  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.
