A new semiclassical approach to the molecular dynamics: Label variable classical mechanics

A new semiclassical approach to molecular collision dynamics is developed. In this approach the state of a system is described by a unit vector in Hilbert space. By choosing a reference unit vector and a continuous mapping, a correspondence between the vector in Hilbert space and the point in a label space is established, i.e., the vector in Hilbert space is parametrized or labeled by complex variables. It is shown that the label variables formally obey classical mechanics. Thus, the time evolution of the vectors in Hilbert space, i.e., the evolution of the states of the system, can be determined by calculating the evolution of the label variables classically. To illustrate this idea, the formalism for calculating the vibrational transition probability in the collinear collision A+BC is presented, and a demonstrative calculation for the collinear Secrest–Johnson model of He+H2 vibrationally inelastic scattering has been carried out. The comparison with the exact quantum results shows that the agreement is encouragingly good.