  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.
