Classical trajectory treatment of inelastic scattering in collisions of ${\mathrm{H}}^{+}$ with ${\mathrm{H}}_2$, HD, and ${\mathrm{D}}_2$

A semiclassical model for vibrational excitation in molecular collisions is proposed, in which three-dimensional classical trajectory calculations are used to evaluate the quantum vibrational transition probabilities. Using an accurate analytic fit to 138 $\mathrm{ab}$ initio points in the important region of the $\mathrm{H}_3^{}{}_{}{}^{+}$ potential surface and a new Monte Carlo interpolation method for averaging over initial conditions, we apply the model to vibrational excitation in collisions of ${\mathrm{H}}^{+}$ with ${\mathrm{H}}_2$, HD, and ${\mathrm{D}}_2$. The calculations reveal the necessity of correcting our previously reported experimental vibrational transition probabilities for rotational contributions. Once this is done, the experimental data and the model calculations are in very good agreement. The theoretical results support our previous conclusion that vibrational excitation in this system is caused primarily by dilution of the molecular bond by the passing proton. The observed maximum in the experimental inelastic differential cross sections at approximately half the rainbow angle is shown to be associated with the second classical rainbow, which results from the anisotropy of the potential.