  Describing the dynamics of nuclei in molecules requires a potential energy
surface, which is traditionally provided by the Born-Oppenheimer or adiabatic
approximation. However, we also need to assign masses to the nuclei. There, the
Born-Oppenheimer picture does not account for the inertia of the electrons and
only bare nuclear masses are considered. Nowadays, experimental accuracy
challenges the theoretical predictions of rotational and vibrational spectra
and requires to include the participation of electrons in the internal motion
of the molecule. More than 80 years after the original work of Born and
Oppenheimer, this issue still is not solved in general. Here, we present a
theoretical and numerical framework to address this problem in a general and
rigorous way. Starting from the exact factorization of the electron-nuclear
wave function, we include electronic effects beyond the Born-Oppenheimer regime
in a perturbative way via position-dependent corrections to the bare nuclear
masses. This maintains an adiabatic-like point of view: the nuclear degrees of
freedom feel the presence of the electrons via a single potential energy
surface, whereas the inertia of electrons is accounted for and the total mass
of the system is recovered. This constitutes a general framework for describing
the mass acquired by slow degrees of freedom due to the inertia of light,
bounded particles. We illustrate it with a model of proton transfer, where the
light particle is the proton, and with corrections to the vibrational spectra
of molecules. Inclusion of the light particle inertia allows to gain orders of
magnitude in accuracy.
