is the bare one-particle electronic transition density between adiabatic states ψ α and ψ β and C a βn is the βn th expansion coefficient for the a th polariton (see Eq. 111), and N and N F indicates the total number of adiabatic states and Fock states, respectively. Further, ψ α (r; R) = ⟨r|ψ α (R)⟩ are the many-electron adiabatic states outside the cavity. Note that since the Pauli-Fierz Hamiltonian (Eq. 104) is purely real, the coefficients {C a αm , C b βn } are also real. In the matter-projected polaritonic transition density ρ M 0a , the photonic DOFs were traced out, leaving only linear combinations of electronic matrix elements of the same photon number (i.e., n = m). Using these sim-ple expressions, one can easily compute any polaritonic observables from the pQED scheme, relying on electronic quantities from widely available electronic structure codes as well as benefiting from the simplicity of the photonic sub-system.