Exchange-correlation functional generalized gradient approximation

Gritsenko, O. V., Schipper, P. R. T., Baerends, E. J., 1997, Exchange and Correlation Energy in Density Functional Theory. Comparison of Accurate DFT Quantities With Traditional Hartree-Fock Based Ones and Generalized Gradient Approximations for the Molecules Li2, N2, F2 , J. Chem. Phys., 107, 5007. 

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. 

The density functional calculations were performed using the Vienna Ab Initio Simulation Package (VASP). ° The spin-polarized generalized gradient approximation, Perdue—Wang exchange correlation function, and ultrasoft pseudopotentials were used. ... 

Three types of exchange/correlation functionals are presently in use (i) functionals based on the local spin density approximation, (ii) functionals based on the generalized gradient approximation, and (iii) functionals which employ the exact Hartree-Fock exchange as a component. The first of these are referred to as local density models, while the second two are collectively referred to as non-local models or alternatively as gradient-corrected models. 

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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