Why Kohn-Sham and Hartree-Fock orbitals are very close to each other. Shell structure of exchange potential

It is found that, in closed-$l$-subshell atoms, the exact local exchange potential of the density functional theory is very well represented, within the region of every atomic shell, by each of the suitably shifted potentials obtained with the non-local Fock exchange operator acting on the Hartree-Fock (HF) orbitals that belong to this shell. This explains the outstanding proximity of Kohn-Sham and HF orbitals and the high quality of the Krieger-Li-Iafrate and localized HF (or, equivalently, common-energy-denominator) approximations to the exchange potential.