such that the cavity-related width α = τ -1 c (see Eq. 30) dominates the rate profile. For the model parameter considered here (ω 0 = 1172 cm -1 , k B T ≈ 200 cm -1 ), we have βω 0 1 and thus n(ω 0 ) ≈ e -βω0 , as we explicitly used in the second line of Eq. 44, indicating the thermal Boltzmann probability of occupying the |ν L state. For larger τ c , it is necessary to use the full FGR expression in Eq. 41 (with convolution), while the approximate expression in Eq. 44 provides a much simpler analytic form for us to analyze basic scaling relations of k VSC . In particular, the resonant behavior can be readily seen because k VSC will reach its maximum when ω c = ω 0 . The broadening of the VSC-modified rate profile in Eq. 44 is dictated by the parameter α (see the expression in Eq. 15), which is the width of J eff (ω) in Eq. 14. Further, when α is much larger than ω c , the Brownian lineshape will gradually become the familiar Drude-Lorentz lineshape. The more general expression in Eq. 41 will predict the same resonant condition if J eff (ω) takes the Markovian limit (Eq. 14), because the function G(ω -ω 0 ) in Eq. A10 only provides additional broadening.