Dimensionality dependence of total energy of closed shells in a bare Coulomb field for large atomic number

For Z particles moving independently in an unscreened Coulomb potential energy −Ze 2 /r, it is known that the total energy E = −(3/2) 1/3 Z 7/3 in nonrelativistic theory as Z→∞. The generalization of this result to d dimensions in the same asymptotice limit is obtained as\documentclass{article}\pagestyle{empty}\begin{document}$$E(Z,d) = - \frac{{2\left({\frac{{d!}}{4}} \right)^{1 - (2/d)} }}{{(d - 1)!(d - 2)}}Z^{3 - (2/d)},$$\end{document}which includes the known result above for the three‐dimensional case.