Local exchange functional

Local exchange and correlation functionals involve only the values of the electron spin densities. Slater and Xa are well-known local exchange functionals, and the local spin density treatment of Vosko, Wilk and Nusair (VWN) is a widely-used local correlation functional. 

We ll now take a brief look at some sample functionals. The local exchange functional is virtually always defined as follows ... 

Hoe WM, Cohen A, Handy NC (2001) Assessment of a new local exchange functional OPTX, Chem Phys Lett, 341 319-328... 

The non-relativistic limit of the electron-electron interaction is the instantaneous Coulomb repulsion, also called the longitudinal part of the electron-electron interaction. The remainder is called the transversal part and mainly covers retardation and magnetic interactions (this separation is somewhat gauge dependent). The longitudinal and transversal part of the relativistic local exchange functional read... 

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. ... 

In an analogous way to the exchange functional we examined earlier, a local correlation functional may also be improved by adding a gradient correction. 

Calculations were done with a full-potential version of the LMTO method with nonoverlapping spheres. The contributions from the interstitial region were accounted for by expanding the products of Hankel functions in a series of atom-ce- -ered Hankels of three different kinetic energies. The corrected tetrahedron method was used for Brillouin zone integration. Electronic exchange and correlation contributions to the total energy were obtained from the local-density functional calculated by Ceperley and Alder " and parametrized by Vosko, Wilk, and Nusair. ... 

Na+/Ca2+ Exchangers. Figure 2 Schematic diagram of NCX localization and function in a pre- and postsynaptic structure (Reproduced from Annunziato L, Pignataro G, Di Renzo GF (2004) Pharmacol Rev 56 633-654). 

BH HLYP Exchange functional with equal-weighted contributions of Hartree-Fock, local spin density, and Becke functionals and Lee, Yang, and Parr s correlation functionals... 

First, the self-energy operator is replaced by a local exchange-correlation potential, which is given by the functional derivative of the exchange-correlation energy with respect to the electron density ... 

However, one feature of the HF potential is that it is not a local potential. In the case of perfect data (i.e. zero experimental error), the fitted orbitals obtained are no longer Kohn-Sham orbitals, as they would have been if a local potential (for example, the local exchange approximation [27]) had been used. Since the fitted orbitals can be described as orbitals which minimise the HF energy and are constrained produce the real density , they are obviously quite closely related to the Kohn-Sham orbitals, which are orbitals which minimise the kinetic energy and produce the real density . In fact, Levy [16] has already considered these kind of orbitals within the context of hybrid density functional theories. 

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] ... 

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