This is a necessary term of the PF Hamiltonian, in order to make sure gauge invariance of the PF Hamiltonian 9,30 and a bounded ground polariton state. In this paper, we do not consider the cavity loss. This can be effectively incorporated by using Lindblad dynamics approaches with the MQC simulations. Recent investigations of polariton photochemistry have been mainly focused on using the Jaynes-Cummings (JC) model or the Tavis-Cummings (TC) model 12 to describe the quantum light-matter interactions. These models usually only consider two electronic states {|g , |e } and the transition dipole µ ge (R) = g| μ|e among them, where the permanent dipole is often ignored. In this context, one can further define the creation and annihilation operators for molecular excitation as σ † ≡ |e g| and σ ≡ |g e|, and thus μ = µ eg (R)•(σ † +σ). The molecule-cavity interaction term in Eq. 5 can now be expressed as