Qualitative models of intramolecular dynamics of acetylene: relation between the bending polyads of acetylene and perturbed Keplerian systems

Abstract

ABSTRACTWe analyse qualitatively the bending vibrational polyads of the acetylene molecule (CH) in the approximation of the resonant oscillator with axial symmetry using an effective vibrational Hamiltonian which reproduces bending vibrational energy levels computed by Michel Herman and coworkers [M. Herman, A. Campargue, M.I.E. Idrissi, and J.V. Auwera, J. Phys. Chem. Ref. Data 32 (3), 921 (2003). doi: 10.1063/1.1531651]. We explain how the classical limit of this quantum system for the total vibrational angular momentum is equivalent to a reduced perturbed Keplerian system on the classical phase space , such as the hydrogen atom in external electric and magnetic fields in the Kustaanheimo–Stiefel formalism. In particular, bending vibrational -polyads of CH correspond to the n-shells of the perturbed hydrogen atom. Within this approach, using the techniques developed for the Keplerian systems and methods of the qualitative theory, we account concisely for all series of bifurcations of the classical nonlinear normal modes and their manifestations in the quantum energy level spectrum described by our predecessors [V. Tyng and M.E. Kellman, J. Phys. Chem. B 110 (38), 18859 (2006). doi: 10.1021/jp057357f]. In addition to local oscillator approximations near stable equilibrium points, notably the local modes discussed by Rose and Kellman [J. Chem. Phys. 105 (24), 10743 (1996). doi: 10.1063/1.472882] and Jacobson et al. [J. Chem. Phys. 109 (1), 121 (1998). doi: 10.1063/1.476529; J. Chem. Phys. 111, 600 (1999). doi: 10.1063/1.479341; J. Chem. Phys. 110, 845 (1999). doi: 10.1063/1.478052], we introduce two new global integrable approximations, and confirm them by constructing respective two regular complementary lattices of quantum states within one -polyad. From the stratification of the phase space, we uncover the geometrical meaning of the corresponding new good quantum numbers and define new kind of wavefunction localisation in the neighbourhood of two spheres in the four-dimensional space. Furthermore, we use our two approximate lattices to uncover quantum monodromy of the system through the evolution of an elementary quantum cell along a closed path in the images of the nearly integrable energy-momentum maps.