The Perdew-Wang 91 Functional

In 1991 Perdew [107,108] modified Becke's functional, constraining it to become like the Sham-Kleinman functional as $x \to 0$, and tend towards zero as $x \to \infty$, while trying to retain the E_x^B88 behaviour for medium range x. The functional form is
$$\begin{align}E_{x}^{PW91}[\rho_{\sigma}] = E_{x}^{D30}[\rho_{\sigma}] - \int \fr... ...\sigma}\sinh^{-1}x_{\sigma}-10^{-6}x_{\sigma}^{4}/\alpha}\, d{\bf r} \end{align}$$
where $\alpha$ is the Dirac coefficient of $\frac{3}{2}\left(\frac{3}{4\pi}\right)^{1/3}$. However, despite the added complexity, energies obtained from E_x^PW91 are seldom an improvement over E_x^B88 [106]. The functional has also been shown to violate the original condition upon which E_x^B88 was developed [109].