where k 0 and k D are directly obtained from HEOM simulations as they are not related to the coupling of the cavity mode. In this work, under the condition η c = 0 (outside the cavity), we find that α ≈ 0.7 will bring the FGR analytic results to quantitatively agree with the numerically exact results from HEOM. This parameter is then fixed for all cases of η c when coupling the RPV mode to the cavity. We aim to develop an analytic rate constant theory that describes the role of {Q j } and q c on the rate constant. To this end, we derived an effective spectral density, J eff (ω), that describes the coupling of ĤRPV + ĤLM + Ĥloss to the reaction coordinate R 0 , based on the effective spectral density theory . The details of the derivation are provided in Supplementary Note 5, and the most general expression is provided in Eq. 34 in Method. A more insightful, approximate expression is