Climbing the ladder of density functional approximations

Kohn-Sham density functional theory is the most widely used method of electronic-structure calculation in materials physics and chemistry because it reduces the many-electron ground-state problem to a computationally tractable self-consistent one-electron problem. Exact in principle for the ground-state energy and electron density, it requires in practice an approximation to the density functional for the exchange-correlation energy. Common approximations fall on the rungs of a ladder, with higher rungs being more complicated to construct and use but potentially more accurate. Each rung of the ladder introduces an additional ingredient to the energy density. From bottom to top, the rungs are (1) the local spin density approximation, (2) the generalized gradient approximation (GGA), (3) the meta-GGA, (4) the hybrid functional, and (5) the generalized random phase approximation. The semi-local rungs (1-3) are important, because they are computationally efficient, they can be constructed non-empirically, they can serve as input to fourth-rung functionals, and the meta-GGA by itself can be accurate for equilibrium properties. Recent and continuing improvements to the meta-GGA are emphasized here.