In Eq. 14, E X ( µ X ) and E A ( µ A ) denote the ground-state and first singlet excited-state energies (dipole moments), respectively, and µ XA denotes the transition dipole moment between state X and A. The ground-state energies and dipole moments for each value of R are calculated at the RHF/cc-pVDZ level and the excited-state energies, dipole moments, and transition dipole moments are calculated at the CIS/cc-pVDZ level (see Ref. 53). The polaritonic surfaces obtained from diagonalizing Eq. ( ) are referred to as the 'Model LP' and 'Model UP' surfaces in Figure . We see with a pure real photon energy, the |LP and |UP surfaces experience a strong splitting in the region where the |X, 1 state (the ground-state plus a photon) crosses the |A, 0 state (the first excited-state without a photon). It should be noted that the NH-CQED-CIS curves are typically slightly stabilized compared to the Model LP and UP curves. We have already seen that the CQED-RHF method can provide orbital relaxation in the presence of cavity coupling that would not be available to the Model LP and UP solutions, and the NH-CQED-CIS wavefunction also includes additional variational flexibility through the excitations coupled through the Ĥdse and Ĥep terms. It is interesting to consider when such deviations between the NH-CQED-CIS and simpler model Hamiltonian's arise; we explore this question with the MgH+ system as a function of fundamental coupling strength λ in Appendix C.. For a strongly dissipative photon, we see that both the model and CQED-CIS curves closely approximate the CIS curves for the lone molecules, which signals that this system is not in the strong-coupling regime. The loss of strong coupling in this case arises because the dissipative energy scale of the photon is significant compared to the interaction energy scale between the photon and the molecular transition . This echos a fundamental condition for strong coupling that the interaction strength g must be large compared to the dissipation energy scale hγ, specifically g > h γ 4 . If this condition is not satisfied, then the energy splitting between the interacting states vanishes. The imaginary part of the frequency giving rise to the curves shown in the bottom panel of Figure was specifically chosen so that g < h γ 4 , leading to the vanishing of the splitting that is observed in the top panel of Figure . As a final illustrative example, we examine the ground-state photon population of the MgH+ system for coupling strengths approaching the ultra-strong coupling regime, defined when the coupling energy scale is commensurate with the bare excitations in the system, e.g. g ≈ Re( ω). We define the groundstate photon occupation as