The canonical picture for the JC cavity QED is that in a molecule-cavity hybrid system, the cavity photons dress the molecular energy level by a photon energy, e.g., |g0 to |g1 state, which can be resonant with the |e0 state and hybridize with it. The transition dipole µ eg causes the mix between |e0 and |g1 states (through the light-matter coupling term in Eq. 9). The NDC between polaritonic PES d 12 (Eq. 51) will be large due to the near degeneracy of the polariton states. When the NDC from other states are smaller than d 12 (Eq. 51), the polariton dynamics is largely confined within the Hilbert subspace {|e0 , |g1 } states. However, this approximation will breakdown under several scenarios, and we just enumerate two commonly encountered ones which we have explored in our numerical results. First, the electronic NDC d eg (Eq. 18) between |e0 and |g0 could play a important role in the dynamics (as shown in Fig. ) and could have a large contribution in the [X] expression (Eq. 39). Note that the effect of electronic NDC d eg is not included in the JC subspace (see Eq. 33). Correctly including the intrinsic NDC is essential to investigate molecular cavity QED. The second scenario is when molecules have large permanent dipole moments. In our case, both SM1 and SM2 have a large permanent dipole µ ee which causes the coupling between |e0 and |e1 through the light-matter coupling in Eq. 5, and this coupling is proportional to the light-matter coupling strength g c . Thus, with an increasing g c , the |e1 state could get a larger population (as shown in Fig. ). Unlike the atomic cavity QED, the dynamical interplay among the electronic, photonic, and nuclear DOFs are often more complicated, thus the truncation to any Hilbert subspace must be done with caution. V. CONCLUSIONS