Figure 1 Estimate on the total number of exchange–correlation
functional approximations created since 1980. The assessment was
obtained by cross-referencing our own functional databases with those
released in the LibXC software library.
PBEsol functionals mentioned above belong to this second rung.
Functionals on the third rung depend on the density, its gradient, and
the Laplacian of the density (the second derivative), or equivalently on
the kinetic energy density τ. These functionals are calledmeta -generalized gradient approximations (m GGAs). Famousm GGA functionals are TPSS39 and
SCAN40 of Perdew and coworkers,
M06-L,41 M11-L,42 and
MN15-L43 of Truhlar and coworkers, and
B97M‑V44 of Mardirossian and Head-Gordon. All
functionals on the first three rungs depend only on local quantities and
are therefore also called “local functionals”. The notation semilocal
is sometimes used in the physics literature for differentiating rung-2
and rung-3 functionals from rung-1 ones. We prefer to avoid it, since it
is mathematically misleading (rung-2 and rung-3 functionals still
depends on variables that are mathematically “local”, such as the
gradient and/or the Laplacian of the density). The additional ingredient
on the fourth rung is a certain percentage of non-local exact
(“Hartree-Fock-like”) exchange. The B3LYP approximation mentioned
above is a hybrid-GGA, while the MN15 and ωB97M-V functionals are hybridm GGAs. The last rung includes contributions from the unoccupied
orbitals, in either double-hybrid45 (or doubly
hybrid46) functionals, or functionals based on
RPA47–51 or other advanced
techniques.52–56 Functionals on both rung-4 and
rung-5 are by definition non-local.