Exchange-correlation

This equation is usually solved self-consistently . An approximate charge is assumed to estimate the exchange-correlation potential and to detennine the Flartree potential from equation Al.3.16. These approximate potentials are inserted in the Kolm-Sham equation and the total charge density is obtained from equation A 1.3.14. The output charge density is used to construct new exchange-correlation and Flartree potentials. The process is repeated nntil the input and output charge densities or potentials are identical to within some prescribed tolerance. 

This ionic potential is periodic. A translation of r to r + R can be acconnnodated by simply reordering the sunnnation. Since the valence charge density is also periodic, the total potential is periodic as the Hartree and exchange-correlation potentials are fiinctions of the charge density. In this situation, it can be shown that the wavefiinctions for crystalline matter can be written as... 

The density is computed as p(r) = 2. n i ). (/ )p. Often, p(r) is expanded in an AO basis, which need not be the same as the basis used for the and the expansion coefficients of p are computed in tenns of those of the It is also connnon to use an AO basis to expand p (r) which, together with p, is needed to evaluate the exchange-correlation fiinctionaTs contribution toCg. 

Professor Axel Becke of Queens University, Belfast has been very actively involved in developing and improving exchange-correlation energy functionals. For a good recent overview, see ... 

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... 

LDA, these effects are modelled by the exchange-correlation potential In order to more accurately... 

In principle, DFT calculations with an ideal exchange-correlation fiinctional should provide consistently accurate energetics. The catch is, of course, that the exact exchange-correlation fiinctional is not known. 

Godby R W, Schluter M and Sham L J 1988 Self-energy operators and exchange-correlation potentials in semiconductors Phys. Rev. B 37 10159-75... 

In Ecjuation (3.47) we have written the external potential in the form appropriate to the interaction with M nuclei. , are the orbital energies and Vxc is known as the exchange-correlation functional, related to the exchange-correlation energy by ... 

The total electron density is just the sum of the densities for the two types of electron. The exchange-correlation functional is typically different for the two cases, leading to a set of spin-polarised Kohn-Sham equations ... 

In addition to the energy terms for the exchange-correlation contribution (which enables the total energy to be determined) it is necessary to have corresponding terms for the potential, Vxc[p(i )]/ which are used to solve the Kohn-Sham equations. These are obtained as the appropriate first derivatives using Equation (3.52). 

L. iitortunately, this simple approach does not work well, but Becke has proposed a strategy which does seem to have much promise [Becke 1993a, b]. In his approach the exchange-correlation energy Exc is written in the following form ... 

Vgiec and Vxc represent the electron-nuclei, electron-electron and exchange-correlation dionals, respectively. The delta function is zero unless G = G, in which case it has lue of 1. There are two potential problems with the practical use of this equation for a croscopic lattice. First, the summation over G (a Fourier series) is in theory over an rite number of reciprocal lattice vectors. In addition, for a macroscropic lattice there effectively an infinite number of k points within the first Brillouin zone. Fortunately, e are practical solutions to both of these problems. 

The first three terms in Eq. (10-26), the election kinetic energy, the nucleus-election Coulombic attraction, and the repulsion term between charge distributions at points Ti and V2, are classical terms. All of the quantum effects are included in the exchange-correlation potential... 

In this equation Exc is the exchange correlation functional [46], is the partial charge of an atom in the classical region, Z, is the nuclear charge of an atom in the quantum region, is the distance between an electron and quantum atom q, r, is the distance between an electron and a classical atom c is the distance between two quantum nuclei, and r is the coordinate of a second electron. Once the Kohn-Sham equations have been solved, the various energy terms of the DF-MM method are evaluated as... 

Her workers to fit the exchange-correlation potential and the charge density (in the Coulomb potential) to a linear combination of Gaussian-typc functions. 

DFT methods compute electron correlation via general functionals of the electron density (see Appendix A for details). DFT functionals partition the electronic energy into several components which are computed separately the kinetic energy, the electron-nuclear interaction, the Coulomb repulsion, and an exchange-correlation term accounting for the remainder of the electron-electron interaction (which is itself... 

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