8 - Functionals for exchange and correlation

Summary

Functional functionals

Summary

Density functional theory is the most widely used method today for electronic structure calculations because of the success of practical, approximate functionals. The crucial quantity in the Kohn–Sham approach is the exchange–correlation energy which is expressed as a functional of the density E_xc[n]. This chapter is devoted to relevant approximate functionals, in particular, the local density approximation (LDA) and examples of generalized-gradient approximations (GGAs). Explicit formulas for certain widely used functionals are given in App. B. Non-local formulations are an active area of research leading to new classes of functionals, in particular, orbital-dependent functionals including the “optimized effective potential” (OEP) method and “hybrid functionals.” Important features are illustrated by a few selected results on atoms and molecules.

As emphasized in the previous chapter, the genius of the Kohn–Sham approach is two-fold: first, the construction of an auxiliary system leads to tractable independent-particle equations that hold the hope of solving interacting many-body problems. The famous Kohn–Sham equations are given in (7.11)–(7.13). Second, and perhaps more important, by explicitly separating out the independent-particle kinetic energy and the long-range Hartree terms, the remaining exchange–correlation functional E_xc[n] can be reasonably approximated as a local or nearly local functional of the density. Even though the exact functional E_xc[n] must be very complex, great progress has been made with remarkably simple approximations. This chapter is devoted to those approximations.