The Structures, Molecular Orbital Properties and Vibrational Spectra of the Homo- and Heterodimers of Sulphur Dioxide and Ozone. An Ab Initio Study

The structures of a number of dimers of sulphur dioxide and ozone were optimized by means of a series of ab initio calculations. The dimer species were classified as either genuine energy minima or transition states of first or higher order, and the most probable structures consistent with the experimental data were confirmed. The molecular orbitals engaged in the interactions resulting in adduct formation were identified and relations between the orbitals of the dimers of the valence isoelectronic monomer species were examined. The vibrational spectra of the most probable structures were computed and compared with those reported in the literature, particularly with spectra observed in cryogenic matrices. The calculations were extended to predict the properties of a number of possible heterodimers formed between sulphur dioxide and ozone.

The SO 2 -O 3 heterodimer has so far eluded investigation, either theoretically or experimentally.

Molecular Structures
A number of trial structures were investigated for each adduct. We used as a template the publication of Hargittai [87], which examined a set of seven likely structural models for the metal dihalides, which we considered candidates as potential structures for the sulphur dioxide dimer. This trial set included three cyclic, two "linear", and two bifurcated models, most of which were also investigated by other workers [3][4][5][6]8,9]. Along with the seven Hargittai structures, we also included two species that involved a S . . . S interaction, with no expectation that they would be strong candidates for the preferred SO 2 dimer structure, but simply for completeness. Our nine possible dimer structures are illustrated in Figure 1 and their symmetries, energies, and Hessian indices are given in Table 1. Figure 2 shows their relative energies, separated according to their classification as genuine minima or transition states. We conclude, in agreement with the previous works [3][4][5][6]9] and with most of the experimental data [9,10,12,13,15,19,20,22,23], that our dimer 5 is the global minimum and is a non-symmetric species of C s symmetry. The centrosymmetric dimer 2 is also a genuine minimum, being less than 1 kJ mol −1 higher in energy. Dimer 2, however, being centrosymmetric, would not be observable by microwave spectroscopy, therefore, there is no possibility of identifying this structure in the gas phase. Table 2 reports the computed geometrical parameters of each dimer and their deviations from the corresponding monomer values. The perturbations of the bond lengths and bond angles from their monomer values are small (less than 0.3 pm and 0.6 • , respectively), indicating a weak interaction in each case.     Optimized geometrical parameters of sulphur dioxide dimers 5 and 2, and their changes relative to the sulphur dioxide monomer. See Figure 1 for numbering of the atoms.  The corresponding set of nine potential dimer structures for ozone are illustrated in Figure 3 and their properties are collected in Table 3. Their relative energies are presented in Figure 4. Only one of our ozone dimers (dimer 2) was found to be a true minimum. This adduct corresponds with the second lowest energy sulphur dioxide dimer, while the counterpart of the C s global minimum of (SO 2 ) 2 (ozone dimer 5) is about 5 kJ mol −1 higher in energy. Our result is in contrast to that of Slanina and Adamowicz [46], who found the C s counterpart of our dimer 5 to be the global minimum species. Part of the difference may be attributed to the use of different basis sets, but it has long been realized that the ozone monomer presents particularly formidable challenges for computation [88,89], and this is even more apparent for its dimer. The parameters of our ozone dimer 2 species are shown in Table 4. Again, the perturbations are minimal (less than 0.25 pm and 0.2 • ). Based on the genuine minimum structures we found for the sulphur dioxide and ozone homodimers, and the C s first order transition state of (O 3 ) 2 , we examined eight structures for the sulphur dioxide-ozone heterodimer, two each corresponding with dimers 2, 3, 5, and 7, with sulphur dioxide and ozone acting as electron donor or electron acceptor in turn (a or b). These eight structures are illustrated in Figure 5 and their properties in Table 5. Structures 2a, 2b, 3a, and 3b were found to be virtually identical; these four structures and complex 5b are all true minima. The relative energies are presented in Figure 6 and the bond lengths and angles and their changes in Table 6. Again, the pattern of very small perturbations is observed; only the free O2O6 bond of the ozone sub-unit in complex 2a shows a substantial increase on complexation.
Molecules 2021, 26, 626 5 of 20 even more apparent for its dimer. The parameters of our ozone dimer 2 species are shown in Table 4. Again, the perturbations are minimal (less than 0.25 pm and 0.2°).    Table 4. Optimized geometrical parameters of ozone dimer 2 and their changes relative to the ozone monomer. See Figure 3 for numbering of the atoms.

Interaction Energies
The interaction energies of the five genuine minimum energy structures are given in Table 7, corrected in each case for BSSE [90] and for zero-point energy differences. Consistent with the relatively insignificant intramolecular structural perturbations discussed above, the interaction energies are all less than 10 kJ mol −1 ; the two SO 2 dimers and the 2a heterodimer all have similar energies, while heterodimer 5b is barely bound at all. Table 6. Optimized geometrical parameters of sulphur dioxide-ozone complexes 2a and 5b, and their changes relative to the sulphur dioxide and ozone monomers. See Figure 5 for numbering of the atoms.

Complex 2a
Complex

Molecular Orbital Properties
The valence molecular orbitals of the sulphur dioxide and ozone monomers are illustrated in Figures 7 and 8, and their descriptions are listed in Tables 8 and 9. The energy ordering of the orbitals follows the conventional sequence, σ < lp(O) ≈ lp(S) < π < π* < σ*. One sulphur and four oxygen lone pairs are expected for the SO 2 monomer, and four terminal and one central oxygen lone pair in the case of O 3 . For both SO 2 and O 3 the π orbitals separate into a bonding orbital delocalized over all three atoms, a non-bonding orbital involving only the out-of-plane p orbitals of the peripheral oxygen atoms, and an antibonding orbital with contributions from the p orbitals of all three atoms. The σ* orbitals, being more diffuse and involving more excited atomic orbitals, are less easy to visualize and to assign.   These monomer orbitals transform readily into those of the dimer and complex species, and the orbitals of the five genuine minima are shown in the Supplementary Material as Figures S1-S5. The corresponding descriptions of the orbitals of the adducts are given in Supplementary Material Tables S1-S5. The major changes in the characters of the orbitals on complexation are that some of the lone pair orbitals of the monomers transform into σ bonding orbitals associated with the intermolecular bonding interactions. Thus, for SO 2 dimers 5 and 2, for example, four oxygen and one sulphur monomer lone pair orbitals go over into six oxygen and two sulphur lone pair orbitals, with two new σ(S . . . O) orbitals.   Further insights into the electronic rearrangements accompanying dimer or complex formation are provided by a consideration of the molecular electrostatic potential maps of the adducts. These plots are shown in Figure 9 for the five associated species. The diagrams indicate the regions of high electron density, shown in red, shading to more electropositive zones, shown in blue, with the peripheral oxygen atoms having the greatest negative potentials and the more positive potentials associated with the sulphur atoms and the central oxygen atoms of the ozone moieties. The potentials cover a range from about −240 to 240 kJ mol −1 .

Vibrational Spectra
The computed wavenumbers of the five associated species, and their shifts relative to the uncomplexed monomers, are shown in Table 10. For the two sulphur dioxide dimers, the antisymmetric SO2 stretching modes tend to be displaced to the red and the symmetric stretching and the SO2 bending to the blue. These shifts are all less than 10 cm −1 in either direction, however, consistent with the very low interaction energies (see Table 7). The comparisons of our calculated intramolecular wavenumbers with experimental values derived from matrix isolation infrared spectroscopic studies [13,15,[19][20][21][22][23] are given in Table 11. As a measure of the level of agreement between the calculated and experimental wavenumbers of SO2 dimer 5, the calculated/experimental ratios derived from the data in Table 11 are found to vary between 1.029 for ν1 (relative to ref. [23]) to 1.060 for ν5 (ref. [22]).
The vibrational data are all in agreement that the observed spectra are compatible with the Cs dimer 5 structure, except for the argon matrix results of Schriver-Mazzuoli et al. [21] and Ito and Hirabayashi [23], who proposed that the Ci isomer 2 more closely fits the experimental data. Indeed, Schriver-Mazzuoli and co-workers were able to assign only one band in each of the fundamental monomer regions with confidence, consistent with only one mode in each of the monomer regions being infrared-active [21]. There are some minor mismatches among the assignments of the stretching modes [20,23], but definitive assignments to the bands of the electron donor and acceptor based on the experimental spectra alone are difficult to achieve.

Vibrational Spectra
The computed wavenumbers of the five associated species, and their shifts relative to the uncomplexed monomers, are shown in Table 10. For the two sulphur dioxide dimers, the antisymmetric SO 2 stretching modes tend to be displaced to the red and the symmetric stretching and the SO 2 bending to the blue. These shifts are all less than 10 cm −1 in either direction, however, consistent with the very low interaction energies (see Table 7). The comparisons of our calculated intramolecular wavenumbers with experimental values derived from matrix isolation infrared spectroscopic studies [13,15,[19][20][21][22][23] are given in Table 11. As a measure of the level of agreement between the calculated and experimental wavenumbers of SO 2 dimer 5, the calculated/experimental ratios derived from the data in Table 11 are found to vary between 1.029 for ν 1 (relative to ref. [23]) to 1.060 for ν 5 (ref. [22]).
The vibrational data are all in agreement that the observed spectra are compatible with the C s dimer 5 structure, except for the argon matrix results of Schriver-Mazzuoli et al. [21] and Ito and Hirabayashi [23], who proposed that the C i isomer 2 more closely fits the experimental data. Indeed, Schriver-Mazzuoli and co-workers were able to assign only one band in each of the fundamental monomer regions with confidence, consistent with only one mode in each of the monomer regions being infrared-active [21]. There are some minor mismatches among the assignments of the stretching modes [20,23], but definitive assignments to the bands of the electron donor and acceptor based on the experimental spectra alone are difficult to achieve.   Ref. [13] Table 10 includes the calculated data for ozone dimer 2. The shifts of the antisymmetric O 3 stretching modes are spectacularly large, given the low interaction energy of this dimer, and are of opposite sign. This result must be viewed in the context of the computed wavenumbers of the ozone monomer, 1157.9, 741.5, and 2244.3 cm −1 for ν 1 , ν 2 , and ν 3 , respectively, which may be compared with the experimental values of 1134.9, 716.0, and 1089.2 cm −1 , reported by Barbe et al. [91]. This assignment admits a most unusual ordering of the stretching vibrations, with ν 1 > ν 3 , which has been confirmed by Lee et al. [88,89]. Slanina and Adamowicz [46] report values of 1135, 726, and 2391 cm −1 for the monomer wavenumbers, in much closer agreement with our results. The antisymmetric stretching mode shifts indicate a significant separation of the two (in-phase and out-of-phase) vibrations of 247.3 cm −1 , compared with separations of only 3.4 cm −1 and 1.7 cm −1 for SO 2 dimers 5 and 2, respectively, and of 17 cm −1 calculated for the C s isomer of (O 3 ) 2 by Slanina and Adamowicz [46]. The anomalous position of the antisymmetric stretching wavenumber confirms the notoriously difficult task of accurately reproducing the experimental wavenumber of the ozone monomer theoretically [88,89,92,93]. These authors pointed out the multiconfigurational nature of the ground state of the ozone monomer, confirmed by a series of high-level computations, including at the CASSCF, MRCI, CCSD, and CCSD(T) levels of theory. In order to examine whether our treatment of the spectra of ozone and its dimer at the MP2/aug-cc-pVTZ level was sufficiently reliable, we repeated those calculations for the monomer at the CASSCF level. We selected the monomer for this test, since the experimental geometry [94] and vibrational spectrum [91] are well-established and are available for comparison. The comparisons of our geometrical parameters and vibrational wavenumbers at the MP2 and CASSCF levels with their experimental counterparts and with the parallel computations of Lee and Scuseria [89] are given in Tables 12 and 13. While the computed bond length and angle are rather insensitive to the level of theory employed, and show fairly similar calculated and experimental differences (see Table 12), the estimation of the antisymmetric OO stretching mode exhibits a large scatter; only the theoretical treatment of Lee and Scuseria using CCSD/TZ+2Pf methodology [89] correctly reproduces the experimental ordering of the ν 1 and ν 3 vibrations (see Table 13). These uncertainties are responsible for the large, computed shifts associated with the monomer ν 3 mode. Our shifts of the symmetric stretching and bending modes of the ozone dimer are much more in line with those of the SO 2 dimers (less than 10 cm −1 ). The computed wavenumber shifts of the SO 2 moieties of SO 2 -O 3 complexes 2a and 5b are quite consistent with those of the SO 2 dimers (less than 5 cm −1 , see Table 10). Similarly, the O 3 shifts of heterodimer 5b are insignificant, but those of the symmetric and antisymmetric O 3 stretching modes of complex 2a are quite substantial and of opposite sign, yielding a separation of 51.6 cm −1 . While this separation is not as dramatic as the corresponding result for the C i ozone dimer, it is quite apparent that the antisymmetric stretching vibrations of ozone molecules in these homo-and heterodimers are extremely sensitive to complexation.

Computational Methodology
The calculations were carried out using Gaussian-16 [95], at the second order level of the Møller-Plesset perturbation theory (MP2) [96] with Dunning's augmented correlationconsistent polarized valence triple-zeta basis sets (aug-cc-pVTZ) [97,98]. Structures were optimized using the verytight keyword, where practicable, and stationary points were identified as genuine minima or transition states by vibrational analysis. The wavenumbers and infrared intensities of the resulting species were determined at the harmonic level. Interaction energies were computed and corrected for basis set superposition error (BSSE) [90], using the Boys-Bernardi full counterpoise procedure [99] and the counterpoise = 2 keyword and for zero-point energy differences. Molecular orbital properties and molecular electrostatic potentials were examined using the Gaussian input keywords pop = full, density = current, and cube(full,orbitals) [95]. The model chemistry employed here is consistent with those represented in a number of similar calculations [4][5][6][7][8] in terms of its ability to yield credible results.

Conclusions
A series of nine structures each of the sulphur dioxide and ozone homodimers and of eight of their heterodimers were investigated. Of these structures, two sulphur dioxide and one ozone dimer and two sulphur dioxide-ozone complexes were found to be genuine minima on their potential surfaces. These five species were all found to be very weakly bound (less than 10 kJ mol −1 ). These low interaction energies resulted in very small perturbations of the bond lengths and bond angles of the monomers (less than 0.36 and 1.39 pm for the SO and OO bond lengths and 0.29 • and 0.58 • for the OSO and OOO angles, respectively). The minimal perturbations of the intramolecular geometries are matched by the small, computed wavenumber shifts (less than 7 cm −1 for the SO 2 dimers and the SO 2 sub-units of the heterodimers, and less than 10 cm −1 for the symmetric stretching and bending of the O 3 moieties of the ozone dimers and the heterodimers). The exceptions to this statement are ozone dimer 2, where the shifts of the antisymmetric O 3 stretching mode are 175.0 and −72.3 cm −1 , and SO 2 -O 3 complex 2a, where the O 3 stretching vibrations undergo shifts of 25.8 cm −1 to the blue (antisymmetric) and the red (symmetric). The ozone moiety appears to be much more sensitive than sulphur dioxide to perturbations of their vibrational spectra due to complexation, but the conclusions regarding the magnitudes of the shifts have to be tempered by recognition of the wellknown susceptibility of ozone to the level of theoretical treatment [88,89,92,93]. Miliordos and Xantheas presented evidence that, while the bonding in SO 2 consists of two σ and two π bonds, that in O 3 is better described as a mixture of a closed shell structure featuring two O-O bonds having bond orders of 1.5 (a delocalized 3-centre-4-electron bond) (82%) and a diradical structure with two σ bonds, a lone pair on the central oxygen atom, and a single electron in a p orbital on each of the terminal atoms (18%) [92]. Takeshita et al. concurred with respect to the description of the SO 2 bonding arrangement (recoupled pair bonds), and the fact that O 3 "has significantly more diradical character than SO 2 " [93]. These fundamental differences are certainly partly responsible for the marked variations in the ability of sulphur dioxide and ozone to form stable dimers, and hence in the differences in the computed vibrational spectra of the dimers. Neither Miliordos and Xantheas [92] nor Takeshita and co-workers [93] offered any insights into the spectra of the monomers, nor into the properties of the dimers.
We acknowledge the limitations of our methodology as they apply to the ozone species, but in the interests of consistency with our other results, we stand by the conclusions presented in this work.

Supplementary Materials:
The following are available online, Figures S1-S5 and Tables S1-S5: titles as indicated in Table of Contents. Acknowledgments: This material is based upon work supported by the National Research Foundation of South Africa. Any opinion, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Research Foundation. The author acknowledges the Centre for High Performance Computing (South Africa) for the use of computing facilities, and Anton Lopis for invaluable technical assistance. The author is also grateful to a reviewer for bringing references 92 and 93 to his attention. He looks back with pleasure on a number of valuable detailed technical discussions and convivial social interactions with Austin Barnes in several countries over many years.