An improved time-dependent harmonic oscillator method for vibrationally inelastic collisions

A quantal solution to vibrationally inelastic collisions is presented based upon a linear expansion of the interaction potential around the time-dependent classical positions of all translational and vibrational degrees of freedom. The full time-dependent wave function is a product of a Gaussian translational wave packet and a multidimensional harmonic oscillator wave function, both centered around the appropriate classical position variables. The computational requirements are small since the initial vibrational coordinates are the equilibrium values in the classical trajectory (i.e., phase space sampling does not occur). Different choices of the initial width of the translational wave packet and the initial classical translational momenta are possible, and two combinations are investigated. The first involves setting the initial classical momenta equal to the quantal expectation value, and varying the width to satisfy normalization of the transition probability matrix. The second involves adjusting the initial classical momenta to ensure detailed balancing for each set of transitions, i→f and f→i, and varying the width to satisfy normalization. This choice illustrates the origin of the empirical correction of using the arithmetic average momenta as the initial classical momenta in the forced oscillator approximation. Both methods are tested for the collinear collision systems CO2–(He, Ne), and are found to be accurate except for near-resonant vibration–vibration exchange at low initial kinetic energies.