Solution of the two-center Dirac equation with 20 digits precision using the finite-element technique

TL;DR

This paper introduces a highly precise numerical solution to the two-center Dirac equation for the hydrogen molecular ion, achieving 20-digit accuracy using finite-element methods, which is crucial for future high-precision experiments.

Contribution

It presents the first fully relativistic, finite-element based solution of the two-center Dirac equation with unprecedented precision of 20 digits.

Findings

Total energies with fractional uncertainties of a few times 10^{-20}

Relativistic contribution uncertainty of 10^{-17}

Results are relevant for future precision measurements

Abstract

We present a precise fully relativistic numerical solution of the two-center Coulomb problem. The special case of unit nuclear charges is relevant for the accurate description of the ${\rm H}_2^+$ molecular ion and its isotopologues, systems that are an active experimental topic. The computation utilizes the 2-spinor minmax approach and the finite-element method. The computed total energies have estimated fractional uncertainties of a few times $10^{-20}$ for unit charges and a bond length of 2 atomic units. The fractional uncertainty of the purely relativistic contribution is $1\times10^{-17}$. The result is relevant for future precision experiments, whereas at present the uncertainties arising from the quantum electrodynamic treatment of the rovibrational transition frequencies. are dominant.