Proof that<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mfrac><mml:mrow><mml:mi>∂</mml:mi><mml:mi>E</mml:mi></mml:mrow><mml:mrow><mml:mi>∂</mml:mi><mml:mrow><mml:msub><mml:mrow><mml:mi>n</mml:mi></mml:mrow><mml:mrow><mml:mi>i</mml:mi></mml:mrow></mml:msub></mml:mrow></mml:mrow></mml:mfrac><mml:mo>=</mml:mo><mml:mi>ε</mml:mi></mml:math>in 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 ($N+1$)-particle systems, which is useful in the calculation of certain excitation energies.