Non-local gradient corrections

We shall discuss the chain termination reaction, Eq. (2), in the present study based on Density Functional Theory at a high level with non-local gradient corrections. A variety of reaction pathways and the associated transition state structures as well as the reaction barriers will be analyzed. 

The molecular calculations are performed in the framework of the density ftinctional theory using the deMon program package (St Amant, 1990 1991). Non local gradient corrections are added for the exchange and correlation terms. 

Finally, in DFT calculations there is the question of the density functional. The accuracy of DFT calculations varies greatly with the choice of functional. The exact functional gives exact results. Very crude functionals give very inaccurate results. Fvmctionals used in the recent past can he grouped into three classes (a) local (b) non-local/gradient-corrected (c) hybrid. Overall, the relative accuracy is... 

DFl BLYP Becke exchange functional (Slater exchange plus correction involving the gradient of electron density) (21). Lee, Yang, Parr correlation functioned with local and non-local (gradient-dependent) terms (22). 

In some cases, the use of the generalized gradient approximation (GGA) for exchange and correlation leads to the appearance of very short-ranged oscillations in the pseudo-potentials (see Fig. 6.4). These oscillations are artifacts of the GGA that usually disappear when non-local core corrections are considered. Nevertheless, they do not pose a real threat for plane-wave calculations, since they are mostly filtered out by the energy cutoff. 

I he function/(r) is usually dependent upon other well-defined functions. A simple example 1)1 j functional would be the area under a curve, which takes a function/(r) defining the curve between two points and returns a number (the area, in this case). In the case of ni l the function depends upon the electron density, which would make Q a functional of p(r) in the simplest case/(r) would be equivalent to the density (i.e./(r) = p(r)). If the function /(r) were to depend in some way upon the gradients (or higher derivatives) of p(r) then the functional is referred to as being non-local, or gradient-corrected. By lonlrast, a local functional would only have a simple dependence upon p(r). In DFT the eiK igy functional is written as a sum of two terms ... 

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. 

Gradient Corrected Density Functional Models See Non-Local Density Functional Models. 

Non-local density functional models or gradient corrected density functional models = density functional models which take explicitly into account the nonuniformity in electron distributions, thus improving on the local density model. 

The simplest exchange-correlation functionals, derived from the density alone, give reasonable results for many molecular properties. However, it has been found that adding terms proportional to the local gradient of the density in both the exchange and correlation integrands leads to much more accurate results. Most modern DFT computations use these so-called non-local or gradient-corrected functionals. 

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