Construction and counting of the number operators of an $n$-degree-of-freedom normalized non-resonant vibrational Hamiltonian

TL;DR

This paper develops an algebraic method for constructing and counting the independent operators of a normalized vibrational Hamiltonian for multi-degree-of-freedom molecular systems, aiding spectroscopic analysis of strongly excited molecules.

Contribution

It introduces a systematic algebraic approach to construct and enumerate operators in vibrational Hamiltonians without resonance, applicable to complex molecular systems.

Findings

Method for constructing Hamiltonians using Lie algebra generators

Enumeration of independent operators up to order N

Application to a triatomic molecule near dissociation limit

Abstract

The present paper is the first of two articles aimed at constructing $n$-degree-of-freedom Hamiltonian systems by an algebraic approach. In molecular spectroscopy, the construction of vibrational Hamiltonian for strongly excited molecular systems by using an algebraic formalism requires the introduction by hand the operators describing the change in energy by numerous quanta and it is tedious to predict in advance the total number of operators appearing in the development. The goal of the two articles is not only to propose in the local limit a systematic method for constructing a normalized vibrational Hamiltonian for a strongly excited $n$-degree-of-freedom molecular system from the generators of the Lie algebra, the algebra of polynomial invariants, but also to enumerate the number of independent operators needed for the construction of the Hamiltonian developed in the base of these generators up to the given order $N$. The first article introduces the theoretical tools used in the both papers (section \ref{norm}), and presents the method of construction in case of absence of resonance (section \ref{const}). Finally, an application for a triatomic non-linear ClOH molecule is considered in case close to the dissociation limit. (section \ref{Appli}).