Rational approximation to Thomas-Fermi equations

TL;DR

This paper introduces a simple rational approximation method for the Thomas-Fermi equation, achieving high accuracy in slope estimation at the origin, outperforming recent complex approaches for both isolated atoms and atoms in magnetic fields.

Contribution

It presents a straightforward rational approximation that significantly improves the accuracy of solutions to the Thomas-Fermi equation over previous methods.

Findings

Rational approximation yields unprecedented accuracy for the slope at the origin.

Small Padé approximants outperform more elaborate recent approaches.

Effective for both isolated atoms and atoms in strong magnetic fields.

Abstract

We show that a simple and straightforward rational approximation to the Thomas-Fermi equation provides the slope at origin with unprecedented accuracy and that relatively small Pad\'e approximants are far more accurate than more elaborate approaches proposed recently by other authors. We consider both the Thomas-Fermi equation for isolated atoms and for atoms in strong magnetic fields.