, where α and β label the spin orbitals as being occupied by spin-up and spindown electrons, respectively. There are three classes of matrix elements that contribute to the Hamiltonian matrix, and we we write each class of matrix elements after shifting the total Hamiltonian in Eq. 1 by E CQED-RHF . The matrix elements involving the CQED-RHF electronic Slater determinant |Φ 0 and photonic states |s and |t , where s,t ∈ {0, 1} involve only the (complex) photonic energy,