Sturmian basis set for the Dirac equation with finite nuclear size: Application to polarizability, Zeeman and hyperfine splitting, and vacuum polarization

We investigate the application of the Sturmian basis set in relativistic atomic structure calculations. We propose a simple implementation of this approach and demonstrate its ability to provide various quantities for hydrogen-like ions, including binding energies, static dipole polarizability, $g$ factor, hyperfine splitting, and nuclear magnetic shielding. Finally, we calculate the all-order (Wichmann-Kroll) vacuum polarization charge density, which was a challenge for the finite-basis-set approach until recently. Comparison of the obtained results with the previously published numerical and analytical calculations is presented. All calculations are performed with the finite size of the nucleus and can in principle be extended to arbitrary binding potentials.