A classical analog for electronic degrees of freedom in nonadiabatic collision processes

It is shown how a formally exact classical analog can be defined for a finite dimensional (in Hilbert space) quantum mechanical system. This approach is then used to obtain a classical model for the electronic degrees of freedom in a molecular collision system, and the combination of this with the usual classical description of the heavy particle (i.e., nuclear) motion provides a completely classical model for the electronic and heavy particle degrees of freedom. The resulting equations of motion are shown to be equivalent to describing the electronic degrees of freedom by the time-dependent Schrödinger equation, the time dependence arising from the classical motion of the nuclei, the trajectory of which is determined by the quantum mechanical average (i.e., Ehrenfest) force on the nuclei. Quantizing the system via classical S-matrix theory is shown to provide a dynamically consistent description of nonadiabatic collision processes; i.e., different electronic transitions have different heavy particle trajectories and, for example, the total energy of the electronic and heavy particle degrees of freedom is conserved. Application of this classical model for the electronic degrees of freedom (plus classical S-matrix theory) to the two-state model problem shows that the approach provides a good description of the electronic dynamics.