Gradient-dependent upper bound for the exchange-correlation energy and application to density functional theory

We propose a simple gradient-dependent bound for the exchange-correlation energy (sLL), based on the recent nonlocal bound derived by Lewin and Lieb. We show that sLL is equivalent to the original Lieb-Oxford bound in rapidly varying density cases, but it is tighter for slowly varying density systems. To show the utility of the sLL bound we apply it to the construction of simple semilocal and nonlocal exchange and correlation functionals. In both cases improved results, with respect to the use of Lieb-Oxford bound, are obtained, showing the power of the sLL bound.

Figure 1

The ratios $LO/E_{xc},LL/E_{xc}$, and $sLL/E_{xc}$ vs $1/r_s$, for $40e^{-}$ jellium spheres (left panel) and for two interacting jellium slabs (right panel). Also shown are $LO/E_{xc}$ and $LL/E_{xc}=sLL/E_{xc}$, for the uniform electron gas (UEG).

Figure 2

Exchange enhancement factors vs $s$, for PBEsol and PBEsol (sLL). Also shown are the asymptotes of sLL and LO bounds.