(\[eq:relax\_me\]), in calculations we instead find the energies $\epsilon_i ^\prime $ from the solution of the equation $$\label{eq:relax1}
(c\boldsymbol{\alpha}\cdot {\bf p} +(\beta -1)c^2 -V_{\rm nuc}-V_{\rm
  Ueh} -V_{\rm HF}^{\rm Ueh})\varphi_i^\prime= \epsilon_i^\prime \varphi_i^\prime \ .$$ The correction is then $$\label{eq:relax2}
\delta \epsilon_i= \epsilon^\prime_i - \epsilon_i \ .$$ It should be noted that the energies from the matrix element Eq. (\[eq:relax\_me\]) and from Eqs. (\[eq:relax1\]), (\[eq:relax2\]) are not equivalent; energies from the solution of the latter equations include higher-order corrections in Uehling to the valence orbitals, while the former equation does not.

In this and the following section we use Cs as our main atom of interest in consideration of the many-body effects core polarization and valence-core correlations, see Tables \[tab:deltaV\] and \[tab:sigma\]. Our final values for Cs, as well as for the other alkali atoms Na through to E119, are presented in Table \[tab:all-order\].

![The Uehling potential $V_{\rm Ueh}$ and the Uehling correction to the direct relativistic Hartree-Fock potential $\delta V_{\rm dir}^{\rm Ueh}$ magnified by $10^5$; the potentials are for Cs.[]{data-label="fig:potentials"}](fig1.pdf){width="\columnwidth"}

In Fig. \[fig:potentials\] we present both the Uehling potential and the Uehling correction to the Hartree-Fock direct potential for Cs. The long range of the relaxation potential makes it possible for 