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Physical Review A

Hq (4) and SUq (2) quantum group symmetries in diatomic molecules

Zhe Chang, Hong Yan

Published Jun 1, 1991• 21 citations

Abstract

We develop a model with ${\mathit{H}}_{\mathit{q}}$(4) and ${\mathrm{SU}}_{\mathit{q}}$(2) quantum group symmetries to describe the vibrational and rotational spectra of diatomic molecules. The representations of ${\mathit{H}}_{\mathit{q}}$(4) and ${\mathrm{SU}}_{\mathit{q}}$(2) and solutions to the quantum systems of q oscillators and q rotators are provided. We emphasize that the energy spectra of the q oscillator and q rotator exhibit remarkably the properties of the infrared, vibrational, and rotational Raman spectra as well as vibrational and rotational structures of electronic transitions of the diatomic molecule. When the parameters are properly selected, the spectra of this model coincide with the observed spectra of diatomic molecules. It is suggested that ${\mathit{H}}_{\mathit{q}}$(4) \ensuremath{\bigotimes}${\mathrm{SU}}_{\mathit{q}}$(2) is an essential symmetry of vibrations and rotations in diatomic molecules.

DOI: 10.1103/physreva.43.6043

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We develop a model with

{\mathit{H}}_{\mathit{q}}

(4) and

{\mathrm{SU}}_{\mathit{q}}

(2) quantum group symmetries to describe the vibrational and rotational spectra of diatomic molecules. The representations of

{\mathit{H}}_{\mathit{q}}

(4) and

{\mathrm{SU}}_{\mathit{q}}

(2) and solutions to the quantum systems of q oscillators and q rotators are provided. We emphasize that the energy spectra of the q oscillator and q rotator exhibit remarkably the properties of the infrared, vibrational, and rotational Raman spectra as well as vibrational and rotational structures of electronic transitions of the diatomic molecule. When the parameters are properly selected, the spectra of this model coincide with the observed spectra of diatomic molecules. It is suggested that

{\mathit{H}}_{\mathit{q}}

(4) \ensuremath{\bigotimes}

{\mathrm{SU}}_{\mathit{q}}

(2) is an essential symmetry of vibrations and rotations in diatomic molecules.

We suggest the quantum Heisenberg group

{\mathit{H}}_{\mathit{q}(\mathit{J})}

(4) as the dynamical symmetry group of a diatomic-molecule system. The deformation parameter q of the

{\mathit{H}}_{\mathit{q}(\mathit{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.

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.

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.

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.

By means of geometric quantization, we discuss the quantum-deformed-symmetric-top model. We show that the quantum-deformed symmetric top and the quantum-standard symmetric top obey the same Heisenberg equations of motion. The quantum-deformed-symmetric-top model possesses quantum group symmetry; thus the quantum group can be dynamical symmetries of the symmetric-top system. The energy levels and wave functions of stationary states of the quantum-deformed-symmetric-top model are obtained exactly by a quantum-group-theoretic approach. Using the exactly solved model, we investigate the rotational spectra of symmetric-top molecules and diatomic molecules when the moment of inertia about the line joining the nuclei is not exactly zero.