We will formulate QED-CASCI for both Eq. 1 and Eq. 5, which we will denote PN-QED-CASCI and CS-QED-CASCI to make reference to the photon number (PN) basis and coherent state (CS) basis for the photonic degrees of freedom, respectively. Although we will consider the reference states to be |R⟩ = |Φ e 0 ⟩ ⊗ |0 p ⟩ in both cases, we note there are two key differences between these formulations. The first difference between these formulations is that the reference for CS-QED-CASCI formally includes an infinite number of photon occupation states through