Analytical solution of the Thomas-Fermi equation for atoms

TL;DR

This paper presents an approximate analytical solution to the Thomas-Fermi equation for neutral atoms using the Ritz variational method, accurately matching numerical results and improving ionization energy calculations for heavy atoms.

Contribution

It introduces a new analytical approximation for the Thomas-Fermi equation that enhances the accuracy of ionization energy predictions for heavy atoms.

Findings

Accurately reproduces numerical solutions of the Thomas-Fermi equation

Provides better ionization energy estimates than previous trial functions

Matches the derivative at the origin with high precision

Abstract

An approximate analytical solution of the Thomas-Fermi equation for neutral atoms is obtained, using the Ritz variational method, which reproduces accurately the numerical solution, in the range $0\leq x\leq50$, and its derivative at $x=0$. The proposed solution is used to calculate the total ionization energies of heavy atoms. The obtained results are in good agreement with the Hartree-Fock ones and better than those obtained from previously proposed trial functions by other authors.