In Figures , and 6, we show both ground state and polariton excitation energies of the HeH + system computed in all three basis sets with the pPF and PF-FCI approaches for varying values of coupling strength. The ground state energies are plotted relative to the uncoupled ground state, E g (λ) -E g (λ = 0). The excitation energies are defined as the energy of a polariton state at a given coupling strength minus the ground state energy at the same coupling strength, E pol (λ) -E g (λ). An interesting feature of this study is that the pPF (ground state and polariton) energies are consistently a lower bound on the PF-FCI energies, and each approaches corresponding PF-FCI energy from below as we approach the complete basis limit. We emphasize that because the projected dipole self-energy is inexact outside of the complete basis limit, pPF and PF-FCI approaches provide a variational approach for two distinct Hamiltonians, and so we cannot make any concrete arguments about which should provide a lower bound in general. We do observe in these cases, that pPF(60,10)/cc-pVDZ ground-state and polariton energies are visibly lower than the PF-FCI-10/cc-pVDZ energies (see Figure ). While the pPF(468,10)/cc-pVTZ and pPF(2880,10)/cc-pVQZ also provide ground state and polariton energies below their PF-FCI-10 counterparts, the difference is not visibly discernible (see Figures and). We also note that progression of the Rabi splitting with increasing coupling strength is very comparable between the pPF and PF-FCI results for all basis sets. The pPF(60,10)/cc-pVDZ result underestimates the Rabi splitting by about 17 meV (∼ 6 • 10 -4 Hartrees) relative to PF-FCI-10/cc-pVDZ, whereas the pPF(2880,10)/cc-pVQZ result underestimates the splitting by less than 5 meV (∼ 2 • 10 -4 Hartrees) relative to PF-FCI-10/cc-pVQZ. We expect that both methods should agree quite closely in the Rabi splitting because this feature of the polariton states is dominated by the bilinear coupling, and the projected bilinear coupling remains the same form as it's exact counterpart. However, the systematic convergence of the Rabi splitting with increasing basis set size is suggestive of the fact that the dipole self energy can play a quantitative role in determining the energies of the polariton states, and the error in the projected dipole self energy does not exactly cancel when taking the difference between the polariton energies. We should also note that the errors in the Rabi splitting could be sensitive to the state truncation for the pPF methods that were chosen based on the converngence of the ground state, not the polariton states. Now that we have shown how the disparity between the pQED and scQED resolves in the complete basis set limit, we can ask a pragmatic question of which requires less computational effort to converge. For this comparison, we use the same direct CI algorithm for the scQED energies as we do for for the N el energies and dipole moments required for the pQED results. Although we are typically only interested in a few low-energy eigenstates of Eq. 1 or Eq. 2, as we saw in Figure , one often needs to project the pPF Hamiltonian on many more states than the number of eigenstates that are of interest when the coupling is strong. Because the scQED approach solves Eq. 1 or Eq. 2 directly in a coupled basis, we need only In Table , we report the timings to solve for N ep = 10 coupled states of HeH + under the same conditions as in Figure using PF-FCI-10/cc-pVXZ compared to the time required to solve for N el = 40, 312, and 1296 states at the FCI/cc-pVDZ, FCI/cc-pVTZ, and FCI/cc-pVQZ levels. We choose these N el as 40% of the total roots of the FCI matrix at each basis set, which is approaching the upper limit of what can be performed with our direct CI approach. We note that for the study in Figure , we performed full diagonalization of each FCI matrix, not direct CI; this is because the iterative eigensolver used in the direct CI approach generally becomes unsuitable for finding 50% or more of the total eigenvalues of a given matrix. We see that in the cc-pVDZ basis set, it is faster to compute the lowest 40 roots of the FCI matrix than to compute the lowest 10 coupled states. However, the time FIG. . Relative energy of the ground-state and excitation energies of polariton states of HHe + as a function of coupling strength computed at the PF-FCI-10/cc-pVQZ and pPF(2880,10)/cc-pVQZ levels required for finding the FCI roots increases by roughly 2 orders of magnitude as we progress from cc-pVDZ to cc-pVTZ and again to cc-pVQZ, whereas the time to solve for the lowest 10 coupled PF-FCI roots increases by only 1 order of magnitude for the same progression. Thus, we see it is approximately 10x faster to find the lowest 10 roots with PF-FCI-10/cc-pVTZ as compared to finding 312 FCI/cc-pVTZ roots, and it is approximately 100x faster to find the lowest 10 roots with PF-FCI-10/cc-pVQZ than to find the lowest 1296 FCI/cc-pVQZ roots. This scaling is suggestive that it will generally be more computationally facile to directly solve for a small number coupled states in a scQED approach than to solve for a large number of uncoupled states to perform subsequent pQED calculations for the same system size. However, as is implied by Fig 3, state truncation does not yield a significant error when the coupling is relatively small, and so there may be many practical cases of moderate coupling when pQED can be more computationally facile. The LiH molecule provides a 4-electron system that has a permanent ground-state dipole moment and a dipoleallowed optical transition (S 0 → S 1 with transition energy of 3.29 eV) refer Figure . Although we can no longer afford to perform full diagonalization to obtain all FCI many-electron states for the cc-pVXZ series, we can perform FCI (and QED-FCI) in a split-valence triple zeta basis set (6-311G) and obtain hundreds of manyelectron states. We will use this system to illustrate the behavior of the pQED and scQED methods in incomplete orbital and many-electron basis limits under strong coupling with λ = (0, 0, 0.05) atomic units.