which is the PF QED Hamiltonian for the model used in this paper. Here, we include only the ground electronic state of the molecule. A detailed derivation of this Hamiltonian from the minimum-coupling Hamiltonian can be found in Ref. 21. In addition, P 2 /2M is the kinetic energy of the nuclear DOF for the molecule, M is the effective mass of the nuclear vibration, V ( R) is the ground electronic state potential energy surface, and R is the reaction coordinate. Further, qc = 1/(2ω c )(â + â † ) and pc = i ω c /2(â † -â) are the photon mode coordinate and momentum operators, respectively, where â † and â are the photon mode creation and annihilation operators, and ω c is the cavity frequency. Finally,