where ĤPF is given in Eq. 2, E j (R) is the Born-Oppenheimer polaritonic potential energy surfaces (which parametrically depend on the nuclear coor-dinates R), and |E j (R)⟩ is the polariton state. We directly diagonalize the polaritonic Hamiltonian ĤPF matrix and obtain the eigenvalues. The basis is constructed using the tensor product of electronic adiabatic states |ψ α (R)⟩ (i.e., eigenstates of the electronic Hamiltonian Ĥel ) and the Fock states |n⟩ (i.e., eigenstates of the photonic Hamiltonian Ĥph ), expressed as |ψ α (R)⟩ ⊗ |n⟩ ≡ |ψ α (R), n⟩. This basis is used to evaluate the matrix elements of ĤPF , and diagonalizing it provides E j (R) and the corresponding polariton states