In the definitions of the TDVP and TDPES, we indicate with the symbol • r an integration over r, and we use the convention that such variable does not appear explicitly in the term in the bra-ket as it is integrated over. Note that the product form of the wavefunction Ψ(r, R, t) is invariant under the phase transformations Φ(r, t; R) = exp [(i/ )θ(R, t)]Φ(r, t; R) and χ(R, t) = exp [(-i/ )θ(R, t)]χ(R, t). Thus, under these transformations, the TDVP and TDPES transform as well, as standard gauge potentials, namely Ãν (R, t) = A ν (R, t) + ∇ ν θ(R, t) and ε(R, t) = ε(R, t) + ∂ t θ(R, t), and Eqs. (4) and ( ) are form-invariant.