Variable dimensionality in atoms and its effect on the ground state of the helium isoelectronic sequence

We calculate binding energies for heliumlike ions of variable dimensionality (D) with the wave functions A=e^(-αR₁-βR₂)+e^(-βR₁-αR₂) and B=A(1+cR₁₂). The binding energy decreases with increasing D. Functions A and B predict "critical binding dimensionalities" at D=3.99 and 4.89, respectively, above which there is no binding in the hydride anion. The exact ground-state binding energy at $D=5$ is shown to be equal to that of the doubly excited 2p² ³Pᵉ state into in three dimensions. By "dimensionality scaling" of atomic units the $D=1$ atom is transformed the δ-function model for which exact energies are known. In the infinite dimensional limit, function A predicts no exchange contribution to binding for nuclear charge Z≥2, with α≠β only for Z<2.