Proof that∂E∂ni=εin density-functional theory

It is shown that the variation of the total energy, as constructed in density-functional theory, with respect to an orbital occupation is equal to the eigenvalue of that orbital, independent of the detailed form of the exchange-correlation functional. This leads to a rigorous connection between the ground-state energies of $N$- and ($N+1\mathrm{}$)-particle systems, which is useful in the calculation of certain excitation energies.