Analytic perturbative classification and assignment of eigenstates of algebraic vibrational Hamiltonians

We show how a classification and assignment for highly excited molecular vibrational states obtained previously by wave function inspection can also be obtained by analytic perturbative methods. This gives additional insight into the meaning of the classification. In addition it allows one to predict beforehand in which region of the spectrum which class of states is to be expected. As examples we present the class of states in DCO which are based on the coupling of the bend with the DC stretch, the class of states of DCO without any coupling and the bend vibrational states of acetylene at both ends of the spectrum within a given polyad. Thereby we learn the exact definition and meaning of the transverse quantum numbers. As a byproduct we get for the nonintegrable original Hamiltonian, integrable approximations which are valid in a restricted region of the phase space classically or for one class of states quantum mechanically.