mechanisms apply.

The structure of the paper is the following. In Section \[sec:finite\] we present the finite-size form of the Uehling potential used in this work. In Section \[sec:first\], the first-order valence shifts are calculated for frozen atomic cores corresponding to core-Hartree, Kohn-Sham, and Hartree-Fock approximations. The many-body effects of core relaxation and valence-core correlations are studied in Sections \[sec:relax\] and \[sec:correlations\], respectively. In these sections, details of the many-body enhancement mechanisms are presented and calculations performed for Na through to E119, with more detailed results given for Cs.

Finite-size Uehling potential {#sec:finite}
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The Uehling potential [@uehling] gives the lowest-order in $Z \alpha$ vacuum polarization correction to the Coulomb potential $V _{\rm nuc}({\bf r})$; $Z$ is the nuclear charge and $\alpha$ is the fine-structure constant. For the point-nucleus case, where $V_{\rm nuc}^{\rm point}(r)=Z/r$, the Uehling potential may be expressed as $$V_{\rm Ueh}^{\rm point} (r) = \frac{2}{3}\Big(\frac{\alpha}{\pi}\Big)\Big(\frac{Z}{r}\Big) \int_1^{\infty}dt
\sqrt{t^2-1}\Big(\frac{1}{t^2}+\frac{1}{2t^4}\Big)e^{-2tr/\alpha} \ .$$ Here and throughout we use atomic units ($\hbar=e=m=1$, $c=1/\alpha$).

To obtain the finite-nucleus expression for the Uehling potential, the expression for the point-like nucleus is folded with the nuclear density $\rho_{\rm nuc}$, $$V_{\rm Ueh}^{\rm fin}({\bf r})=\frac{1}{Z}\int d^3r' V_{\rm Ueh}^{\rm point}(|{\bf
    r}-{\bf r}'|) \rho_{\rm nuc} ({\bf r}') \ ,$$ where 