Comparison of the WKB and purely classical methods for vibrational transition probabilities

A rigorous comparison of the WKB evaluation of the quantal approach, P_WKB, with the purely classical theory, P_C, of vibrational energy transfer is presented. For the Lennard-Jones (12-6) potential, P_C≈(7/6) Γ(6/7)]^?2P_WKB, when the collision energy is appropriately symmetrized. The first two terms in the exponent of P_WKB are identical to those of P_C. For the (m?6) potential, the relation is P_C ≈ (m + 2)/m] > 62; Γm/(m + 2)] > 62;^?2P_WKB, which shows that P_C ≈ P_WKB for n > 62;2. For exponential potentials, P_WKB = P_C. In the WKB approach the perturbation integral is evaluated by use of the trajectory determined in the immediate neighborhood of the pole. Both the most probable distance and the pole lie in the region of strong repulsive interaction.