We applied the semiclassical initial value representation method to calculate energies, lifetimes and wave functions of scattering resonances in a two-dimensional potential for O + O₂ collision. Such scattering states represent the metastable O₃^* species and play a central role in the process of ozone formation. The focus was on the ¹⁶O¹⁶O¹⁸O isotopomer and the anomalous isotope effect associated with formation of this molecule, either through the ¹⁶O¹⁶O + ¹⁸O or the ¹⁶O + ¹⁶O¹⁸O channels. An interesting correlation between the local vibration mode character of the metastable states and their lifetimes was observed and explained. New insight is obtained into the mechanism by which the long-lived resonances in the DZPE part of spectrum produce the anomalously large isotope effect.

First, we showed that the semi-classical IVR method can be successfully applied to calculate energies, lifetimes and wave functions of long-lived scattering resonances (metastable states) in a barrierless potential with a deep attractive well and long range interaction tails in the channels. Using a new cut-off procedure for chaotic trajectories, we achieved stable propagation of wave packets for up to t = 4.0 ps. The number of trajectories needed to obtain converged results was, as usual, large (N = 10⁶ - 10⁸ at t = 0) but, since the trajectories are totally independent, the parallel processing was also very efficient. The intrinsic massive parallelization of the method allowed us to achieve the wall-clock-time acceleration by several orders of magnitude. This demonstrated that the IVR approach to scattering resonances is computationally appealing. The results of semi-classical wave packet propagation agreed well with fully quantum results. Autocorrelation functions were computed and then analyzed using the Prony method which permits one to extract energies and widths of the resonances. Further improvement of accuracy for widths of very narrow resonances seems to be possible by employing a method of analysis (other then Prony) that would be less sensitive to the noise present in the semiclassical autocorrelation function at long propagation times.

Second, we used this approach to study recombination reaction which forms an asymmetric ozone isotopomer ¹⁶O¹⁶O¹⁸O. We demonstrated that wave functions of the metastable O₃^* states are highly localized in one or another channel. This interesting behavior was explained by introducing two independent progressions of highly excited vibrational states for ¹⁶O¹⁶O--¹⁸O and ¹⁶O--¹⁶O¹⁸O local stretch modes. We showed that within the DZPE energy range, when the ¹⁶O¹⁶O channel is still energetically closed, the metastable states of the ¹⁶O¹⁶O--¹⁸O progression are very long-lived (t ~ 50 - 500 ps) while the states of the ¹⁶O--¹⁶O¹⁸O progression decay much faster (t ~ 0.1 - 0.3 ps). This property correlates with the local mode character found in the wave functions. The nature of such narrow resonances in the DZPE energy range is somewhat similar to the Feshbach resonances which occur due to interaction between open and closed electronic channels. The difference is that in our case only one electronic state is involved and the two channels are the ¹⁶O¹⁶O + ¹⁸O and the ¹⁶O + ¹⁶O¹⁸O channels of a chemical reaction. The two channels are coupled by the PES, one is open and one is closed due to quantum DZPE.

Finally, we studied kinetics of this reaction and showed that these long-lived resonances in the DZPE energy range are responsible for the anomalous isotope effect found in ozone. We studied two methodologically important limiting cases and then demonstrated that the isotope effect is extremely sensitive to lifetimes of these states. Thus, accurate determination of widths of narrow resonances (in the range between 0.4 cm^-1 and 0.003 cm^-1) becomes crucial for correct prediction of the isotope effect.

In this work we considered a simplified 2D model of ozone formation where the angle q in the Hamiltonian was fixed. This was certainly an approximation, though it is justified by the fact that the contribution of the insertion reactions of type ^YO + ^XO^ZO → ^XO^YO^ZO is known to be very small in ozone. Of course, for the accurate prediction of the isotope effect an extension of this work onto a 3D problem with a full dimensional accurate PES is necessary. Using the semi-classical IVR approach and the methodology developed here such an extension should be relatively straightforward and we plan to explore this opportunity in the future.