Limitations of the LDA

One limitation of the LDA is that it is blind to derivatives of the charge density. Improvements have been proposed by developing functionals that depend not just on $\rho$ but also on $\nabla \rho$. The functionals based on this idea are known as generalised gradient approximations (GGA). They are often an improvement over the LDA, but there are cases where the LDA still performs better than GGA functionals.

A more serious limitation of the LDA, or indeed of any approximation based on a sum of local contributions (including GGA functionals), i.e. terms that only depend on the value of density at the point where they are calculated, is related to their short-sightness. They cannot deal with long range interactions, such as van der Waals (vdw) or any other electrostatic interaction that is not already coded in the Coulomb term. In the particular case of London dispersive interactions, which arise from the coupling of dynamically induced dipoles, appearing when a spontaneous charge fluctuation in one region of space induces the appearance of a charge fluctuation in a different region of space, information coming only from the static value of the density is not sufficient, but one also needs to relate to how the charge density changes with time in response to external perturbations.