Note here that the matrix elements of R Ĥph are zero in the adiabatic-Fock basis and that the matrix elements of R Ĥel will lead to the usual electronic gradient of the adiabatic PESs, R E α (R), as well as the non-adiabatic couplings, d αβ (R) (Eq. 4). Additional terms arise in the nuclear force that are proportional to the gradient on the molecular dipole matrix elements R μ, and these terms may contribute large amounts to the nuclear force in regions of large light-matter coupling λ.