Hydrogen negative ion: Semiclassical quantization and weak-magnetic-field effect

Dynamics starting from zero velocities in the classical Coulomb three-body problem

A qualitative geometrical analysis of the classical Coulomb three-body problem is developed. Three ions and atom are investigated: namely, H-, He, and Li+. The dynamics of these atom and ions is treated in the approximation that the nucleus has infinite mass. Geometrical structures of the dynamics starting from the initial conditions with zero velocities are elucidated. In particular, the distribution of the final states in the initial condition space, its fractal structure, and binary collision orbits are specified. As a result, it is found that binary collision orbits form a lobelike structure. It is shown that the lobelike structure is, in fact, fractal. The lobelike structure is continuously connected to the collinear eZe configuration space, in which a well-known binary symbolic dynamics exists. With these observations, it is expected that as well as triple-collision orbits, binary-collision orbits also serve us some clue to find a symbolic dynamics for the full dynamics with zero angular momentum.

Semiclassical analysis of real atomic spectra beyond Gutzwiller's approximation

Real atomic systems, like the hydrogen atom in a magnetic field or the helium atom, whose classical dynamics are chaotic, generally present both discrete and continuous symmetries. In this paper, we explain how these properties must be taken into account in order to obtain the proper (i.e., symmetry projected) h expansion of semiclassical expressions like the Gutzwiller trace formula. In the case of the hydrogen atom in a magnetic field, we shed light on the excellent agreement between present theory and exact quantum results.

Planck's over 2 pi corrections in semiclassical formulas for smooth chaotic dynamics

The validity of semiclassical expansions in the power of Planck's over 2 pi for the quantum Green's function have been extensively tested for billiards systems, but in the case of chaotic dynamics with smooth potential, even if formulas are existing, a quantitative comparison is still missing. In this paper, extending the theory developed by Gaspard et al. [Adv. Chem. Phys. 90, 105 (1995)], based on the classical Green's functions, we present an efficient method allowing the calculation of Planck's over 2 pi corrections for the propagator, the quantum Green's function, and their traces. In particular, we show that the previously published expressions for Planck's over 2 pi corrections to the traces are incomplete.

Simple dynamical system with discrete bound states

We study numerically the dynamical system of a two-electron atom with the Darwin interaction as a model to investigate scale-dependent effects of the relativistic action-at-a-distance electrodynamics. This dynamical system consists of a small perturbation of the Coulomb dynamics for energies in the atomic range. The key properties of the Coulomb dynamics are (i) a peculiar mixed-type phase space with sparse families of stable nonionizing orbits and (ii) scale-invariance symmetry, with all orbits defined by an arbitrary scale parameter. The combination of this peculiar chaotic dynamics [(i) and (ii)], with the scale-dependent relativistic corrections (Darwin interaction), generates the phenomenon of scale-dependent stability: We find numerical evidence that stable nonionizing orbits can exist only for a discrete set of resonant energies. The Fourier transform of these nonionizing orbits is a set of sharp frequencies. The energies and sharp frequencies of the nonionizing orbits we study are in the quantum atomic range.

Chaotic dynamics in classical s-wave helium

Classical dynamics of s-wave helium in the case of E<0 is investigated by the geometric method. The ambiguousness of the orbits after two-electron critical collision (TECC) is eliminated by confining the motion to the fundamental domain defined by r1>r2. Scattering orbits are classified into undelayed and delayed (resonant) according to whether they can avoid the recurrence of reaction. A global Smale horseshoe on the surface of section is constructed for the case of Z=2 which implies that the bounded motion is purely chaotic and explains the onset of the chaotic (resonant) scattering when the incident energy is below a threshold. Immediately below the threshold, the probability of resonant scattering increases linearly with the energy difference. Moreover, we found that the permitted code sequences for the scattering orbits at a given incident energy are determined by the principal TECC orbit. Compared with the e-Ze- collinear helium, the s-wave helium is less chaotic and exhibits a more intricate threshold behavior.