Expectation values of atoms and ions: The Thomas-Fermi limit

The Thomas-Fermi model and the $Z^{-1\mathrm{}}$ perturbation expansion for ions with nuclear charge $Z$ and $N$ electrons are considered in the limits of large $Z$ and $N$ where the Thomas-Fermi model becomes exact. It is shown that the Baker expansion of the Thomas-Fermi function $\phi \mathrm{}\left(x\right)={\Sigma }_{n=0\mathrm{}}^{\infty }{C}_n{x}^{\frac{n}{2}}$ may be rearranged into $\phi \mathrm{}\left(x\right)={\Sigma }_{n=0\mathrm{}}^{\infty }{B}_n\mathrm{}\left(y\right)\mathrm{}{x}^{\frac{3n}{2}}$, where $y=C_{2}x$ and the $B_n$'s are polynomials in $y$. The functions $B_n$ are obtained through a recursive set of differential equations where $B_0=1+y$ is known. It is then shown that the function $f\left(q\right)\mathrm{}$, $q=\frac{N}{Z}$, which determines the total binding energy by means of $E(N,Z)=Z^{\frac{7}{3}}f\left(q\right)\mathrm{}$ in both the Thomas-Fermi theory and the $Z^{-1\mathrm{}}$ perturbation theory, is given by $f\left(q\right)=q^{\frac{1}{3}}{\Sigma }_{n=0\mathrm{}}^{\infty }{a}_n{q}^n$. The first few coefficients in this series are determined. The function $f\left(q\right)\mathrm{}$ is then computed through numerical integrations of the Thomas-Fermi equation with different initial slopes. The results are tabulated for $0\le q\le 1$ and analytically continued beyond $q=1\mathrm{}$. Finally the ratio $\frac{V_{\mathrm{ne}}}{E}$ ($V_{\mathrm{ne}}$ being the nuclear-electron attraction energy) is obtained in terms of $f\left(q\right)\mathrm{}$ for both positive and negative ions and found to be in agreement with the corresponding Hartree-Fock ratios. It is an extension of the well-known ratio of $\frac{7}{3}$ for neutral atoms.