second-order case $\delta \epsilon_i^{(2)}=\langle
\varphi_i|\Sigma^{(2)}({\bf r}_1,{\bf r}_2,\epsilon_i)|\varphi_i\rangle$ while for the all-order case $\delta \epsilon_i^{(\infty)}=\langle
\varphi_i|\Sigma^{(\infty)}({\bf r}_1,{\bf
  r}_2,\epsilon_i)|\varphi_i\rangle$. This potential, referred to as the [*correlation potential*]{}, may be added to the Hartree-Fock potential in the relativistic Hartree-Fock equations for the valence electrons. This yields Brueckner orbitals $\varphi_{{\rm Br},i}$ and energies $\epsilon_{{\rm Br},i}$. Such a procedure also takes into account the higher orders in $\Sigma$ in the Brueckner orbitals and energies.

In this work we calculate $\Sigma^{(2)}$ for all atoms considered and we calculate $\Sigma^{(\infty)}$ for Cs. We calculate $\Sigma^{(2)}$ using a B-spline basis set [@johnson] obtained by diagonalizing the relativistic Hartree-Fock operator on a set of 40 splines of order $k=9$ within a cavity of radius $40\,{\rm a.u.}$ The exchange part of $\Sigma^{(\infty)}$ is also considered in the second-order, with (multipolarity-dependent) factors used to screen the Coulomb interaction. For the direct part of $\Sigma^{(\infty)}$, the Feynman diagram technique is used for inclusion of the core polarization and hole-particle classes of diagrams; see Ref. [@dzuba] for further details about the method. The correlation potential method, using all-orders $\Sigma^{(\infty)}$, has proven to be remarkably successful in high-precision atomic structure calculations for heavy alkali atoms; notably, this method was used to obtain one of 