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arXiv:quant-ph/9701025 · The Journal of Chemical Physics

Coupled Q-oscillators as a model for vibrations of polyatomic molecules

Dennis Bonatsos, C. Daskaloyannis, P. Kolokotronis

Published Jan 8, 1997• 30 citations
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Abstract

The system of two Q-deformed oscillators coupled so that the total Hamiltonian has the suQ(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 Q-oscillators by deformed anharmonic oscillators. The generalization to n oscillators is straightforward. The applicability of the formalism to highly symmetric polyatomic molecules is discussed.

DOI
10.1063/1.473189

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The system of two Q-deformed oscillators coupled so that the total Hamiltonian has the suQ(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 Q-oscillators by deformed anharmonic oscillators. The generalization to n oscillators is straightforward. The applicability of the formalism to highly symmetric polyatomic molecules is discussed.
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.
A generalized deformed oscillator giving the same spectrum as the Morse potential is constructed through the use of quantum-algebraic techniques. The model of n coupled anharmonic oscillators of Iachello and Oss [Phys. Rev. Lett. 66, 2976 (1991)], suitable for the description of vibrational spectra of polyatomic molecules, is subsequently written in terms of such generalized deformed oscillators. In addition to clarifying the relation of the model of Iachello and Oss to other models using coupled oscillators for the description of vibrational molecular spectra, the present formalism allows for the construction of a large class of exactly soluble models with no extra computational effort. As an example, the way of including a coupling of the Darling-Dennison type is shown.
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.
By introducing appropriately generalized annihilation and creation operators for oscillators interacting by Morse potentials and generalized coherent states, we have derived the quantal equations of motion for displacements in a one-dimensional chain of Morse oscillators. The equations of motion derived have the same mathematical structures as their classical analogs which have been studied before. Since they are essentially identical in structure, it is possible to infer from the classical studies that the quantal system also would exhibit solitary waves and solitons. It is pointed out in the paper that the formalism used here for anharmonic systems should also be useful for other fields of study related to high amplitude vibrational motions of polyatomic molecules for which the usual normal mode analysis is not suitable.