  We construct exact Kohn-Sham potentials for the ensemble density-functional
theory (EDFT) from the ground and excited states of helium. The
exchange-correlation (XC) potential is compared with the quasi-local-density
approximation and both single determinant and symmetry eigenstate
ghost-corrected exact exchange approximations. Symmetry eigenstate
Hartree-exchange recovers distinctive features of the exact XC potential and is
used to calculate the correlation potential. Unlike the exact case, excitation
energies calculated from these approximations depend on ensemble weight, and it
is shown that only the symmetry eigenstate method produces an ensemble
derivative discontinuity. Differences in asymptotic and near-ground-state
behavior of exact and approximate XC potentials are discussed in the context of
producing accurate optical gaps.
