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

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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 Veff(r) provides a very good starting point. A sufficient, if not necessary, condition for there to be such a Veff is for the ratio {ρ}(Z){≡}V_e-e(Z)/V_e-ν(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_TF(Z)=-c₇Z7/3 for all Z, with c₇ a known constant, while {ρ}(Z) is approximated by ρTF(Z)=(1/7) for all Z. A simple proof that {ρ}(Z){~}(1/7) as Z{~}{∞}, by Rau (unpublished) and by Thirring [Quantum Mechanics of Large Systems (Springer-Verlag, New York, 1983)], used only the assumption (built into TF theory) that the binding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.We use that fact, the (recently developed) corrected TF result E(Z) {}-(c₇{Z}7/3$+${c}₆Z6/3+c₅{X}5/3$) (with ${c}₇$, ${c}₆$, and ${c}₅$ theoretically determined), and the virial theorem, and we show that ${V}_{e{{-}}{ν}}(Z)Z(dE/dZ) with an error that is at most of order{Z}4/3$; we thereby obtain an expression for {ρ}(Z) which approaches (1/7) as Z{~}{∞}, the error for large Z being of order 1/Z, and is reasonably accurate even for moderate values of Z. One finds {ρ}(Z){}J(n-6)sₙ/Jnsₙ, where each sum is over n=7, 6, and 5, and where sₙ=cₙ{Z}^{(n{{-}}7)/3}$. The value of {ρ}(Z) for heavy atoms in ``strong'' magnetic fields, those describable by a one-dimensional (statistical) TF-like theory, is discussed briefly. Some numerical studies are included. The results obtained are of interest in atomic theory itself, and in low-energy {β}-decay processes for which the difference in atomic binding energies is important.