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

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