Nonlocal exchange functional

The situation at present is that the nonlocal exchange functional suggested by Becke (B) in 198821 has been established as a standard expression in DFT calculations. The choice of the best correlation functional is less obvious than the choice of the exchange functional. The presently most popular correlation functionals are those of Perdew (P86)22, Lee et al. (LYP)23, Perdew and Wang (PW91)24 and Vosko et al. (VWN)25. 

This argument as well as (2.14) and (2.15) indicate that one needs a rather nonlocal exchange functional to reproduce the —1/r behavior The component of which cancels the self-interaction in the Hartree potential must be as nonlocal as h itself, which is a quite nonlocal Coulomb integral. 

The form of F(p, Vp) varies and often contains empirical parameters. F(p, Vp) is frequently termed a gradient or non-local correction, since the potential is computed not only as a funcion of the location but also as a function of the Laplacian of the charge density, Vp(r). Of course, even these nonlocal functionals are perfectly local in a mathematical sense. The development of nonlocal exchange functionals is dominated by Becke, who has published a number of increasingly refined mathematical expressions for F(p, Vp) since 1983 (B). Nonlocal correlation functionals have been proposed by Perdew (P), Lee, Yang, and Parr (LYP), and Perdew and Wang (PW). The most commonly used nonlocal functional combinations are BP, BLYP and BPW. Earlier correction schemes like the self-interaction correction by Stoll, PreuB, and Pavli-dou (SPP) have been found to be inferior to the gradient-corrected functionals in most cases and seldom appear in the literature. 

The first term is the familiar one-electron operator, the second term represents the Coulomb potential, and the third term is called exchange-correlation potential. HF and DFT differ only in this last term. In HF theory there is only a nonlocal exchange term, while in DFT the term is local and supposed to cover both exchange and correlation. It arises as a functional derivative with respect to the density ... 

Glossman, M. D., L. C. Baibas, A. Rubio, and I. A. Alonso. 1994. Nonlocal Exchange and Kinetic Energy Density Functionals with Correct Asymptotic Behavior for Electronic Systems. Int. I. Q. Chem. 49, 171. 

Glossman, M. D., Baibas, L. C., Rubio, A. and Alonso, J. A. Nonlocal exchange and kinetic energy density functionals with correct asymptotic behavior for electronic systems, Int.J. Quantum ( hem., 49 (1994), 171-184... 

E.g. Perdew JP (1995) Nonlocal density functionals for exchange and correlation theory and application. In Ellis DE (ed) Density functional theory of molecules, clusters, and solids. Kluwer, Dordrecht, The Netherlands... 

Langreth, D.C. and Mehl, M.J. (1981). Easily implementable nonlocal exchange-correlation energy functional, Phys. Rev. Lett. 47, 446 50. 

The asymptotic structure of the exchange potential vx(r) was derived via the relationship between density functional theory and many-body perturbation theory as established by Sham26. The integral equation relating vxc(r) to the nonlocal exchange-correlation component Exc(r, rf ) of the self-energy (r, r7 >) is... 

Rushton PP, Tozer DJ, Clark SJ (2002) Nonlocal density-functional description of exchange and correlation in silicon, Phys Rev B, 65 235203... 

In a recent paper [58] we try to investigate the role of the local and nonlocal exchange and correlation terms. One-electron energies of S04 have been obtained from local and non-local density functional theory (DFT). The Gaussian92 program [59] was used, in which the general form of a hybrid density functional has the following form ... 

J. P. Perdew, Nonlocal Density Functionals for Exchange and Correlation Theory and Applications in Density Functional Theory of Molecules, Clusters, and Solids, edited by D. E. Ellis (Kluwer, Netherlands, 1995), p. 47-66. 

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