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.