Assuming a single-reference (i.e., single-determinant) wavefunction for the electronic DOFs |HF⟩ (taken as Hartree-Fock determinant) for the polaritonic ground state in the GCS basis (see Sec. 3.2.1), then the deviation in the molecular dipole ∆ μ = μ -⟨ μ⟩ GS=HF = 0 since ⟨HF| μ|HF⟩ = ⟨ μ⟩ GS=HF , effectively removing the bilinear coupling term in the GCS light-matter Hamiltonian (Eq. 29). However, the DSE term remains since ⟨ (∆ μ ) 2 ⟩ HF 0 even for the singledeterminant approximation. To be clear, for approaches with multiple determinants (i.e., the use of excited Slater determinants in a CIS-like expansion), the direct coupling term will survive. For a single-reference ground state, Eq. 29 can then be simplified to,