Gradient dependent functionals

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]. 

One of the main shortcomings of local density functionals successfully addressed by the gradient dependent functionals is the overestimate of the binding energy. Bond length is predicted rather well with DFT methods, with a tendency to underestimate the weaker bonds. 

This chapter is intended to give the reader an idea of what types of functionals exist, and to describe what their main features are, separately for local functionals (TF, LDA and Xa Sec. 5.1), semilocal, or gradient-dependent, functionals (GEA and GGA Sec. 5.2), and nonlocal functionals (hybrids, orbital functionals such as meta-GGAs, EXX and SIC, and integral-dependent functionals such as ADA Sec. 5.3). This chapter does deal only most superficially with the actual construction of these functionals. For more... 

All three terms are again functionals of the electron density, and functionals defining the two components on the right side of Equation 57 are termed exchange functionals and correlation functionals, respectively. Both components can be of two distinct types local functionals depend on only the electron density p, while gradient-corrected functionals depend on both p and its gradient, Vp. ... 

Chemokines, a superfamily of small (8 to 14 kd) chemoattractant cytokines, were originally identified by their capacity to mediate gradient-dependent cell migration, accompanied by Ca2+ flux, in vitro. Sequence homology of mammal, bird, and fish chemokines indicates that this family of proteins is highly conserved throughout evolution. Chemokines are multifunctional molecules with a vast repertoire of specialized functions in different organs. Broadly, chemokine actions are associated with cell adhesion, cytokine secretion, cellular... 

For an experiment concerning the investigation of the dependence on the concentration of a catalyst Y, [Y], the molar concentration of the catalyst could be varied in a linear way, that is, M [Y] = (3t, where p is the concentration gradient. If the dependence function is known, for example, D kobs = /cy[Y], the mathematical model will assume the forms... 

The functional gradient of 5tot indicates the function pointing in the direction of fastest increase. Gradients depend on an inner product because it provides a measure of distance for functions [2]. One choice of an inner product for functions is the L2 inner product, defined by... 

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