Assigning vibrational polyads using relative equilibria: application to ozone

Abstract

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 E(n) 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 wellknown 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.

Quantum monodromy and its generalizations and molecular manifestations

“…It is possible to find fractional monodromy in systems with special axial symmetries, such as a nondiagonal S 1 symmetry, which acts on the two subspaces in a way similar the action of the 1:ðÀ2Þ resonant oscillator. Another possibility may occur in the study of localized oscillations about stable nonlinear normal modes (such as local modes) in a system with three or more vibrational degrees of freedom, see [22,[43][44][45][46][47] and references therein.…”

Fundamental studies of vibrational resonance phenomena by multivalued resummation of the divergent Rayleigh–Schrödinger perturbation theory series: deciphering polyad structures of three H216O isotopologues

“…4–6 In this connection, an important theoretical concept is the vibrational polyad quantum number that labels manifolds of significantly coupled states. 7–17 For a typical semi-rigid molecule, when the basis functions with the same polyad number are assembled together, the corresponding matrix representation of the Hamiltonian breaks down into a (quasi-) block diagonal form that holds up to several quanta of excitation in each normal mode.…”

A Lagrangian fibration of the isotropic 3-dimensional harmonic oscillator with monodromy

“…We note that perturbations of isotropic harmonic oscillators have many applications. Examples where similar techniques as in this paper are used include assigning rovibrational spectra of molecules using polyads, 35 ion traps, 36 and galactic dynamics. 37 It is to be expected that the results of our paper has applications along the same line which deserves further investigation.…”