Three-electron coalescence points in two and three dimensions

TL;DR

This paper investigates the non-analytical behavior of three-electron wave functions at coalescence points in 2D and 3D systems, revealing logarithmic singularities and providing explicit forms for specific spin states.

Contribution

It introduces an alternative method to analyze wave function singularities at three-electron coalescence points, highlighting the presence of logarithmic terms in 2D and 3D systems.

Findings

Logarithmic singularities characterize wave functions at coalescence points.

Explicit forms of singularities are provided for certain spin states.

Non-analytical behavior is similar to that near the helium nucleus.

Abstract

The form of the wave function at three-electron coalescence points is examined for several spin states using an alternative method to the usual Fock expansion. We find that, in two- and three-dimensional systems, the non-analytical nature of the wave function is characterized by the appearance of logarithmic terms, reminiscent of those that appear as both electrons approach the nucleus of the helium atom. The explicit form of these singularities is given in terms of the interelectronic distances for a doublet and two quartet states of three electrons in a harmonic well.