XXII WORKSHOP ON GEOMETRIC METHODS IN PHYSICS
29 JUNE - 5 JULY 2003, BIAŁOWIEŻA, POLAND
Frederic Faure - Topological indices in molecular spectra
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.