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 $A=e^{(-\alpha R_1-\beta R_2)}+e^{(-\beta R_1-\alpha R_2)}$ and $B=A(1+c{R}_{12})$. The binding energy decreases with increasing $D$. Functions $A$ and $B$ predict "critical binding dimensionalities" at $D=3.99\mathrm{} \mathrm{and} 4.89$, respectively, above which there is no binding in the hydride anion. The exact ground-state binding energy at $D=5\mathrm{}$ is shown to be equal to that of the doubly excited $2p^2$ ${}_{}{}^3{}_{}{}^{}P_{}^e{}_{}{}^{}$ state into in three dimensions. By "dimensionality scaling" of atomic units the $D=1\mathrm{}$ atom is transformed the $\delta$-function model for which exact energies are known. In the infinite dimensional limit, function $A$ predicts no exchange contribution to binding for nuclear charge $Z\ge \surd 2$, with $\alpha \ne \beta$ only for $Z<\mathrm{}\surd 2$.