where the right hand side of Eq. 6 defines a coherent state wavefunction for a photon. The second difference is that the electronic reference determinant is written in terms of canonical molecular orbitals for the PN-QED-CASCI formulation and in terms of QED-HF orbitals in the CS-QED-CASCI formulations. This latter difference represents a particular choice of a singleelectron orbital basis; there are many other valid choices of this orbital basis that can be explored in future work. One implication of this choice, however, is that the orbital basis of CS-QED-CASCI depends on the details of the cavity, while the orbital basis of PN-QED-CASCI does not.