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Simple analytical formulas for the number of energy levels in the vibrational polyads are given. These formulas account for the resonances between the vibrational modes, and for the symmetry of the problem, so that the number of states of a particular symmetry type can be computed. The formulas are used to estimate the differential and integral densities of states from the minimum initial information about the molecule. Examples of the vibrational structure of triatomic molecules A3, tetrahedral molecules AB4, and linear molecules AB2 are considered. The analytical formulas are compared to the ab initio results for H3+ [J. R. Henderson et al., J. Chem. Phys. 98, 7191 (1993)]. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The structure of bending overtones of the H3+ and D3+ molecular ions at the energies below the barrier to linearity is analyzed using energies and wave functions from full three-dimensional discrete variable representation calculations. The lowest-in-energy states of the vibrational polyads ν2=4,5,6 are shown to follow the localization pattern of local bending modes, three equivalent-by-symmetry principal periodic trajectories of the corresponding classical two-mode system near the equilibrium. [ABSTRACT FROM AUTHOR]

Intra- and interpolyad structures are investigated in the vibrational energy pattern of acetylene, using the spectroscopic Hamiltonian presented in previous papers in this series [see El Idrissi et al., J. Chem. Phys. 110, 2074 (1999)]. The existence of two constants of the motion is shown to generate very regular patterns in the manifold of vibrational energy levels. Distinct regular and oscillatory contributions are evidenced in the number of vibrational levels in the main polyads, which are fully reproduced using the generating function presented in Sadovskií and Zhilinskií [J. Chem. Phys. 103, 10520 (1995)]. Further developments of this approach are outlined. © 2000 American Institute of Physics. [ABSTRACT FROM AUTHOR]

We demonstrate how relative equilibria of a vibrating molecule, which are families of principal periodic orbits otherwise known as nonlinear normal modes, can be used to describe the global polyad structure of vibrational energy levels. The classical action integral n(E) computed along these orbits at different energies E corresponds to the polyad quantum number n so that the energy En of different relative equilibria describes the splitting of n-polyads. Further information on the internal polyad structure can be driven from the stability analysis of relative equilibria. We use the ozone molecule as a concrete example where n-polyads or "hyperpolyads" should be distinguished from the well-known polyads of the 1:1 stretching mode resonance; the stretching polyads are structural elements of hyperpolyads. We give dynamical interpretation of the relation between relative equilibria and n-polyads based on the normal form reduction in the limit of small vibrations near the equilibrium.

We consider the wide class of systems modeled by an integrable approximation to the 3 degrees of freedom elastic pendulum with 1:1:2 resonance, or the swing-spring. This approximation has monodromy which prohibits the existence of global action-angle variables and complicates the dynamics. We study the quantum swing-spring formed by bending and symmetric stretching vibrations of the CO2 molecule. We uncover quantum monodromy of CO2 as a nontrivial codimension 2 defect of the three dimensional energy-momentum lattice of its quantum states.