Application of U(2)⊃O(2) and U(2)⊃U(1) dynamical symmetries to diatomic molecular vibrations

Abstract

Algebraic Hamiltonians based on the Casimir invariants of U(1) and O(2) dynamical symmetries of U(2) are used to describe the vibrational energy level spectrum of a diatomic molecule. The set of vibrational eigenstates arising from the U(1) chain remains closed under the operation of the appropriate ladder operators, whereas no such closure occurs in the case of the O(2) chain. This result is shown to be a consequence of the differing roles of the dimension of the irreducible representation of U(2), the so-called vibron number, in the two dynamical symmetry chains. A simple procedure is proposed to effect closure in the O(2) chain, as required for the eigenstates of a Morse oscillator Hamiltonian, to which the algebraic Hamiltonian based on the quadratic Casimir invariant of O(2) reduces within a coordinate representation. For each symmetry chain, application of the Holstein-Primakoff transformation leads to a set of ladder operators that act on one-dimensional harmonic oscillator eigenstates. Explicit comparison of the two dynamical symmetry chains is made with respect to the vibrational energy spectrum of the hydrogen molecule in its ground electronic state.