DFT Calculations, Atomic Ionization Potentials -- Which Exchange-Correlation Functional to Use, to Preserve Koopmans Theorem?
I think the most widely-used approach at the moment comes from Roi Baer and Leeor Kronik and others, e.g.


Phys. Rev. Lett. (2010) 105, art. 266802
Phys. Rev. B (2011) 84, art. 075144


Basically the idea is to tune the range-separation parameter $\gamma$ between short-range and long-range electrostatic effects in a range-separated hybrid functional. You find a match such that Koopmans' theorem holds.

The result is an "optimally-tuned" range-separated hybrid DFT functional (OT-RSH).

It's not perfect, for example, it violates size consistency: J. Chem. Phys. (2013) 138, 204115