where the r and R are the position of the electron and the nucleus, respectively. We choose the same parameters used in Ref. 48, which is L = 19 a.u., a + = 3.1 a.u., a -= 4.0 a.u., a f = 5.0 a.u. and the proton mass is M = 1836 a.u. For the polariton dynamics simulation when coupling this molecule with cavity, the photon energy of the cavity mode is chosen as ω c = 2.721 eV (≈ 0.1 a.u.). Fig. presents the two lowest adiabatic electronic states of SM2 (red and blue curves). Fig. presents the NDC between them (green curve). The matrix elements of the dipole moment under the adiabatic representation (Eq. 31) of SM2 are presented in Fig. .