11} manifold. The spectroscopic constants obtained for the five low-lying electronic states Table I Spectroscopic constants (equilibrium distances R e in/~, harmonic frequencies o9¢ in cm I and adiabatic excitation energies Te in eV) for the three low-lying AS-coupled states of PtH from CAS-SCF calculations
Re w~
2A
2E+
2H
1.559 2235 0.00
1.579 2130 0.13
1.624 2002 0.54
Re coe Te
1.559 2190 0.00
are in remarkably good agreement with two recently reported uncorrelated four-component Dirac-Fock calculations which include spin-orbit coupling at the level of the self-consistent procedure [4,5]. Here, we will compare our results to the more comprehensive data of Dyall [4]. Further theoretical analyses of PtH have been discussed elsewhere in the literature [4,5,21]. First, we find the correct qualitative state ordering with relatively small excitation energies from the [2 = 5 / 2 ground state to the lowest ,(2 = 1 / 2 (I) and 12 = 3 / 2 (I) excited states and the larger energetic gap to the two higher excited states. The overall energetic splitting of the considered state manifold increases with larger Z~ff(Pt). Increasing the scaling parameter also shortens the bond lengths and raises the frequencies of the second pair of excited states (12 = 1/2(II), ,(2 = 3/2(II)), while the opposite applies to their lower counterparts. Trends in the calcu-
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lated frequencies and bond distances parallel those from Dyall's four-component study [4], except for the bond length o f the 1-2 = 1 / 2 (I) state, which is a bit overestimated in our calculation. Quantitatively, our frequencies are slightly lower and our bond distances somewhat larger than the respective Dirac-Fock (DF) values and the absolute average deviations (50 cm -1 and 0.013 ,~) are almost independent of Zeff(Pt) in the considered range. These differences are partly due to our use of state-averaged natural CAS orbitals while the DF treatment includes individual orbital optimization for each state. As far as the excitation energies are concerned, the best agreement between the method employed here and the D i r a c - F o c k calculations is found for Zeff(Pt) = 1200 (absolute average deviation: 0.03 eV). For smaller scaling parameters, the excitation energies as derived from the present study are smaller than the four-component results. Independent of Zeff(Pt), the absolute splitting between the lowest two excited states is always somewhat underestimated. Finally, we note that with respect to the experimental parameters of PtH ( g 2 = 5 / 2 : R e = 1 . 5 2 8 A, we = 2 3 7 8 cm -1", O = 3 / 2 (I): R e = 1.519 A, o~e = 2265 cm -I [22]) our calculations show the typical behavior of dynamically uncorrelated wavefunctions for transition-metal systems (overestimation of bond lengths by = 0.05 A, underestimation of frequencies by = 250 c m - ~ for Zeff(Pt) = 1200). On the other hand, the calculated electronic excitation energies exactly match the experimentally observed ones (g2 = 5 / 2 ---, g2 = 3/2(1): 0.40 eV; /2 - 5 / 2 --*/2 = 3/2(11): 1.45 eV [23]) for Zeef(Pt)= 1200. The quantitative side of this agreement is probably due to a fortuitous cancellation of errors. 3.2. P l a t i n u m a t o m P t
Recent studies [4,5,24] have shown that the relative energies of the lowest 14 atomic energy levels of the platinum atom depend strongly on the relative weights of the three electron configurations 5dl°6s °, 5d96s 1 and 5da6s 2 in the optimization of the orbitals from which these levels are constructed. In order to approach agreement with the experimental spectrum [25], both electron correlation and quantum electrodynamic corrections were found to be important. In view of these difficulties, we restrict ourselves primarily to a comparison to an available numerical
Table 3 Relative energies of the low-lying states of the Pt atom (in cm- i ) from CAS-SCF calculations. The four-component Dirac-Fock results are from Ref. [4] J
zat(Pt) = 950
Zeff(Pt)= 1200 Dirac-Fock
Exp.[25]
3 4 2 2 1 3 2 2 0 1 0 4 2 0
0 1010 1217 7021 8334 8141 12752 16357 16662 20736 21471 25151 26812 53579
0 342 1201 7856 10528 9457 14833 17092 17052 22054 22556 26259 29345 55614
0 824 776 6567 10132 10116 15502 13496 18567 6140 21967 26639 -
0 941 1312 6910 9227 10287 15055 17473 17673 22418 24350 25054 29210 54687
four-component Dirac-Fock calculation [4] without additional corrections (the basis set error with a one-particle space of approximately valence TZ2P quality was found to be small [5]). Similar to this study, singlet and triplet orbitals were optimized separately for CAS-SCF wavefunctions (10 electrons in 6 active orbitals; 5s, 5p always doubly occupied) with equal weights for the 5d96s I and 5dS6s 2 configurations and a relative weight of 10% for the 5d1°6s ° configuration. After a corresponding orbital transformation [26], the respective natural orbitals were used in a spin-orbit calculation which yielded the atomic spectrum of platinum as listed in Table 3. The so-obtained level structure agrees well with the four-component result. Qualitatively, only the first J = 1 and the second J = 3 levels have the incorrect ordering for Ze,(Pt) = 950. Using a scaling parameter Z e f f ( P t ) = 1200, also the fourth J = 2 and the first J = 0 level are interchanged but their splitting is small (40 c m - J ) and they are also close in the Dirac-Fock treatment (200 c m - l ) . The absolute average deviation in the excitation energies between the present results and the Dirac-Fock study is 1224 cm-1 for Zeff(Pt)= 950 and 726 cm -1 for Zeff(Pt) = 1200. However, the energy difference between the J = 3 ground state and the J = 4 first excited state is closer to the Dirac-Fock result for Z a e ( P t ) = 950. The experimentally observed atomic energy levels for platinum are also compiled in Table 3. Absolute average deviations between calculated and observed
C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
excitation energies amount to 2847 c m - l (Z~ff(Pt) = 950) and 3129 cm - l (Z~ff(Pt)= 1200), compared to 3110 cm -1 for the D i r a c - F o c k calculation. The errors are smaller (1599 cm -I for Zeff(Pt)=950, 1800 cm -1 for Z~ff(Pt)= 1200, 1601 cm - l for Dirac-Fock) if the first J = 0 level, which depends strongly on the choice of the one-particle basis [4], is omitted. The operator Hso used with a scaling parameter Zeff(Pt) between 950 and 1200 reproduces more rigorous treatments of spin-orbit effects on spectroscopic constants and low-lying excited state energies in platinum-containing species in a semi-quantitative manner. The large magnitude of the optimum parameter arises from the missing nodal structure of the valence pseudo-orbitals in the region near the nucleus from which spin-orbit effects predominantly arise [13,15,16]. The main purpose of the present investigation is not a rigorous first-principles treatment of spin-orbit coupling in platinum, but to establish the applicability of this simple semi-empirical approach. Thus, after the successful calibration of the theoretical four-component Dirac-Fock results for PtH and Pt, we will now continue with applications to cases in which spin-orbit effects are expected to be important but where, to the best of our knowledge, no rigorous theoretical reference data are available. It is important to reiterate that in spite of being called an effective nuclear charge, no physical relevance should be attributed to the scaling parameter Zeff(Pt). Its sole purpose is to correct for errors in the approximation. 3.3. P l a t i n u m c a t i o n P t +
The platinum cation was chosen as the first test case because comparison of the calculated energy levels to the low-lying part of the experimentally observed spectrum should be less difficult than for the neutral platinum atom: only two LS terms (2D, 4F) arising from two electronic configurations (5d96s °, 5d86s t ) dominate the spectrum of Pt + up to excitation energies of 2 eV [25] and, thus, the individual J-levels are expected to be less mixed than in the neutral platinum atom. For Pt +, CAS-SCF wavefunctions were obtained separately for the 2D and 4F states with 9 active electrons in 6 active orbitals (6s, 5d) and the 5s and 5p orbitals always doubly occupied. Neglecting spin-orbit coupling, the ZD-~ 4F
513
Table 4 Relative energies of the low-lying states of the Pt + cation (in cm -~ ) from CAS-SCF calculations. The experimental values are from Ref. [25] J 5/2 9/2 3/2 7/2 5/2 3/2
Zeff(Pt) Zeff(Pt) = 950 = 950 a
Zeff(Pt)
Zcff(Pt)
= 1200
= 1200 a
0 2897 7671 8164 12320 15405
0 2357 9653 9010 14267 18200
0 3516 9653 10169 15426 19359
0 4056 7671 9323 13479 16564
Exp. 0 4787 8420 9356 13329 15791
a The calculated levels deriving from the 4F term are shifted upwards by 1159 cm-~ to correct for the disagreement between the experimental and calculated ~D -~ 4F excitation energies. excitation energy is calculated to be 4968 cm - I , 1159 cm -l lower than the experimental value (obtained as the weighted average of the corresponding J-levels [25]). Part of this disagreement reflects the larger dynamic correlation energy in the low-spin 5d96s ° configuration. After a corresponding orbital transformation [26] //so was diagonalized in the basis of the CAS-SCF wavefunctions and the obtained level spectrum of Pt ÷ is listed in Table 4, along with the experimental data. To approximately correct for the neglect of dynamic correlation in the CAS-SCF wavefunctions, the calculated spin-orbit levels deriving from the 4F state have been shifted upwards by 1159 cm - l . After this manipulation, the absolute average deviation between the calculated and the observed excitation energies amounts to 487 cm -1 for Zeff(Pt)= 950 and 1796 cm -1 for Zeff(Pt) = 1200. Larger than average errors occur in particular for the two highest J = 5 / 2 and J = 3 / 2 levels which are expected to interact with higher-lying terms that are not included in the present calculation. The final deviation between computed and observed excitation energies for Z~ff(Pt) = 950 are of the order of magnitude of quantum electrodynamic corrections in the spectrum of neutral atomic platinum [4,5]. Thus, we can conclude that for similar charges Z~ff(Pt) the chosen semi-empirical approach yields consistent results for spin-orbit interaction in both the neutral and the cationic platinum atom. 3.4. P l a t i n u m o x i d e P t O
The ground state of PtO has been under some discussion in recent years [27]. The latest spectro-
514
C. Heinemann et aL / Chemical Physics Letters 245 (1995) 509-518 4or E
T
2¢J
Fig. 1. Valence molecular orbital diagram for PtO and PtO +. The electron configuration corresponds to the 3E- ground state of PtO. Removal of the spin-down electron from the 3s orbital affords the 4 ~ - ground state of PtO +. Orbitals inactive in the CAS-SCF calculations (Is(O), 5s(Pt) and 5p(Pt)) are not shown.
scopic experiments [28] suggest a Hund's case (c) [6,29] description with an O = 0 ground state arising from a 3 E state (valence configuration: 1842"rr23tr2, see Fig. 1) via spin-orbit coupling with electronically excited states, in particular the ~E+ state (valence configurations: 18427r43tr°/ 1842 xr 23tr 2). The excitation energy from the ground state to the first excited state ( O = 1) was found to be 0.12 eV [28]. Density functional calculations [30] have investigated the lowest state arising from the 1842 ~ 23tr 2 configuration but a definitive theoretical answer to the question of the ground state symmetry could not be given due to the multi-configurational character of the problem. Thus, we have calculated CAS-SCF wavefunctions for the lowest AS-coupled singlet and triplet states of PtO distributing 16 electrons in 10 active orbitals as shown in Fig. 1 (5s, 5p orbitals of platinum and is orbital of oxygen doubly occupied in all CSFs). The lowest CAS-SCF energies were always obtained for the 3 ~ - state and the lowest singlet states were of ~A and ~E + symmetry. Thus, the electronic situation in PtO bears some similarity to the dioxygen molecule 02 [31], which also has a 3E- ground state and two low-lying excited IA and i E + states [29]. The diagonalization of /-/so was performed using the natural orbitals for
the 3~- state. The reason for using the ground state's natural orbitals in the spin-orbit treatment was that the corresponding orbital transformation [26], which is necessary prior to the evaluation of matrix elements with two different sets of orbitals, failed for PtO. However, reoptimization of the excited states' natural orbitals lowers their total energies by only 2-3 mhartree and using different sets of singlet or triplet orbitals affects the computed spinorbit matrix elements by less than 10%. These effects are considerably smaller as compared to other approximations made in this study. Spectroscopic constants for the three lowest AScoupled states of PrO and their four spin-orbit levels are given in Table 5. In the AS-coupling scheme, the 3E - ground state of PtO has a shorter P t - O distance and higher vibrational frequency as compared to the two excited singlet states. Within the {3~ - , 1A, l~ +} manifold, a nonvanishing spin-orbit matrix element occurs only between the M s = 0 component of the 3 ~ - state and the 1E+ state, giving rise to an /2 = 0 ground state, a low-lying O = 1 excited state (unperturbed with respect to 3E-), the unperturbed IA state ( O = 2) and an /2 = 0 state deriving mainly from 1E +. Similar to the PtH case (see above), spin-orbit coupling increases the overall energetic manifold splitting and mixes the states such that the bond length of the lower O = 0 state is slightly increased and its frequency decreased while the opposite applies for the higher /2 = 0 state. The calculated adiabatic excitation energies between the 12 = 0 ground state and the /-2 = 1 first excited state (0.06 eV with Zeff(Pt)=950, 0.10 eV with Zeff(Pt)= 1200) compare reasonably well with the experimental result of 0.12 eV [28] thus confirming the interpretation of the PtO ground state as derived from the spectroscopic experiments [28]. Since the CAS-SCF method accounts for near-degeneracy effects in the valence region we expect that the inclusion of dynamic correlation will not alter the qualitative ordering of states as obtained here. However, these effects are expected to shorten the P t - O bond length and to enhance the stretching frequency relative to the CAS-SCF results, which deviate from the experimental data (R~ = 1.727 c m - l , toc = 851 cm-1 for the /-2 = 0 ground state [28]) within the expected range. In addition, the calculated trends for the bond lengths and frequencies of the 12 = 0 ground state and the
C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518 Table 5 Spectroscopic constants (equilibrium distances R e in ,~, harmonic frequencies oJ~ in c m - t and adiabatic excitation energies Te in eV) for low-lying states of PtO from CAS-SCF calculations. The experimental values are from Ref. [28] R,~
¢oe
Te
a]£ IA i y, +
1.791 1.819 1.849
723 648 588
0.00 0.59 0.94
Zc,(Pt) = 950 ~,h O = 0 ~(2 = I -(2 = 2 .(2 = 0
1.792 1.791 1.819 1.848
722 723 648 595
0.00 0.06 0.65 1.06
Zcrf(Pt) = 1200 a.h ~f2 = 0 1.792 .(2 = 1 1.791 .(2 = 2 1.819 .(2 = 0 1.848
719 723 648 597
0.00 0.10 0.69 1.14
Zcfr(Pt) = 1200 a,c f2 = 0 u 1.797 -(2 = I 1.801 .(2 = 2 1.843 g2 = 0 1.880
705 658 635 484
0.00 0.03 0.52 0.92
expt. ,(2 = 0 O = 1
851 832
0.00 0.12
1.727 1.729
Z~tr(O) = 5.6, as recommended in Ref. [14]. S p i n - o r b i t operator diagonalized in the space of the states {35~ - ,
IA, 1~+}.
c S p i n - o r b i t operator diagonalized in the space of the states { 3 ~ -
iA, 1~;+, 3rl ' 117, 34,}. a This state is stabilized by 0.12 eV with respect to the 3 ~ - state.
12 = 1 first excited state are opposite to the experimentally observed ones [28]. Besides dynamic correlation effects, spin-orbit interaction with higher excited states might constitute the reason for this disagreement. To investigate the influence of the next excited states, a second diagonalization of H~o including two additional triplet states (3II and 3 ~ ) and one additional singlet state (l I I ) was performed 2. This enlargement of the basis for the representation of H~o increases the P t - O distances and lowers the frequencies of all four low-lying states. Moreover, the 12 = 1 first excited state is stabilized with respect to the
515
12 = 0 ground state and the calculated excitation energy (0.03 eV) is reduced to only 20% of the experimental value. On the other hand, inclusion of the higher excited states affords the correct trend for the bond lengths and frequencies of these two states. These effects indicate a drawback of the employed method in treating spin-orbit effects: Since spiqorbit coupling is neglected during the self-consistent field procedure, it might be neccesary to include a large number of low-lying electronically excited states in the spin-orbit treatment to obtain meaningful results for the properties under study. Thus, we regard the presented methodology as a semi-quantitative treatment, which allows one to derive the magnitude of spin-orbit effects within the correct ballpark but should be used with caution when quantitative agreement with spectroscopic data is intended. 3.5. Platinum oxide cation PtO +
This diatomic molecule is of interest in organometallic gas-phase ion chemistry since it constitutes an important intermediate in the Pt+-cata lyzed gas-phase oxidation of methane by molecular oxygen [32,33]. Based on the electronic spectrum of neutral PtO (see above) one can expect low-lyhlg states of the AS symmetries 4 ~ - (l~42,rr23o.J configuration, see Fig. I), 21] (l~42"rr33cr°), 2A (l~42q'r23cr 1) or 2~+ (l~42,rr23crl). There should also be a large number of additional low-lying states with a hole in the l~-shell. Using CAS-SCF wavefunctions with 15 active electrons in 10 active orbitals (similar to the treatment of neutral PtO, see above), we find a high-spin 4E_ ground state for PtO +, similar to the isoelectronic NiO + molecule [8]. The natural orbitals for this state were also used to calculate the potential energy curves for the lowest low-spin doublet state ( 2 E - ) and the three lowest spin-orbit coupled states (12 = 3 / 2 , 12 = 1/2(I) and 12= 1/2(II)), which arise from the 2/4~- couple
2 These states have vertical excitation energies of 1.02 (3I/), 1.16 ( l i d and 1.42 eV (3alP) from the ground state at a P t - O distance of 1.80 A. The next excited state has a vertical excitation energy of 1.71 eV and is of I I-I symmetry.
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C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
Table 6 Spectroscopic constants (equilibrium distances R e in ~,, harmonic frequencies toe in c m - t and adiabatic excitation energies Te in eV) for low-lying states of PtO + from CAS-SCF calculations
4~2y..,Zetr(Pt) = 1200 a,b ~O = 3 / 2 c /'2 = 1/2(1) O = 1/2(1I)
Re
toe
:re
1.805 1.815
675 671
0.00 0.51
1.815 1.807 1.815
683 728 610
0.00 0.05 0.55
a Zeff(O) = 5.6, as recommended in Ref. [14]. b Spin-orbit operator diagonalized in the space of the states {4~ - , 2 Z - ' 21-i, 2A, 4A, 2~.+, 2[i ' 2A 41-i}.
4, E / eV ~
Pt
l.,OOX/H 1.084A \H
1
2b 1 (-5.6)
4a 1 (-10.8)
T
3al (-18.9) 1~2 (-1;'.1) 2b2 (-18.2) lb, (-18.5)
TJ. Tl T,I. TI
2a I (-21.1)
T J,
lb 2 (-23,9)
T,L
la I (-31,6)
T,I,
c This state is stabilized by 0.20 eV with respect to the 4 ~ - state.
via spin-orbit interaction with excited states 3 (see Table 6). Since the 4 E - state has the angular momentum quantum number A = 0, it does not undergo a finestructure splitting [6]. However, the splitting into the two /2 = 3 / 2 and O = 1 / 2 components occurs by inclusion of the excited states via nonvanishing offdiagonal elements of the spin-orbit operator. The /2 = 3 / 2 state is found to be the ground state of PtO +, as evident from the two matrix elements for the 4E-/Ell interaction (R = 1.80 .~): ( 4 E - , M s = 3 / 2 , /2=3/21/-/,ol21-I, Ms=l~2, / 2 = 3 / 2 ) = 1823+488i cm -1 and ( 4 E - , M , = l / 2 , /2 M, = - 1 / 2 , /2 = 1 / 2 ) = 1053-281i cm -1 The excitation energy to the first excited O = 1 / 2 state (0.05 eV) is of a similar order of magnitude as the /2 = 0 / / 2 = 1 state splitting in neutral PtO (see above). The next excited state deriving from the AS-coupled 2 E - state has also /2 = 1 / 2 and the respective excitation energy from the ground state is only marginally affected by spin-orbit interaction. Thus, we conclude that the interesting gas-phase chemistry of PtO ÷ [32,33] is dominated by the high-spin quartet ground state. An experimen-
1/21nsol2rl,
3 Individual orbital optimization for the 2 ~ - state lowers its total energies by typically 1 mhartree. In the spin-orbit treatment, the following states were included (symmetry, vertical excitation energy at 1.80 .~): 4 ~ - , 0.00 eV; 2 ~ - , 0.51 eV, 21I, 0.83 eV; 2A, 0.92 eV; 4A, 1.08 eV; 2~+, 1.22 eV; 21I, 1.24 eV; 2A, 1.60 eV; 41-[, 1.80 eV.
Fig. 2. Geometry employed in the calculations of PtCH~ and valence molecular orbital diagram obtained from a restricted open-shell Harttee-Fock (ROHF) calculation for the 2A I ground state. Orbitals inactive in the CAS-SCF calculations (1 s(C), 5s(Pt) and 5p(Pt)) are not shown.
tal challenge would be to try to differentiate between the two O = 3 / 2 and O = 1 / 2 components of the 4]~- ground state as monitored by different chemical behavior.
3.6. Cationic platinum carbene complex PtCH2+ This organometallic fragment constitutes the key intermediate in the Pt÷-mediated activation [33] and subsequent oxidation of methane [32] in the gas phase. It has been shown that scalar relativistic effects (mass-velocity and Darwin terms) strengthen the Pt+-CH2 bond by as much as 55 kcal/mol [34], and it, therefore, appears interesting to investigate spin-orbit effects in this species with the method proposed in this study. Here, we report on the energetic stabilization of the 2A 1 ground state by spinorbit effects at its minimum geometry [33,34], which is shown in Fig. 2 together with a valence molecular orbital diagram. The CAS-SCF active space comprised 9 orbitals for 15 active electrons (5s, 5p orbitals of Pt and ls orbital of C always doubly occupied) and three sets of natural orbitals were optimized, one for the ground state (CAS-1), one for an average over the lowest five doublet states with
C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
517
Table 7 Energetic stabilization AEso of the 2A l ground state of PtCH~ by interaction with excited doublet and quartet states CAS treatment a
States considered in spin-orbit calculation b
A E~o (eV)
CAS- 1
five lowest doublets five lowest doublet and two lowest quartets
0.10 0.13
CAS-2
five lowest doublets five lowest doublet and two lowest quartets
0.18 0.18
CAS-3
five lowest doublets five lowest doublet and two lowest quartets
0.14 0.18
a See
text.
"ZL~,.(Pt) = 1200; ZeIT(C) = 3.6, as recommended in Ref. [14].
equal weights for each state (CAS-2) and one for the lowest quartet state (CAS-3). Contrary to the isoelectronic PtO + cation with its 4 ~ - ground state, a perfect-pairing situation prevails in the PtCH~ molecule: If CAS-SCF calculations are performed with individual orbital optimization for each state, the four lowest excited states have doublet multiplicity (vertical excitation energies: 2A l 2A2:0.70 eV; 2A 1 ~ 2A l (II): 0.87 eV; 2A 1 ~ 2 B2:1.36 eV; 2A 1 --* 2A2:1.55 eV) and the vertical excitation energies to the accidentally degenerate lowest two quartet s t a t e s ( 4 B 1, 2 8 2, arising from the one-electron excitations 3a I ~ 2b I and la 2 ~ 2b 1, respectively) amount to 1.81 eV. Nonvanishing spin-orbit matrix elements with the ground state are found lor all excited states except for 2A1(II). As evident from the results shown in Table 7, the absolute spin-orbit stabilization of the ground state depends somewhat on the sets of natural orbitals and the choice of excited states in the spin-orbit treatment, but its order of magnitude can be recognized as 0.14 eV. Further refinement by optimizing the orbital basis would be possible; however, the uncertainty of the result is within the expected overall error of the employed theoretical model for the treatment of spin-orbit effects. More important is our finding that the spin-orbit stabilization of the 'bare' Pt ÷ cation (0.48 eV 4) is considerably larger
4 According to Table 4, the J = 5 / 2 ground state of Pt + lies 0.48 eV below the 2D state (weighted average of the first J = 5 / 2 and J = 3 / 2 states) for Zen.(Pt)= 1200.
than for the cationic carbene complex PtCH~-. As a consequence, earlier theoretical values for the binding energy of CH 2 towards Pt ÷, which included only the mass-velocity and Darwin terms for the relativistic treatment, would have to be lowered by = 6130.34 eV (8 kcal/mol). Dirac-Fock- and spin-free Douglass-Kroll calculations [35] corroborate this conclusion from both its qualitative and quantitative side and we expect similar effects also for the binding energies of other unsaturated open-shell platinum species. Overall, spin-orbit effects weaken the interaction between Pt + and CH 2 to a much smaller extent than the scalar relativistic terms strengthen it. Thus, our earlier conclusion [34] that large relativistic effects stabilize the PtCH~- molecule to an extent that exothermic methane activation by Pt ÷ cations in the gas-phase becomes thermochemically feasible, remains valid also upon inclusion of spin-orbit effects.
4. Summary Spin-orbit effects in platinum species can be treated in a semi-empirical manner by use of the operator /-/so with a scaling parameter Zeff(Pt) between 950 and 1200. In this manner the qualitative ordering of excited states is usually well reproduced. Using a pseudo-potential approximation for the treatment of the 60 core electrons of platinum and CASSCF wavefunctions expanded in basis sets of TZ2P quality, errors in calculated excitation energies between spin-orbit coupled energy levels are kept to
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C. Heinemann et al. / Chemical Physics Letters 245 (1995) 509-518
the order o f 0.1 e V w h e n f o u r - c o m p o n e n t D i r a c F o c k calculations are taken as a reference. C o m p a r i son to e x p e r i m e n t a l data yields s e m i - q u a n t i t a t i v e a g r e e m e n t d e p e n d i n g on the d e g r e e o f d y n a m i c electron correlation, a point w h i c h should be considered further. W e h o p e this p r a g m a t i c way o f handling valence-shell s p i n - o r b i t effects in platinum c o m pounds can be applied to m o r e c o m p l i c a t e d p o l y atomic potential e n e r g y surfaces in the near future. The ultimate goal is to understand the influence o f s p i n - o r b i t c o u p l i n g on the activity o f p l a t i n u m - c o n taining catalysts on a m o l e c u l a r basis.
Acknowledgements W e thank P r o f e s s o r M a r k S. G o r d o n , Professor Shiro Koseki and Dr. M i c h a e l W. S c h m i d t for a copy o f the latest v e r s i o n o f G A M E S S , for a preprint o f their recent w o r k on s p i n - o r b i t c o u p l i n g [15] and for useful discussions. F i n a n c i a l support by the Deutsche Forschungsgemeinschaft (Schwerpunktprogramm " R e l a t i v i s t i s c h e E f f e k t e " ) , the F o n d s der C h e m i s c h e n Industrie ( K 6 k u l 6 - f e l l o w s h i p for CH), the V o l k s w a g e n - S t i f t u n g and the E u r o p e a n S c i e n c e F o u n d a t i o n ( R E H E p r o g r a m ) are gratefully acknowledged.
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