Given the modifications to the polaritonic potential energy surfaces observed in Figure in the main text, it is interesting to consider if the ground-state potential energy surfaces can also be modified. While such modifications would be discernible from ground-state ab initio polaritonic structure methods, we suspect the onset would not be observed until the onset of ultra-strong coupling and perhaps still under special conditions like the field polarization being oriented relative to a permanent dipole moment in particular ways. Here we consider only the case of the equilibrium geometry of the ground-state of MgH+ computed at the RHF/cc-pVDZ, CQED-RHF/cc-pVDZ, and CQED-CIS/cc-pVDZ levels of theory with the same coupling parameters considered in the main text for this system, namely hω = 4.75 eV and λ = (0, 0, 0.0125 atomic units. The potential energy curves from each of these methods is shown in Figure , and we find from numerical location of the minima that the equilibrium bondlength predicted by all levels of theory is approximately 1.65 Angstroms.