Evaluation of Approximations Used in the Calculation of Excitation by Collision. I. Vibrational Excitation of Molecules

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Abstract
Approximate procedures for calculating transition probabilities for excitation by molecular collision have been examined in terms of a simplified model, vibrational excitation of a harmonic vibrator by collinear collision with an atom. Digital‐computer solutions were obtained for the coupled differential equations resulting from time‐dependent perturbation theory with an expansion of the total wavefunction in terms of eigenfunctions of the target molecule. Calculations with the expansion truncated at 2, 3, ..., 10 states show that, as the collision velocity is increased, more states must be included to obtain an ``exact'' solution. For single quantum transitions j→j+1 at low velocities, the first‐order perturbation approximation results are reliable up to probabilities of about 0.1; at higher velocities, they greatly exceed the ``exact'' results. The two‐state approximation always yields drastic underestimates. For multiple quantum transitions at all velocities, the first‐order perturbation calculation greatly underestimates the transition probability because it includes only the direct transition j→k and excludes the stepwise transitions during a single collision j→j+1→...→k−1→k. To evaluate this dominant process, a multiple‐step perturbation theory was developed. The resulting equation for a harmonic oscillator, P0→n = (P0→1)n/n!, gives excellent predictions in the low velocity range. For higher collision velocities, a computer calculation employing many eigenfunctions is required. The relationship of these results to calculation of other inelastic collision processes is discussed briefly.