where M is the mass of the molecule, N is the total number of molecules, E g (R J ) is the ground-state potential energy surface of the J-th molecule, A 0 characterizes the strength of the light-matter coupling strength, ϵ is the polarization vector of the cavity field, and µ(R J ) is the dipole moment of the J-th molecule. When one molecule is in the transition state, the Hamiltonian can be rewritten in an effective 3-mode expression Ĥ(x), where x = {R ‡ , R B , q c } represents the coordinates of the reactive molecule, the collective bright mode, and the photon mode. To compute the normal mode frequencies, the 3-mode Hessian matrix is written as,