A local-normal description of vibrational excitations of pyramidal molecules in terms of Morse oscillators

A general description of vibrational excitations of pyramidal molecules in both local and normal representations is presented. This study is restricted to the case when no tunneling motion is allowed. The Hamiltonian is first written in terms of curvilinear internal coordinates. The Wilson's G matrix as well as the potential are expanded in terms of Morse variables, which allows the identification of a set of six Morse oscillators as zeroth-order Hamiltonian. An algebraic realization of the Hamiltonian is obtained by introducing a linear expansion of the coordinates and momenta in terms of creation and annihilation operators of Morse functions. This algebraic realization provides in natural form the representation of the Hamiltonian in terms of local interactions. The normal interactions are constructed by successive couplings of tensors defined as linear combinations of the ladder operators. The matrix transformation between the local and normal interactions is obtained for the complete Hamiltonian. This analysis provides the spectroscopic parameters in both local and normal schemes in explicit form as functions of the force constants and structure parameters. To exemplify, the analysis of the vibrational excitations of stibine and arsine is presented. Force constants as well as the corresponding x,K relations are given. A comparison with the results obtained using the U(ν+1) unitary group approach is included.