Approximations generalized gradient

The correlation functional requires a slightly different scheme, as, on the one hand, the RLDA is not known completely, and, on the other, some GGAs for Ec (Lee et al. 1988) are not based on the LDA. Therefore, only one overall correction factor for the complete correlation part of the GGA has been used, 

In contrast to the rather inaccurate RLDA, the RGGA allows an examination of the importance of relativistic corrections to Exc[n] for the properties of molecules and solids. This question has been investigated both for noble-metal compounds (Mayer etal. 1996 Varga et al. 1999) and for metallic gold and platinum (Schmid etal. 1998) within the framework of LAPW calculations (Blaha et al. 1995). Prototype results for CU2 and AU2 are given in Table 4.4. 

It turns out that even for Au, which usually exhibits the effects of relativity most clearly (Pyykkb 1988), the impact of the correction factors PX/C on the molecular binding properties is marginal, i.e. smaller than usual differences between two basis sets. It seems worthwhile pointing out that in the case of the dissociation energy the 

On the other hand, a comparison of the (R)GGA results in Tables 4.4 and 4.5 with experiment reveals the limitations of GGAs for heavy elements. For both Au2 and the Au metal the gradient terms overcorrect the errors of the LDA. It seems worthwhile noting that the particularly large deviations in the case of the metal are not due to the usual neglect of the spin-orbit coupling for the valence electrons within the LAPW 

While spin-orbit coupling contributes significantly to the cohesive energy, its effect is too small to explain the differences between GGA and experimental data in Table 4.5. Thus, on the one hand, the results in Tables 4.4 and 4.5 illustrate the role of error cancellation, in particular for the LDA. On the other hand, they indicate the need for fundamentally new concepts for Exc[ri] (such as implicit functionals) in the relativistic regime. 

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Massobrio C, Pasquarello A and Corso A D 1998 Structural and electronic properties of small Cu clusters using generalized-gradient approximations within density functional theory J. Chem. Phys. 109 6626... 

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. 

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. 

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., Wang, Y., 1986, Accurate and Simple Density Functional for the Electronic Exchange Energy Generalized Gradient Approximation , Phys. Rev. B, 33, 8800. 

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] ... 

Their use in homogeneous catalysis has nowadays a mostly qualitatively value, being replaced by the more accurate generalized gradient approximation (GGA), wich will be discussed in the next section. 

If the main limitations of HF theory are overcome by the introduction of electron correlation, those of density functional theory are expanded by the use of more accurate functionals. These functionals, that improve the uniform gas description of the LDA approach, are labeled as non-local or Generalize Gradient Approximation (GGA). 

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