Gradient expansion approximation

Later on Cahn and Hilliard presented some thermodynamic estimates for the nucleation of liquid in vapour. Values of AO and the composition profiles c(r) of the embryos have been estimated using the mean-field and gradient expansion approximations for the free energy functional F c(7 ). A number of qualitative features in variation... 

To improve upon this, from a chemical point of view rather crude assumption, the most widely employed corrections are based on using not only the density, but also its gradient. These corrections form the so-called generalized gradient approximation, GGA, or gradient expansion approximation (GEA) methods ... 

For the kinetic energy functional, regular gradient expansion approximation (GEA)74 takes the following form ... 

Fig. 2. Accuracy of approximations to —9 spa PB ° a ne(l fr°m gradient expansion approximation (zeroth- or second-order GEA) for the HF...HF dimer at various intermolecular separations (d(FF)). The difference between the Coulomb energy (J) obtained using approximate -Ta in Eqs. 31-32 and that (Jref) obtained from supermolecular Kohn-Sham calculations, which do not...

We attribute the failure of the gradient expansion approximation in approximating Tgad[pA, Pb] in the case where pa and pB do not overlap significantly as it is the case of two weakly interacting molecules to the artificial attraction arising from the second-order GEA contribution to T ad[pA, Pb - Interestingly, in the recent ar-... 

Opposite to its exchange-correlation counterpart, the functional Ts p can be very reasonably approximated by means of gradient expansion approximation truncated to the second order86. The accuracy of the corresponding T ad[pA, Pb functional is, however, very disappointing as discussed in the previous section. The encouraging experience with GGA route for Exc[p] suggests a similar solution for T d pA,pB. ... 

The main issue involved in using DFT and the KS scheme pertains to construction of expressions for the XC functional, Exc[n], containing the many-body aspects of the problems (1.38). The main approaches to this issue are (a) local functionals the Thomas Fermi (TF) and LDA, (b)semilocal or gradient-dependent functionals the gradient-expansion approximation (GEA) and generalized gradient approximation (GGA), and (c) nonlocal functionals hybrids, orbital functionals, and SIC. For detailed discussions the reader is referred to the reviews [257,260-272]. 

The approximations most often employed in atomic and molecular studies are the local density approximation (LDA), in which the energy density is a function of the density only, and its extension, the gradient expansion approximation and/or generalized gradient approximation (GEA/GGA), where besides the local density, the nonlocal density gradient is included in the energy density function formulation that is,... 

There are approximations that go beyond the LDA. They consider that the dependence ExcLp] may be non-local i.e., E c may depend on p at a given point (locality), but also on p nearby (non-locality). When we are at a point, what happens further off depends not only on p at that point, but also the gradient of p at the point, etc. This is how the idea of the gradient expansion approximation (GEA) appeared... 

GEA Gradient Expansion Approximation A class of the DFT functionals that take... 

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