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Physics of Atomic Nuclei

Algebraic approach to thermodynamic properties of diatomic molecules

Maia N Angelova, Alejandro Frank

Published Oct 1, 2005• 15 citations
Publisher

Abstract

A simple model extending Lie algebraic techniques is applied to the analysis of thermodynamic vibrational properties of diatomic molecules. Local anharmonic effects are described by means of a Morse-like potential and the corresponding anharmonic bosons are associated with the SU (2) algebra. The total number of anharmonic bosons, fixed by the potential shape, is determined for a large number of diatomic molecules. A vibrational high-temperature partition function and the related thermodynamic functions are derived and studied in terms of the parameters of the model. The idea of a critical temperature is introduced in relation to the specific heat. A physical interpretation in terms of a quantum deformation associated with the model is given.

DOI
10.1134/1.2121908

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A simple model extending Lie algebraic techniques is applied to the analysis of thermodynamic vibrational properties of diatomic molecules. Local anharmonic effects are described by means of a Morse-like potential and the corresponding anharmonic bosons are associated with the SU (2) algebra. The total number of anharmonic bosons, fixed by the potential shape, is determined for a large number of diatomic molecules. A vibrational high-temperature partition function and the related thermodynamic functions are derived and studied in terms of the parameters of the model. The idea of a critical temperature is introduced in relation to the specific heat. A physical interpretation in terms of a quantum deformation associated with the model is given.
In this paper, we consider the one-dimensional anharmonic oscillator, which represents well the anharmonic vibrations in diatomic molecules. For the description of the associate potential we use the Morse potential, which gives a good approximation of the experimentally observed vibrational modes of molecules and hence contributes to the realistic description of the spectrum of diatomic molecules. The generalized and Gaussian coherent states are thus constructed and compared in terms of the localization of the particle in those states. We apply these results to the example of the sodium chloride molecule, 1 H 35 Cl.
We model an anharmonic diatomic molecule using deformed creation and annihilation operators such that the energy spectrum generated by a Hamiltonian of the harmonic oscillator's form written in terms of deformed operators is similar to that of a Morse potential. We construct an approximate time evolution operator and evaluate transition probabilities which are compared with those obtained by an expansion in a basis of Morse eigenfunctions. The algebraic results compare favorably with the numerical results.
q-deformation of the U(1) dynamical symmetry of U(2) group is introduced for the first time for vibrations of diatomic molecules. The question of pure real and pure imaginary value of q-deformation parameter, often raised in the literature, is found to be related to the particular choice of the constants of the model. The complex q-deformation parameter is allowed, with pure real and pure imaginary values as its special cases. Compared to the other dynamical symmetry Oq(2) of Uq(2), the result of Uq(1) of Uq(2) remains better even when the vibron number in Oq(2) is varied. The role of the vibron number in the two dynamical symmetries of U(2) is shown to be different. In Uq(1) symmetry, the vibron number is fixed by the number of observed states of the system since Uq(1) represents the q-deformation of a truncated anharmonic oscillator. A possible physical meaning to the q-deformation parameter is suggested in terms of higher order anharmonicities of the truncated anharmonic oscillator.
The system of two QQQ-deformed oscillators coupled so that the total Hamiltonian has the suQ_QQ​(2) symmetry is proved to be equivalent, to lowest order approximation, to a system of two identical Morse oscillators coupled by the cross-anharmonicity usually used empirically in describing vibrational spectra of triatomic molecules. The symmetry also imposes a connection between the self-anharmonicity of the Morse oscillators and the cross-anharmonicity strength, which can be removed by replacing the QQQ-oscillators by deformed anharmonic oscillators. The generalization to nnn oscillators is straightforward. The applicability of the formalism to highly symmetric polyatomic molecules is discussed.
We show that a direct connection can be drawn, based on fundamental quantum principles, between the Morse potential, extensively used as an empirical description for the atomic interaction in diatomic molecules, and the harmonic potential. This is conceptually achieved here through a non-additive translation operator, whose action leads to a perfect equivalence between the quantum harmonic oscillator in deformed space and the quantum Morse oscillator in regular space. In this way, our theoretical approach provides a distinctive first principle rationale for anharmonicity, therefore revealing a possible quantum origin for several related properties as, for example, the dissociation energy of diatomic molecules and the deformation of cubic metals.