operator. We take the nuclear density to correspond to the two-parameter Fermi distribution, with the nuclear thickness (90% to 10% fall-off) taken to be $2.3\, {\rm fm}$ for all atoms and the half-density radii found from the root-mean-square radii of Ref. [@rms]. For E119, we take $r_{\rm rms}=6.5\, {\rm fm}$, within the range predicted from Hartree-Fock-BCS theory [@BCS].

  ------ ------------ --------------------- -------------- --------------------- ----------- --------------------------------- ------------------ --------------
  Atom   State         $\epsilon_{\rm CH}$                  $\epsilon_{\rm KS}$               $\delta\epsilon_{\rm HF}^{(1)}$                     
                                              This work      Ref. [@sapcheng]                            This work              Ref. [@sapcheng]    This work
  Na     $3s_{1/2}$         -0.173341        -7.056\[-7\]      -6.888\[-7\]       -0.178764            -6.678\[-7\]               -6.573\[-7\]     -5.559\[-7\]
  K      $4s_{1/2}$         -0.139522        -1.497\[-6\]      -1.472\[-6\]       -0.144032            -1.431\[-6\]               -1.416\[-6\]     -1.224\[-6\]
  Rb     $5s_{1/2}$         -0.131786        -5.274\[-6\]      -5.224\[-6\]       -0.135854            -5.202\[-6\]               -5.167\[-6\]     -4.648\[-6\]
  Cs     $6s_{1/2}$         -0.120057        -1.131\[-5\]      -1.123\[-5\]       -0.124015            -1.133\[-5\]               -1.128\[-5\]     -1.054\[-5\]
  Fr     $7s_{1/2}$         -0.122284        -4.556\[-5\]      -4.533\[-5\]       -0.125432            -4.642\[-5\]               -4.628\[-5\]     -4.997\[-5\]
  ------ ------------ --------------------- -------------- --------------------- ----------- --------------------------------- ------------------ --------------

In this work we consider three different electronic potentials $V_{\rm
  el}$. One is the relativistic Hartree-Fock potential, which we use as our starting point for many-body perturbation theory. We also consider the simpler core-Hartree and Kohn-Sham potentials as points of reference. Labzowsky [*et al.*]{} [@labzowsky] and Sapirstein and Cheng [@sapcheng] have calculated the $s$-wave vacuum polarization (and self-energy) corrections for the 