In this work, we use the Shin-Metiu (SM) model as the "ab-initio" model molecular system to investigate strong and ultra-strong light-matter interactions between a molecule and an optical cavity. We choose two different parameter sets of the SM model, and hereafter we refer to them as SM1 and SM2, respectively. The SM1 is the model-I documented in Ref. 47. The distance between two fixed ions is L = 18.897 a.u., and the cut-off for the modified Coulomb interaction R c = 2.8345 a.u. Fig. presents the two lowest adiabatic electronic states (defined in Eq. 3) of SM1 model (red and blue curves), and Fig. presents the NDC between them (green curve). The energy gap at R = 0 a.u. between these two states is about 1.281 eV. The NDC is in the order of 0.2 a.u., and has a peak at R = 0 a.u. as well. The matrix elements of the dipole moment under the adiabatic representation (Eq. 31) of SM1 are presented in Fig. . Note that in some nuclear configurations, the magnitude of the permanent dipoles (µ gg and µ ee ) could be even larger than the transition dipole µ eg , making the atomic JC model invalid. This can significantly change the polaritonic PES when the light-matter coupling is strong enough and cause some strange dynamical behaviors as we will discuss later.