On the Leading Energy Correction for the Statistical Model of the Atom: Non-Interacting Case

It is shown that the infima of the Hellmann and the Hellmann-Weizsäcker functional without electron-electron interaction can be written in terms of the nuclear charge Z and the particle number N as E(Z,N) = Z^2([alpha]1N^1/3 + [alpha]2 + ... ). In the case of Hellmann functional we calculate both [alpha]1 and [alpha]2, in the case of the Hellmann-Weizsäcker functional we calculate [alpha]1. We compare our results with Thomas-Fermi theory. Finally, we apply our result to bound the quantum mechanical ground state energy ofthe system