Phase-space derivation of propensity rules for energy transfer processes between Born-Oppenheimer surfaces

We consider a simple method for calculating weak Franck-Condon factors. The Wigner transform of the wave function on an initial Born-Oppenheimer state is calculated for the donor potential surface and projected onto the acceptor energy surface energy shell. The integrated projection yields an approximation for the relevant Franck-Condon factors, while phase-space integrand shows where in phase space the leakage occurs between the donor and acceptor states. This in turn determines the initial conditions on the acceptor surface for subsequent IVR and energy flow. Propensity rules are obtained by recognizing phase-space points of closest approach of the initial-state Wigner function and the final-state energy surface. The example of two coupled harmonic oscillators is explicitly solved to demonstrate the power of this phase-space approach.