the nucleus. This leads to larger Uehling shifts for all waves and is more important for the orbitals with $l>0$.

The difference between the results using $\Sigma^{(2)}$ or $\Sigma^{(\infty)}$ is significant, affecting the second digit in the results. However, the fitted results shown in columns five and eight of Table \[tab:sigma\], agree very well, to about 2% for $6s$ and significantly better for the other waves. That is, [*as long as the correlation potential is fitted to reproduce the experimental energies, the correlation corrections to the Uehling shifts are insensitive to the details of $\Sigma$*]{}.

Therefore, we have performed calculations for Na through to E119 with the second-order correlation potential $\Sigma^{(2)}$ fitted to reproduce experimental energies (for E119, we use fitting factors from Fr). These results, which also include the important core-relaxation effects, are presented in the sixth column of Table \[tab:all-order\].

In the final column of Table \[tab:all-order\] we present, along with the results of Ref. [@derev_relax], the vacuum polarization calculations performed by Thierfelder and Schwerdtfeger [@thierfelder] in the relativistic Hartree-Fock approach. Their results are in reasonable agreement with our relaxed core results $\delta \epsilon$. We are not aware of other many-body vacuum polarization results for the alkali atoms.

