Local exchange

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 ... 

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. 

If no correlation is introduced (ec = 0), the KS equations reduce to the well known Xa method proposed by Slater22 as a simplification of the Hartree-Fock scheme with a local exchange operator ... 

Heidin, L., and B. I. Lundqvist. 1971. Explicit local exchange-correlation potentials. J. Phys. C 4, 2064. 

Von Barth, U. and Hedin, L. 1972. A local exchange-correlation potential for the spin polarized case I. J. Phys. C Solid State Phys. 5 1629-1642. 

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] ... 

Gruschus J, Kuki A, multiconfigurational computation of superexchange including correlation and non-local exchange (in progress)... 

Ellegren H (2002) Dosage compensation do birds do it as well Trends Genet 18 25—28 Farris SD, Rubio ED, Moon JJ, Gombert WM, Nelson BH, Krumm A (2005) Transcription-induced chromatin remodeling at the c-myc gene involves die local exchange of histone H2A.Z. J Biol Chem... 

The Xa multiple scattering method generates approximate singledeterminant wavefunctions, in which the non-local exchange interaction of the Hartree-Fock method has been replaced by a local term, as in the Thomas-Fermi-Dirac model. The orbitals are solutions of the one-electron differential equation (in atomic units)... 

An alternative way to see this effect is to consider the enhancement factor over local exchange in PW91. We may write any GGA energy, for spin-unpolarized systems, as... 

Fig. 11. Enhancement factor over local exchange, F (fs,s), for spin-unpolariad systems in the PW91 GGA...

In the remainder of this section, we give a brief overview of some of the functionals that are most widely used in plane-wave DFT calculations by examining each of the different approaches identified in Fig. 10.2 in turn. The simplest approximation to the true Kohn-Sham functional is the local density approximation (LDA). In the LDA, the local exchange-correlation potential in the Kohn-Sham equations [Eq. (1.5)] is defined as the exchange potential for the spatially uniform electron gas with the same density as the local electron density ... 

In addition to the C4V structure of the M(CO)3E9 ions shown in Fig. 5, a second isomer has been characterized for both the tin and lead clusters. Sn9W(CO)3" and Pb9Mo(CO)3 have both been isolated in a Cs structure in which the M(CO)3 fragment occupies a 5-coordinate waist vertex site instead of the 4-coordinate capping site shown in Fig. 5 [55, 60]. Both of these isomers give the same NMR spectra, which is consistent with the higher symmetry 4 structures. We interpret these data to indicate that there is local exchange between the 4-coordinate and 5-coordinate sites, but this interpretation is controversial [34, 55]. 

Fig. 3.7 The Heitler-London configuration A(1) B(2) and A(2) B(1) (a) and (b) respectively, where 0A and represent the atomic 1s orbitals centred on atoms A and respectively, and 1 and 2 represent the coordinates of the two (indistinguishable) electrons, (c) The molecular orbital basis function in the singlet state where electrons 1 and 2 have opposite spin, (d) The up and down spin eigenfunctions corresponding to local exchange fields of opposite sign on A and B.

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