Here, we briefly review some basic knowledge of the molecular Hamiltonian, which will be useful for our discussions of molecular cavity QED. We begin by defining the matter Hamiltonian as follows ĤM = T + V (x) = j 1 2m j p2 j + V (x j ), (19) where j is the index of the j th charged particle (including all electrons and nuclei), with the corresponding mass, m j , and canonical momentum, pj = -iℏ∇ j . We denote electronic coordinate with r, and nuclear coordinate with R, and use xj ∈ {r j , R j } to represent either the electron or nucleus, with x being the coordinate operator for all charged particles. Further, T = TR + Tr is the kinetic energy operator for all charged particles, where TR and Tr represent the kinetic energy operator for nuclei and for electrons, respectively. Further, V (x) is the potential operator that describes the Coulombic interactions among the electrons and nuclei. The electronic Hamiltonian is often defined as Ĥel = ĤM -TR = Tr + V (x), (20) which includes the kinetic energy of electrons, electronelectron interactions, electron-nuclear interactions, and nuclear-nuclear interactions. The essential task of the electronic structure community is focused on solving the eigenstates of Ĥel at a particular nuclear configuration R as follows