  In approximate Kohn-Sham density-functional theory, self-interaction
manifests itself as the dependence of the energy of an orbital on its
fractional occupation. This unphysical behavior translates into qualitative and
quantitative errors that pervade many fundamental aspects of density-functional
predictions. Here, we first examine self-interaction in terms of the
discrepancy between total and partial electron removal energies, and then
highlight the importance of imposing the generalized Koopmans' condition --
that identifies orbital energies as opposite total electron removal energies --
to resolve this discrepancy. In the process, we derive a correction to
approximate functionals that, in the frozen-orbital approximation, eliminates
the unphysical occupation dependence of orbital energies up to the third order
in the single-particle densities. This non-Koopmans correction brings physical
meaning to single-particle energies; when applied to common local or semilocal
density functionals it provides results that are in excellent agreement with
experimental data -- with an accuracy comparable to that of GW many-body
perturbation theory -- while providing an explicit total energy functional that
preserves or improves on the description of established structural properties.
