The collinear helium atom and two-electron ions

TL;DR

This paper investigates collinear configurations of helium-like atoms, deriving mathematical representations and calculating expectation values for various configurations, revealing unusual properties and constructing accurate wave functions.

Contribution

It introduces new mathematical formulas and highly accurate wave functions for collinear two-electron atomic systems, expanding understanding of their physical properties.

Findings

Unusual properties of n-e-e configurations identified.

Explicit formulas for expectation values derived.

Accurate ground state wave functions constructed.

Abstract

Collinear configurations of the helium-like atomic systems, relevant, e.g., for the quasifree mechanism of the double photoionization of helium, are studied, parameterized by the single scalar parameter $-1\leq \lambda\leq1$ ("collinear parameter") where $\lambda=0$ corresponds to the electron-nucleus ($\textbf{e-n}$) coalescence and $\lambda=1$ corresponds to the electron-electron ($\textbf{e-e}$) coalescence. In general, $\lambda>0$ corresponds to the \textbf{n-e-e} configuration, and $\lambda<0$ to the \textbf{e-n-e} configuration. Simple mathematical representations of the expectation values of the Dirac delta function relevant for the collinear configurations are derived and calculated from fully three-body dynamics without approximation for the two-electron atomic wave functions with nuclear charge $1\leq Z\leq5$. Simple formulas for calculating the expectation values of the kinetic and potential energy operators in collinear configurations are derived. Unusual physical properties of the \textbf{n-e-e} collinear configurations found for certain ranges of $\lambda$ are presented. The first few angular Fock coefficients for collinear configurations are derived as functions of $\lambda$. Highly accurate model wave functions describing the ground states of the two-electron atoms with collinear arrangement of the particles are constructed. All results are illustrated by tables and figures.