Accurate calculation of the solutions to the Thomas-Fermi equations

TL;DR

This paper presents highly accurate numerical solutions to the Thomas-Fermi equations for atoms and atoms in strong magnetic fields using advanced approximation methods, significantly improving precision in key boundary parameters.

Contribution

The paper introduces a combination of numerical techniques to achieve unprecedented accuracy in solving the Thomas-Fermi equations for various atomic conditions.

Findings

Precise boundary slope at origin for atomic solutions

Accurate boundary location in magnetic field cases

Implementation of multiple approximation methods for enhanced accuracy

Abstract

We obtain highly accurate solutions to the Thomas-Fermi equations for atoms and atoms in very strong magnetic fields. We apply the Pad\'e-Hankel method, numerical integration, power series with Pad\'e and Hermite-Pad\'e approximants and Chebyshev polynomials. Both the slope at origin and the location of the right boundary in the magnetic-field case are given with unprecedented accuracy.