Kohn-Sham potentials in exact density-functional theory at noninteger electron numbers

Within exact electron density-functional theory, we investigate Kohn-Sham (KS) potentials, orbital energies, and noninteracting kinetic energies of the fractional ions of Li, C, and F. We use quantum Monte Carlo densities as input, which are then fitted, interpolated at noninteger electron numbers $N$, and inverted to produce accurate KS potentials $v_{\mathrm{s}}^N\left(\mathbf{r}\right)$. We study the dependence of the KS potential on $N$, and in particular we numerically reproduce the theoretically predicted spatially constant discontinuity of $v_{\mathrm{s}}^N\left(\mathbf{r}\right)$ as $N$ passes through an integer. We further show that, for all the cases considered, the inner orbital energies and the noninteracting kinetic energy are nearly piecewise linear functions of $N$. This leads us to propose a simple approximation of the KS potential $v_{\mathrm{s}}^N\left(\mathbf{r}\right)$ at any fractional electron number $N$ which uses only quantities of the systems with the adjacent integer electron numbers.

Figure 1

KS potentials $v_{\mathrm{s}}^{Z+c}\left(r\right)$ (hartree) as a function of the radial distance $r$ and the fractional excess electron number $c=N-Z$ for Li (left), C (middle), and F (right). Colors indicate height (from red to green for $v_{\mathrm{s}}^{Z+c}\left(r\right)<0$, white for $v_{\mathrm{s}}^{Z+c}\left(r\right)\approx 0$, blue for $v_{\mathrm{s}}^{Z+c}\left(r\right)>0$).

Figure 2

Change in Kohn-Sham potentials $v_{\mathrm{s}}^{Z+c}\left(r\right)-v_{\mathrm{s}}^{Z^{-}}\left(r\right)$ as a function of $c>-1$ for Li (top), C (middle), and F (bottom), for selected values of $r$.

Figure 3

Change in Kohn-Sham orbital energies for all orbitals ${\varepsilon }_i^{Z+c}-{\varepsilon }_i^{Z^{-}}$ as a function of $c>-1$ for Li (top), C (middle), and F (bottom).

Figure 4

Change in Kohn-Sham kinetic energy $T_{\mathrm{s}}^{Z+c}-T_{\mathrm{s}}^Z$ and kinetic energy $T^{Z+c}-T^Z$ as a function of $c$.