and the dipole moment in the cavity-coupled case takes on an orientation dependence compared to the lone molecular case, the oscillator strengths in the cavity case would be modified in turn. Turning to the NH-CQED-CIS Hamiltonian, it can be seen that unlike the canonical CIS method, the NH-CQED-CIS method couples single excitations of the electronic and photonic terms to the CQED-RHF reference state. Specifically, it can be seen in Eq. ( ) that the CQED-RHF wavefunction can couple to states which involve singly-excited electronic configurations and singly-occupied photon states. This coupling can lower the energy of the lowest energy eigenstate of the NH-CQED-CIS Hamiltonian relative to the ground-state determined by the CQED-RHF method (see Table ). The cavity-induced modification to the symmetry of the CQED-RHF wavefunction has important consequences for which singly-excited configurations can contribute to the ground state. We examine the impacts of this cavity effect again with formaldehyde coupled to a lossless photon with ω = 0.382 atomic units (10.4 eV), which is approximately resonant with the first two dipole allowed transitions at the CIS/cc-pVDZ level of theory (see Ref. 52). We consider the same coupling magnitudes as before, this time focusing only on the λ z and λ yz cases. The polarization vector in the λ z case belongs to the A 1 irrep of the C 2v point group, while the polarization vector in the λ yz case belongs to the A irrep of the C s point group. In Figure , we present the dominant singlyexcited contributions to the NH-CQED-CIS ground-state in the λ z and λ yz cases. We indeed see slightly more permissive mixing of singly-excited configurations into the NH-CQED-CIS ground-state in the λ yz case due to the lower symmetry of the wavefunction. We report the changes in the ground-state energy as predicted by the NH-CQED-CIS method relative to the canonical RHF method and the CQED-RHF method in As a second illustrative example of the NH-CQED-CIS method, we consider the upper-polariton (|UP ) and lowerpolariton (|LP ) states that emerge from coupling MgH + to a photon resonant with the ground to first singlet excited state (|X → |A ) transition . We consider the photon to be polarized along the z axis, in alignment with the relevant transition dipole moment, with λ z = 0.0125 atomic units. This time, we allow the photon to have a complex energy where the imaginary part accounts for photonic dissipation, which can also be related to the energy uncertainty of the photonic mode . We consider the real part of the photon energy to be 4.75 eV, and the imaginary part to be either 0 eV or 0.45 eV. This value of λ leads to a coupling energy scale hg ≈ h ω 2 λ µ ≈ 0.1eV , which is comparable to the single molecule coupling strength reported by Baumberg and co-workers . Similarly, the dissipation energy scale of 0.45 eV corresponds to approximately a 10 fs lifetime, which is shorter but on a similar order to the lifetimes of the plasmonic resonances reported in Ref. 4. We specifically chose this value because it represents the γ = 4g threshold at which the Rabi splitting of polaritonic surfaces associated with strong coupling dissapears