Elimination of the asymptotic fractional dissociation problem in Kohn-Sham density-functional theory using the ensemble-generalization approach

Many approximations within density-functional theory spuriously predict that a many-electron system can dissociate into fractionally charged fragments. Here, we revisit the case of dissociated diatomic molecules, known to exhibit this problem when studied within standard approaches, including the local spin-density approximation (LSDA). By employing our recently proposed [E. Kraisler and L. Kronik, Phys. Rev. Lett. 110, 126403 (2013)] ensemble generalization we find that asymptotic fractional dissociation is eliminated in all systems examined, even if the underlying exchange correlation (xc) is still the LSDA. Furthermore, as a result of the ensemble-generalization procedure, the Kohn-Sham potential develops a spatial step between the dissociated atoms, reflecting the emergence of the derivative discontinuity in the xc energy functional. This step, predicted in the past for the exact Kohn-Sham potential and observed in some of its more advanced approximate forms, is a desired feature that prevents any fractional charge transfer between the system's fragments. It is usually believed that simple xc approximations such as the LSDA cannot develop this step. Our findings show, however, that ensemble generalization to fractional electron densities automatically introduces the desired step even to the most simple approximate xc functionals and correctly predicts asymptotic integer dissociation.

Figure 1

Total energy of the Li and F atoms as a function of $N$ (top) and energy of the dissociated $\mathrm{Li}\cdots \mathrm{F}$ molecule as a function of $q$ (bottom), computed with LSDA (dash-dotted line) and eLSDA (solid line), compared to the expected piecewise-linear (dotted line) behavior (the latter is obtained by linear interpolation of the eLSDA energies at integer electron values).

Figure 2

Total energy of the (${\mathrm{H}\cdots \mathrm{H})}^{+}$ and (${\mathrm{F}\cdots \mathrm{F})}^{+}$ molecules as a function of $q$ computed with the LSDA (dash-dotted line) and eLSDA (solid line) functionals, compared to the expected piecewise-linear (dotted line) behavior.

Figure 3

Dependence of ${\mu }_A\left(q\right)-{\mu }_B\left(q\right)$ on $q$ for the $\mathrm{Li}\cdots \mathrm{F}$ molecule, computed with LSDA (dash-dotted line) and eLSDA (solid line), compared to the expected discontinuous stair-step (dotted line) behavior.

Figure 4

The potential $w\left(\mathbf{r}\right)$ for the system ${\mathrm{C}}^{1-q}\cdots {\mathrm{F}}^{1+q}$ plotted against the interatomic axis $z$: (a) with LSDA, at $q=q_0=-0.273$; (b) with eLSDA, at $q=-0.001$; (c) with eLSDA, at $q=+0.001$. The ho levels of both ions are marked with horizontal lines.