To go beyond the Born-Oppenheimer approximation 101 and to consider nonadiabatic, excited-state effects in the coupled dynamics of electrons and nuclei in molecular systems, the standard theoretical construction is the Born-Huang representation 102 of the molecular wavefunction . This is an exact representation of the electron-nuclear wavefunction as a linear combination of the so-called adiabatic states, i.e., the eigenstates of the electronic Hamiltonian. An alternative to the Born-Huang representation is the exact factorization, which expresses the electron-nuclear wavefunction as the product of a marginal nuclear amplitude and a conditional electronic amplitude, parametrically dependent on the nuclear configuration . The exact factorization can be extended to any multicomponent many-body wavefunction, since no assumption is made on the physical properties of the subsystems to identify the marginal and the conditional amplitudes, e.g., the small electron-nuclear mass ratio usually invoked in the Born-Oppenheimer approximation.