Collinear configuration of the 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≤λ≤1 (“collinear parameter”) where λ=0 corresponds to the electron–nucleus (e −−n) coalescence and λ=1 corresponds to the electron–electron (e −−e) coalescence. In general, λ>0 corresponds to the n–e–e configuration, and λ<0 to the 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≤Z≤5. Simple formulas for calculating the expectation values of the kinetic and potential energy operators in collinear configurations are derived. Unusual physical properties of the n-e-e collinear configurations found for certain ranges of λ are presented. The first few angular Fock coefficients for collinear configurations are derived as functions of λ. Highly accurate model wave functions describing the ground states of the two-electron atoms with a collinear arrangement of the particles are constructed. All results are illustrated in tables and figures.