The Gill Functional

With the knowledge that satisfying limiting constraints had not helped in the E_x^PW91 functional and that the E_x^B88 functional was a (successful) attempt to reduce the x² behaviour of the Sham-Kleinman functional, Gill [106] introduced a simple exchange functional
$$\begin{align}E_{x}^{G96}[\rho_{\sigma}] = E_{x}^{D30}[\rho_{\sigma}] - b \int \rho_{\sigma}^{4/3} x_{\sigma}^{3/2}\, d{\bf r} \end{align}$$
where the parameter b = 1/137 was chosen by fitting the exchange energy of the Argon atom. The E_x^G96 functional has incorrect behaviour for both high and low x, yet it is similar to E_x^B88 for mid-range x values. The fact that this far simpler functional performs comparably to E_x^B88 [106] shows again that the limiting behaviour of functionals is of less importance than the behaviour for mid-range values of x.