On the Time-dependent Born-oppenheimer Approximation with Smooth Potential

We give a general reduction scheme for the study of the quantum propagator of molecular Schrr odinger operators with smooth potentials. This reduction is made up to innnitely (resp. exponentially) small error terms with respect to the inverse square root of the mass of the nuclei, depending on the C 1 (resp. analytic) smoothness of the interactions. Then we apply this result to the case when an electronic level is isolated from the rest of the spectrum of the electronic Hamiltonian.