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Electric properties of the ozone molecule
Volume 190, number 3,4
CHEMICAL PHYSICS LETTERS
6 March 1992
Electric properties of the ozone molecule K. Andersson ~, P. Borowski i, P.W. Fowler 2, P.-A. Malmqvist 1, B.O. Roos 1 and A.J. Sadlej Theoretical Chemistry, Chemical Center, University of Lund, P.O. Box 124, S-221 O0 Lund, Sweden 2 DepartmentofChemistry, UniversityofExeter, StockerRoad, ExeterEX44QD, UK
Received 19 December 1991
The dipole moment, quadrupole moment and dipole polarisability of 03 are calculated by the self-consistent field (SCF), many-body perturbation theory (MBPT), complete active space SCF (CASSCF), restricted active space SCF (RASSCF), multireference CI (MRCI) and CASSCF-perturbation theory (CASPT) methods, using a 5s3p2d polarised basis set. At the CASPT-2 level, the dipole moment is #b=0.2136 au. The quadrupole moments are O,~=- 1.158, Obb=--0.260, Oct= + 1.418; and the polarisabilityanisotropies are ct~-abb = 16.2 and aaa-otto= 18.6, all in atomic units. Experimental data for these quantities are 0.2100( I ), - 1.03(12), -0.52(16), + 1.55(19), and 19.2(7), 17.9(7), respectively. Similar results are obtained by CASSCF, RASSCF and MRCI, and by MBPT when taken to fourth order. The transverse in-plane polarisability obtained by the singlereference methods shows an erratic behaviour, which is similar to known variations in computed antisymmetric stretching frequencies.
1. Introduction Aside from its e n v i r o n m e n t a l a n d chemical importance, the ozone molecule has a significant role in the calibration o f theoretical methods. The experimental vibrational frequencies o f ozone are not well r e p r o d u c e d at the S C F level, a n d some difficulty and controversy has a t t e n d e d their calculation at correlated levels [ 1-6 ]. Single-reference methods can give erratic a n d even unphysical results for the antisymmetric stretching frequency unless taken to high o r d e r [ 3 ], a n d small changes in the way that higher excitations are included can give frequency shifts o f the o r d e r o f 100 c m - ~ [ 6 ]. A n o t h e r way o f probing this problem, which has its roots in the electronic structure o f ozone, is the calculation o f electric properties such as the polarisability. In the presence o f a transverse in-plane electric field the terminal a t o m s o f ozone b e c o m e inequivalent, as during the antisymmetric vibration, and similar difficulties in calculation o f correlated transverse polarisabilities Olaa and a n t i s y m m e t r i c stretching frequencies oJ3 are to be expected; both are second derivatives o f the energy with respect to a perturbation o f B~ symmetry ~. This p a p e r presents a c o m p a r i s o n o f electric properties (dipole m o m e n t / t , q u a d r u p o l e m o m e n t O a n d
dipole polarisability a ) calculated at various levels o f theory. It is shown that correlated m e t h o d s are needed to give a qualitatively correct description o f the properties, and that the m a i n difficulties lie with the transverse in-plane polarisability. Multi-reference methods and the single-reference M B P T method - when taken to the fourth o r d e r - give a consistent picture o f electric properties, and are in reasonable agreement with experimental evidence.
2. Method The ozone molecule was fixed at the experimental equilibrium geometry ( R o o = 1.2717 A, z _ O O O = 116.8 °) [7,8], and the polarised oxygen basis ( 1 0 s 6 p 4 d ) ~ [ 5s3p2d ] from the Sadlej c o m p i l a t i o n [ 9 ] was used. The success o f these basis sets for dipole moments, polarisabilities [9 ] and q u a d r u p o l e m o m e n t s [ 10 ] at S C F and correlated levels for conventional closed-shell molecules is well d o c u m e n t e d . ~J The axis convention used in the present paper puts the molecule in the xz plane, where z is the C2 axis (b principal axis) and x is the a principal axis of 1603./zx---/zatransforms a s l 1 in this setting of C2v.
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367
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Spherical harmonic combinations of the d-type GTOs were used. Methods of calculation employed in the present work included SCF, MBPT, CASSCF, RASSCF, MRCI and CASPT. For methods which satisfy the Hellmann-Feynman theorem (SCF, CASSCF and RASSCF), the dipole (#) and quadrupole (O) moments were calculated as expectation values. The corresponding dipole polarisabilities (or) follow as the numerical derivatives of induced dipoles. For the perturbative calculations (MBPT, CASPT), and for the MRCI calculation, the electric properties were calculated as numerical first or second derivatives of the energy with respect to the electric field (or field gradient for O). The strength of the perturbing field was IFI =0.0005 au. All correlated calculations used either the MOLCAS- 1 or the MOLCAS-2 packages [ 11 ] #2, and some of the uncorrelated calculations used the SYSMO coupled perturbed Hartree-Fock program [ 12 ].
3. Results The calculated electrical properties from each method are listed in table 1 and are now discussed in turn. 3. I. S C F results
In the SCF approximation, the polarised basis gives the usual exaggerated polarity of the electron cloud, with #b=0.3070 e ao(~0.7803 D) in comparison with the experimental equilibrium dipole moment of ~0.2100 e a o (0.5337(1) D) [13]. The la 2 SCF ground state is ionic, whereas more accurate wavefunctions for ozone are biradical in character [ 14 ]. The pattern of the quadrupole moment is in quali#2 G. KadstrOm, P.-,~. Malmqvist, B.O. Roos, A.J. Sadlej and P.-O. Widmark, MOLCAS-I System of Quantum Chemistry Programs, Theoretical Chemistry, Chemical Center, University of Lund, P.O. Box 124, S-221 00 Lund, Sweden. The original version of the MBPT codes included in MOLCAS was written by M. Urban, V. Kellr, I. (~ernusakand J. Noga, Department of Physical Chemistry, Comenius University, CS84215 Bratislava, Czechoslovakia. See also e.g. ref. [I1 ]; MOLCAS version 2, K. Andersson, M.P. Fiilscher, R. Lindh, P.-A. Malmqvist, J. Olsen, B.O. Roos, A.J. Sadlej and P.-O. Widmark, University of Lund, Sweden ( 1991 ). 368
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tative agreement with results from molecular beam electric resonance (MBER) spectroscopy [ 13 ] (see table 1 ), in that a positive out-of-plane component Oct is approximately cancelled by the transverse inplane component Oa~. The sign of the small axial quadrupole Obb = --(Oaadl-Occ) depends on the balance of the two large components, and SCF and experimental values are of opposite sign. Although this component is not well determined in the MBER experiment, the various correlated calculations that we have performed also give negative values of Obb and support the experimental sign. More surprising is the pattern of polarisability components in the uncorrelated SCF approximation (equivalent to the coupled Hartree-Fock CHF method). The components abb and ac~ are, as the experimental evidence suggests [ 13 ], approximately equal. However, the transverse in-plane component, aaa, is about three times larger than the other two, which leads to large anisotropies (Olaa--Olbb), ( a o a - a ~ ) and a large rotational average t~. Comparison with the MBER experimental data and the various correlated theoretical results (table 1 ) confirms this unusual pattern as a failure of the SCF wavefunction. Normally, the S C F / C H F method underestimates the molecular polarisability, typically by 5%-15% for a closed-shell neutral molecule. Internal evidence that the S C F / C H F results are in error are provided by the orbital energies. Problems with estimation of ionisation potentials of ozone by the use of Koopmans' theorem have been noted before [ 3 ]. In the present basis, the highest occupied orbitals (bl, al, a2) have energies -0.5659, -0.5550 and - 0 . 4 9 0 2 Eh, respectively, the LUMO (b2) also has a negative orbital energy ( - 0 . 0 4 9 4 Eh), and the next unoccupied orbital (al) is only slightly higher in energy, a warning of significant non-dynamical correlation effects caused by near-degeneracy of ground and excited states. Transitions from H O M O - n to L U M O + n for n = 0 , l, 2 are all dipole allowed and contribute to aaa. Analysis of the orbital polarisabilities with the SYSMO program shows that 97% of aaa comes from the two occupied ~ orbitals (94% from 1a2), whereas the smaller polarisability components have significant a contributions (68% for Olbb, 45% for a~c). Any shift in occupation between the H O M O and LUMO of the ground-state
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CHEMICAL PHYSICS LETTERS
6 March 1992
Table 1 Electric properties of ozone (in au) calculated with different correlated and uncorrelated wavefunctionsa, b and c are the principal axes where b is the symmetry axis, a the transverse in-plane axis, and c the normal to the molecular plane. In this coordinate system the nuclear positions are (0, + 0.839483, 0 ) and ( +_2.046841, - 0.419742, 0 ), in au. A positive dipole moment corresponds to the polarity 6-O2:O6+. O is referred to the molecular centre of mass. Details of each type of calculation are given in the text
Method
ltb
O~
Obb
O~
a~a
Olbb
Olcc
Olaa -- Olbb
Olaa -- Olcc
Ol
SCF MBPT-2 MBPT-3 MBPT-4 TCSCF CASSCF(p) CASSCF RASSCF TCSCF-PT CASPT MRCI-2 a)
0.3070 0.1938 0.2249 0.2179 0.0680 0.2175 0.2146 0.2126 0.2514 0.2136 0.2090
- 1.374 - 1.113 - 1.211 - 1.136 -0.746 - 1.226 - 1.209 -1.112 - 1.167 - 1.158 -1.122
+0.238 -0.358 -0.096 -0.283 -0.547 -0.141 -0.172 -0.289 -0.150 -0.260 -0.247
+ 1.135 + 1.471 + 1.306 + 1.419 + 1.293 + 1.367 + 1.381 +1.401 + 1.317 + 1.418 +1.369
38.58 21.56 29.26 31.54 24.54 28.40 28.32 30.64 35.18 30.40 30.00
12.71 14.26 13.59 14.78 12.59 12.57 12.56 13.60 14.63 14.24 14.12
11.26 11.88 11.68 12.06 10.78 10.64 10.64 11.68 12.00 11.84 11.84
25.88 7.30 15.67 16.76 11.95 15.83 15.76 17.04 20.55 16.16 15.88
27.33 9.68 17.38 19.48 13.76 17.76 17.68 18.96 23.18 18.56 18.16
20.85 15.90 18.18 19.46 15.97 17.20 17.17 18.64 20.60 18.83 18.65
exp. b)
0.2100(1)
- 1.03(12)
-0.52(16)
+ 1.55(19)
19.2(7)
17.9(7)
(19.3) c~
a) Including the Davidson correction. bl MBER results from ref. [ 13]. Polarisability anisotropies and errors as quoted in the discussion section of that paper. c~ Extrapolated from refractive index measurements on ozone gas [ 17].
configuration will therefore have drastic effects on Olaa.
3.2. M B P T results As a simple way of incorporating some degree of electron correlation in an SCF reference, m a n y - b o d y perturbation theory is a popular m e t h o d for calculating corrections to molecular electrical properties. The usual pattern of results is a n overlarge increase in polarisability and an exaggerated decrease in dipole m o m e n t at the MBPT-2 level (where double excitations are brought in to the w a v e f u n c t i o n ) , modified by oscillatory a n d plausibly convergent corrections at higher orders of p e r t u r b a t i o n theory, as multiple excitations are introduced. The behaviour of ltb, Otbb a n d c~cc for ozone follow this pattern (table 1 ), but Olaa is erratic. Falling by ~ 17 au between SCF a n d M B P T calculations, this c o m p o n e n t then rises by ~ 8 - 1 0 au in the MBPT-3 a n d MBPT4 approximations. The role of double excitations can be followed to higher order by using the linear coupled cluster LCCD approximation. Each iteration in this m e t h o d is equivalent to taking the double-excitation contrib u t i o n to one order higher in m a n y - b o d y perturba-
tion theory. As fig. 1 shows, the second order wildly overestimates the change in aaa, but in the limit the double excitations by themselves tend to increase ct~ over the SCF value. Smaller changes are seen for the 'well-behaved' components of ot and/~.
3.3. Two-configuration S C F results An obvious strategy for improving the SCF reference function is the inclusion of the first excited configuration, where 2b2 is doubly occupied in place of la2. A previous calculation on ozone by Yamaguchi et al. [2 ] used a two-configuration selfconsistent-field ( T C S C F ) wave-function in which an optimal mixture of configurations I...4bl26a121a22 [ and I...4b2ga22b2 [ was formed. A two-configuration generalised valence b o n d ( G V B ) function has also been used to achieve the same result [ 1 ]. The equivalent procedure was carried out for the present basis with the CASSCF program of the MOLCAS package, by using two active electrons in an active space composed of the la2 and 2b 2 orbitals (the H O M O a n d L U M O of the SCF configuration). The SCF configuration retained a weight of 79% in the T C S C F function. Table 1 shows that this approach exaggerates correlation effects on properties, underestimating the 369
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CHEMICALPHYSICSLETTERS
0.4
u/a.u. 0.3 0.2
~tb
0.1 0 C~/a.U. 45 (~aa
40 35 30 25 20
C~bb
15 10
IXc c
5 0
i
i
i
i
i
i
i
i
i
i
i
i
2
4
6
8
10
12
14
16
I8
20
22
24 n
Fig. 1. Variation of computed electric properties of ozone with order of approximation (n) of the linear coupled-clusterdoubles (LCCD) method. See text for a discussion of the curves. dipole moment by 68% compared to experiment and decreasing aaa by ~ 14 au. from the high SCF result. Other components of a have plausible, though probably slightly too low, values. The TCSCF dipole moments for ozone in a variety of basis sets are systematically low, as are the average polarisabilities quoted by Yamaguchi et al. [ 2 ]. This approach is known to be unsatisfactory for ozone; its failure seems to confirm that several configurations are of comparable importance in correcting the SCF wavefunction for this molecule. Although two configurations may dominate the wavefunction in C2v geometries (and by implication in fields to the symmetry axis), at least four configurations arising from redistribution within the n MOs are important for C~ geometries [3] (and, by implication, for C~aa). 3.4. CASSCF results A CASSCF calculation with all valence orbitals active is always a safe method for including all possible 370
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near-degeneracy effects, but is often too expensive. In the present case, the smallest meaningful active space would comprise only the I a2 and 2b2 orbitals, giving the two-configuration SCF already described. Its natural extension is to include then the 2p-derived occupied and unoccupied molecular orbitals. This choice of twelve electrons in nine orbitals generates 666 configurations in C2v symmetry, and 1280 and 1296 in Cs(xz) and Cs(yz), respectively. The SCF configuration has a weight of 83%, and the replacements a22--,b2 and aEblb2-~al 2 2 2 l a2blb I l I have weights of 9% and 1%, respectively. While the populations of the most strongly occupied active orbitals exceed 1.95, and those of the least occupied active orbitals are less than 0.06, the population of la2 falls to 1.79 and that of 2b2 rises to 0.25. Thus, the very large weight (21%) of the b~ configuration of the TCSCF has now been redistributed among a number of different configurations, explaining the large overcorrection of some properties when going from SCF to TCSCF. The properties are shown in table l by the label "CASSCF(p)", and it is seen that even this limited CASSCF function gives reasonable electric properties, with polarisability components 1.5-3 au below the MBPT-4 values. However, this nine orbital active space, while by no means large, is unnecessary for our purposes. Exclusion of the two most strongly occupied 2p orbitals gives a CASSCF with seven active orbitals, having 136 configurations in C2v, and 254 in both Cs(xz) and C,(yz). This calculation, labelled simply 'CASSCF' in table 1, gives the zeroth-order wavefunction used for the CASPT calculation described later. It is seen from table l to be in no way inferior to the CASSCF(p); in fact, though rather fortuitously, it compares slightly better with experiment. When the large near-degeneracy effects within the n space have been accounted for, the next most important correlations involve single electron promotions to the t~* orbitals. 3.5. R A S S C F results While the conceptually simple CASSCF method can be very efficient in including the most important intravalence correlation effects, the number of configurations and the computation time rises steeply with the size of the active space. An attempt to in-
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clude more correlation by increasing the active space is successful only in the rare cases where a very small number of active electrons have to be correlated. However, when the most important configurations have been included in a suitable CASSCF space, the most important additional configurations are the single and double excitations to extra-valence orbitals. Such configurations can be included by the RASSCF method, which is an extension of CASSCF where a large active orbtial space is subdivided into three parts: the RAS-I, RAS-2, and RAS-3 spaces. The first contains strongly occupied orbitals, from which it is allowed to remove only a limited number of electrons; the third contains weakly occupied orbitals, to which only a limited number of electrons may be promoted. The RAS-2 space plays a r61e similar to the CASSCF active space, since it can be occupied and spin-coupled in any possible way consistent with point group and spin symmetry, and consistent with the restrictions applied to the RAS1 and RAS-3 spaces. Typically, we allow at most two electrons to be removed from the RAS-1 orbitals, or excited to the RAS-3 space, in effect performing an MR-SDCI calculation with a complete reference, in a small but optimised orbital space. In the ozone case, any reasonable CI calculation shows that the simultaneous excitation of a a and a electron is an important effect. This can be qualitatively understood in terms of a dynamic polarisation of the a orbitals, in response to charge fluctuations in the n space. In the qualitative valencebond picture, the wavefunction is dominated by one biradical and two symmetry-equivalent ionic structures. These have quite different ~ charge distributions, and the most important refinement to this simple picture is therefore to define for each such structure an individual set of polarised sigma orbitals. To describe this effect in an MO picture requires inclusion of simultaneously aa*- and nn*-excitated configurations in the CI. This is not only important in the sense of lowering the electronic energy. In order to describe the response of the wavefunction to symmetry-breaking perturbations, the energetic balance between these three wavefunction fragments must be properly modelled. To include this dynamic polarisation we performed RASSCF calculations with the seven highest occupied a orbitals (four al and three b~ in C2v) in
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RAS-1; the five lowest unoccupied a* orbitals (three a~ and two b~) in RAS-3; and a RAS-2 space consisting of the three most important n orbitals (one a2 and two b2). This partitioning of the active space generated 3363 configurations in C2v symmetry, and 6664 and 4498 in Cs(xz) and Cs(yz), respectively. Though much larger, in this respect, than the CASSCF calculations, the RASSCF calculations are still quite cheap, and rather modest when compared to common practice. A complete unrestricted active space of the same size would produce more than a million configurations in a CASSCF calculation. The results are in good agreement with experiment, as is seen in table 1. 3.6. M R C I results The MRCI calculation uses orbitals optimised by the RASSCF procedure. The smallest reasonable reference space must include the SCF configuration and the a22--,b22 configuration, and the results of such a calculation are included in table 1 by the label 'MRCI-2'. The weights of the two reference configurations are 78.4% and 9.7%, respectively. The total reference weight, 88.2%, is lower than one would usually accept. However, from a calculation of the potential surface [15 ], it was known that the reference weight increases extremely slowly with the number of additional reference functions (thirteen reference functions instead of two increases the weight by less than 1%), so by this criterion it is not worth using a larger reference space, unless one is determined to make it very large. Energies was corrected for unliked quadrupole excitations by the Davidson procedure. The results are essentially in agreement with RASSCF, CASSCF and MBPT-4, and with experiment. The differences are too small to allow judgement of the relative merits of these methods. However, the problem with the reference space suggests that MRCI may be troublesome in similar cases, unless a large reference space is used. 3.7. CASPT2 results One way to avoid the problem of near-degeneracy of the H O M O and LUMO of the SCF configuration is to incorporate these effects already at the zeroth371
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CHEMICALPHYSICSLETTERS
order level and then apply perturbation theory to calculate the contributions from the dynamical correlation. Second-order CASPT results have been obtained using two different CASSCF wavefunctions as zeroth-order wavefunctions, with a non-diagonal Fock-type operator as the unperturbed Hamiltonian [ 16 ]. In a first set of calculations the two-configuration SCF wavefunction, described above, was used as the reference function to investigate further the quality of this wavefunction. The results are in table 1 under TCSCF-PT. (In this case, a field strength of IFI =0.001 au instead of 0.0005 was used to obtain numerical derivatives. The effects of this deviation are negligible. ) Those properties for which TCSCF is inaccurate are overcorrected when the perturbation is added. This is most pronounced for the dipole moment and the transverse in-plane polarisability. The polarisability anisotropies are almost doubled on going from the TCSCF to the TCSCF-PT level of theory. In a second set of calculations a more extensive reference function was used. This function, with seven of the 2p-derived orbitals as active space, was described in section 3.4 and appears by the label "CASSCF" in table 1. The corresponding second-order CASPT results are labelled "CASPT". Except for the dipole moment, all CASPT results lie between the MBPT-3 and MBPT-4 results, indicating an oscillating behaviour of the MBPT series. A comparison of the TCSCF, MRCI-2 and CASPT methods shows that the MRCI-2 results are between TCSCF and CASPT results for most of the calculated properties. The single and double excitations are evidently not able to correct the rather poor two-configuration description, in spite of an individual optimisation of all CI coefficients with a formally exact Hamiltonian and good orbitals. The CASPT, on the other hand, handicapped by the use of a precontracted single (but multiconfiguration) reference and a rather crude approximation to the Hamiltonian, gives a very good result. This indicates that an accurate description of the dynamic correlation is less important than the extensive inclusion of such configurations, which differ by more than double excitations from not only the HF reference, but even from the TCSCF. This is in agreement with the results of Watts et al. [ 6 ], who find significant truncation errors even when including all triples cl/asters in a cou372
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pied-cluster calculation based on the HF reference.
4. Conclusions The general picture of/t, O and ot from all but the least sophisticated calculations is similar. It is clear that the single-determinant SCF approach overestimates /tb and especially Olaa for ozone, but that MBPT-2 and TCSCF swing too far in the opposite direction. It is interesting to note that Stanton et al. [ 3 ] found that the SCF value o f o 9 3 doubled on going to the MBPT-2 level, and that Yamaguchi et al. [ 2 ] found an incorrect ordering of symmetric and antisymmetric frequencies in TCSCF when compared to SCF. 093 is still somewhat low at the MBPT-4 level according to the calculations described in ref. [ 3 ]. As mentioned in section 1, this parallel between transverse polarisation and antisymmetric stretching motion is not unexpected. Calculation of Olaa is another way of responding to the "great challenge to theory posed by the Cs subspace of the potential energy surface" [ 3 ]. Multi-reference methods of the CASSCF and RASSCF types are found to give satisfactory results for or, as also does the MBPT-4 method. The RASSCF gives a particularly compact representation of the correlated wavefunction. Unfortunately, in view of the erratic behaviour of the MBPT series, the reliability of MBPT-4 is apparent only after post hoc comparison with more powerful correlated methods. This difficulty apart, the computed electric properties of ozone are now on the same sound footing as those of many other small closed-shell molecules.
Acknowledgement The work reported in this article was to a large extent conducted as part of a joint Study Project supported by IBM Sweden and by the Swedish Natural Science Research Council.
References [ 1] S.M. Adler-Golden, S.R. Langhoff,C.W. Bauschlicher Jr., and G.D. Carney, J. Chem. Phys. 83 ( 1985 ) 255.
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[2] Y. Yamaguchi, M.J. Frisch, T.J. Lee, H.F. Schaefer III and J.S. Binkley, Theoret. Chim. Acta 69 (1986) 337. [ 3 ] J.F. Stanton, W.N. Lipscomb, D.H. Magers and R.J. Bartlett, J. Chem. Phys. 90 (1989) 1077. [4] K. Raghavachari, G.W. Trucks, J.A. Pople and E. Replogle, Chem. Phys. Letters 158 (1989) 207. [5 ] T.J. Lee and G.E. Scuseria, J. Chem. Phys. 93 (1990) 489. [6] J.D. Watts, J.F. Stanton and R.J. Bartlett, Chem. Phys. Letters 178 ( 1991 ) 471. [ 7 ] T. Tanaka and Y. Morino, J. Mol. Spectry. 33 (1970) 538. [8] M. Lichtenstein, J.J. Gallagher and S.A. Clough, J. Mol. Spectry. 40 ( 1971 ) 10. [ 9 ] A.J. Sadlej, Collection Czech. Chem. Commun. 53 ( 1988 ) 1995; A.J. Sadlej, Theoret. Chim. Acta 79 ( 1991 ) 123.
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[ 10] K. Woliitski, A.J. Sadlej and G. Karlstr6m, Mol. Phys. 72 ( 1991 ) 425. [ 11 ] M. Urban, I. Huba~, V. Kell6 and J. Noga, J. Chem. Phys. 72 (1980) 3378. [ 12 ] P. Lazzeretti and R. Zanasi, J. Chem. Phys. 72 (1980) 6768. [13] K.M. Mack and J.S. Muenter, J. Chem. Phys. 66 (1977) 5278. [ 14 ] R.P. Messmer and D.R. Salahub, J. Chem. Phys. 65 (1976) 779. [ 15 ] K. Andersson, P. Borowski, P.-A Malmqvist and B.O. Roos, to be published. [16] K. Andersson, P.-/k Malmqvist and B.O. Roos, J. Chem. Phys. 96 (1992), in press. [ 17 ] C. Cuthbertson, Phil. Trans. Roy. Soc. A 213 ( 1913 ) 1.
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Electric properties of the ozone molecule K. Andersson ~, P. Borowski i, P.W. Fowler 2, P.-A. Malmqvist 1, B.O. Roos 1 and A.J. Sadlej Theoretical Chemistry, Chemical Center, University of Lund, P.O. Box 124, S-221 O0 Lund, Sweden 2 DepartmentofChemistry, UniversityofExeter, StockerRoad, ExeterEX44QD, UK
Received 19 December 1991
The dipole moment, quadrupole moment and dipole polarisability of 03 are calculated by the self-consistent field (SCF), many-body perturbation theory (MBPT), complete active space SCF (CASSCF), restricted active space SCF (RASSCF), multireference CI (MRCI) and CASSCF-perturbation theory (CASPT) methods, using a 5s3p2d polarised basis set. At the CASPT-2 level, the dipole moment is #b=0.2136 au. The quadrupole moments are O,~=- 1.158, Obb=--0.260, Oct= + 1.418; and the polarisabilityanisotropies are ct~-abb = 16.2 and aaa-otto= 18.6, all in atomic units. Experimental data for these quantities are 0.2100( I ), - 1.03(12), -0.52(16), + 1.55(19), and 19.2(7), 17.9(7), respectively. Similar results are obtained by CASSCF, RASSCF and MRCI, and by MBPT when taken to fourth order. The transverse in-plane polarisability obtained by the singlereference methods shows an erratic behaviour, which is similar to known variations in computed antisymmetric stretching frequencies.
1. Introduction Aside from its e n v i r o n m e n t a l a n d chemical importance, the ozone molecule has a significant role in the calibration o f theoretical methods. The experimental vibrational frequencies o f ozone are not well r e p r o d u c e d at the S C F level, a n d some difficulty and controversy has a t t e n d e d their calculation at correlated levels [ 1-6 ]. Single-reference methods can give erratic a n d even unphysical results for the antisymmetric stretching frequency unless taken to high o r d e r [ 3 ], a n d small changes in the way that higher excitations are included can give frequency shifts o f the o r d e r o f 100 c m - ~ [ 6 ]. A n o t h e r way o f probing this problem, which has its roots in the electronic structure o f ozone, is the calculation o f electric properties such as the polarisability. In the presence o f a transverse in-plane electric field the terminal a t o m s o f ozone b e c o m e inequivalent, as during the antisymmetric vibration, and similar difficulties in calculation o f correlated transverse polarisabilities Olaa and a n t i s y m m e t r i c stretching frequencies oJ3 are to be expected; both are second derivatives o f the energy with respect to a perturbation o f B~ symmetry ~. This p a p e r presents a c o m p a r i s o n o f electric properties (dipole m o m e n t / t , q u a d r u p o l e m o m e n t O a n d
dipole polarisability a ) calculated at various levels o f theory. It is shown that correlated m e t h o d s are needed to give a qualitatively correct description o f the properties, and that the m a i n difficulties lie with the transverse in-plane polarisability. Multi-reference methods and the single-reference M B P T method - when taken to the fourth o r d e r - give a consistent picture o f electric properties, and are in reasonable agreement with experimental evidence.
2. Method The ozone molecule was fixed at the experimental equilibrium geometry ( R o o = 1.2717 A, z _ O O O = 116.8 °) [7,8], and the polarised oxygen basis ( 1 0 s 6 p 4 d ) ~ [ 5s3p2d ] from the Sadlej c o m p i l a t i o n [ 9 ] was used. The success o f these basis sets for dipole moments, polarisabilities [9 ] and q u a d r u p o l e m o m e n t s [ 10 ] at S C F and correlated levels for conventional closed-shell molecules is well d o c u m e n t e d . ~J The axis convention used in the present paper puts the molecule in the xz plane, where z is the C2 axis (b principal axis) and x is the a principal axis of 1603./zx---/zatransforms a s l 1 in this setting of C2v.
0009-2614/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
367
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Spherical harmonic combinations of the d-type GTOs were used. Methods of calculation employed in the present work included SCF, MBPT, CASSCF, RASSCF, MRCI and CASPT. For methods which satisfy the Hellmann-Feynman theorem (SCF, CASSCF and RASSCF), the dipole (#) and quadrupole (O) moments were calculated as expectation values. The corresponding dipole polarisabilities (or) follow as the numerical derivatives of induced dipoles. For the perturbative calculations (MBPT, CASPT), and for the MRCI calculation, the electric properties were calculated as numerical first or second derivatives of the energy with respect to the electric field (or field gradient for O). The strength of the perturbing field was IFI =0.0005 au. All correlated calculations used either the MOLCAS- 1 or the MOLCAS-2 packages [ 11 ] #2, and some of the uncorrelated calculations used the SYSMO coupled perturbed Hartree-Fock program [ 12 ].
3. Results The calculated electrical properties from each method are listed in table 1 and are now discussed in turn. 3. I. S C F results
In the SCF approximation, the polarised basis gives the usual exaggerated polarity of the electron cloud, with #b=0.3070 e ao(~0.7803 D) in comparison with the experimental equilibrium dipole moment of ~0.2100 e a o (0.5337(1) D) [13]. The la 2 SCF ground state is ionic, whereas more accurate wavefunctions for ozone are biradical in character [ 14 ]. The pattern of the quadrupole moment is in quali#2 G. KadstrOm, P.-,~. Malmqvist, B.O. Roos, A.J. Sadlej and P.-O. Widmark, MOLCAS-I System of Quantum Chemistry Programs, Theoretical Chemistry, Chemical Center, University of Lund, P.O. Box 124, S-221 00 Lund, Sweden. The original version of the MBPT codes included in MOLCAS was written by M. Urban, V. Kellr, I. (~ernusakand J. Noga, Department of Physical Chemistry, Comenius University, CS84215 Bratislava, Czechoslovakia. See also e.g. ref. [I1 ]; MOLCAS version 2, K. Andersson, M.P. Fiilscher, R. Lindh, P.-A. Malmqvist, J. Olsen, B.O. Roos, A.J. Sadlej and P.-O. Widmark, University of Lund, Sweden ( 1991 ). 368
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tative agreement with results from molecular beam electric resonance (MBER) spectroscopy [ 13 ] (see table 1 ), in that a positive out-of-plane component Oct is approximately cancelled by the transverse inplane component Oa~. The sign of the small axial quadrupole Obb = --(Oaadl-Occ) depends on the balance of the two large components, and SCF and experimental values are of opposite sign. Although this component is not well determined in the MBER experiment, the various correlated calculations that we have performed also give negative values of Obb and support the experimental sign. More surprising is the pattern of polarisability components in the uncorrelated SCF approximation (equivalent to the coupled Hartree-Fock CHF method). The components abb and ac~ are, as the experimental evidence suggests [ 13 ], approximately equal. However, the transverse in-plane component, aaa, is about three times larger than the other two, which leads to large anisotropies (Olaa--Olbb), ( a o a - a ~ ) and a large rotational average t~. Comparison with the MBER experimental data and the various correlated theoretical results (table 1 ) confirms this unusual pattern as a failure of the SCF wavefunction. Normally, the S C F / C H F method underestimates the molecular polarisability, typically by 5%-15% for a closed-shell neutral molecule. Internal evidence that the S C F / C H F results are in error are provided by the orbital energies. Problems with estimation of ionisation potentials of ozone by the use of Koopmans' theorem have been noted before [ 3 ]. In the present basis, the highest occupied orbitals (bl, al, a2) have energies -0.5659, -0.5550 and - 0 . 4 9 0 2 Eh, respectively, the LUMO (b2) also has a negative orbital energy ( - 0 . 0 4 9 4 Eh), and the next unoccupied orbital (al) is only slightly higher in energy, a warning of significant non-dynamical correlation effects caused by near-degeneracy of ground and excited states. Transitions from H O M O - n to L U M O + n for n = 0 , l, 2 are all dipole allowed and contribute to aaa. Analysis of the orbital polarisabilities with the SYSMO program shows that 97% of aaa comes from the two occupied ~ orbitals (94% from 1a2), whereas the smaller polarisability components have significant a contributions (68% for Olbb, 45% for a~c). Any shift in occupation between the H O M O and LUMO of the ground-state
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Table 1 Electric properties of ozone (in au) calculated with different correlated and uncorrelated wavefunctionsa, b and c are the principal axes where b is the symmetry axis, a the transverse in-plane axis, and c the normal to the molecular plane. In this coordinate system the nuclear positions are (0, + 0.839483, 0 ) and ( +_2.046841, - 0.419742, 0 ), in au. A positive dipole moment corresponds to the polarity 6-O2:O6+. O is referred to the molecular centre of mass. Details of each type of calculation are given in the text
Method
ltb
O~
Obb
O~
a~a
Olbb
Olcc
Olaa -- Olbb
Olaa -- Olcc
Ol
SCF MBPT-2 MBPT-3 MBPT-4 TCSCF CASSCF(p) CASSCF RASSCF TCSCF-PT CASPT MRCI-2 a)
0.3070 0.1938 0.2249 0.2179 0.0680 0.2175 0.2146 0.2126 0.2514 0.2136 0.2090
- 1.374 - 1.113 - 1.211 - 1.136 -0.746 - 1.226 - 1.209 -1.112 - 1.167 - 1.158 -1.122
+0.238 -0.358 -0.096 -0.283 -0.547 -0.141 -0.172 -0.289 -0.150 -0.260 -0.247
+ 1.135 + 1.471 + 1.306 + 1.419 + 1.293 + 1.367 + 1.381 +1.401 + 1.317 + 1.418 +1.369
38.58 21.56 29.26 31.54 24.54 28.40 28.32 30.64 35.18 30.40 30.00
12.71 14.26 13.59 14.78 12.59 12.57 12.56 13.60 14.63 14.24 14.12
11.26 11.88 11.68 12.06 10.78 10.64 10.64 11.68 12.00 11.84 11.84
25.88 7.30 15.67 16.76 11.95 15.83 15.76 17.04 20.55 16.16 15.88
27.33 9.68 17.38 19.48 13.76 17.76 17.68 18.96 23.18 18.56 18.16
20.85 15.90 18.18 19.46 15.97 17.20 17.17 18.64 20.60 18.83 18.65
exp. b)
0.2100(1)
- 1.03(12)
-0.52(16)
+ 1.55(19)
19.2(7)
17.9(7)
(19.3) c~
a) Including the Davidson correction. bl MBER results from ref. [ 13]. Polarisability anisotropies and errors as quoted in the discussion section of that paper. c~ Extrapolated from refractive index measurements on ozone gas [ 17].
configuration will therefore have drastic effects on Olaa.
3.2. M B P T results As a simple way of incorporating some degree of electron correlation in an SCF reference, m a n y - b o d y perturbation theory is a popular m e t h o d for calculating corrections to molecular electrical properties. The usual pattern of results is a n overlarge increase in polarisability and an exaggerated decrease in dipole m o m e n t at the MBPT-2 level (where double excitations are brought in to the w a v e f u n c t i o n ) , modified by oscillatory a n d plausibly convergent corrections at higher orders of p e r t u r b a t i o n theory, as multiple excitations are introduced. The behaviour of ltb, Otbb a n d c~cc for ozone follow this pattern (table 1 ), but Olaa is erratic. Falling by ~ 17 au between SCF a n d M B P T calculations, this c o m p o n e n t then rises by ~ 8 - 1 0 au in the MBPT-3 a n d MBPT4 approximations. The role of double excitations can be followed to higher order by using the linear coupled cluster LCCD approximation. Each iteration in this m e t h o d is equivalent to taking the double-excitation contrib u t i o n to one order higher in m a n y - b o d y perturba-
tion theory. As fig. 1 shows, the second order wildly overestimates the change in aaa, but in the limit the double excitations by themselves tend to increase ct~ over the SCF value. Smaller changes are seen for the 'well-behaved' components of ot and/~.
3.3. Two-configuration S C F results An obvious strategy for improving the SCF reference function is the inclusion of the first excited configuration, where 2b2 is doubly occupied in place of la2. A previous calculation on ozone by Yamaguchi et al. [2 ] used a two-configuration selfconsistent-field ( T C S C F ) wave-function in which an optimal mixture of configurations I...4bl26a121a22 [ and I...4b2ga22b2 [ was formed. A two-configuration generalised valence b o n d ( G V B ) function has also been used to achieve the same result [ 1 ]. The equivalent procedure was carried out for the present basis with the CASSCF program of the MOLCAS package, by using two active electrons in an active space composed of the la2 and 2b 2 orbitals (the H O M O a n d L U M O of the SCF configuration). The SCF configuration retained a weight of 79% in the T C S C F function. Table 1 shows that this approach exaggerates correlation effects on properties, underestimating the 369
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0.4
u/a.u. 0.3 0.2
~tb
0.1 0 C~/a.U. 45 (~aa
40 35 30 25 20
C~bb
15 10
IXc c
5 0
i
i
i
i
i
i
i
i
i
i
i
i
2
4
6
8
10
12
14
16
I8
20
22
24 n
Fig. 1. Variation of computed electric properties of ozone with order of approximation (n) of the linear coupled-clusterdoubles (LCCD) method. See text for a discussion of the curves. dipole moment by 68% compared to experiment and decreasing aaa by ~ 14 au. from the high SCF result. Other components of a have plausible, though probably slightly too low, values. The TCSCF dipole moments for ozone in a variety of basis sets are systematically low, as are the average polarisabilities quoted by Yamaguchi et al. [ 2 ]. This approach is known to be unsatisfactory for ozone; its failure seems to confirm that several configurations are of comparable importance in correcting the SCF wavefunction for this molecule. Although two configurations may dominate the wavefunction in C2v geometries (and by implication in fields to the symmetry axis), at least four configurations arising from redistribution within the n MOs are important for C~ geometries [3] (and, by implication, for C~aa). 3.4. CASSCF results A CASSCF calculation with all valence orbitals active is always a safe method for including all possible 370
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near-degeneracy effects, but is often too expensive. In the present case, the smallest meaningful active space would comprise only the I a2 and 2b2 orbitals, giving the two-configuration SCF already described. Its natural extension is to include then the 2p-derived occupied and unoccupied molecular orbitals. This choice of twelve electrons in nine orbitals generates 666 configurations in C2v symmetry, and 1280 and 1296 in Cs(xz) and Cs(yz), respectively. The SCF configuration has a weight of 83%, and the replacements a22--,b2 and aEblb2-~al 2 2 2 l a2blb I l I have weights of 9% and 1%, respectively. While the populations of the most strongly occupied active orbitals exceed 1.95, and those of the least occupied active orbitals are less than 0.06, the population of la2 falls to 1.79 and that of 2b2 rises to 0.25. Thus, the very large weight (21%) of the b~ configuration of the TCSCF has now been redistributed among a number of different configurations, explaining the large overcorrection of some properties when going from SCF to TCSCF. The properties are shown in table l by the label "CASSCF(p)", and it is seen that even this limited CASSCF function gives reasonable electric properties, with polarisability components 1.5-3 au below the MBPT-4 values. However, this nine orbital active space, while by no means large, is unnecessary for our purposes. Exclusion of the two most strongly occupied 2p orbitals gives a CASSCF with seven active orbitals, having 136 configurations in C2v, and 254 in both Cs(xz) and C,(yz). This calculation, labelled simply 'CASSCF' in table 1, gives the zeroth-order wavefunction used for the CASPT calculation described later. It is seen from table l to be in no way inferior to the CASSCF(p); in fact, though rather fortuitously, it compares slightly better with experiment. When the large near-degeneracy effects within the n space have been accounted for, the next most important correlations involve single electron promotions to the t~* orbitals. 3.5. R A S S C F results While the conceptually simple CASSCF method can be very efficient in including the most important intravalence correlation effects, the number of configurations and the computation time rises steeply with the size of the active space. An attempt to in-
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clude more correlation by increasing the active space is successful only in the rare cases where a very small number of active electrons have to be correlated. However, when the most important configurations have been included in a suitable CASSCF space, the most important additional configurations are the single and double excitations to extra-valence orbitals. Such configurations can be included by the RASSCF method, which is an extension of CASSCF where a large active orbtial space is subdivided into three parts: the RAS-I, RAS-2, and RAS-3 spaces. The first contains strongly occupied orbitals, from which it is allowed to remove only a limited number of electrons; the third contains weakly occupied orbitals, to which only a limited number of electrons may be promoted. The RAS-2 space plays a r61e similar to the CASSCF active space, since it can be occupied and spin-coupled in any possible way consistent with point group and spin symmetry, and consistent with the restrictions applied to the RAS1 and RAS-3 spaces. Typically, we allow at most two electrons to be removed from the RAS-1 orbitals, or excited to the RAS-3 space, in effect performing an MR-SDCI calculation with a complete reference, in a small but optimised orbital space. In the ozone case, any reasonable CI calculation shows that the simultaneous excitation of a a and a electron is an important effect. This can be qualitatively understood in terms of a dynamic polarisation of the a orbitals, in response to charge fluctuations in the n space. In the qualitative valencebond picture, the wavefunction is dominated by one biradical and two symmetry-equivalent ionic structures. These have quite different ~ charge distributions, and the most important refinement to this simple picture is therefore to define for each such structure an individual set of polarised sigma orbitals. To describe this effect in an MO picture requires inclusion of simultaneously aa*- and nn*-excitated configurations in the CI. This is not only important in the sense of lowering the electronic energy. In order to describe the response of the wavefunction to symmetry-breaking perturbations, the energetic balance between these three wavefunction fragments must be properly modelled. To include this dynamic polarisation we performed RASSCF calculations with the seven highest occupied a orbitals (four al and three b~ in C2v) in
6 March 1992
RAS-1; the five lowest unoccupied a* orbitals (three a~ and two b~) in RAS-3; and a RAS-2 space consisting of the three most important n orbitals (one a2 and two b2). This partitioning of the active space generated 3363 configurations in C2v symmetry, and 6664 and 4498 in Cs(xz) and Cs(yz), respectively. Though much larger, in this respect, than the CASSCF calculations, the RASSCF calculations are still quite cheap, and rather modest when compared to common practice. A complete unrestricted active space of the same size would produce more than a million configurations in a CASSCF calculation. The results are in good agreement with experiment, as is seen in table 1. 3.6. M R C I results The MRCI calculation uses orbitals optimised by the RASSCF procedure. The smallest reasonable reference space must include the SCF configuration and the a22--,b22 configuration, and the results of such a calculation are included in table 1 by the label 'MRCI-2'. The weights of the two reference configurations are 78.4% and 9.7%, respectively. The total reference weight, 88.2%, is lower than one would usually accept. However, from a calculation of the potential surface [15 ], it was known that the reference weight increases extremely slowly with the number of additional reference functions (thirteen reference functions instead of two increases the weight by less than 1%), so by this criterion it is not worth using a larger reference space, unless one is determined to make it very large. Energies was corrected for unliked quadrupole excitations by the Davidson procedure. The results are essentially in agreement with RASSCF, CASSCF and MBPT-4, and with experiment. The differences are too small to allow judgement of the relative merits of these methods. However, the problem with the reference space suggests that MRCI may be troublesome in similar cases, unless a large reference space is used. 3.7. CASPT2 results One way to avoid the problem of near-degeneracy of the H O M O and LUMO of the SCF configuration is to incorporate these effects already at the zeroth371
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order level and then apply perturbation theory to calculate the contributions from the dynamical correlation. Second-order CASPT results have been obtained using two different CASSCF wavefunctions as zeroth-order wavefunctions, with a non-diagonal Fock-type operator as the unperturbed Hamiltonian [ 16 ]. In a first set of calculations the two-configuration SCF wavefunction, described above, was used as the reference function to investigate further the quality of this wavefunction. The results are in table 1 under TCSCF-PT. (In this case, a field strength of IFI =0.001 au instead of 0.0005 was used to obtain numerical derivatives. The effects of this deviation are negligible. ) Those properties for which TCSCF is inaccurate are overcorrected when the perturbation is added. This is most pronounced for the dipole moment and the transverse in-plane polarisability. The polarisability anisotropies are almost doubled on going from the TCSCF to the TCSCF-PT level of theory. In a second set of calculations a more extensive reference function was used. This function, with seven of the 2p-derived orbitals as active space, was described in section 3.4 and appears by the label "CASSCF" in table 1. The corresponding second-order CASPT results are labelled "CASPT". Except for the dipole moment, all CASPT results lie between the MBPT-3 and MBPT-4 results, indicating an oscillating behaviour of the MBPT series. A comparison of the TCSCF, MRCI-2 and CASPT methods shows that the MRCI-2 results are between TCSCF and CASPT results for most of the calculated properties. The single and double excitations are evidently not able to correct the rather poor two-configuration description, in spite of an individual optimisation of all CI coefficients with a formally exact Hamiltonian and good orbitals. The CASPT, on the other hand, handicapped by the use of a precontracted single (but multiconfiguration) reference and a rather crude approximation to the Hamiltonian, gives a very good result. This indicates that an accurate description of the dynamic correlation is less important than the extensive inclusion of such configurations, which differ by more than double excitations from not only the HF reference, but even from the TCSCF. This is in agreement with the results of Watts et al. [ 6 ], who find significant truncation errors even when including all triples cl/asters in a cou372
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pied-cluster calculation based on the HF reference.
4. Conclusions The general picture of/t, O and ot from all but the least sophisticated calculations is similar. It is clear that the single-determinant SCF approach overestimates /tb and especially Olaa for ozone, but that MBPT-2 and TCSCF swing too far in the opposite direction. It is interesting to note that Stanton et al. [ 3 ] found that the SCF value o f o 9 3 doubled on going to the MBPT-2 level, and that Yamaguchi et al. [ 2 ] found an incorrect ordering of symmetric and antisymmetric frequencies in TCSCF when compared to SCF. 093 is still somewhat low at the MBPT-4 level according to the calculations described in ref. [ 3 ]. As mentioned in section 1, this parallel between transverse polarisation and antisymmetric stretching motion is not unexpected. Calculation of Olaa is another way of responding to the "great challenge to theory posed by the Cs subspace of the potential energy surface" [ 3 ]. Multi-reference methods of the CASSCF and RASSCF types are found to give satisfactory results for or, as also does the MBPT-4 method. The RASSCF gives a particularly compact representation of the correlated wavefunction. Unfortunately, in view of the erratic behaviour of the MBPT series, the reliability of MBPT-4 is apparent only after post hoc comparison with more powerful correlated methods. This difficulty apart, the computed electric properties of ozone are now on the same sound footing as those of many other small closed-shell molecules.
Acknowledgement The work reported in this article was to a large extent conducted as part of a joint Study Project supported by IBM Sweden and by the Swedish Natural Science Research Council.
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[2] Y. Yamaguchi, M.J. Frisch, T.J. Lee, H.F. Schaefer III and J.S. Binkley, Theoret. Chim. Acta 69 (1986) 337. [ 3 ] J.F. Stanton, W.N. Lipscomb, D.H. Magers and R.J. Bartlett, J. Chem. Phys. 90 (1989) 1077. [4] K. Raghavachari, G.W. Trucks, J.A. Pople and E. Replogle, Chem. Phys. Letters 158 (1989) 207. [5 ] T.J. Lee and G.E. Scuseria, J. Chem. Phys. 93 (1990) 489. [6] J.D. Watts, J.F. Stanton and R.J. Bartlett, Chem. Phys. Letters 178 ( 1991 ) 471. [ 7 ] T. Tanaka and Y. Morino, J. Mol. Spectry. 33 (1970) 538. [8] M. Lichtenstein, J.J. Gallagher and S.A. Clough, J. Mol. Spectry. 40 ( 1971 ) 10. [ 9 ] A.J. Sadlej, Collection Czech. Chem. Commun. 53 ( 1988 ) 1995; A.J. Sadlej, Theoret. Chim. Acta 79 ( 1991 ) 123.
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[ 10] K. Woliitski, A.J. Sadlej and G. Karlstr6m, Mol. Phys. 72 ( 1991 ) 425. [ 11 ] M. Urban, I. Huba~, V. Kell6 and J. Noga, J. Chem. Phys. 72 (1980) 3378. [ 12 ] P. Lazzeretti and R. Zanasi, J. Chem. Phys. 72 (1980) 6768. [13] K.M. Mack and J.S. Muenter, J. Chem. Phys. 66 (1977) 5278. [ 14 ] R.P. Messmer and D.R. Salahub, J. Chem. Phys. 65 (1976) 779. [ 15 ] K. Andersson, P. Borowski, P.-A Malmqvist and B.O. Roos, to be published. [16] K. Andersson, P.-/k Malmqvist and B.O. Roos, J. Chem. Phys. 96 (1992), in press. [ 17 ] C. Cuthbertson, Phil. Trans. Roy. Soc. A 213 ( 1913 ) 1.
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