Vacuum polarization and Wichmann-Kroll correction in the finite basis set approximation

TL;DR

This paper explores the use of the finite basis set method to calculate vacuum polarization effects and the Wichmann-Kroll correction in hydrogen-like ions, addressing convergence issues and validating results through Green's function integration.

Contribution

It applies the finite basis set approach to vacuum polarization and Wichmann-Kroll correction calculations, which are underexplored in this context, and validates the method with cross-checks.

Findings

Convergence of the method depends on basis set type and size.

Results for Wichmann-Kroll correction are consistent with Green's function integration.

Vacuum polarization corrections are evaluated for multiple heavy hydrogen-like ions.

Abstract

The finite basis set method is commonly used to calculate atomic spectra, including QED contributions such as bound-electron self-energy. Still, it remains problematic and underexplored for vacuum-polarization calculations. We fill this gap by trying this approach in its application to the calculation of the vacuum-polarization charge density and the Wichmann-Kroll correction to the electron binding energy in a hydrogen-like ion. We study the convergence of the method with different types and sizes of basis sets. We cross-check our results for the Wichmann-Kroll correction by direct integration of the Green's function. As a relevant example, we consider several heavy hydrogen-like ions and evaluate the vacuum polarization correction for $S$ and $P$ electron orbitals.