Semiclassical Calculation of Rotational Excitation

In the preceding article the Schrödinger equation and its solution are developed for the problem of translation—rotation energy exchange in a representation of rotor functions separately quantized along the moving interparticle axis. The quantum-mechanical time-dependent differential equations for the coefficients in this representation are obtained by the method of variation of constants with a time-dependent potential calculated from a classical trajectory and a symmetrized, initial relative kinetic energy. These equations are solved numerically for a limited number of basis functions, with intermultiplet transitions neglected, for some atom—diatomic systems. The isotropic part of the potential is taken to be a Lennard-Jones (12, 6) potential, with parameters appropriate for the cases K–HBr, Cs–HBr, and Li–HBr. For the present exploratory calculations the anisotropic part is assumed to be proportional to the inverse sixth power of the interparticle distance and to have P2 symmetry. Probabilities of inelastic collisions along given trajectories and total inelastic cross sections are calculated for the j=0→2, 2→4, 0→4, and 1→3 transitions. The total cross section for the 0→2 transition in the case representative of K–HBr is 5.74 Å2 for a reasonable estimate of the anisotropy of the assumed potential. A test calculation is made for the system He–H2 and agreement is found to 20% with Roberts' first-order distorted wave calculation in which the translational motion is treated quantum mechanically. The effect of intermultiplet transitions is examined and it is estimated that their neglect may incur an error in the total cross section of about 50%.