The term ⟨ d⟩ = λ • ⟨ μ⟩ in Eq. 4 is computed from the expectation value of the molecular dipole moment which typically comes from a modified Hartree-Fock calculation that includes cavity effects, e.g. QED-HF. Applying this transformation with this choice of z to Eq. 1 yields the Pauli-Fierz Hamiltonian in the coherent state basis: