Remarks on the existence and accuracy of theZ−13expansion of the nonrelativistic ground-state energy of a neutral atom

It is known that the nonrelativistic ground-state energy $E$ of a neutral atom composed of $Z$ electrons can be rather well represented, especially for large $Z$, by $E\approx -(c_7{Z}^{\frac{7}{3}}+c_6{Z}^{\frac{6}{3}}+c_5{Z}^{\frac{5}{3}})$ Ry, where $c_7$ and $c_6$, the Thomas-Fermi and Scott coefficients, can be determined theoretically. (Recently $c_5$ was also determined theoretically.) We argue that $E$ or its derivative is a discontinuous function of $Z$—a reflection of the shell structure of atoms—so that $E$ cannot be expanded as an infinite power series in $Z^{-\frac{1}{3}}$, though it may well have an expansion through the three terms noted above, or possibly even a fourth. We also provide some insight into the remarkable accuracy of the above three-term expansion for all $Z$ down to $Z=1\mathrm{}$. We do this by examining three models [($a$), ($b$), and ($c$)] of an atom. In model ($a$) electron screening is neglected, in model ($b$) an electron is screened by electrons in lower shells but not by electrons in its own shell, and in model ($c$) there is also screening by some electrons in the same shell. We show that if $E^{\alpha }$ is the ground-state energy of the atom in model ($\alpha$), we can write $E^{\alpha }=E_{\mathrm{smth}}^{\alpha }+\Delta E^{\alpha }$, where $E_{\mathrm{smth}}^{\alpha }$ is a smooth continuous function of $Z$ which can be expanded as a convergent power series in $Z^{-\frac{1}{3}}$ for $Z>\mathrm{}{\left(18\mathrm{}\sqrt{3}\right)}^{-1\mathrm{}}$, rapidly convergent for $Z\ge 1$, and where $\Delta E^{\alpha }$ or its derivative is a discontinuous function which cannot be so expanded. We have