Hyper GGA functionals

Importantly, even though the hyper-GGA functionals appear as the last rung in the Perdew s ladder (Fig. 2.2), there is stiU another rung before reaching the chemical accuracy. This last level corresponds to fully non-local functionals, which includes the exact exchange and refines the correlation part by evaluating part of it exactly. 

A critical feature of this quantity is that it is nonlocal, that is, a functional based on this quantity cannot be evaluated at one particular spatial location unless the electron density is known for all spatial locations. If you look back at the Kohn-Sham equations in Chapter 1, you can see that introducing this nonlocality into the exchange-correlation functional creates new numerical complications that are not present if a local functional is used. Functionals that include contributions from the exact exchange energy with a GGA functional are classified as hyper-GGAs. 

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... 

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. 

See Refs [64-67]. GGA is followed by meta-GGA, hyper-GGA, and hybrid functionals (see Refs [63,67, 68] and references therein). 

Exchange-correlation functionals, which depend explicitly on Kohn-Sham orbitals such as meta-GGAs, hybrid- and hyper functionals discussed before fall also in this cathegory. 

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