Piecewise Linearity of Approximate Density Functionals Revisited: Implications for Frontier Orbital Energies

In the exact Kohn-Sham density-functional theory, the total energy versus the number of electrons is a series of linear segments between integer points. However, commonly used approximate density functionals produce total energies that do not exhibit this piecewise-linear behavior. As a result, the ionization potential theorem, equating the highest occupied eigenvalue with the ionization potential, is grossly disobeyed. Here, we show that, contrary to conventional wisdom, most of the required piecewise linearity of an arbitrary approximate density functional can be restored by careful consideration of the ensemble generalization of density-functional theory. Furthermore, the resulting formulation introduces the desired derivative discontinuity to any approximate exchange-correlation functional, even one that is explicitly density dependent. This opens the door to calculations of the ionization potential and electron affinity, even without explicit electron removal or addition. All these advances are achieved while neither introducing empiricism nor changing the underlying functional form. The power of the approach is demonstrated on benchmark systems using the local density approximation as an illustrative example.

Figure 1

Energy of the ${\mathrm{H}}_2$ molecule (top) and of the C atom (bottom) as a function of fractional charge $q$ for various functionals. The EXX results for ${\mathrm{H}}_2$ have been shifted upward by 0.4 Ry for clarity. The straight solid lines connect the energies obtained at the integer value as a reference for complete piecewise linearity.

Figure 2

Deviation from piecewise linearity in the density, $D(\overrightarrow{r})$, obtained for the ${\mathrm{H}}_2$ molecule for $q=-0.5$ using (a) LSDA and (b) eLSDA.

Figure 3

Frontier orbital energy ${\epsilon }_{\mathrm{ho}}$, the energy derivative $\partial E/\partial q$ as a function of $q$, and the negative of the IP $-I$, calculated for ${\mathrm{H}}_2$ with the LSDA and eLSDA functionals.