  We evaluate the accuracy of electron densities and quasiparticle energy gaps
given by hybrid functionals by directly comparing these to the exact quantities
obtained from solving the many-electron Schrodinger equation. We determine the
admixture of Hartree-Fock exchange to approximate exchange-correlation in our
hybrid functional via one of several physically justified constraints,
including the generalized Koopmans' theorem. We find that hybrid functionals
yield strikingly accurate electron densities and gaps in both
exchange-dominated and correlated systems. We also discuss the role of the
screened Fock operator in the success of hybrid functionals.
