Thus, using the ab-initio QED approach outlined in Sec. 3, one can obtain |Ψ a (R(t))⟩ and directly solve Eq. 156 using the propagation of R(t). Note that a similar expression can be obtained for the c αn (t) when using the adiabatic-Fock representation instead. Note that the derivative couplings in this basis (adiabatic-Fock) are sparse since ⟨n|∇ R |m⟩ = 0, as the photonic Fock states have no dependence on the nuclear coordinates unlike the electronic adiabatic ones 23). This is not true for the generalized coherent state (GCS) 148 or polarized Fock state (PFS) bases, which intrinsically entangle the electronic and photonic DOFs. Note here that the adiabatic polaritonic states can be obtained through any of the excited state scQED schemes discussed in Sec. 3.2 while the adiabatic electronic and photonic basis can be computed from bare electronic structure calculations dressed with a photonic basis via the pQED scheme (see Sec. 3.1).