  For properties of interacting electron systems, Kohn-Sham (KS) theory is
often favored over many-body perturbation theory (MBPT) owing to its low
computational cost. However, the exact KS potential can be challenging to
approximate, for example in the presence of localized subsystems where the
exact potential is known to exhibit pathological features such as spatial
steps. By modeling two electrons, each localized in a distinct potential well,
we illustrate that the step feature has no counterpart in MBPTs (including
Hartree-Fock and GW) or hybrid methods involving Fock exchange because the
spatial non-locality of the self-energy renders such pathological behavior
unnecessary. We present a quantitative illustration of the orbital-dependent
nature of the non-local potential, and a numerical demonstration of Kohn's
concept of the nearsightedness for self energies, when two distant subsystems
are combined, in contrast to the KS potential. These properties emphasize the
value of self-energy-based approximations in developing future approaches
within KS-like theories.
