Improvement of the Basis for the Solution of the Dirac Equation in Cassini Coordinates

TL;DR

This paper enhances the basis functions for solving the two-centre Dirac equation in Cassini coordinates, replacing B-splines with step-like functions to improve accuracy in finite-basis-set methods.

Contribution

It introduces a new basis incorporating step-like functions, significantly improving the accuracy of Dirac equation solutions compared to previous B-spline methods.

Findings

Increased accuracy of energy eigenfunctions

Improved efficiency of the basis representation

Enhanced convergence of the solution

Abstract

We propose an improvement of the basis for the solution of the stationary two-centre Dirac equation in Cassini coordinates using the finite-basis-set method presented in Ref. [1]. For the calculations in Ref. [1], we constructed the basis for approximating the energy eigenfunctions by using smooth piecewise defined polynomials, called B-splines. In the present article, we report that an analysis of the employed representation of the Dirac matrices shows that the above approximation is not efficient using B-splines only. Therefore, we include basis functions which are defined using functions with step-like behaviour instead of B-splines. Thereby, we achieve a significant increase of accuracy of results as compared to Ref. [1].