Foundation 2: How many functionals exist?
KS-DFT is based on the existence of the exact exchange-correlation functional ,7 i.e. a mathematical object that returns a number (the xc energy) given the electron density of the system (a three-dimensional function) as input. Unfortunately, Kohn and Sham did not give any indication on the form of the exact functional, and to this day, the exact functional itself is unknown and essentially unknowable.21,24 The practical use of KS-DFT relies on the development of xcfunctional approximations to the exact functional (simply called functionals from now on).16,19,21,22,25,26 More than 300 approximations have been proposed to date,27 with the total number growing substantially every year. According to our own estimate—shown in Figure 1 —the functionals development machine is at full production strength, having produced at least 10 new functionals in each of the last 15 years. If the current trend holds for the next decades, we will reach a total of 500 functionals right around 2030. At present, there is no apparent sign of slow down.
As the name suggest, an xc functional is usually composed of an exchange part and a correlation part. Some of the functionals have been developed by optimizing for a specific property, for example, the BB1K28 and BMK29 functionals have been developed for kinetics, while the PBEsol functional30 was introduced for solid state calculations. Other functionals, like B3LYP,31–34PBE,35 MN15,36 or ωB97M-V,37 have been developed to be applicable to a broader range of problems, at least in chemistry. In 2001, Perdew and Schmidt introduced the Jacob’s Ladder of density functional approximations,38 as an attempt to classify the functionals according to the “ingredients” used in their definitions. The main idea behind Jacob’s Ladder is that functionals belonging to a higher rung yield more accurate results than functionals from lower rungs, at the expense of a more complex functional form (and to some degree, a slightly increased computational cost). The Jacob’s Ladder is composed of five rungs. The first rung corresponds to the local spin density approximation, or LSDA. These functionals only depend on the density of the system. On the second rung, we find functionals that use the density and its gradient, called generalized gradient approximations (GGAs). The PBE and