Meta-GGA functionals

Therefore, the effect of the density Laplacian is included implicitly in the kinetic energy density. It is natural that the next step in density gradient correction is the kinetic energy density correction on Jacob s ladder (see Sect. 5.1). Major meta-GGA functionals include the van Voorhis-Scuseria 1998 (VS98) meta-GGA exchange-correlation (van Voorhis and Scuseria 1998), the Perdew-Kurt-Zupan-Blaha (PKZB) meta-GGA exchange-correlation (Perdew et al. 1999), and the Tao-Perdew-Staroverov-Scuseria (TPSS) meta-GGA exchange-correlation (Tao et al. 2003) functionals. 

As far as I know, the first meta-GGA functional was the Lap-series metacorrelation functional (Proynov et al. 1995). This functional is also a CS-type correlation functional (see Sect. 5.3). In this functional, the size of the exclusion volume is represented by a momentum, which is written by the kinetic energy density to derive a CS-type correlation functional that depends on the kinetic energy density. 

The VS98 meta-GGA exchange-correlation functional (van Voorhis and Scuseria 1998) is the first exchange functional incorporating the kinetic energy density. Similarly to the PF exchange functional (see Sect. 5.2), this functional is derived from the analytical expansion of the density matrix in Eq.(5.7) (Negele and Vautherin 1972), 

This functional contains 7 semiempirical parameters (a through / and a) for each exchange, parallel-spin or opposite-spin pair correlation functional, and consequently has 21 semiempirical parameters in total (for the parameter values, see van Voorhis and Scuseria 1998). Therefore, this functional is also classified as a semiempirical functional (see Sect. 5.6) and is actually taken as the first meta-GGA semiempirical functional using Za- This functional is also characteristic in associating the correlation functional with the exchange one through the same hg. 

The term, meta-GGA functional, first appeared in the PKZB meta-GGA exchange-correlation functional (Perdew et al. 1999). The PKZB exchange functional intends to enhance the PBE-GGA functional using the kinetic energy density on the basis of the fundamental conditions extended to the density Laplacian, 

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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. 

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... 

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... 

In order to make improvements over the LSDA, one has to assume that the density is not uniform. The approach that has been taken is to develop functionals that are dependent on not only the electron density but also derivatives of the density. This constitutes the generalized gradient approximation (GGA) and is the second rung on Jacob s Ladder. The third rung, meta-GGA functional, includes a dependence on the Laplacian of the density (V p) or on the orbital kinetic energy density (t). The fourth row, the hyper-GGA or hybrid functionals, includes a dependence on exact (HF) exchange. Finally, the fifth row incorporates the unoccupied Kohn-Sham orbitals. This is most widely accomplished within the so-called double hybrid functionals. 

In order to assess the performance of the OS global hybrid functionals from a different point of view, we also compare the orbital energies and IPs of valence and core orbitals for OCS molecule in a sense of Koopmans theorem. IPs obtained by the OS hybrid functionals are shown in Table 14.16. The deviations from experimental IPs [53] and values of a, are shown in parentheses and square brackets, respectively. For HOMO, the OS global hybrid functionals provide comparatively similar IPs 11.45,10.99, 11.18, and 11.17 eV for SVWN5, BLYP, PBE, and TPSS functionals, and the corresponding deviations are at most 0.25 eV. The OS hybrid functionals also reproduce Ols and Sis IPs within the deviation of 2.5 eV for the LDA, GGA, and meta-GGA functionals, though the accurate estimation of large IPs is rather difficult. 

The above assessment reveals that the LCOE improves FON dependence and estimation of IPs significantly for all global hybrid functionals, which bases S VWN5, BLYP, PBE, and TPSS XC functionals and an added HFx term. Finally, let us compare the results of the OS functional based on LC-BLYP. For core orbitals, the global hybrid-based OS functionals basically perform slightly better than the OS functional of LC-BLYP does, although the obtained a, values are relatively different. For valence orbitals, all OS functionals provide MAEs less than 0.5 eV. The MAE of the conventional LC-BLYP is the smallest among all functionals, which is consistent with the previous reports [100, 101]. The overall MAEs of the OS functional of LC-BLYP are comparable to those of the LDA, GGA, and meta-GGA functionals. 

Inclusion of either the Lapladan or orbital kinetic energy density as a variable leads to the so-called meta-GGA functionals, and functionals which in general use orbital information may also be placed in this category. Calculation of the orbital kinetic energy density is numerically more stable than calculation of the Laplacian of the density, and the two t functions in eq. (6.44) are common components of meta-GGA functionals. 

Meta-GGA functionals Functionals correcting GGA functionals with the kinetic energy density t. 

The TPSS meta-GGA exchange-correlation functional (Tao et al. 2003) intends to remove the semiempirical parameters from the PKZB functional to construct a nonempirical meta-GGA functional. 

For meta-GGA functionals, correlation functionals have been chiefly developed e.g., the Filatov-Thiel 1998 (FT98) (Filatov and Thiel 1998) and the Krieger-Chen-lafrate-Savin (KCIS) (Krieger et al. 1999) meta-GGA correlation functionals. 

Meta-GGA functionals incorporate the Laplacian of the electron density and generally depend on the electron kinetic energy density. These features allow a systematic improvement of results for many quantum chemical calculations. We have used the exchange functional BR89 (Becke-Roussel 1989 represented in analytic form) [87,90] and the correlation functional B94 (Becke 1994) [90, 91]. Minnesota functionals tested are M05 [92], M05-2X [93], M06-L [94], M06-HF [95], M06 [96], M06-2X [96], and Ml 1 [97]. Another global hybrid studied is BMK [98]. A hybrid extension of the nonempirical exchange-correlation TPSS (Tao, Perdew, Staroverov, and Scuseria) [99] and functional TPSSh [100] is also examined. 

The meta-GGA functional M06L performs very well, with MUEs of 0.11 eV for CEBES and 0.19 eV for chemical shifts, while the BR89B94 functional gives a high MUE of 0.55 eV for the CEBEs. 

Becke s B95 meta-GGA correlation functional, containing two parameters whose values were fitted to atomic correlation energy data, is often used. The TPSS (Tan, Perdew, Staroverov, Scuseria) meta-GGA functional has given good results for many properties. A reparametrization of TPSS gave the oTPSS functional, where o is for optimized [L. Goerigk and S. Grimme, J. Chem. Theory CompuL, 6,107 (2010)]. 

GGA, meta-GGA, hybrid-GGA, and hybrid-meta-GGA functionals give not only good equilibrium geometries, vibrational frequencies, and dipole moments, but also generally accurate molecular atomization energies. Eor example, B3LYP/6-31 H-G(2d,p) calculations on the G2 data set gave an MAE of 3.1 kcal/mol (Foresman and Frisch, Chapter 7). 

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