Meta-GGA methods

GGA DFT can be improved in several ways here, we will briefly discuss the hybrid methods and the meta-GGA methods. In the meta-GGA approach, the molecular XC energy at each integration point is described by the density, the gradient of the density, and the orbital-dependent kinetic energy density ... 

In the hybrid approach, the exchange and correlation energies are calculated by the LSDA, GGA, or meta-GGA methods, and are then combined with the FIF exchange energy ... 

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 further reduction of computational complexity is achieved by not evaluating the orbital functional self-consistently, via Eq. (91), but only once, using the orbitals and densities of a converged self-consistent LDA or GGA calculation. This post-GGA or post-LDA strategy completely avoids the OEP and has been used both for hybrid functionals and Meta-GGAs [108, 109, 112, 113]. A drawback of post methods is that they provide only approximations to the selfconsistent total energies, not to eigenvalues, effective potentials, orbitals or densities. 

In the case of hybrid functionals, still another mode of implementation has become popular. This alternative, which also avoids solution of Eq. (91), is to calculate the derivative of the hybrid functional with respect to the singleparticle orbitals, and not with respect to the density as in (91). The resulting single-particle equation is of Hartree-Fock form, with a nonlocal potential, and with a weight factor in front of the Fock term. Strictly speaking, the orbital derivative is not what the HK theorem demands, but rather a Hartree-Fock like procedure, but in practice it is a convenient and successful approach. This scheme, in which self-consistency is obtained with respect to the singleparticle orbitals, can be considered an evolution of the Hartree-Fock Kohn-Sham method [6], and is how hybrids are commonly implemented. Recently, it has also been used for Meta-GGAs [2]. For occupied orbitals, results obtained from orbital selfconsistency differ little from those obtained from the OEP. 

Systematic procedures for optimizing GGA and meta-GGA exchange-correlation functionals have been developed by Becke [215,216], Adamson et al. [217], Handy and coworkers [218-221]. In Becke s method, the total exchange-correlation energy is written as... 

This is a common feature of aU LDA and GGA functionals, not excluding even the most advanced meta-GGA functionals. Many recent DFT methods provide much better results for dispersion-controlled complexes (Zhao and Truhlar 2008). 

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