By Parshin Shafarevich

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1. I. Zhilinskin& / Physics Reports 341 (2001) 85}171 energy levels into rotational multiplets associated with individual vibrational components we start with more formal de"nition of such a decomposition. We "rst remark that all rotational levels of the group of vibrational states which span the vibrational representation of the symmetry group G at a given J form a reducible representa tion of the group G which can be written symbolically as (36) " ;D (E , where D (E stands for the decomposition of the irreducible representation (J ) of the O(3) group E with respect to the symmetry group G of the problem.

Thus, the number of vibrational degrees of freedom which are to be studied in molecular models simultaneously may vary considerably from molecule to molecule and surely depends on the accuracy needed. The dimension of the phase space in any case is twice the number of degrees of freedom. The presence of resonance condition between harmonic frequencies ensures the formation of a group of quasi-degenerate vibrational levels, the so-called vibrational polyads. In the simplest case of the vibrational problem with K quasi-degenerate modes a natural possibility is to introduce an approximate integral of motion associated with the total number of vibrational quanta.

1) N(N, K)" N! 1)! (49) (50) or in the form of a generating function depending on an auxiliary variable 1 . g ( )" ) (1! )) (51) To obtain N(N, K) from the generating function g ( ) we expand the latter in the power series ) g ( )"C #C #C #2#C ,#2 . (52) ) , The coe$cient before , gives the number of states in the polyad with polyad quantum number N, C "N(N, K) . (53) , The two alternative representations of N(N, K) in Eqs. (49) and (50) are equivalent; the form in Eq. 1). The generalization to the system of harmonic oscillators with the resonance condition : : 2 : "d : d : 2 : d is straightforward.

### Algebraic Geometry 3 Curves Jaobians by Parshin Shafarevich

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