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The Journal of Chemical Physics

U q (4)⊇O q(4) description of diatomic molecules: Vibrational spectra

Raj K. Gupta, Ian L. Cooper

Published Feb 22, 1995• 7 citations
Publisher

Abstract

Quantum-deformed algebra (q-DA) of the O(4) dynamical symmetry of the U(4) group is studied in respect of the vibrational spectra of diatomic molecules. It is shown that, as in the case of the U(6) group for nuclei, the q-deformation parameter in Uq(4)⊇Oq(4) must also be complex for any meaningful gain in going from the classical to the q-deformed algebra. Application is made to the ground electronic state (X 1Σ+g) of the H2 molecule.

DOI
10.1063/1.468622

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Physical Review A•Dec 1, 1991•21 citations
Quantum-deformed algebra (q-DA) of the O(4) dynamical symmetry of the U(4) group is studied in respect of the vibrational spectra of diatomic molecules. It is shown that, as in the case of the U(6) group for nuclei, the q-deformation parameter in Uq(4)⊇Oq(4) must also be complex for any meaningful gain in going from the classical to the q-deformed algebra. Application is made to the ground electronic state (X 1Σ+g) of the H2 molecule.
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.
We suggest a model of q-deformed rotating oscillator which possesses the symmetry of quantum group and point out that this model can be used to describe the vibrational and rotational structures of many microscopic systems. We construct the representation of q-oscilla or algebra and give energy values of q-oscillaor in this representation. It is shown that the q-oscillator spectra can be used to describe infrared spectra, Raman spectra and vibrational structure of electronic transitions of diatomic molecules.
A deformed version of the vibron model for diatomic molecules is constructed. Both the O(4) and U(3) dynamical symmetries of the model are rewritten, using the concept of complementary subalgebras, in a more convenient form, which is subsequently deformed. The present model unifies the so far independent successful quantum-algebraic approaches to rotational and to vibrational spectra of diatomic molecules. In addition, the method can be used for the construction of deformed versions of the U(5) and O(6) limits of the interacting boson model of nuclear structure.
The SU q (1,1) quantum algebra is used for the description of vibrational spectra of diatomic molecules for which SU(1,1) is used in the classical approach. While in the classical case only the first two terms of the empirical Dunham expansion occur, in the quantum case all terms are obtained. The improved description of the empirical data is obtained with q being a phase (and not a real number), in agreement with the situation occurring in the q-rotor (having the symmetry SU q (2)) used for the description of rotational spectra of molecules and nuclei.
We suggest the quantum Heisenberg group Hq(J){\mathit{H}}_{\mathit{q}(\mathit{J})}Hq(J)​(4) as the dynamical symmetry group of a diatomic-molecule system. The deformation parameter q of the Hq(J){\mathit{H}}_{\mathit{q}(\mathit{J})}Hq(J)​(4) symmetry group is a function of J (the quantum number of the angular momentum of the rigid rotation) and characterizes the interaction between vibration and rotation. The vibrational and rotational structures of diatomic molecules are compactly expressed using the quantum-group-theoretic approach. Taylor expansions of the analytic formulas from this approach reproduce the results of the anharmonic-oscillator and the nonrigid-rotator models. For some randomly selected molecules in particular states, the parameters are computed to fit the phenomenological formulas to high accuracy.