Fig. presents the VSC-modified rate profile as a function of the cavity frequency ω c . Here, the light-matter coupling strength η c and the cavity frequency ω c are variables. The temperature is set to be T = 300 K, the characteristic frequency of the photon-loss bath is γ c → ∞, and the cavity lifetime is chosen to be τ c = 100 fs. Fig. presents the time-dependent rate constant k(t) (see discussions below Eq. 37) inside a resonant cavity (with ω c = ω 0 = 1172 cm -1 ). As one gradually increases the light-matter coupling strength η c (see legend in panel a) from 0 (black, outside the cavity) to 0.0125 (violet), 0.01875 (blue), 0.025 (cyan), 0.0375 (green), 0.05 (orange), 0.075 (red) and 0.1 (brown), there is always a well-defined plateau value that provides a rate constant (associated with a rate process), with the plateau time t p ≈ 2 ps. Fig. presents the resonance effect of the VSCmodified rate constant when a molecular vibration is coupled to a cavity. A resonantly enhanced sharp peak of the rate constant is exhibited when ω c = ω 0 . To characterize cavity modification effects, we present the rate constant enhancement factor k/k 0 from Fig. , after reaching the dynamic plateau time. As η c increases, the resonance peak intensity also becomes stronger. This sharp resonance enhancement of the rate constant has been observed in recent VSC experiments, such as Fig. of Ref. 4 or Fig. of Ref. 5. However, we need to remind the reader that in experiments there are at least 10 6 molecules collectively coupled to the cavity, whereas the current studies only considered a single molecule strongly coupled to the cavity mode. Further, in that experiment, the cavity mode is tuned to be resonant with solvent vibrations, while, in our model system, the cavity frequency is tuned to match the |ν L → |ν L transition for the reactant well. Future studies will be needed that will be based on a model system that more accurately reflects the experimental setup. Note that because there is a clear frequency separation between the quantum transition frequency ω 0 (Eq. 25) and the classical bottom of the well frequency ω cl 0 (all of which are labeled as the vertical dashed lines in Fig. ), the results clearly demonstrate that the VSC rate enhancement is only related to ω 0 (Eq. 25), rather than ω cl 0 (Eq. 26) or ω b (Eq. 21). Interestingly, experimental observation always suggests that the VSC-modified rate profile and the transmission spectra of the bare molecule peaks at ω 0 (e.g., Fig. of Ref. 4 and Fig. of Ref. 5). This is because, in the spectra measurements, the laser field causes the transition |ν L → |ν L , resulting in a maximum intensity of the signal at the transition frequency ω 0 . For the VSC-modified rate profile, the cavity mode qc will promote the transition |ν L → |ν L , which reaches its highest magnitude when ω c = ω 0 . This explains why the molecular transmission spectra always show the same peak position as the VSC-modified rate profile. This resonance structure is also predicted and explained by the FGR expression in Eq. 44, where the VSC-modified rate profile will peak at ω c = ω 0 . The rate profile should also have a finite width around ω 0 , including both contributions from both α in J eff (ω) and G(ω -ω 0 ). We will return to the quantitative comparison of k VSC using the full expression in Eq. 41 and the HEOM results at the end of the paper. when ω c = ω 0 , as a function of the light-matter coupling strength η c . The data points obtained from the exact HEOM simulations are depicted with circles, and the solid line provides a fitting curve using the first 5 data points (for the range of η c < 0.05). The fitting suggests that k/k 0 has an almost perfect quadratic dependence to η c for the range of η c < 0.05, such that k/k 0 ∝ 1 + C • η 2 c , where C = 4|∆ x | 2 ω 0 τ c (see Eq. 44). This scaling relation can also be intuitively understood using the FGR expression in Eq. 41. Note that η c in-