Violation of a local form of the Lieb-Oxford bound

In the framework of density-functional theory, several popular density functionals for exchange and correlation have been constructed to satisfy a local form of the Lieb-Oxford bound. In its original global expression, the bound represents a rigorous lower limit for the indirect Coulomb interaction energy. Here we employ exact-exchange calculations for the G2 test set to show that the local form of the bound is violated in an extensive range of both the dimensionless gradient and the average electron density. Hence, the results demonstrate the severity in the usage of the local form of the bound in functional development. On the other hand, our results suggest alternative ways to construct accurate density functionals for the exchange energy.

Figure 1

Count histogram of exact enhancement factors for the electronic exchange as a function of the dimensionless gradient in the G2 test set. The results are compared to the enhancement factors of several density functionals. The line shows the local Lieb-Oxford bound that is violated for ${\mathcal{F}}_x>2.27$.

Figure 2

Same as Fig. 1 but as a function of the Wigner-Seitz radius $r_s=3^{1/3}{\left(4\pi n\right)}^{-1/3}$.

Figure 3

Enhancement factor as a function of the dimensionless gradient $s$ for four diatomic molecules.

Figure 4

Spatial behavior of the enhancement factor across the molecular axis of CO${}_2$. The density $n$ and dimensionless gradient $s$ are also shown (in arbitrary units). The arrows mark the position of the atoms.