Presence of energy bands in quantum energy spectra of molecules reflects
the existence of ``slow`` and ``fast`` motions in corresponding classical
problem. I will discuss generic qualitative modifications of energy bands
under the variation of some strict or approximate integrals or motion
considered as control parameters. I compare purely quantum description
with semi-quantum one (slow dynamical variables are classical; fast
variables are quantum) and with 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.

As preliminary example I will treat rotational bands (one slow degree
of freedom, classical phase space is two-dimensional sphere).
Generic modification of bands consists in this case in the redistribution of
only one energy level between two bands.

Problem with two slow variables (vibrational polyads of triply degenerate
oscillator, classical phase space is complex projective space CP2)
shows nontrivial rearrangement in the case of presence of three quantum
(fast, electronic) states. The modification consists in rearrangement of
three bands into two new bands. One of these new bands is in fact a pair
of topologically coupled initial bands with nontrivial second Chern class
which cannot be decomposed by any perturbation into two bands without mixing
with an extra band.

In conclusion I will formulate some tentative conjectures about
relation between topological invariants in semi-quantum model,
generating functions giving the numbers of states in bands for quantum
problems, and Hamiltonian monodromy and singularities of toric
fibrations for corresponding purely classical problems.

Relevant references:
Europhys.Lett. 6, 573-578 (1988); J.Mol.Spectrosc. 163, 326-338 (1994);
J.Mol.Spectrosc. 169, 1-17 (1995); Phys.Lett. A 256, 235-244 (1999);
Phys.Rev.Lett. 85, 960-963 (2000); Lett.Math.Phys. 55, 219-238 (2001);
Phys.Rev.A 65,012105 (2001); Phys.Lett.A 302, 242-252 (2002);
Acta Appl.Math. 70, 265-282 (2002); Phys.Rep. 341, 11-84, 85-171 (2001).