Local density approximations, exchange correlation functionals

Exchange-Correlation Energy Functionals Local Density Approximation... 

A review of the approximations in any time-depedendent density functional calculation of excitation energies is given. The single-pole approximation for the susceptibility is used to understand errors in popular approximations for the exchange-correlation kernel. A new hybrid of exact exchange and adiabatic local density approximation is proposed and tested on the He and Be atoms. 

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. 

The local density approximation is highly successful and has been used in density functional calculations for many years now. There were several difficulties in implementing better approximations, but in 1991 Perdew et al. successfully parametrised a potential known as the generalised gradient approximation (GGA) which expresses the exchange and correlation potential as a function of both the local density and its gradient ... 

In principle, the KS equations would lead to the exact electron density, provided the exact analytic formula of the exchange-correlation energy functional E was known. However, in practice, approximate expressions of Exc must be used, and the search of adequate functionals for this term is probably the greatest challenge of DFT8. The simplest model has been proposed by Kohn and Sham if the system is such that its electron density varies slowly, the local density approximation (LDA) may be introduced ... 

Proynov, E. I., A. Vela, and D. R. Salahub. 1994. Gradient-free exchange-correlation functional beyond the local-spin-density approximation. Phys. Rev. A 50, 2421. 

Density-functional theory, developed 25 years ago (Hohenberg and Kohn, 1964 Kohn and Sham, 1965) has proven very successful for the study of a wide variety of problems in solid state physics (for a review, see Martin, 1985). Interactions (beyond the Hartree potential) between electrons are described with an exchange and correlation potential, which is expressed as a functional of the charge density. For practical purposes, this functional needs to be approximated. The local-density approximation (LDA), in which the exchange and correlation potential at a particular point is only a function of the charge density at that same point, has been extensively tested and found to provide a reliable description of a wide variety of solid-state properties. Choices of numerical cutoff parameters or integration schemes that have to be made at various points in the density-functional calculations are all amenable to explicit covergence tests. 

The term Exc[p] is called the exchange-correlation energy functional and represents the main problem in the DFT approach. The exact form of the functional is unknown, and one must resort to approximations. The local density approximation (LDA), the first to be introduced, assumed that the exchange and correlation energy of an electron at a point r depends on the density at that point, instead of the density at all points in space. The LDA was not well accepted by the chemistry community, mainly because of the difficulty in correctly describing the chemical bond. Other approaches to Exc[p] were then proposed and enable satisfactory prediction of a variety of observables [9]. 

The problem is that the exchange correlation functional Exc is unknown. Approximate forms have to be used. The most well-known is the local density approximation (LDA) in which the expressions for a uniform electron gas are... 

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