Trajectory-based schemes have also been derived from the exact factorization , which is essentially a rewriting of the molecular Schrödinger equation in terms of coupled electronic and nuclear equations. This formalism introduces the new concepts of time-dependent vector potential and timedependent scalar potential, or time-dependent potential energy surface . They govern the nuclear quantum-mechanical evolution and encode the full dynamical effect of the electrons, including nonadiabatic effects . In turn, the electron-nuclear coupling is explicitly given in terms of the spatial variation of the nuclear wavefunction . Evolving classicallike nuclei according to the force determined from the time-dependent potentials and reconstructing the nuclear distribution from an ensemble of trajectories yields the coupled-trajectory mixed quantum-classical (CT-MQC) algorithm . Simplification of this numerical procedure have been proposed, by combining the adiabatic evolution of the trajectories and the hopping idea of TSH with the reconstruction of the nuclear distribution, either by using coupled trajectories, i.e., CT-TSH , or by using auxiliary trajectories, i.e., SHXF .