  Topological and geometrical methods are used to calculate the
pseudorotational part of the vibronic spectrum for an electronic triplet of an
octahedral, tetrahedral, or icosahedral molecule. The calculations take into
account the nontrivial geometry inherent in the Jahn-Teller effect. It is shown
that the Jahn-Teller effect gives rise to a geometry, which is related to the
isoparametric geometry of E. Cartan. The pseudorotational spectra correspond to
the spectra of connection Laplacians on nontrivial line bundles over base
spaces with this geometry. Globally, the isoparametric submanifolds form a
totally geodesic foliation of S^4 and the spectral flow of these connection
Laplacians on this foliation is computed.
