The Generalized Gradient Approximation

The argument of the function F is the reduced density gradient for spin a 

For the function F two main classes of realizations have been put forward (see in particular Adamo, di Matteo, and Barone, 1999). The first one is based on a GGA exchange functional developed by Becke, 1988b. As outlined above, this functional is abbreviated simply as B (sometimes one also finds B88) 

P is an empirical parameter that was determined to 0.0042 by a least-squares fit to the exactly known exchange energies of the rare gas atoms He through Rn. In addition to the sum rules, this functional was designed to recover the exchange energy density asymptotically far from a finite system. 

Functionals which are related to this approach include among others the recent FT97 functional of Filatov and Thiel, 1997, the PW91 exchange functional (Perdew, 1991, and Burke, Perdew, and Wang, 1998) and the CAM(A) and CAM(B) functionals developed by Handy and coworkers (Laming, Termath, and Handy, 1993). 

The second class of GGA exchange functionals use for F a rational function of the reduced density gradient. Prominent representatives are the early functionals by Becke, 1986 (B86) and Perdew, 1986 (P), the functional by Lacks and Gordon, 1993 (LG) or the recent implementation of Perdew, Burke, and Emzerhof, 1996 (PBE). As an example, we explicitly write down F of Perdew s 1986 exchange functional, which, just as for the more recent PBE functional, is free of semiempirical parameters  

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Philipsen, P.H.T. and Baerends, E.J. (2000) Relativistic calculations to assess the ability of the generalized gradient approximation to reproduce trends in cohesive properties of solids. Physical Review B - Condensed Matter, 61, 1773-1778. 

Patton, D. C., Pederson, M. R., 1997, Application of the Generalized-Gradient Approximation to Rare-Gas Dimers , Phys. Rev. A, 56, R2495. 

Perdew, J. P., Kurth, S., Zupan, A., Blaha, P., 1999, Accurate Density Functional With Correct Formal Properties A Step Beyond the Generalized Gradient Approximation , Phys. Rev. Lett., 82, 2544. 

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. 

Unless otherwise stated, all of our calculations used the generalized gradient approximation as defined by the Perdew-Wang 91 functional. In shorthand, we used PW91-GGA calculations. Unless otherwise stated k points were placed in reciprocal space using the Monkhorst-Pack method. Some specific details for the calculations from each section of the chapter are listed below. 

P. H. T. Philipsen and E. J. Baerends, Relativistic Calculations to Assess the Ability of the Generalized Gradient Approximation to Reproduce Trends in Cohesive Properties of Solids, Phys. Rev. B 61 (2000), 1773. 

M. Fuchs, J. F. F. Da Silva, C. Stampfl, J. Neugebauer, and M. Scheffler, Cohesive Properties of Group-Ill Nitrides A Comparative Study of All-Electron and Pseudopotential Calculations Using the Generalized Gradient Approximation, Phys. Rev. B 65 (2002), 245212. 

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. 

Density-Functional Theory. Transition metals pose a problem for classical quantum chemical methods like self-consistent field (SCF), perturbation theory, configuration interaction (Cl), and variations on these methods, because of the very large electron correlation. SCF underestimates binding substantially, and post-SCF methods are so expensive for transition metals that one can do a calculation only on models with few atoms. DFT on the other hand is relatively cheap it is about as expensive as SCF. Moreover, with the development of the generalized-gradient approximations it is also reasonably accurate. A large majority of quantum chemical... 

More recently, Zhao et al. (2001a) employed the FLAPW method within the generalized gradient approximation (GGA) to investigate further electronic and magnetic properties of ordered Gai xMnxAs alloys with low and high Mn concentrations, 0.031 x 0.5. 

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