$Z$ expansions of the total electronic energy for isoelectronic atoms: A method for determining ${\overline{V}}_{\mathrm{en}}$ and ${\overline{V}}_{\mathrm{ee}}$ from experimental total electronic energies

The $Z$ dependence of the total electronic energy for an atomic isoelectronic series, was investigated in several different ways, first by the consideration of a Taylor-series expansion of ${\overline{V}}_{\mathrm{en}}$, the average electron-nuclear attractive potential energy, used in conjunction with the virial and Hellmann-Feynman theorems and, secondly, a Taylor-series expansion of ${\overline{V}}_{\mathrm{ee}}$, the average electron-electron repulsive potential energy, used in conjunction with the same two theorems. A direct Taylor-series expansion of the total energy $E$ was also studied. Convergence of these expansions was inferred from the rate of convergence of least-squares fits of tabulated values of the total electronic energies as a function of the increasing number of terms in the expansion. Theoretical nonrelativistic Hartree-Fock energies, accurate theoretical nonrelativistic energies, and experimental total energies, the latter approximately corrected for relativistic and quantum electrodynamic effects, were used in the fits. These expansions were compared with $\frac{1}{Z}$ expansion results. All Taylor-series variants studied showed comparable or superior convergence properties to the $\frac{1}{Z}$ expansion. The choice of a ${\overline{V}}_{\mathrm{en}}$ or a ${\overline{V}}_{\mathrm{ee}}$ expansion forces certain behavior on the total electronic energy $E$ as a function of $Z$. From a least-squares rate-of-convergence criterion and other considerations it appears that the Taylor-series expansion of ${\overline{V}}_{\mathrm{en}}$ offers slightly better results. It is pointed out that by the use of the virial and Hellmann-Feynman theorems it is possible to obtain accurate estimates of ${\overline{V}}_{\mathrm{ee}}$ and ${\overline{V}}_{\mathrm{en}}$ from a knowledge of total electronic energies of isoelectronic series alone. The technique is applied to accurate values for the He isoelectronic series and experimental data corrected for relativistic and radiative contributions for the He through Ne isoelectronic series in order to determine the effect of electron correlation on the quantities ${\overline{V}}_{\mathrm{en}}$ and ${\overline{V}}_{\mathrm{ee}}$. Comparisons are made with available experimental and theoretical results in the case of Ne. The $Z$ expansions for the energy were also used to fit individual ionization potentials across an isoelectronic series. These fits were used to predict electron affinities and improved values for certain ionization potentials.