where χ eq ≡ P R / P P denotes the ratio of equilibrium population between the reactant and product. The time derivative ṖR (t) is evaluated numerically. A simple derivation is provided in Appendix B. For the symmetric DW potential model (Eq. 20) considered in this work, χ eq = 1, and for more general cases, it can be obtained either by path-integral Monte Carlo approaches or by the imaginary time evolution of HEOM. The limit t → t p represents that the dynamics have already entered into the rate process regime (linear response regime) and t p represents the "plateau" time of the time-dependent rate (which is equivalent to a flux-side time correlation function formalism ). A detailed discussion is provided in Appendix B. Without taking the limit t → t p , one can view Eq. 37 as the flux-side correlation function that provides the time-dependent rate constant k(t), which captures both the initial transient dynamics (the oscillatory behaviors of k(t)) and the longer time rate process (plateau of k(t)). An example of this k(t) is provided in Fig. . With the above preparations, we applied HEOM to solve the quantum dynamics and compute the rate constants. The bare reaction kinetics and VSC "resonance effect" under the influence of various parameters are investigated.