Higher order gradient or meta-GGA methods

The logical extension of GGA methods is to allow the exchange and correlation functionals to depend on higher order derivatives of the electron density, with the Laplacian (V ) being the second-order term. Alternatively, the functional can be taken to depend on the orbital kinetic energy density t, which for a single orbital is identical to the von Weizsacker kinetic energy Tw (eq. (6.3)). 

The orbital kinetic energy density and the Lapladan of the density essentially carry the same information, since they are related via the orbitals and the effective potential (all potential terms in the KS equation). 

This may also be seen from the gradient expansion of rfor slowly varying densities.  

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. 

One of the earliest attempts to include kinetic energy functionals was by Becke and Roussel (BR), who proposed the exchange functional shown in eq. (6.47).  

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