Density functional theory reduces the quantum mechanical groundstate many-electron problem to self-consistent one-electron form, through the Kohn-Sham equations [1]. This method is formally exact, but for practical calculations, the exchange-correlation energy as a functional of the density must be approximated. In doing so, the local (spin-) density approximation (LDA) has long been the standard choice [2]. Although simple, the LDA results in a realistic description of the atomic structure, elastic, and vibrational properties for a wide range of systems. Yet the LDA is generally not accurate enough to describe the energetics of chemical reactions (heats of reaction and activation energy barriers), leading to an overestimate of the binding energies of molecules and solids in particular. Also, there are several examples where the LDA puts molecular conformations or crystal bulk phases in an even qualitatively wrong energetic order [3,4]

Recent generalized gradient approximations (GGA's) have overcome such deficiencies to a considerable extent [1,5,6], giving for instance a more realistic description of energy barriers in the dissociative adsorption of hydrogen on metal and semiconductor surfaces [7,8]. Gradient corrected or GGA functionals depend on the local density as well as on the spatial variation of the density. Computationally they as simple to use as the LDA. On the other hand it is becoming increasingly clear that such GGA functionals are still too limited to provide not only a consistent improvement over the LDA, but also to achieve the desired chemical accuracy of (better than 1 kcal/mol or 50 meV/atom) in general.

In this lecture we review present formulations of the LDA and the GGA. We compare various functionals and discuss general trends which illustrate what one should or should not expect from them. To complement this ``phenomenological'' view, we discuss sum rules and other constraints that hold for the exact exchange-correlation functional, and consider their implications for approximate functionals like the LDA and GGA [9].

Finally, we briefly look at ongoing developments in orbital dependent
exchange-correlation functionals (exact Kohn-Sham exchange,
Meta-GGA's) [10,11,12,13], which are hoped to
be more accurate and more widely applicable than the LDA or GGA's.