Relative contributions of the electron-electron and electron-nuclear interactions to the ground-state energy of a neutral atom

One can easily obtain a good estimate of the total energy E(Z) of a ground-state neutral atom with Z (≫1) electrons, primarily because an effective central potential $V_{\mathrm{eff}\left(\mathrm{r}\right)}$ provides a very good starting point. A sufficient, if not necessary, condition for there to be such a $V_{\mathrm{eff}}$ is for the ratio ρ(Z)≡$V_{e\mathrm{-}e}$(Z)/$V_{e\mathrm{-}\nu }$(Z) of the electron-electron and electron-nuclear contributions to E(Z) to be small, and it is found to be only about (1/7) for large Z. In the (statistical) nonrelativistic Thomas-Fermi (TF) model, which becomes exact as Z∼∞, E(Z) (in rydbergs) is approximated by $E_{\mathrm{TF}\left(\mathrm{Z}\right)=\mathrm{-}{c}_7{Z}^{7/3}}$ for all Z, with $c_7$ a known constant, while ρ(Z) is approximated by ${\rho }_{\mathrm{TF}\left(\mathrm{Z}\right)=(1/7)\mathrm{}}$ for all Z. A simple proof that ρ(Z)∼(1/7) as Z∼∞, by Rau (unpublished) and by Thirring [Quantum Mechanics of Large Systems (Springer-Verlag, New York, 1983)], used only the assumption (built into TF theory) that the binding energy of the last electron in a neutral atom is negligible.