Notably, the projected dipole self energy is missing the quadrupole terms that are present in the exact dipole self energy operator. Thus, in addition to the discretization error that arises from a finite orbital basis, the exact structure of the dipole self energy is lost upon projection onto an incomplete electronic subspace. We will see that this generally leads to a different variational problem whereby the pQED energies can converge to different solutions than the scQED approach.