The OH -anion is a charged model system represented within the 6-31G basis set with a bond length of 0.9 Angstroms coupled to a photon with frequency ℏω cav = 5.96 eV (0.219 Hartrees) polarized along the zaxis with λ z = 0.05 atomic units. We note that this field does not couple directly to a transition in the molecule; while there is a dipole allowed transition at 5.96 eV, in this coordinate system, it does not have a transition dipole moment along the polarization axis of the field. Therefore, the coupling occurs through the permanent dipole moment of the molecule. The number of electronic states used for pCS and pPF calculations are N el = 50 and we consider this molecule both with its center of mass located at the cavity origin and displaced by 20 Å from the cavity origin along the z axis, which aligns with the permanent dipole moment and the polarization axis of the field (see Figure ). Charged molecules have origin dependent dipoles and this property can leads to numerical difficulties wherein PF-FCI can show origindependent energies when the photonic Fock space is incomplete. Our motivation is to examine the behavior of the pPF and PF-FCI methods with systematically increasing photonic Fock spaces, and also to investigate if the pCS method is fully origin invariant just like the CS-FCI method. In Figure , we show the origin-dependent energy error for different levels of photonic Fock space truncation using PF-FCI-N p , pPF(50,N p ), CS-FCI-N p and pCS(50,N p ) methods all in the 6-31G basis set. In particular, we define the origin dependent error as the ground state energy of the displaced system minus the ground state energy of the system at the origin for the same level of truncation. We observe that both PF-FCI and pPF approaches show a remarkably similar origin dependence that converges at the same rate as the photonic Fock space is increased (see Figure . 12). Furthermore, we observe that the pCS and CS-FCI methods are robustly origin invariant (see Figure ).