Aiming to develop on-the-fly procedures to solve the exactfactorization equations by exploiting quantum chemistry, the electronic wavefunction is expanded in the adiabatic basis, as shown in Eq. (11). Therefore, the expression Φ(r,t; R) = ∑ l C l (R,t)ϕ l (r; R) is inserted in Eq. ( ) to derive a set of evolution equations for the expansion coefficients C l (R,t). Following from the above observations, when the idea of trajectories is introduced, the electronic equation ( ) is affected as well, and the evolution of the coefficients is ultimately expressed as a total time-derivative Ċl (R α (t),t) = Ċα l (t). The key quantity arising in such a quantum-classical formulation of the electron-nuclear dynamics is the so-called quantum momentum. In Eq. ( ), the operator Û[Φ, χ] depends on the nuclear wavefunction, and, when it is expressed in polar form in terms of its modulus |χ| (or |χ| 2 ) and phase S, one gets (14) where the term in square brackets is the classical momentum P ν , that depends on the TDVP as anticipated above, and it is summed to the quantum momentum P ν , which appears as a purely imaginary correction. The quantum momentum encloses information about the spatial delocalization of the nuclear density, and this information has to be recovered even when trajectories are used to mimic the nuclear dynamics. Therefore, the non-local character of the quantum nuclear density is encoded in the quantum momentum, that is why the algorithms derived from the exact factorization require either a coupled-trajectory scheme or an auxiliarytrajectory scheme .