Presence of energy bands in quantum energy spectra of molecules reflects the existence of “slow” and “fast” motions in corresponding classical problem. Generic qualitative modifications of energy bands under the variation of some strict or approximate integrals or motion considered as control parameters are analyzed within purely quantum description, within semi-quantum one (slow dynamical variables are classical; fast variables are quantum) and within purely classical one. In quantum approach the reorganization of bands is seen from the redistribution of energy levels between bands. In semi-quantum approach the system of bands is represented by a complex vector bundle with the base space being the classical phase space for slow variables. The topological invariants (Chern classes) of the bundle are related to the number of states in bands through Fedosov deformation quantization. In purely classical description the reorganization of energy bands is manifested through the presence of Hamiltonian monodromy.
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