which has a clear resonance behavior at ω c = ω 0 due to the effective spectral density J eff (ω 0 ) (c.f. Eq. 6). The analytic rate constant produces semiquantitative agreement with the exact HEOM simulations, and needs to be further corrected with phonon broadening (see Appendix A in Ref. 9) when τ c → ∞. These early mechanistic studies suggest that the key to obtaining the correct resonance behavior in k/k 0 is to have the quantum description of the vibrational state |ν L ⟩ and |ν ′ L ⟩, such that the quantum vibratinoal frequency ω 0 ≡ (E ′ -E)/h is explicitly incorporated in the simulations. The quantum vibrational frequency ω 0 in the current model is different than the barrier frequency ω b or the classical vibrational frequency. Note that in most of the TST-type rate theory or RPMD simulations, there is no explicit information of ω 0 , and the partition function in principle contains all vibrational frequencies. This is the reason why these existing rate theories can not give rise to the correct resonance condition in Eq. 1. The recently derived analytic theory in Eq. 8, on the other hand, explicitly contains ω 0 and is thus capable of correctly describing the correct resonance condition.