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A Physical Expression of Electrons in Density Functional Theory

We present a formally exact density functional theory (DFT) determination of the average electron energy. Our orbital-free theory, which is based on a different accounting of energy functional terms, partially solves one well known downside of Kohn-Sham (KS) DFT: that orbital energies have but tenuous connections to physical quantities. Our computed average electron energies are close to experimental ionization potentials in one-electron systems, demonstrating a surprisingly small effect of self-interaction and other exchange- correlation errors in established DFT methods. We argue for the use of the average electron energy as a physics condition on otherwise fictitious KS energy levels and as a design criterion for density functional approximations.