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It is found that, in closed-$l$-shell atoms, the exact local exchangepotential $v_{\text{x}}(\bf r)$ of the density functional theory (DFT) is verywell represented, within the region of every atomic shell, by each of thesuitably shifted potentials obtained with the non-local Fock exchange operatorfor the individual Hartree-Fock (HF) orbitals belonging to this shell.Consequently, the continuous piecewise function built of shell-specificexchange potentials, each defined as the weighted average of the shiftedorbital exchange potentials corresponding %the HF orbitals from to a givenshell, yields another highly-accurate representation of $v_{\text{x}}(\bf r)$.These newly revealed properties are {\em not} related to the well-knownstep-like shell structure in the response part of $v_{\text{x}}(\bf r)$, butthey result from specific relations satisfied by the HF orbital exchangepotentials. These relations explain the outstanding proximity of the occupiedKohn-Sham and HF orbitals as well as the high quality of the Krieger-Li-Iafrateand localized HF (or, equivalently, common-energy-denominator) approximationsto the DFT exchange potential $v_{\text{x}}(\bf r)$. The constant shifts addedto 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 correspondingKS and HF orbitals. It is discussed why these differences are positive and growwhen the respective orbital energies become lower for inner orbitals.

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