Density functional theory exchange-correlation holes

The concept of the exchange-correlation hole is widely used in density functional theory and its most relevant properties are the subject of the following section. 

Pollet R, Colonna F, Leininger T, Stoll H, Werner HJ, Savin A (2003) Exchange-correlation energies and correlation holes for some two- and four-electron atoms along a nonlinear adiabatic connection in density functional theory, Int J Quant Chem, 91 84-93... 

A system of fundamental theoretical importance in many-body theory is the uniform-density electron gas. After decades of effort, exchange-correlation effects in this special though certainly not trivial system are by now well understood. In particular, sophisticated Monte Carlo simulations have provided very useful information (5) and have been conveniently parametrized by several authors (6). If the exchange-correlation hole function at a given reference point r in an atomic or molecular system is approximated by the hole function of a uniform electron gas with spin-densities given by the local values of p (r) and Pp(C obtain an... 

Currently, research in our laboratory continues on real-space models of exchange and correlation hole functions in inhomogeneous systems. We anticipate that this work will ultimately generate completely non-empirical parameter-free beyond-LDA density functional theories. The quality of molecular dissociation energies and related properties obtainable with existing semi-empirical gradient-corrected DFTs approaches chemical accuracy, and we hope these future theoretical developments will continue this trend. 

The exchange-correlation hole is of considerable interest in density functional theory, as the exact exchange-correlation energy may be expressed in terms of this hole. By use of the Hellmann-Feynman theorem, one may write the exchange-correlation energy as the electrostatic interaction between the density and the hole, averaged over coupling constant[13], i.e.,... 

Pistol, M. E. Ahnbladh, C. O. Adiabatic connections and properties of coupling-integrated exchange-correlation holes and pair densities in density functional theory. Chem. Phys. Lett. 2009, 480, 136-139. 

Density-functional theory, even with rather crude approximations such as LDA and GGA, is often better than Hartree-Fock LDA is remarkably accurate, for instance, for geometries and frequencies, and GGA has also made bond energies quite reliable. Therefore, the aura of mystery appeared around DFT (see discussion of this by Baerends and Gritsenko [367]). The simple truth is not that LDA/GGA is particularly good, but that Hartree-Fock is rather poor in the two-electron chemical-bond description. This becomes clear when one considers the statistical two-electron distribution, which is usually cast in terms of the exchange-correlation hole the decrease in probabihty to find other electrons in the neighborhood of a reference electron, compared to the (unconditional) one-electron probabihty distribution [337]. 

Fe—S dimers, 38 443-445 map, four-iron clusters, 38 458 -functional theory, 38 423-467 a and b densities, 38 440 broken symmetry method, 38 425 conservation equation, 38 437 correlation for opposite spins and Coulomb hole, 38 439-440 electron densities, 38 436 exchange energy and Fermi hole, 38 438-439... 

Fig. 11.12. The long-chased electron correlation dragon is finally found in its correlation hole, and we have an ratceptional opportunity to see what it looks like. Correlation potential-efficiency analysis of various DFT methods and cranpariscxi with the exact theory for the harmonic helium atom (with the force constant k = ) according to Kais et al. Panel (a) shows correlation potential Vc (which is less important than the exchange potential) as a function of the radius r (a) and of density p (b). The same notation is used as in Fig. 11.10. The DFT potentials produce plots that differ widely from the exact correlation potential...

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