-4.605\[-10\]                        -4.421\[-10\]

  $5d_{5/2}$      -0.076590           -0.079296                 -1.135\[-10\]                        -1.015\[-10\]                    -0.077501                 -1.059\[-10\]                        -1.018\[-10\]
  ------------ ---------------- --------------------- ---------------------------------- -------------------------------------- --------------------- ---------------------------------- --------------------------------------

In this section we consider the effect on the valence Uehling shift due to account of higher orders in many-body perturbation theory. The many-body perturbation theory is expanded in the residual Coulomb interaction, $1/|{\bf r}_i-{\bf r}_j|+V_{\rm el}$, where $V_{\rm el}$ is the electronic potential used for the zeroth-order calculations. When calculations are performed in zeroth-order in the relativistic Hartree-Fock approximation, $V_{\rm HF}=V_{\rm el}$, as they have been in this work, then the first-order many-body corrections cancel exactly.

The lowest-order many-body corrections then arise in the second order in the Coulomb interaction. The corresponding diagrams, in the Goldstone and Feynman formalisms, may be found, e.g., in Refs. [@review; @dzuba]. There is a method, developed by Dzuba, Flambaum, and Sushkov [@DFS1989], for taking into account the higher orders of many-body perturbation theory using the Feynman diagram technique; the Coulomb lines are modified by including an infinite series of core polarization loops and an infinite series of hole-particle interactions in each loop.

A non-local, energy-dependent potential may be determined, with its averaged value corresponding to the correlation correction to the energy. For example, for the 