What's right and what's wrong with the density-gradient expansions for the exchange and correlation energies?

In a system of slowly-varying electron density, the exchange and correlation hole surrounding an electron may be expanded in gradients of the density. The zero-order term of this expansion, the local density approximation, satisfies important properties of the exact hole: The density of the exchange hole is everywhere negative, and integrates to a deficit of one electron, while the density of the correlation hole integrates to zero. Here the second-order term of the expansion is shown to improve the description of the hole near the electron, at the cost of worsening the description far away and violating these exact properties. The gradient expansion may be used to improve upon the local density approximation for the energy of a real system with long-range interactions, but only after this expansion has been appropriately cut off. The cut-off or generalized gradient approximation, which involves only the local density and its first derivative, provides qualitative and quantitative information about many-body effects in inhomogeneous systems