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

It is found that for closed-l-shell atoms, the exact local exchange potential v(x)(r) calculated in the exchange-only Kohn-Sham (KS) scheme 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 nonlocal Fock exchange operator for the individual Hartree-Fock (HF) orbitals belonging to this shell. This newly revealed property is not related to the well-known steplike shell structure in the response part of v(x)(r), but it results from specific relations satisfied by the HF orbital exchange potentials. These relations explain the outstanding proximity of the occupied HF and exchange-only KS 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(x)(r). Another highly accurate representation of v(x)(r) is given by the continuous piecewise function built of shell-specific exchange potentials, each defined as the weighted average of the shifted orbital exchange potentials corresponding to a given shell. The constant shifts added to the HF orbital exchange potentials, to map them onto v(x)(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.