Often, ⟨d ˆ⟩ is computed for a given electronic reference state in, for example, the QED-HF reference in scQED formulations. Here we will also investigate the CS transformation of the projected Pauli-Fierz Hamiltonian for a pQED formulation, where ⟨d ˆ⟩ will be computed in the adiabatic many-electron basis. We note that within the Born-Oppenheimer approximation, the nuclear contribution in ⟨d ˆ⟩ exactly cancels with the nuclear contribution to d ˆ, hence we write Eq. 2 with d ˆe and ⟨d ˆe⟩ to denote only the electronic contribution to both terms. We can approach the variational solution to Eq. 1 or Eq. 2 in two complementary ways. The approach denoted pQED will first find the adiabatic eigenstates that define H ˆe|ψ α (R)⟩ = E α (R)|ψ α (R)⟩ using standard quantum chemistry tools, where R denotes the coordinates of the nuclei which are fixed within the Born-Oppenheimer approximation. In a subsequent step, one builds a Hamiltonian matrix from Eq. 1 or Eq. 2 in the basis of direct products between these adiabatic eigenstates and photonic Fock states. The approach denoted scQED will seek the eigenstates of Eq. 1 or Eq. 2 directly in a product basis of many-electron states (Slater determinants in this work) and photonic Fock states.