Advances over the last twenty years are reported, with the emphasis on describing correlations through the distribution of an electron pair's wavefunction in its configuration space. The symmetry of the Schrodinger equation and its eigenfunctions are thus viewed and found to be important. The radial correlations are more significant for the mechanics of the pair than the angular correlations. Connections with other approaches to the subject are described. Earlier studies concerning doubly excited helium and the hydrogen negative ion are related to Wannier's threshold law (1953) for ionisation by electron collision, but these two subjects are not yet fully connected. Recent studies on the alkaline earths and helium negative ion are described and viewed as a prelude to a renewed analysis and interpretation of the optical spectra and electron collisions for atoms with partially filled valence shells, which afford ample opportunity for the manifestation of strong correlations.

https://iopscience.iop.org/article/10.1088/0034-4885/46/2/001/pdf

Correlations of two excited electrons

A comprehensive review is given on photoionization of rare gas atoms using monochromatized synchrotron radiation. Emphasis is put upon the general experimental and theoretical background, and illustrative examples are presented in order to show the present status and the progress in the field during the last decade.

https://iopscience.iop.org/article/10.1088/0034-4885/55/9/003/pdf

Photoionization of atoms using synchrotron radiation

The authors attempt to give a comprehensive discussion of observations of atomic negative-ion resonances throughout the periodic table. A review of experimental and theoretical approaches to the study of negative-ion resonances is given together with a consideration of the various schemes that are used for their classification. In addition to providing, where possible, tabulated data for the energies, widths, and symmetries of these states, the authors also attempt to highlight regularities in their behavior both within groups of the periodic table and along isoionic sequences.

Atomic negative-ion resonances

Hyperspherical coordinate approach to atomic and other Coulombic three-body systems

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 Schr\"odinger 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.

/pdf/the-theory-of-two-electron-atoms-between-ground-state-and-59rw1mn6bo.pdf

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

For comparison with our recent group-theoretical predictions of configuration-mixing coefficients, we report extensive configuration-interaction calculations for doubly excited states in the helium isoelectronic sequence below the $N=2$ ($S, P, D$ states), $N=3$ ($S, P, D, F, G$ states), $N=4$ ($S, P, D$ states), and $N=5$ ($^{1}P^{\mathrm{o}}$ states) ionization thresholds. Two new quantum numbers label the Rydberg series, and are used to predict three "selection rules" for dipole excitation and autoionization processes which agree with experiment. New autoionization widths are reported for the helium states and confirm our group-theoretically predicted selection rules. Our width of 0.151 eV for the $^{1}P^{\mathrm{o}}$ state at 62.92 eV is in agreement with the recent experimental value 0.132 \ifmmode\pm\else\textpm\fi{} 0.014 eV. Calculations of helium energies and autoionization widths are strongly affected by "avoided crossings" of the doubly excited states as $Z$ is varied continuously. The new quantum numbers project out states in ${\mathrm{H}}^{\ensuremath{-}}$ which correspond to observed $^{1}P^{\mathrm{o}}$ shape resonances above the $N=2 \mathrm{and} 3$ ionization thresholds.

Comparison of doubly-excited helium energy levels, isoelectronic series, autoionization lifetimes, and group-theoretical configuration-mixing predictions with large-configuration-interaction calculations and experimental spectra

A group theoretical method which predicts Coulomb repulsion mixing coefficients of doubly−excited atomic states is presented. Good agreement with calculated mixings in helium is found. Two new quantum numbers and three rules classify and predict the energy level orderings. The exact mathematical construction of 1/r12 in the group theoretical configuration−mixed basis is given. It leads naturally to a formulation of the corresponding many−electron problem of configurational mixings.

David R. Herrick

Papers

Comparison of doubly-excited helium energy levels, isoelectronic series, autoionization lifetimes, and group-theoretical configuration-mixing predictions with large-configuration-interaction calculations and experimental spectra

Group theoretic prediction of configuration mixing effects due to Coulomb repulsions in atoms with applications to doubly‐excited spectra

Algebra of the noncompact group o(3,2) and the hydrogen-atom radial functions.