Exchange energy density of inhomogeneous electron gas near a metal surface

Using the first-order density matrix of an infinite barrier model of a metal surface, the exchange energy density can be evaluated exactly as a function of the distance z from the barrier. This result is compared with the local approximation $\text{?}\text{3}\text{4}\text{e}{}^2\text{(}\text{3}\text{π}\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>3</mtext>}}\text{[?(z)]}{}^{\mathrm{<mtext>4</mtext><mtext>3</mtext>}}$ where the electron density in the model. The local approximation follows the exact result quite closely at all z but leads to a large percentual error in the integrated surface exchange energy.