Uniform electron gas model

The uniform electron gas model gives us the first approximation for Exc,... 

Gradient-Corrected and Hybrid Functionals. The LDA and LSDA are based on the uniform-electron-gas model, which is appropriate for a system where p varies slowly with position. The integrand in the expression (15.126) for E is a function of only p, and the integrand in is a function of only p and pP. Functionals that go... 

Where g"(0), the curvature of the one-matrix model at the origin, equals -1/5 for the uniform electron gas models. Notice that this formula does not depend on the value of p in Equation 1.104. 

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... 

To make matters worse, the use of a uniform gas model for electron density does not enable one to carry out good calculations. Instead a density gradient must be introduced into the uniform electron gas distribution. The way in which this has been implemented has typically been in a semi-empirical manner by working backwards from the known results on a particular atom, usually the helium atom (Gill, 1998). It has thus been possible to obtain an approximate set of functions which often serve to give successful approximations in other atoms and molecules. As far as I know, there is no known way of yet calculating, in an ab initio manner, the required density gradient which must be introduced into the calculations. 

At the center of the approach taken by Thomas and Fermi is a quantum statistical model of electrons which, in its original formulation, takes into account only the kinetic energy while treating the nuclear-electron and electron-electron contributions in a completely classical way. In their model Thomas and Fermi arrive at the following, very simple expression for the kinetic energy based on the uniform electron gas, a fictitious model system of constant electron density (more information on the uniform electron gas will be given in Section 6.4) ... 

In this section we introduce the model system on which virtually all approximate exchange-correlation functionals are based. At the center of this model is the idea of a hypothetical uniform electron gas. This is a system in which electrons move on a positive background charge distribution such that the total ensemble is electrically neutral. The number of elec-... 

This is a very drastic approximation since, after all, the density in our actual system is certainly anything but constant and does not even come close to the situation characteristic of the uniform electron gas. As a consequence, one might wonder whether results obtained with such a crude model will be of any value at all. Somewhat surprisingly then, experience tells us that the local (spin) density approximation is actually not that bad, but rather deliv-... 

The distinction is probably best indicated by example. Following from Eq. (8.7) and the discussion in Section 8.1.2, the exchange energy for the uniform electron gas can be computed exactly, and is given by Eq. (8.23) with the constant a equal to. However, the Slater approach takes a value for a of 1, and the Xa model most typically uses j. All of these models have the same local dependence on the density, but only the first is typically referred to as LDA, while the other two are referred to by name as Slater (S) and Xa. ... 

Density functional approaches to molecular electronic structure rely on the existence theorem [10] of a universal functional of the electron density. Since this theorem does not provide any direction as to how such a functional should be constructed, the functionals in existence are obtained by relying on various physical models, such as the uniform electron gas and others. In particular, the construction of an exchange-correlation potential that depends on the electron density only locally seems impossible without some approximations. Such approximate exchange-correlation potentials have been derived and applied with some success for the description of molecular electronic ground states and their properties. However, there is no credible evidence that such simple constructions can lead to either systematic approximate treatments, or an exact description of molecular electronic properties. The exact functional that seems to... 

Mathematical expressions for the functionals which are found in the Kohn-Sham operator are usually derived either from the model of a uniform electron gas or from a fitting procedure to calculated electron densities of noble gas atoms. Two different functionals are then derived. One is the exchange functional Fx and the other the correlation functional Fc, which are related to the exchange and correlation energies in ab initio theory. We point out, however, that the definition of the two terms in DFT is slightly different from ab initio theory, which means that the corresponding energies cannot be directly compared between the two methods. 

Furthermore, since the 2-integration in Equation 4 extracts only the spherical average of the hole function with respect to the reference point 1, details of its angular dependence are unimportant. Spherically symmetric hole-function models are therefore perfectly justified. These constraints, among others discussed elsewhere (4), are satisfied in any many-electron system, whether a helium atom, a transition-metal cluster, a uniform electron gas, etc.. To the extent that properties such as these contain the essential physics of exchange and correlation phenomena, hole-function models provide a simple and convenient alternative to traditional ab initio technology. 

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