those performed in Hartree-Fock, see Table \[tab:CHKS\]. It illustrates how sensitive the vacuum polarization corrections are to the details of atomic structure.

Core relaxation {#sec:relax}
===============

The Uehling potential affects the binding energies of the valence electron not only through the direct first-order shift $\delta
\epsilon ^{(1)}$, given above, but also indirectly through Uehling corrections to the core electrons. All-orders account of the Uehling correction in the core may be easily determined by adding the Uehling potential to the nuclear potential, $V_{\rm nuc}+V_{\rm Ueh}$, in the relativistic Hartree-Fock equations. Self-consistent solution for the electrons of the core then leads to a new Hartree-Fock potential $V^{\rm Ueh}_{\rm HF}$ which contains a correction due to Uehling $\delta V_{\rm HF}^{\rm Ueh}=V_{\rm HF}^{\rm Ueh}-V_{\rm HF}$, where $V_{\rm HF}^{\rm Ueh}$ and $V_{\rm HF}$ are, respectively, the self-consistent relativistic Hartree-Fock potentials with and without the Uehling potential included. This correction, $\delta V_{\rm HF}^{\rm Ueh}$, is referred to as a [*relaxation*]{} correction. The correction to the energy of the valence electron may then be expressed as $$\label{eq:relax_me}
\delta \epsilon_i=-\langle \varphi_i |V_{\rm Ueh}+\delta
V_{\rm HF}^{\rm Ueh}|\varphi_i\rangle =\delta
\epsilon_i^{(1)}+ \delta \epsilon_i^{\rm relax} \ .$$

While the source of the relaxation correction is made more transparent from inspection of Eq. 