where the slope is related to ∆H ‡ and the y-intercept is related to ∆S ‡ . The rate constants were obtained from HEOM simulations (filled circles), and fitted by the least square to obtain linearity (thin lines), as well as from FGR rate theory (open circles). One can see that as Ω R increases, both ∆H ‡ and ∆S ‡ changes. Here, both the HEOM simulation and the FGR rate theory predict the same trend. The previous work based on the classical Grote-Hynes rate theory can only explain the change in ∆S ‡ . The current FGR-based theory can explain changes in both ∆H ‡ and ∆S ‡ , which has been observed in experiments . Again, we emphasize that the VSC reaction mechanism is not related to modification of the ∆H ‡ nor ∆S ‡ , but rather how cavity can mediate vibrational excitations (see Eq. 7) by coupling to Q. However, if one chooses to interpret the change of k/k 0 through the effective changes of ∆H ‡ and ∆S ‡ , then they will exhibit features depicted in Fig. . Another factor that could influence the VSC-rate constant is the cavity lifetime τ c , which also explicitly shows up in the analytic expression of Γ Q (ω) (see Eq. 10) and R(ω) (see Eq. 12), however, we find that k/k 0 is not particularly sensitive to τ c , and the magnitude of suppression will be maximized under τ c → ∞ limit, but start to converge when τ c ≥ 300 fs (see Supplementary .