A novel connection between algebraic spectroscopic parameters and force constants in the description of vibrational excitations of linear triatomic molecules

Abstract

A connection between an algebraic approach to the dynamics of triatomic molecules based on the U (2) × U (3) × U (2) Lie algebra and the traditional description in configuration space is presented. The connection is established in four steps. First, the molecular Hamiltonian is expanded in symmetrized local coordinates. Second, the Hamiltonian is transformed into an algebraic representation by introducing the realization of coordinates and momenta in terms of bosonic creation and annihilation operators of normal character. The third step is to perform a canonical transformation applied to the bosons associated with the stretching degrees of freedom in order to obtain a unified representation in a local scheme. Finally, an anharmonization procedure is applied to identify the U (2) × U (3) × U (2) dynamical algebra. The main advantage of the proposed approach is that it provides relations between the spectroscopic parameters and the molecular structure and force constants. As an application, the analysis of the vibrational excitations of CO 2 in its ground electronic state is considered. In this scheme, each stretching degree of freedom is identified as an interacting Morse oscillator, with an associated U (2) dynamical algebra, and the doubly degenerate bending degree of freedom is modelled with a U (3) dynamical algebra, obtaining as a final result a reasonable set of force constants.

A study of the Raman spectrum of CO2 using an algebraic approach

“…The CO 2 molecule has a strong normal-mode behavior and an intense Fermi resonance. Due to its normal-mode character, its direct description from a local-mode scheme is not correct [1]. This issue can be overcome introducing the canonical transformation…”

“…A convenient improvement is achieved with the introduction of the anharmonization procedure [1,2,10] …”

“…but now the operators b † ± (b ± ) are generators of a U (3) dynamical algebra [13], with matrix elements [1] …”

“…An algebraic description of the vibrational excitations of carbon dioxide (CO 2 ) in its ground electronic state has been recently reported [1,2]. The novel contribution of that work is the use of a polyad-preserving local-mode model to describe a molecule with a strong normal-mode behavior, taking advantage of the inclusion of anharmonic effects from the outset [3].…”

“…In Refs. [1,2] a set of 101 experimental vibrational energies were fitted with a root mean square deviation of 0.53 cm −1 . This description can be considered accurate for the standards of vibrational spectroscopy.…”