Algebraic Methods Applied To Highly Excited Vibrational Spectrum In Molecules

The development and the accuracy of measurement in modern spectroscopy have motivated the rapid progress in atomic and molecular spectrum. Many theoretical models have been introduced and developed for studying atomic and molecular spectrum, including ab initio calculatins, Dunham expansion, normal-mode model, local-mode model, and so on. In recent years, the local-mode model has been further developed not only for molecular vibrations but also for rotations. That model can be realized in terms of algebraic schemes. Algebraic methods can be employed to describe the collision between atoms and molecules, electronic spectrum in molecules, and excitations of atomic chain in one dimension. Among those algebraic methods, U(2) model is more typical one who considers vibrations and rotations separately, and describes vibrations of single bond in molecules. Thus U(2) model is extensively applied.In this thesis, we employ two kinds of U(2) model to molecular vibrations. Firstly, we use the unvaried operators of SO(2) to construct Hamiltonian for stretching vibrations in XY4 molecules, which has an advantage that its matrix is directly given. The Hamilton matrix is calculated in terms of symmetrized bases. The model is applied to the experimental spectrum of ¹²⁰SnD₄ . Three model parameters are determined via a least-squares fit with the smallest standard deviation between the experimental and the calculated values. Secondly, we take bending vibrations into account within U(2) model. We directly use creation and annihilation operators of U(2) to construct Hamiltonian for bend-CP and stretch vibrations in HCP, where Fermi resonances between the stretch and the bend are considered. The model has advantage that it can be reduced to other model in a limit. Fitting the experimental data shows that the standard deviation in the U(2) model is smaller than that in the reduced model, which indicates that the U(2) model is more suitable for highly excited vibrations in HCP. With our calculations and other studies, we can conclude that U(2) model becomes a potential effective model for molecules, especially for polyatomic molecules and highly excited states, where the anharmonic interactions between vibrations are dominant. Finally, we discuss other possible applications of algebraic methods to investigate other properties in molecules.