Meta-GGA

Adamo, C., Ernzerhof, M., Scuseria, G. E., 2000, The meta-GGA Functional Thermochemistry with a Kinetic Energy Density Dependent Exchange-Correlation Functional , J. Chem. Phys., 112, 2643. 

Table III The ratio of approximate to exact exchange energy per electron of a uniform electron gas. For the approximations listed in Tables I and II but not listed here, this ratio is exactly 1. The Becke-Roussel exchange functional is a non-empirical meta-GGA based upon the hydrogen atom.

The third rung of Jacob s ladder is defined by meta-GGA functionals, which include information from n r), Vn(r), and V2n(r). In practice, the kinetic energy density of the Kohn-Sham orbitals,... 

The classes of functionals shown in Fig. 10.2 do not exhaust the kinds of functionals that can be constructed. Any functional that incorporates exact exchange is called a hybrid functional. The hypcr-GGA functionals listed above can therefore also be referred to as hybrid-GGA methods. Hybrid-meta-GGA functionals also exist—these combine exact exchange with meta-GGA functionals this group of functionals is not shown in Fig. 10.2. 

A simple physical example to illustrate dispersion interactions are the dimers of rare-gas atoms such as He, Ne, and Ar. These atoms are well known for their lack of chemical reactivity, but the fact that these gases can be liquefied at sufficiently low temperatures makes it clear that attractive interactions between rare-gas atoms exist. Zhao and Truhlar examined the performance of a large number of meta-GGA functionals for describing He-He, He-Ne, He-Ar, and Ne-Ar dimers with localized basis set calculations.15... 

Hybrid and meta-GGA DFT functionals usually offer some improvement over corresponding pure DFT functionals. 

The hybrid exchange-correlation functionals discussed so far apply one of the possible GGA functionals as their orbital-free component. Using meta-GGA for this purpose leads to the functionals branded as hyper-GGA by Perdew - the convention adopted also in this review. Such functionals, take the following general... 

The TPPSh functional42 is a one-parameter combination of the parameter-free meta-GGA (TPSS) with exact exchange . Other functional of this type were also proposed.55-57... 

In the previous sections, the following nonempirical functionals were mentioned LDA, PBE, and TPSS. These functionals represent the lowest three rungs on the Jacob s ladder LDA, GGA, and meta-GGA. At the GGA-and meta-GGA levels, the approximate exchange-correlation functionals were obtained, not directly, but through consttuction of approximations to the exchange-correlation hole (nxc(r, r ))... 

Ernzerhof and Scuseria53 analyzed the electron affinities and ionization potentials obtained using the LDA-, PBE-, and PBE1PBE functionals with the reference date taken from the G2-1 data set.99 Whereas the PBE functional leads to significantly more accurate results than does LDA, no noticeable improvement occurs at the meta-GGA level. 

The discrepancies between the calculated values of —eHOMO and experimental ionization potentials are frequently attributed to self-interaction error. Indeed, applying the Perdew-Zunger technique4 to correct this error of the LDA-, GGA-(PBE), and meta-GGA (TPSS) functionals improves the numerical values of —8HOmo-98 Interestingly, these studies showed that the Perdew-Zunger correction does not improve ionization potentials and electron affinities if calculated as energy differences (ASCF). 

The term non-local is used sometimes in die literature in association with gradient-dependent (GGA) functionals. This nomenclature is not applied in this work. The LDA-, GGA-, and meta-GGA functionals are referred to as semi-local as they do not account for any long-range non-locality of the exchange-correlation energy density excseim local(r) = exc(p(r), Vp(r), V2p(r), r(r)) whereas... 

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