Construction and counting of the number of operators included in a normalized vibrational Hamiltonian with n degrees of freedom with a p:q resonance

We propose a method of construction of a normalized vibrational Hamiltonian of a highly excited molecular system with $n$ degrees of freedom in the case of a a $p:q$ resonance. We present also the counting of all the independent operators and the counting of all the parameters included in the Hamiltonian (Counting theorems 1 to 8). The method introduces, on a systematic way, all the operators, in particular the coupling operators, that can be built from the polynomials formed by products of powers of the generators of a Lie algebra: the algebra of the invariant polynomials built in classical mechanics from the the kernel $Ker \,ad_{\mathcal{H}_{0}}$ of the adjoint operator $ad_{\mathcal{H}_{0}}$ (see [6] or [4],[5]). Application to the non-linear triatomic molecule ClOH is then given, taking into account the Fermi resonance between the O-Cl stretching oscillators and the bending motion. The study of this molecular system in highly excited vibrational states (until almost the dissociation limit) has been realized in [2], with a fit of 725 levels of energy. On the 86 coefficients (among which 31 coupling coefficients) that we count, and completely compatible with [2], the smallest rms value leads to keep only 28 non-zero coefficients. In the appendix, we explain the vocabulary and the strategy employed in order to demonstrate the theorems of coupling operators included in the Hamiltonian.