The potential V (q, x c ) has three critical points with (q ‡ , x ‡ c ) ≡ (0, 0) being the saddle point and (q ± , x± c ) ≡ (±q m , -λ c d(±q m )/ω c ) denoting the reactant and product minima respectively, where we have denoted q m ≡ (|A 2 |/2A 4 ) 1/2 . The classical barrier height V b ≈ 2030 cm -1 and is independent of the coupling strength. In addition, as noted by Fischer et al , the derivative of the dipole d ′ (q) vanishes for q = (2δ) -1/2 ≈ q m , and hence the Hessian at the minimum is nearly independent of the cavity-system coupling. Thus, a double harmonic approximation is fairly good and yields the two real frequencies ω