It is important to note that simply using the GSC transformation does not yield a fully origin invariant Fock matrix or single-particle orbital energies. Ref. 68 discusses such drawbacks and proposes the so-called strong coupling variant of the Hartree-Fock theory (SC-QED-HF), which transforms the Pauli-Fierz Hamiltonian in a similar way as the GSC rotation but now contains orbital specific rotations, similar to the polarized Fock state basis (see Sec. 3.1.2).