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We present and test a new approximation for the exchange-correlation (xc)energy of Kohn-Sham density functional theory. It combines exact exchange witha compatible non-local correlation functional. The functional is byconstruction free of one-electron self-interaction, respects constraintsderived from uniform coordinate scaling, and has the correct asymptoticbehavior of the xc energy density. It contains one parameter that is notdetermined ab initio. We investigate whether it is possible to construct afunctional that yields accurate binding energies and affords other advantages,specifically Kohn-Sham eigenvalues that reliably reflect ionization potentials.Tests for a set of atoms and small molecules show that within our local-hybridform accurate binding energies can be achieved by proper optimization of thefree parameter in our functional, along with an improvement in dissociationenergy curves and in Kohn-Sham eigenvalues. However, the correspondence of thelatter to experimental ionization potentials is not yet satisfactory, and if wechoose to optimize their prediction, a rather different value of thefunctional's parameter is obtained. We put this finding in a larger context bydiscussing similar observations for other functionals and possible directionsfor further functional development that our findings suggest.

A self-interaction-free local hybrid functional: Accurate binding energies vis-\`a-vis accurate ionization potentials from Kohn-Sham eigenvalues