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 ${E}_{N}$ 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 ${E}_{M}\ensuremath{-}{E}_{M\ensuremath{-}1}$ to ${E}_{M+1}\ensuremath{-}{E}_{M}$. The exchange-correlation potential $\frac{\ensuremath{\delta}{E}_{\mathrm{xc}}}{\ensuremath{\delta}n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}$ jumps by the same constant, and $\frac{{\mathrm{lim}}_{r\ensuremath{\rightarrow}\ensuremath{\infty}}\ensuremath{\delta}{E}_{\mathrm{xc}}}{\ensuremath{\delta}n(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})}>~0$.