where P 2 0 /2M is the kinetic energy of the nuclear DOF for the molecule, M is the effective mass of the nuclear vibration, V ( R0 ) is the ground electronic state potential energy surface, and R0 is the reaction coordinate. Further, qc = h/(2ω c )(â + â † ) and pc = i hω 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. The light-matter coupling strength η c is expressed as η c = 1/(2hω c ϵ 0 V), which characterizes the light-matter coupling strength, where ϵ 0 is the permittivity, and V is the quantization volume inside the cavity. In Eq. 2, we had also explicitly assumed that the ground state dipole moment µ( R0 ) is linear and always aligned with the cavity polarization direction, such that µ( R0 ) = R0 . Finally, the phonon bath is expressed as