Grote–Hynes rate theory

In multidimensional transition state theory, the reactant to product rate constant is given as^40,44–46

where {Ωi0} are normal-mode frequencies of the Hamiltonian in the reactant well, and {Ω2‡,...,ΩN‡} are the stable normal-mode frequencies at the barrier, such that Ωi‡2>0 for i > 1, and Ω1‡2<0 is the imaginary frequency of the transition state.

Considering a simplified (classical) model of the molecule-cavity hybrid system, H−Hvib=P22M+E(R)+pc22+12ωc2(qc+2ℏωc3χ⋅μ(R))2 which only contains two DOFs {q_c, R} (and are viewed as classical DOFs), the normal-mode frequencies at R₀ are Ω±2=12(ω02+C02ωc2+ωc2)±12(ω02+C02ωc2+ωc2)2−4ωc2ω02, where C0=2ωcMℏχ⋅μ0′. The normal-mode frequencies at R_‡ are Ω±‡2=12(−ωb2+C‡2ωc2+ωc2)±12(−ωb2+C‡2ωc2+ωc2)2+4ωc2ωb2, where C‡=2ωcMℏχ⋅μ‡′. Details of the derivation of these normal-mode frequencies are provided in Supplementary Note ². Using these normal-mode frequencies, the rate constant in Eq. (¹⁰) for the H^−H^vib model is expressed as k=12πΩ+Ω−Ω+‡e−βEb, where E_b is the energy barrier. Using the fact that (Ω+‡Ω−‡)2=−ωb2ωc2 and (Ω+Ω−)2=ω02ωc2 (see the general proof in ref. ⁴²), the rate constant can be further expressed as follows

where kTST=ω02πe−βEb, κGH=λωb is the transmission coefficient in the GH theory, λ=−(Ω−‡)2, which is Eq. (⁶). Same procedure can also be used to derive the expressions⁴⁶ of the κ_GH for the model Hamiltonian in Eq. (¹). Alternatively, one can derive the transmission coefficient κ_GH from the equation of motion^39,56, with the details provided in Supplementary Note ⁴.

When considering the phonon bath H^vib under the Markovian limit (while consider q_c as the non-Markovian coordinate), λ can be obtained by solving the following equation

where κGH=λωb. We consider a bath friction coefficient ζ = 400 cm⁻¹ according to the spectral density J(ω). The detailed derivations of Eq. (¹²), as well as the assessment of the validity of the Markovian limit of the phonon bath are provided in Supplementary Note ⁴.