Surface jumping: Franck-Condon factors and condon points in phase space

We generalize the concept of Tully-Preston surface hopping to include larger jumps in the case that the surfaces do not cross. Instead of identifying a complex hopping point, we specify a jump between two locales in phase space. This concept is used here to find propensity rules for the accepting vibrational mode(s) in a radiationless vibronic relaxation transition. A model inspired by the S₂ → S₀ vibronic relaxation transition of the benzene molecule in which 30 modes of vibration compete for the electronic energy is studied within this approach. For this model, we show that almost all of the energy must go to a single C-H local stretching. The initial conditions for vibrations of this mode are a coordinate jump of the hydrogen atom toward the ring. All of the other modes undergo an almost vertical transition, in which the energy that they take is determined by their equilibrium displacement between the two surfaces. We observe that for a large energy gap the masses and frequencies become the defining parameters for choosing the accepting mode. Anharmonicities are very important when a competition between degenerate modes occurs. These conclusions are demonstrated by the specific model considered here but apply in general to any weak internal conversion process.