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 $E_M-E_{M-1\mathrm{}}$ to $E_{M+1\mathrm{}}-E_M$. The exchange-correlation potential $\frac{\delta E_{\mathrm{xc}}}{\delta n\left(\overrightarrow{\mathrm{r}}\right)}$ jumps by the same constant, and $\frac{{\lim }_{r\to \infty }\delta E_{\mathrm{xc}}}{\delta n\left(\overrightarrow{\mathrm{r}}\right)}>~0$.