We will briefly overview the recent work to integrate the PF QED Hamiltonian (Eq. 55) into a variety of electronic structure methods to provide a selfconsistent solution to the ground and excited polaritonic states. 156 Note that in the previous section for the pQED approach we chose a basis for polaritons that cannot change in a variational sense, while for the self-consistent methods (Approach (II)), the basis is variationally updated to minimize the energy of the entire Hamiltonian. In this sense, the variational procedure may require a smaller number of overall electronic/photonic states than the pQED procedure; however, the scQED method requires knowledge of the lowlevel basis of the electronic system (e.g., atomic orbitals, plane waves, etc.) while the pQED method only requires the resulting many-body state energies and transition dipoles (i.e., as solved by CIS, TD-DFT, EOM-CC). In this way, the computational cost of the variational scheme becomes more complicated after adding the additional DOFs of the photonic sub-system to the low-level basis. In contrast to this, in the pQED procedure, the convergence of the basis becomes an important consideration due to the lack of response of the basis to the presence of the photon field. As is usually done, we first approach this problem by way of meanfield Hartree-Fock (HF) theory. Thus far, we are not aware of any theoretical work on scQED method that explicitly solves many molecules coupled to many cavity modes beyond long wavelength approximation, such as described by the ĤG d•E Hamiltonian in Eq. 101. Since this is highly relevant to the description of the actual molecule-cavity coupling in most of the experimental set up, future theoretical works should focus towards this direction to achieve a more direct comparison with experiments. Nevertheless, the on-going ab initio scQED approaches layout the ground work toward that goal.