Furthermore, one should note that this choice of Z (Eq. 28) or the case of SC-QED-HF in Ref. 68 only applies in the limit of infinite light-matter coupling. Recent works have performed a variational optimization procedure on the shift parameter Z, e.g., choosing Z OPT under the condition ∂ E (R) ∂Z = 0, which further minimizes the energy of the ground state. This minimization is done on top of the self-consistent iterations and can be interpreted as an additional DOF for the self-consistent procedure. Given this rotation of the Hamiltonian, both the DSE as well as the bilinear interaction term would remain, even for single-determinant approaches, but would be scaled (or partially shifted away) by some optimal factor proportional to ∆λ = λ -Z OPT and (∆λ) 2 , respectively. This partial transformation can be interpreted as a rotation that pushes the maximal amount of complicated many-body electron-photon and photon-mediated electron-electron correlations into the mean-field solution of the QED problem.