Density-Functional Theory for Fractional Particle Number: Derivative Discontinuities of the Energy

The Hohenberg-Kohn theorem is extended to fractional electron number N, for an isolated open system described by a statistical mixture. The curve of lowest average energy E₍ versus N is found to be a series of straight line segments with slope discontinuities at integral N. As N increases through an integer M, the chemical potential and the highest occupied Kohn-Sham orbital energy both jump from E₌-E₌-₁ to E₌+₁-E₌. The exchange-correlation potential {Eₗ₂} ({r) } jumps by the same constant, and limₑ{Eₗ₂} ({r) }>~0.