High-precision minmax solution of the two-center Dirac equation

TL;DR

This paper introduces a highly precise numerical method using finite element techniques to solve the two-center Dirac equation, providing benchmark results for molecular ions with extremely low uncertainties.

Contribution

The paper develops a minmax finite element method for solving the two-center Dirac equation with unprecedented accuracy, applicable to systems with various nuclear charges.

Findings

Achieved fractional uncertainties of ~10^{-23} for H2+

Achieved fractional uncertainties of ~10^{-21} for Th2+

Provided benchmark solutions for molecular ions

Abstract

We present a high-precision solution of Dirac equation by numerically solving the minmax two-center Dirac equation with the finite element method (FEM). The minmax FEM provide a highly accurate benchmark result for systems with light or heavy atomic nuclear charge $Z$. A result is shown for the molecular ion ${\rm H}_2^+$ and the heavy quasi-molecular ion ${\rm Th}_2^{179+}$, with estimated fractional uncertainties of $\sim 10^{-23}$ and $\sim 10^{-21}$, respectively. The result of the minmax-FEM high-precision of the solution of the two-center Dirac equation, allows solid control over the required accuracy level and is promising for the application and extension of our method.