the Kohn-Sham approximation, Eq. (\[eq:rel\]) is solved self-consistently for the core and valence electrons together.

In Table \[tab:CHKS\] we present our results for the first-order Uehling corrections to the $s$-wave valence binding energies for the alkali atoms Na through to Fr performed in the core-Hartree and Kohn-Sham approximations; we also present our first-order Hartree-Fock results for comparison (a more comprehensive list of Hartree-Fock results is presented later in the paper in Table \[tab:all-order\]). Our core-Hartree and Kohn-Sham results are presented alongside the vacuum polarization calculations of Sapirstein and Cheng [@sapcheng] performed in the same potentials. As well as taking into account all-orders in $Z\alpha$ in the vacuum polarization, the authors of Ref. [@sapcheng] include electronic screening through the use of an effective charge. Both effects act to reduce the size of the Uehling correction. This may explain the deviation between the results.

The core-Hartree and Kohn-Sham results are in fairly good agreement with each other, though reference to the results performed in other atomic potentials [@sapcheng; @labzowsky] shows that there is a sizeable spread in the values, on the order of 10% the size of the shifts. There is also a significant difference between the local atomic potential results and 