In the previously discussed QED approach, one is required to compute the many-body electronic states { |ψ α ⟩ } for use in a direct diagonalization procedure of the Pauli-Fierz Hamiltonian (Eq. 18) coupled with some basis for the photonic DOFs (e.g., Fock/number states, polarized Fock states) to arrive at a description of the polaritonic states in these choices of basis. This method, in the infinite basis limit, provides the exact results for the polaritonic states, capturing the exact correlation between the electronic and photonic DOFs. The primary limitation of this approach is that the electronic basis converges very slowly for strong light-matter coupling strengths, requiring the calculation of highly excited electronic states (∼100) to converge even the lowest polaritonic energies. Contrary to this, the photonic basis only requires a relatively small number of basis states to converge for realistic, ab initio systems. Due to this limitation of the direct diagonalization approach, the community seeks other approximations in line with previous electronic structure approaches that now include the photonic interactions to some degree to arrive at a self-consistent description of the problem. In principle, this approach will require many fewer basis states than a single, direct diagonalization but with similar approximations for the electron-photon and photon-photon interactions that are intrinsic to the many-body method of choice. In this way, these approaches will be able to capture the cases where strong light-matter coupling is too strong to utilize the direct diagonalization approach with a frozen basis. We define these self-consistent analogues to the commonly used electronic structure approaches as scQED-X, where X can be any of the standard approaches, such as Hartree-Fock theory (HF), density functional theory (DFT), linear response time-dependent DFT (LR-TD-DFT), coupled cluster (CC), configuration interaction (CI), full configuration interaction (FCI), , second-order Møller-Plesset perturbation theory (MP2), and density matrix renormalization group (DMRG). 80