Uniform electron gas

If the time-dependent wave function of electrons Vl(t) is known, the current density approach outlined in Section 2 can be used directly to calculate the energy loss and also to analyze the spatial distribution of related effects. This is the case when the energy loss to a free electron is considered. Assuming that the electron is initially at rest, we can describe its state in the projectile frame as a state of scattering in the projectile Coulomb field. The wave function in the laboratory frame is then given by the expression 

With the wave function (19), equation (16) for the current density can be transformed to the form 

After the integration over the longitudinal coordinate qp this equation acquires a familiar form 

Expression (26) permits to perform a detailed analysis of the energy transfer to a free electron as a function of the impact parameter . The hyper-geometric function, entering in this equation, can be expressed 

The main result of this treatment, equations (29) and (30), can be used, particularly, to present the perturbation and classical results as corresponding approximations for the corrective factor 3(17, In the former approach, the parameter 17 in equation (30) should be taken to be zero. Then, the energy loss (equation (29)) will be proportional to Zj. To derive the expression for 2(0, we just replace the Coulomb wave function Fq in equation (30) by the solution of equation (28) for 17 = 0  

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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... 

Early attempts at deducing functionals for the kinetic and exchange energies considered a non-interacting uniform electron gas. For such a system it may be shown that T[p] and K[p] are given as... 

In the Local Density Approximation (LDA) it is assumed that the density locally can be treated as a uniform electron gas, or equivalently that the density is a slowly varying... 

The correlation energy of a uniform electron gas has been determined by Monte Carlo methods for a number of different densities. In order to use these results in DFT calculations, it is desirable to have a suitable analytic interpolation formula. This has been constructed by Vosko, Wilk and Nusair (VWN) and is in general considered to be a very accurate fit. It interpolates between die unpolarized ( = 0) and spin polarized (C = 1) limits by the following functional. 

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. 

By carrying out this combination of semi-empirical procedures and retreating from the pure Thomas-Fermi notion of a uniform electron gas it has actually been possible, somewhat surprisingly, to obtain computationally better results in many cases of interest than with conventional ab initio methods. True enough, calculations have become increasingly accurate but if one examines them more closely one realizes that they include considerable semi-empirical elements at various levels. From the purist philosophical point of view, or what I call "super - ab initio" this means that not everything is being explained from first principles. 

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-... 

Here, exc(p(r)) is the exchange-correlation energy per particle of a uniform electron gas of density p( ). This energy per particle is weighted with the probability p(r) that there is in fact an electron at this position in space. Writing Exc in this way defines the local density approximation, LDA for short. The quantity exc(p(r)) can be further split into exchange and correlation contributions,... 

The exchange part, ex, which represents the exchange energy of an electron in a uniform electron gas of a particular density is, apart from the pre-factor, equal to the form found by Slater in his approximation of the Hartree-Fock exchange (Section 3.3) and was originally derived by Bloch and Dirac in the late 1920 s ... 

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-... 

Now suppose the energy density of the electrons in bulk jellium is given by nf(n), so that the density functional f(n) is the energy per electron of a uniform electron gas of density n in the positive background. (The integral of nf(n), plus inhomogeneity terms, is the quantity one minimizes to obtain the density profile.) The pressure in bulk jellium is then n(df/dn)9 so that... 

The inherent problems associated with the computation of the properties of solids have been reduced by a computational technique called Density Functional Theory. This approach to the calculation of the properties of solids again stems from solid-state physics. In Hartree-Fock equations the N electrons need to be specified by 3/V variables, indicating the position of each electron in space. The density functional theory replaces these with just the electron density at a point, specified by just three variables. In the commonest formalism of the theory, due to Kohn and Sham, called the local density approximation (LDA), noninteracting electrons move in an effective potential that is described in terms of a uniform electron gas. Density functional theory is now widely used for many chemical calculations, including the stabilities and bulk properties of solids, as well as defect formation energies and configurations in materials such as silicon, GaN, and Agl. At present, the excited states of solids are not well treated in this way. 

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... 

On the practical side, we note that nature provides a number of extended systems like solid metals [29, 30], metal clusters [31], and semiconductors [30, 32]. These systems have much in common with the uniform electron gas, and their ground-state properties (lattice constants [29, 30, 32], bulk moduli [29, 30, 32], cohesive energies [29], surface energies [30, 31], etc.) are typically described much better by functionals (including even LSD) which have the right uniform density limit than by those that do not. There is no sharp boundary between quantum chemistry and condensed matter physics. A good density functional should describe all the continuous gradations between localized and delocalized electron densities, and all the combinations of both (such as a molecule bound to a metal surface a situation important for catalysis). 

Table I The essentially-exact PW92 exchange-correlation energy per electron (in hartree) in a spin-unpolarized = 0) uniform electron gas of density parameter (in bohr), and the deviation (in hartree) of other approximations from PW92. (1 hartree = 27.21 eV = 627.5 kcal/moi.]...

Table III The ratio of approximate to exact exchange energy per electron of a uniform electron gas. For the approximations listed in Tables I and II but not listed here, this ratio is exactly 1. The Becke-Roussel exchange functional is a non-empirical meta-GGA based upon the hydrogen atom.

The LSD and PW91-GGA system-averaged holes agree at zero interelec-tronic separation, where both are nearly exact. In Sect. 2.3, we discuss how the near-universality of this on-top hole density provides the missing link between real atoms and molecules and the uniform electron gas. Except very close to the nucleus, the local on-top hole density is also accurately represented by LSD, even in the classically-forbidden tail region of the electron density [18]. 

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