Note that this quantum frequency ω 0 (Eq. 25) is different from the classical bottom of the well frequency ω cl 0 (Eq. 26) or top of the barrier frequency ω b (Eq. 21) for an anharmonic potential. Previous work that uses classical rate theory to study VSC effects cannot find the resonance behavior of ω c = ω 0 , because these theories rely on the classical description of the system (such as trajectories) which are sensitive to the classical frequencies of the potential. More detailed discussions of these previous classical rate theories for VSC can be found in Appendix E. In this sense, the proper description of the VSC resonance effect needs a quantum description that provides the quantum frequency information ω 0 (by providing the eigenenergies of the vibrational states).