Here, we give an interesting example that has been extensively explored, which is a LiF molecule coupled to a single mode cavity. We will only focus on the polariton potential energy surfaces and not consider the time-dependent polariton dynamics (which will be discussed in Sec. 4.1). In addition, we will only focus on two electronic states of the LiF molecule (N = 2). We emphasize that one should treat the number of electronic states N as a convergence parameter in pQED calculations. Fig. presents the polariton potential energy surfaces predicted by various quantum optics model Hamiltonians for the model LiF molecule shown in Fig. (the details of the model can be found in Ref. 41). Here, only two diabatic states are considered, which are denoted as the ionic state |I⟩, and covalent state |C⟩. These two diabatic states are coupled through a diabatic coupling V IC (R) (dotted yellow line in Fig. ) that causes a splitting (avoided crossing) near the anti-crossing of the diabatic potentials V C (R) and V I (R) (solid red and blue line in Fig. , respectively). The adiabatic electronic states, ground |g(R)⟩ and excited |e(R)⟩ states can be obtained by diagonalizing the electronic Hamiltonian Ĥel