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Direct mapping between exchange potentials of Hartree-Fock and Kohn-Sham schemes as origin of orbitals proximity

It is found that, in closed-$l$-shell atoms, the exact local exchange potential $v_{\text{x}}(\bf r)$ of the density functional theory (DFT) 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 for the individual Hartree-Fock (HF) orbitals belonging to this shell. Consequently, the continuous piecewise function built of shell-specific exchange potentials, each defined as the weighted average of the shifted orbital exchange potentials corresponding %the HF orbitals from to a given shell, yields another highly-accurate representation of $v_{\text{x}}(\bf r)$. These newly revealed properties are {\em not} related to the well-known step-like shell structure in the response part of $v_{\text{x}}(\bf r)$, but they result from specific relations satisfied by the HF orbital exchange potentials. These relations explain the outstanding proximity of the occupied Kohn-Sham and HF orbitals as well as the high quality of the Krieger-Li-Iafrate and localized HF (or, equivalently, common-energy-denominator) approximations to the DFT exchange potential $v_{\text{x}}(\bf r)$. The constant shifts added to the HF orbital exchange potentials, to map them onto $v_{\text{x}}(\bf r)$, are nearly equal to the differences between the energies of the corresponding KS and HF orbitals. It is discussed why these differences are positive and grow when the respective orbital energies become lower for inner orbitals.