Discontinuous Nature of the Exchange-Correlation Functional in Strongly Correlated Systems

Standard approximations for the exchange-correlation functional have been found to give big errors for the linearity condition of fractional charges, leading to delocalization error, and the constancy condition of fractional spins, leading to static correlation error. These two conditions are now unified and extended to states with both fractional charge and fractional spin to give a much more stringent condition: the exact energy functional is a plane, linear along the fractional charge coordinate and constant along the fractional spin coordinate with a line of discontinuity at the integer. Violation of this condition underlies the failure of all known approximate functionals to describe the gaps in strongly correlated systems. It is shown that explicitly discontinuous functionals of the density or orbitals that go beyond these currently used smooth approximations is the key for the application of density functional theory to strongly correlated systems.

Figure 1

Exact energy (a.u.) for densities with fractional charges and fractional spins in $\mathrm{H}[n_{\alpha },n_{\beta }]$. Fractional charges occur along the edge lines and fractional spins arise at the intersection $n_{\alpha }+n_{\beta }=1$. The line for $n_{\alpha }=n_{\beta }$ is highlighted.

Figure 2

The same as Fig. 1 for approximate functionals. All calculations are self-consistent using a cc-pVQZ basis set.