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

The Thomas-Fermi model and the Z^-1 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 φ(x)=Σₙ₌₀^∞Cₙxn/2 may be rearranged into φ(x)=Σₙ₌₀^∞Bₙ(y)x3n/2, where y=C₂x and the Bₙ's are polynomials in y. The functions Bₙ are obtained through a recursive set of differential equations where B₀=1+y is known. It is then shown that the function $f(q)$, q=N/Z, which determines the total binding energy by means of E(N,Z)=Z7/3f(q) in both the Thomas-Fermi theory and the Z^-1 perturbation theory, is given by f(q)=q1/3Σₙ₌₀^∞aₙqⁿ. The first few coefficients in this series are determined. The function $f(q)$ is then computed through numerical integrations of the Thomas-Fermi equation with different initial slopes. The results are tabulated for 0≤q≤1 and analytically continued beyond $q=1$. Finally the ratio VₙₑE (Vₙₑ being the nuclear-electron attraction energy) is obtained in terms of $f(q)$ 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 7/3 for neutral atoms.