High-precision solution of the Dirac Equation for the hydrogen molecular ion by an iterative method

TL;DR

This paper presents a highly precise numerical solution to the Dirac equation for H₂⁺ using an iterative method and a new basis set, achieving unprecedented accuracy and enabling nonperturbative relativistic calculations.

Contribution

The authors introduce a kinetically balanced exponential basis set and an iterative method to solve the Dirac equation with extremely high accuracy for the hydrogen molecular ion.

Findings

Achieved relativistic energy accuracy of 10^{-20}

Wavefunctions in good agreement with perturbation theory

First step towards nonperturbative self-energy calculations

Abstract

The Dirac equation for H$_2^+$ is solved numerically using an iterative method proposed by Kutzelnigg [Z. Phys. 11, 15 (1989]. The four-component wavefunction is expanded in a newly introduced kinetically balanced exponential basis set. The ground-state relativistic energy is obtained with an accuracy of $10^{-20}$, which represents an improvement by several orders of magnitude, and is shown to be in good agreement with results obtained from perturbation theory.Highly accurate relativistic wavefunctions are obtained, which is a first step towards nonperturbative calculations of the one-loop self-energy correction in hydrogen molecular ions.