The Born-Oppenheimer approximation can be used when a quantum system is coupled to a comparatively slower quantum system, which can then be treated classically. For example, this is the case of the vibrationnal-rotationnal coupling in some small molecules. This situation has a clear manifestation in the exact energy spectrum, which possess groups of energy levels (so called energy bands).

Using a semi-classical approach, we show that the exact number of energy levels in each band can be expressed with the Atiyah-Singer index formula. This formula depends on topological indices related to a vector bundle, and can be computed within the Born-Oppenheimer approximation. The modification of these topological indices is related to degeneracy manifolds, and level exchange between consecutive bands. All this will be explained on simple examples.