Universite du Littoral, Dunkerque, France.

REORGANIZATION OF ENERGY BANDS IN MOLECULES.

WHAT FOLLOWS FROM SYMMETRY AND TOPOLOGY?

Abstract :

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).