The relative importance of different terms in the PF Hamiltonian and the consequences of ignoring them is illustrated in Fig. . For example, the dipole selfenergy (DSE) plays a crucial role in molecular polaritons to guarantee a bounded ground state and excited states, even though DSE is a constant in atomic polaritons and are dropped out in most of the atomic cavity QED models. This unphysical bending of polariton potential is demonstrated in Fig. . Without DSE, the gauge invariance between the minimal coupling Hamiltonian and the electric-dipole Hamiltonian will break down. Further, without the DSE, the Dipole (a.u.) ground state is no-longer bounded and becomes dissociative (and unbounded) at a large nuclear distance. The Rabi model, which explicitly ignores the presence of the permanent dipole, explicitly breaks down when electronic states have a large permanent dipole difference. Neglecting the permanent dipole, as commonly done for most of the current molecular cavity QED studies, can cause unphysical dips in the polariton potentials, 105 as demonstrated in Fig. . The JC model which assumes RWA, explicitly breaks down in the recently emerged ultra-strong coupling regime, and also gives unphysical dips of the potential (Fig ). Thus, one need to use the most rigorous Hamiltonian to describe the light-matter interactions and try to avoid unnecessary approximations.