Virtual transitions in nonadiabatic condensed-phase reactions

We calculate the rate of a reaction proceeding through a virtual intermediate for arbitrary vibrational potentials in three electronic states, expressing it in a form amenable to saddle-point approximations. Treating the mode-specific case, in which a small number of vibrational modes are coupled to the electronic transitions, dissipation is included by vibrational coupling to a large collection of harmonic oscillators. A semiclassical saddle-point expansion at large energy denominator allows us to identify the criteria for the validity of the two-state approximation, in which the intermediate state is ignored, aside from generating an effective coupling between the initial and final states. We discuss the physical meaning of these criteria, which define a high-energy intermediate. In this case, virtual transfer may dominate over the two-step process. In the limit of a small energy denominator, we find that the virtual transfer rate is not necessarily negligible compared to the first step of a two-step transfer. In this case, the rate is found to depend on the damping of the vibrationally coupled modes over a certain range.