Functional form of the generalized gradient approximation for exchange: The $\mathrm{PBE}\alpha$ functional

A functional form for the exchange enhancement in the generalized gradient approximation within density-functional theory is given. It satisfies the constraints used to construct the Perdew-Burke-Ernzerhof (PBE) functional but can be systematically varied using one parameter called $\alpha$. This gives the possibility to estimate the reliability of a computational result or to fit the parameter for a certain problem. Compared to other semiempirical functionals, the present has the advantage of using only one physically transparent parameter and the fitted functional will obey the same exact conditions as PBE functional. One functional is constructed by fixing $\alpha$ using the exchange enhancement for the slowly varying electron gas. This functional is free of empirical parameters and gives good results for densely packed solids. The simple form of the exchange enhancement means that oscillating terms in the exchange potential are avoided.

Figure 1

(Color online) Derivatives of the exchange enhancement that enter the exchange potential [Eq. (7)]. Lines for PBE, WC, and mPBE $F_x$ are shown. The thick lines are for the $s^{-1}\frac{d{F}_x}{ds}$ term and the thin lines for the $\frac{d}{ds}\left(s^{-1}\frac{d{F}_x}{ds}\right)$ term.

Figure 2

(Color online) Exchange enhancements $F_x$ due to density gradients. The PBE, RPBE, and WC functionals are shown with thick lines. Two $\mathrm{PBE}\alpha$ functionals are shown with thin lines and marked with the $\alpha$ used.