Gradient corrected approximations

Because electron densities vary rather strongly spatially near the nuclei, corrections to the above approximations to T p] and 7 are needed. One of the more commonly used so-called gradient-corrected approximations is that invented by Becke [M], and referred to as the Becke88 exchange functional ... 

Fig. 13. Total eneigy of ferromagnetically and antifenomagnetically ordered states of gadolinium with gradient correction approximations in the local spin density approximation (LSDA) and in LSDA + U. (From Harmon et aL 1995.)...

Our first example is the calculation of optical properties of long conjugated molecular chains [54]. For these systems, the local or gradient-corrected approximations can give overestimations of several orders of magnitude. The problem is related to a non-local dependence of the xc potential In a system with an applied electric field, the exact xc potential develops a linear part that counteracts the applied field [54,55]. This term is completely absent in both the LDA and the GGA, but is present in more non-local functionals like the EXX. 

On the Performance of Gradient-Corrected Approximation Functional and Polarizable Continuum Model in the Study of 1,2,3-Triazine in Water. 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Nonlocal density gradient corrections (GC)-local spin density (LDA) approximation. 

Bray, M. R., Deeth, R. J., Paget, V. J., Sheen, P. D., 1996, The Relative Performance of the Local Density Approximation and Gradient Corrected Density Functional Theory for Computing Metal-Ligand Distances in Werner-Type and Organometallic Complexes , Int. J. Quant. Chem., 61, 85. 

All calculations presented here are based on density-functional theory [37] (DFT) within the LDA and LSD approximations. The Kohn-Sham orbitals [38] are expanded in a plane wave (PW) basis set, with a kinetic energy cutoff of 70 Ry. The Ceperley-Alder expression for correlation and gradient corrections of the Becke-Perdew type are used [39]. We employ ah initio pseudopotentials, generated by use of the Troullier-Martins scheme [40], The coreradii used, in au, were 1.23 for the s, p atomic orbitals of carbon, 1.12 for s, p of N, 0.5 for the s of H, and 1.9, 2.0, 1.5, 1.97,... 

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