Some considerations on the description of vibrational excitations in terms of U ( ν + 1) unitary groups

Abstract

An alternative formulation of the unitary group approach based on the dynamical algebra u ( ν + 1) to describe vibrational excitations of ν equivalent oscillators is proposed. Instead of providing the expansion of the Hamiltonian in terms of invariant Casimir operators associated with chains of groups, we introduce in addition to the bosonic operators a ˆ i † ( a ˆ i ) a set of ν operators b ˆ i † ( b ˆ i ) , in terms of which any dynamical variable can be expanded. The introduction of these operators has the advantage that in the harmonic limit the familiar creation and annihilation operators a ˆ i † ( a ˆ i ) for the harmonic oscillator are recovered. This approach allows to establish a one to one correspondence with the interactions in configuration space in the harmonic limit. In the framework of this formalism, a representation of the Hamiltonian in terms of both normal and local operators can be established from the outset. With the purpose of showing the advantages of our formulation the case of pyramidal molecules is illustrated in detail. We compare our analytical description to previous spectroscopic studies for the specific case of stibine.