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