We start by introducing the Pauli-Fierz (PF) nonrelativistic QED Hamiltonian to describe the light-matter interaction. The PF Hamiltonian can be rigorously derived by applying the Power-Zienau-Woolley (PZW) Gauge transformation and a unitary phase transformation on the minimal-coupling Hamiltonian in the Coulomb gauge (i.e. the "p • A" Hamiltonian) under the long-wavelength limit. A rigorous derivation is provided in the Appendix. For a molecule coupled to a single photon mode inside an optical cavity, the PF Hamiltonian is