Recursion theory for nonrelativistic ground-state atomic energies and expectation values of $r^{-1\mathrm{}}$

A recursion theory for the determination of binding energies and expectation values of $r^{-1\mathrm{}}$ is presented and discussed for neutral atoms. The derived recursion relations are parameter free and provide accurate estimates of $E$ and $〈r^{-1\mathrm{}}〉$ in terms of just the atomic number $Z$ and the corresponding zero-order perturbation energy ${\epsilon }_0$. By construction the formulas yield exact values for $Z=1\mathrm{}$ and for $Z\to \infty$ (in a relative sense). It is shown that for large $Z$ the model energy corresponds to a $Z^{-\frac{1}{3}}$ series; i.e., $E=C_0{Z}^{\frac{7}{3}}+C_1{Z}^2+\cdots$, where $C_1=0.4701\mathrm{}$. Various modifications of the theory are considered; the most accurate one is manifested in a single-parameter formula that yields energies of all atoms, in the range $1\le Z\le 86$, with an average deviation of 0.09% compared to the corresponding Hartree-Fock values. Similarly, the single-parameter counterpart for $〈r^{-1\mathrm{}}〉$ gives results, for these atoms, with an average deviation of 0.2% compared to the corresponding Hartree-Fock values. Finally, it is noticed that the recursion theory is based to some extent on the local properties of a hypothetical two-dimensional surface, $E(Z,N)\mathrm{}$, that contains the energies of neutral atoms as a particular subset.