Generalized gradient approximation GGA , exchange-correlation

Exchange-correlation functionals, which determine the reliability of Kohn-Sham calculations, are compared in terms of the basic concepts in their development, and for their features and problems, in Chap. 5. This chapter uses as examples the major local density approximation (LDA) and generalized gradient approximation (GGA) exchange-correlation functionals and meta-GGA, hybrid GGA, and semi-empirical functionals to enhance the degree of approximation in terms of their concepts, applicabilities, and problems. 

Local exchange-correlation functionals such as generalized gradient approximations (GGA) are continuum approximations, which can, at best, average over the discontinuity. In regions where the HOMO and LUMO are significant, they provide an approximate average description [39—41] ... 

The wave functions are expended in a plane wave basis set, and the effective potential of ions is described by ultrasoft pseudo potential. The generalized gradient approximation (GGA)-PW91, and local gradient-corrected exchange-correlation functional (LDA)-CAPZ are used for the exchange-correlation functional. 

We use generalized gradient approximations (GGA) to the exchange-correlation (XC) functional of DFT for all NMR calculations that are reported here. All current dependent terms (28) in the XC functional are neglected. The M(CO)6 and [MO4]2-, M = Cr, Mo, W, calculations (7) employed the BP86 functional (29,30). All other calculations were performed with the PW91 GGA (57). 

Here Ho is the kinetic energy operator of valence electrons Vps is the pseudopotential [40,41] which defines the atomic core. V = eUn(r) is the Hartree energy which satisfies the Poisson equation ArUn(r) = —4nep(r) with proper boundary conditions as discussed in the previous subsection. The last term is the exchange-correlation potential Vxc [p which is a functional of the density. Many forms of 14c exist and we use the simplest one which is the local density approximation [42] (LDA). One may also consider the generalized gradient approximation (GGA) [43,44] which can be implemented for transport calculations without too much difficulty [45]. Importantly a self-consistent solution of Eq. (2) is necessary because Hks is a functional of the charge density p. One constructs p from the KS states Ts, p(r) = (r p r) = ns Fs(r) 2, where p is the density matrix,... 

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