Study of the density-gradient expansion for the exchange energy

It is well known that the density-gradient expansion of the Hartree-Fock exchange energy for the bare-Coulomb interaction contains divergent terms of order $e^4$ where $e$ is the electronic charge. We argue that the exchange energy evaluated with Kohn-Sham orbitals (i.e., those derived from a local effective potential) is purely of order $e^2$ and therefore its gradient expansion is well defined. This density-gradient expansion, with the a priori coefficient of Sham, is shown to converge by comparison with numerically refined values for the exact exchange energy of a metal surface in the linear-potential model. As the electron density profile becomes more slowly varying, the relative error of the zeroth-order (local-density) term tends to zero. We present here the first demonstration that, in addition, the absolute error is increasingly canceled by the second-order (gradient) term. Like the gradient expansion for the kinetic energy but unlike the one for correlation, the gradient expansion for exchange gives useful results even for "physical" surface profiles. One- and many-electron atoms are also discussed. It is observed that, as the atomic number increases, the relative errors of the local-density and gradient-expansion approximations decrease in magnitude, but the gradient term corrects only a small fraction of the error of the local-density approximation. This is a consequence of the fact that the convergence condition $\frac{\left|\nabla n\right|}{2}{k}_{F}n\lesssim 1$ is increasingly satisfied as the atomic number increases but the second convergence condition $\frac{\left|{\nabla }^{2}n\right|}{2k_F\left|\nabla n\right|}\ll 1$ is not so well satisfied.