Exchange-correlation functionals density functional theory

Key words Density functional theory - Exchange-correlation functional - Fractional number of electrons -Self interaction error - Derivative discontinuity... 

B. Santra, A. Michaelides, M. Fuchs, A. Tkatchenko, C. Filippi, M. Scheffler, On the accuracy of density-functional theory exchange-correlation functionals for H bonds in small water clusters. II. The water hexamer and Van der Waals interactions. J. Chem. Phys. 129(19), 194111 (2008)... 

Current Density Functional Theory (CDFT). - The Kohn-Sham density functional expression has to be modified when magnetic fields are present so that the density functional depends on the paramagnetic current density. A local exchange-correlation term then has the form... 

Density frmctional theory is also derived from first principles but is fundamentally different in that it is not based on the wavefunction but instead on the electron density of the A -particle system 1 1. Hohenberg and Kohnl showed that the energy for a system is a unique frmctional of its electron density. The true exchange-correlation functional necessary to provide the exact DFT solution, however, is unknown. The accuracy of density functional theory (DFT) is then limited to quality of the exchange-correlation functional that is used. 

Becke A D 1995 Exchange-correlation approximations in density-functional theory Modern Eiectronic Structure Theory vol 2, ed D R Yarkony (Singapore World Scientific) pp 1022-46... 

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

There is no systematic way in which the exchange correlation functional Vxc[F] can be systematically improved in standard HF-LCAO theory, we can improve on the model by increasing the accuracy of the basis set, doing configuration interaction or MPn calculations. What we have to do in density functional theory is to start from a model for which there is an exact solution, and this model is the uniform electron gas. Parr and Yang (1989) write... 

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 ... 

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. 

Encl[p] is the non-classical contribution to the electron-electron interaction containing all the effects of self-interaction correction, exchange and Coulomb correlation described previously. It will come as no surprise that finding explicit expressions for the yet unknown functionals, i. e. T[p] and Encl[p], represents the major challenge in density functional theory and a large fraction of this book will be devoted to that problem. 

We have repeatedly indicated that usually the exchange contributions are significantly larger in absolute numbers than the corresponding correlation effects. Therefore, an accurate expression for the exchange functional in particular is a prerequisite for obtaining meaningful results from density functional theory. However, we have seen in Chapters 1 and 5 that the... 

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