Photoabsorption spectra of the diamagnetic hydrogen atom in the transition regime to chaos: Closed orbit theory with bifurcating orbits

With increasing energy the diamagnetic hydrogen atom undergoes a transition from regular to chaotic classical dynamics, and the closed orbits pass through various cascades of bifurcations. Closed orbit theory allows for the semiclassical calculation of photoabsorption spectra of the diamagnetic hydrogen atom. However, at the bifurcations the closed orbit contributions diverge. The singularities can be removed with the help of uniform semiclassical approximations which are constructed over a wide energy range for different types of codimension one and two catastrophes. Using the uniform approximations and applying the high-resolution harmonic inversion method we calculate fully resolved semiclassical photoabsorption spectra, i.e., individual eigenenergies and transition matrix elements at laboratory magnetic field strengths, and compare them with the results of exact quantum calculations.Comment: 26 pages, 9 figures, submitted to J. Phys.

Perturbations of the 1:1:1 resonance with tetrahedral symmetry: a three degree of freedom analogue of the two degree of freedom Henon-Heiles Hamiltonian

We study a class of three degree of freedom (3-DOF) Hamiltonian systems that share certain characteristics with the 2-DOF Henon-Heiles Hamiltonian. Our systems represent a 1:1:1 resonant three-oscillator whose principal nonlinear perturbation is the cubic potential term xyz with tetrahedral symmetry. After normalizing and reducing the 1:1:1 oscillator symmetry, we show that near the limit of linearization all our systems can be described as a one-parametric family. Such reduced systems have been suggested earlier by Hecht (1960 J. Mol. Spectrosc. 5 355) and later by Patterson (1985 J Chem. Phys. 83 4618) to model triply degenerate vibrations of tetrahedral molecules. We describe relative equilibria (RE) of these systems, classify all qualitatively different family members, and discuss bifurcations of RE involved in the transitions from one region of regular parameter values to the other

Monodromy in perturbed Kepler systems: Hydrogen atom in crossed fields

We demonstrate that an integrable approximation to the hydrogen atom in orthogonal electric and magnetic fields has monodromy, a fundamental dynamical property that makes a global definition of action-angle variables and of quantum numbers impossible. When the field strengths are sufficiently small, we find our integrable approximation using a two step normalization procedure. One of dynamically invariant sets of the resulting integrable system is a doubly pinched torus whose existence proves the presence of monodromy

Global bending quantum number and the absence of monodromy in the HCN-CNH molecule

We introduce and analyze a model system based on a deformation of a spherical pendulum that can be used to reproduce large amplitude bending vibrations of flexible triatomic molecules with two stable linear equilibria. On the basis of our model and the recent vibrational potential [ J. Chem. Phys. 115, 3706 (2001) ], we analyze the HCN/CNH isomerizing molecule. We find that HCN/CNH has no monodromy and introduce the second global bending quantum number for this system at all energies where the potential is expected to work. We also show that LiNC/LiCN is a qualitatively different system with monodromy

Local modes of silane within the framework of stretching vibrational polyads

We define stretching relative equilibria (RE) of silane and other similar tetrahedral molecules in terms of the dynamical polyad symmetry which assumes the resonance condition 1:1 between the two stretching vibrational modes ν1 and ν3 of the molecule. Exploiting symmetry and topology arguments and reducing the dimension of the classical mechanical system, we find these RE. One of them, with local symmetry C3v and minimal energy within a polyad, corresponds to the local modes. We give the upper energy limit of the local mode localization within a polyad

Classification of perturbations of the hydrogen atom by small static electric and magnetic fields

We consider perturbations of the hydrogen atom by sufficiently small homogeneous static electric and magnetic fields of all possible mutual orientations. Normalizing with regard to the Keplerian symmetry, we uncover resonances and conjecture that the parameter space of this family of dynamical systems is stratified into zones centred on the resonances. The 1 : 1 resonance corresponds to the orthogonal field limit, studied earlier by Cushman & Sadovskif (Cushman & Sadovskif 2000 Physica 142, 166-196). We describe the structure of the 1 : 1 zone, where the system may have monodromy of different kinds, and consider briefly the 1 : 2 zone

Qualitative analysis of molecular rotation starting from inter-nuclear potential

We study how qualitative features of the molecular rotational dynamics can be derived directly from the internuclear (vibrational) potential. This approach is presented on the example of a tetrahedral molecule A4 using several increasingly elaborated models of the potential

Dynamical manifestation of Hamiltonian monodromy

Hamiltonian monodromy —a topological property of the bundle of regular tori of a static Hamiltonian system which obstructs the existence of global action-angle variables— occurs in a number of integrable dynamical systems. Using as an example a simple integrable system of a particle in a circular box with quadratic potential barrier, we describe a time-dependent process which shows that monodromy in the static system leads to interesting dynamical effects