The collinear helium atom and two-electron ions

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.