  We reconsider the time-dependent Born-Oppenheimer theory with the goal to
carefully separate between the adiabatic decoupling of a given group of energy
bands from their orthogonal subspace and the semiclassics within the energy
bands. Band crossings are allowed and our results are local in the sense that
they hold up to the first time when a band crossing is encountered. The
adiabatic decoupling leads to an effective Schroedinger equation for the
nuclei, including contributions from the Berry connection.
