Homogeneous electron-gas

The uniform electron gas is perhaps the simplest of all the many-fermion systems that are important to chemistry. It is the basic ingredient of density functional (DF) approximations, both at the local density level and beyond, as 

QMC methods were first applied to the case of the electron gas by Ceper-ley in the late 1970s,and the results have been widely used in density functional theory. Only recently have these early calculations been extended by others to provide greater detail. Pickett and Broughton carried out VQMC calculations for the spin-polarized gas. Ortiz and Ballone used both VQMC and fixed-node DQMC for the spin-polarized gas in the density range most important to density functional theory. Kenny et al. performed VQMC and DQMC calculations for the nonpolarized homogeneous electron gas, incorporating relativistic effects via first-order perturbation theory. 

The 1994 calculations of Ortiz and Ballone yielded improved correlation energies, which can provide input for density functional theory computations. In particular, the DQMC results have been fitted to give improved parameters for the Perdew-Wang ° and the Perdew-Zunger computer codes. 

---

On the other hand the Thomas-Fermi method, which treats the electrons around the nucleus as a perfectly homogeneous electron gas, yields a mathematical solution that is universal, meaning that it can be solved once and for all. This feature already represents an improvement over the method which seeks to solve Schrodinger equation for every atom separately. This was one of the features that made people go back to the Thomas-Fermi approach in the hope of... 

Inserting equation (6-14) into equation (6-12) retrieves the p4/3 dependence of the exchange energy indicated in equation (3-5). This exchange functional is frequently called Slater exchange and is abbreviated by S. No such explicit expression is known for the correlation part, ec. However, highly accurate numerical quantum Monte-Carlo simulations of the homogeneous electron gas are available from the work of Ceperly and Alder, 1980. 

One obvious drawback of the LDA is that, when we replace unknown exchange-correlation energy by the known form of the exchange-correlation for a homogeneous electron gas in Equation (17), we have a problem in that cancelation of self-Coulomb... 

The development of the method started in the mid 1920 s with the work of Thomas and Fermi [8, 9]. The aim was to formulate an electronic structure theory for the solid state, based on the properties of a homogeneous electron gas, to which we introduce a set of external potentials (i.e. the atomic nuclei). The original formulation, with later additions by Dirac [10] and Slater [11], was, however, inadequate for accurate description of atomic and molecular properties, and it was not until the ground-breaking work of Kohn and coworkers in the mid 1960 s that the theory was put in a form more suited to computational chemistry [12,... 

Starting from a homogeneous electron gas and the above theorems, Kohn and Sham in 1965 proposed a solution to the problem of electronic interaction in many-electron systems based on defining and iteratively solving a set of coupled one-electron equations [13]. With this development DFT was put on similar... 

The initial implementation of DFT employed the so-called local density approximation, LDA (or, if we have separate a and [i spin, the local spin density approximation, LSDA). The basic assumption is that the density varies only slowly with distance -which it is locally constant. Another way of visualizing the concept of LDA is that we start with a homogeneous electron gas and subsequently localize the density around each external potential - each nucleus in a molecule or a solid. That the density is locally constant is indeed true for the intermediate densities, but not necessarily so in the high- and low-density regions. To correct for this, it was rec-... 

The Thomas-Fermi (TF) model (1927) for a homogeneous electron gas provides the underpinnings of modern DFT. In the following discussion, it will be shown that the model generates several useful concepts, relates the electron density to the potential, and gives a universal differential equation for the direct calculation of electron density. The two main assumptions of the TF model are as follows ... 

Replace (AN/l3) by p, the finite density of the homogeneous electron gas. Taking AV > 0, p can be locally replaced by p(r). Using atomic units and summing the... 

An exact expression for the correlation energy per particle ec( o) of a homogeneous electron gas does not exist, but good approximations to this nevertheless do exist.24 Also, nearly exact correlation energies have been obtained numerically for different densities25 and the results have been parametrized as useful functions ec n).26 The corresponding LDA correlation potential... 

In Eq. [47], epc ( ) and exc (n) are the exchange-correlation energy densities for the nonpolarized (paramagnetic) and fully polarized (ferromagnetic) homogeneous electron gas. The form of both exc(n) and exc(n) has been conveniently parameterized by von Barth and Hedin. Other interpolations have also been proposed24,33 for eKC(n, J ). The results for the homogeneous electron gas can be used to construct an LSDA... 

There is, however, an exception in an infinite homogeneous electron gas, electrons are delocalized. Neglecting their orbital motion does not contradict quantum mechanics, and lacking localization is unproblematic when only cross... 

As in the TFD method, this equation has to be solved with the boundary conditions (t)(0) = 1 andXcCt) (Xc) = < Xc) where = bXc is the cutoff point where the pressure of the electron gas becomes zero. The value of ( )(Xc)/Xcnecessary for this differs from the non-relativistic case and can be found from similar grounds [27], just by making zero the pressure of a homogeneous electron gas, given by... 

Of course, Eq. (2.65) reduces to the standard result in the case of the homogeneous electron gas [51-53]. Eq. (2.65) thus provides an alternative to Eq. (2.49) for all systems without a gap at the Fermi surface. In practice, a combination of the RPA with the second order functionai (2.49) suggests itseif as a rather universal form for E c-... 

The explicit form derived by Pernal for the effective nonlocal potential allows one to establish one-electron equations that may be of great value for the development of efficient computational methods in NOF theory. Although recent progress has been made, NOF theory needs to continue its assessment. Some other essential conditions such as the reproduction of the homogeneous electron gas should be utilized in the evaluation of approximate implementations. 

---