Numerical solution of the relativistic single-site scattering problem for the Coulomb and the Mathieu potential

TL;DR

This paper discusses numerical methods for solving the relativistic single-site scattering problem in the Korringa-Kohn-Rostoker Green function method, focusing on Coulomb and Mathieu potentials, with analytical insights into asymptotic behaviors.

Contribution

It provides a comprehensive analysis of numerical solutions for the relativistic scattering problem, including implementation details and tests for Coulomb and Mathieu potentials.

Findings

Validated numerical methods for Coulomb and Mathieu potentials

Analytical insights into irregular scattering solutions near the origin

Demonstrated applicability for full-potential calculations

Abstract

For a reliable fully-relativistic Korringa-Kohn-Rostoker Green function method, an accurate solution of the underlying single-site scattering problem is necessary. We present an extensive discussion on numerical solutions of the related differential equations by means of standard methods for a direct solution and by means of integral equations. Our implementation is tested and exemplarily demonstrated for a spherically symmetric treatment of a Coulomb potential and for a Mathieu potential to cover the full-potential implementation. For the Coulomb potential we include an analytic discussion of the asymptotic behaviour of irregular scattering solutions close to the origin ($r\ll1$).