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

TL;DR

This paper introduces a systematic method for constructing and counting the independent operators in a normalized vibrational Hamiltonian for molecules with p:q resonance, demonstrated on the ClOH molecule.

Contribution

It provides a novel systematic approach to build and count operators in vibrational Hamiltonians with p:q resonance, including coupling operators, using Lie algebra techniques.

Findings

Developed counting theorems for operators and parameters in the Hamiltonian.

Applied the method to the ClOH molecule with Fermi resonance, fitting 725 energy levels.

Reduced the Hamiltonian to 28 significant coefficients with minimal rms error.

Abstract

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.