where ω 0 is the reactant frequency, ω ‡ is the barrier frequency, µ ′ 0 is the slope of the permanent dipole at the reactant well, µ ′ ‡ is the slope of the permanent dipole at the transition state, ⟨...⟩ denotes the ensemble average, and C 0 = A 0 2ω 3 c ℏ ⟨µ ′2 0 ⟩ N -1 represents the coupling between the cavity mode and the collective bright mode.