The theory of two-electron atoms: between ground state and complete fragmentation

Abstract

Since the first attempts to calculate the helium ground state in the early days of Bohr-Sommerfeld quantization, two-electron atoms have posed a series of unexpected challenges to theoretical physics. Despite the seemingly simple problem of three charged particles with known interactions, it took more than half a century after quantum mechanics was established to describe the spectra of two-electron atoms satisfactorily. The evolution of the understanding of correlated two-electron dynamics and its importance for doubly excited resonance states is presented here, with an emphasis on the concepts introduced. The authors begin by reviewing the historical development and summarizing the progress in measuring the spectra of two-electron atoms and in calculating them by solving the corresponding Schrö dinger equation numerically. They devote the second part of the review to approximate quantum methods, in particular adiabatic and group-theoretical approaches. These methods explain and predict the striking regularities of two-electron resonance spectra, including propensity rules for decay and dipole transitions of resonant states. This progress was made possible through the identification of approximate dynamical symmetries leading to corresponding collective quantum numbers for correlated electron-pair dynamics. The quantum numbers are very different from the independent particle classification, suitable for low-lying states in atomic systems. The third section of the review describes modern semiclassical concepts and their application to two-electron atoms. Simple interpretations of the approximate quantum numbers and propensity rules can be given in terms of a few key periodic orbits of the classical three-body problem. This includes the puzzling existence of Rydberg series for electron-pair motion. Qualitative and quantitative semiclassical estimates for doubly excited states are obtained for both regular and chaotic classical two-electron dynamics using modern semiclassical techniques. These techniques set the stage for a theoretical investigation of the regime of extreme excitation towards the three-body breakup threshold. Together with periodic orbit spectroscopy, they supply new tools for the analysis of complex experimental spectra.

Non-dispersive wave packets in periodically driven quantum systems

“…As a matter of fact, the classical and quantum dynamics of the three-body Coulomb problem generated by Hamiltonian (265) has been a largely unexplored "terra incognita" until very recently [164], since the dimensionality of the phase space dynamics increases from effectively two to effectively eight dimensions when a second electron is added to the familiar Kepler problem. Furthermore, the exact quantum mechanical treatment of the helium atom remains a formidable task since the early days of quantum mechanics, and considerable advances could be achieved only very recently, with the advent of modern semiclassical and group theoretical methods [165][166][167][168].…”

Intermanifold similarities in partial photoionization cross sections of helium

“…1). For a more complete comparison between the different quantum numbers see [6,18]. The approximate constants of motion for correlated two-electron dynamics expressed through the approximate quantum numbers imply a nodal structure for the respective resonance states [33].…”

Energy-level statistics of interacting trapped bosons

“…P(s) distribution follows the Gaussian orthogonal ensemble (GOE) or the Gaussian unitary ensemble (GUE) of random matrices depending on whether the Hamiltonian has time-reversal symmetry or not [2,4]. The spectral properties of many different many-fermion quantum systems like atoms and atomic nuclei and also quantum billiards have already been studied [3][4][5][6][7][8][9][10][11]. The P (s) distribution of the nuclear data ensemble agrees very well with the GOE and in the atomic spectra the nearest-neighbor spacing distribution is of the Wigner type.…”