Here, we have defined the "weighted" R-TDVP over the photonic wavefunction, i.e., A α (q, R,t) η = η(q,t)| A α (q, R,t) |η(q,t) q , where the the symbol • q indicates an integration over q, and the "weighted" q-TDVP over the nuclear wavefunction, i.e., S ν (q, R,t) ζ = ζ (R,t)| S ν (q, R,t) |ζ (R,t) R , where the the symbol • R stands for an integration over nuclear positions. In addition, the effect of the photons in Eq. ( ) is encoded in the photonic mean-field potential V P m f (R,t),