Generalized effective Hamiltonians: Time scale separation within a semiclassical formalism

A theory for collision-induced transitions in the subset of quantum states |_e〉 is developed, where the full molecular basis is |a _ea_u〉. The only assumption is that changes in a _u occur much more quickly than those in a_e. The choice of a_e and a_u is in general arbitrary, and it is shown how to make this separation based on the type of experimental measurement under consideration. For example, an evaluation of an atom-vibrotor problem (i.e., a_e=n, a_u=jm_j) is considered. In conjuction with this illustration an analysis of the breathing sphere (BS) approximation is carried out. It is concluded that a modified BS method (i.e., use of the diagonal rotational potential) should be valid when the random phase approximation holds for the rotational scattering amplitude.