Functional Thomas-Fermi approximation

Within the Thomas-Fermi approximation, the linear response function is independent of the wavevector q, since from eqn (6.20) it is given by... 

The change in potential energy AV is first order in the nuclear displacements, and of course, Ap is obtained correctly to the same order from equation (122). Handler and March show that the Thomas-Fermi approximation to the linear response function F has the form... 

Sections 13-1.5 are then concerned with relating the above model to atomic ions in a hot, non-degenerate plasma in an external electric field. The first step is to add an atomic-like potential energy V(r) to the model. Strictly, V(r) should be calculated self-consistently as a function of p, F and the plasma density. While this has not been achieved numerically at the time of writing, a model potential F(r) is incorporated into the treatment of Sect. 7.3 by means of the semiclassical Thomas-Fermi approximation. The second step taken by Amovilli et al. [41] is to connect the strength of the harmonic potential with the plasma density (Sect. 7.4). 

In theory it should be possible to calculate all observables, since the HK theorem guarantees that they are all functionals of no(r). In practice, one does not know how to do this explicitly. Another problem is that the minimization of Ev[n is, in general, a tough numerical problem on its own. And, moreover, one needs reliable approximations for T[n] and U[n] to begin with. In the next section, on the Kohn-Sham equations, we will see one widely used method for solving these problems. Before looking at that, however, it is worthwhile to recall an older, but still occasionally useful, alternative the Thomas-Fermi approximation. 

The Thomas-Fermi approximation (34) for T[n is not very good. A more accurate scheme for treating the kinetic-energy functional of interacting electrons, T[n], is based on decomposing it into one part that represents the kinetic energy of noninteracting particles of density n, i.e., the quantity called above Ts[n], and one that represents the remainder, denoted Tc[n (the sub-... 

Ts[n is not known exactly as a functional of n [and using the LDA to approximate it leads one back to the Thomas-Fermi approximation (34)], but it is easily expressed in terms of the single-particle orbitals fair) of a noninteracting system with density n, as... 

The lack of a theoretical framework certainly did not promote the development of density functionals. This situation changed radically in 1964 with the paper of Pierre Hohenberg and Walter Kohn. Hohenberg and Kohn established a one-to-one correspondence between electron densities of nondegenerate ground states and external local potentials, v r), which differ by more than a constant. All physical properties obtainable with v can therefore be expressed in terms of the electron density. It was thus established that, for example, the Thomas-Fermi approximation to the kinetic energy can in principle be refined to yield arbitrary precision. Hohenberg and Kohn defined the density functional F[p]... 

Note that the total energy is not given literally in terms of a functional of the charge density as it is, e.g., in the Thomas-Fermi approximation. Instead, the kinetic energy is given by the form (12) in terms of the wave functions. 

For most metal-oxide interfaces, however, the Fermi level does not coincide with Ezcp- A charge transfer takes place, which aligns the chemical potentials, and induces an interfadal dipole potential, which bends the bands. It is possible to estimate the self-consistent charge density in the vicinity of the interface, within a Thomas-Fermi approximation, if the MIGS density at mid-gap is taken equal to a single exponential function Af( zcp,z) = noexp(—z//p). The potential V z) due to the mean charge density p(z) is related to p(z) by Poisson s equation ... 

The Thomas-Fermi approximation consists in combining (39) with (42) and minimizing the resulting energy functional... 

The Thomas-Fermi approximation (42) for T ri is not very good. A more accurate scheme for treating the kinetic-energy functional of... 

In the LDA one exploits knowledge of the density at point r. Any real system is spatially inhomogeneous, i.e. it has a spatially varying density (r), and it would clearly be useful to also include information on the rate of this variation in the functional. A first attempt at doing this was the so-called GEAs. In this class of approximation one tries to systematically calculate gradient corrections of the form V (r), V (r)p, V n(r), etc. to the LDA. A famous example is the lowest-order gradient correction to the Thomas-Fermi approximation for Ts n],... 

In this section we will approach the question which is at the very heart of density functional theory can we possibly replace the complicated N-electron wave function with its dependence on 3N spatial plus N spin variables by a simpler quantity, such as the electron density After using plausibility arguments to demonstrate that this seems to be a sensible thing to do, we introduce two early realizations of this idea, the Thomas-Fermi model and Slater s approximation of Hartree-Fock exchange defining the X(/ method. The discussion in this chapter will prepare us for the next steps, where we will encounter physically sound reasons why the density is really all we need. 

Since the early days of quantum mechanics, the wave function theory has proven to be very successful in describing many different quantum processes and phenomena. However, in many problems of quantum chemistry and solid-state physics, where the dimensionality of the systems studied is relatively high, ab initio calculations of the structure of atoms, molecules, clusters, and crystals, and their interactions are very often prohibitive. Hence, alternative formulations based on the direct use of the probability density, gathered under what is generally known as the density matrix theory [1], were also developed since the very beginning of the new mechanics. The independent electron approximation or Thomas-Fermi model, and the Hartree and Hartree-Fock approaches are former statistical models developed in that direction [2]. These models can be considered direct predecessors of the more recent density functional theory (DFT) [3], whose principles were established by Hohenberg,... 

The shape function had a role in theoretical chemistry and physics long before it was named by Parr and Bartolotti. For example, in x-ray measurements of the electron density, what one actually measures is the shape function—the relative abundance of electrons at different locations in the molecule. Determining the actual electron density requires calibration to a standard with known electron density. On the theoretical side, the shape function appears early in the history of Thomas-Fermi theory. For example, the Majorana-Fermi-Amaldi approximation to the exchange potential is just [3,4]... 

Various reasons have been advanced for the relative accuracy of spin-polarized Kohn-Sham calculations based on local (spin) density approximations for E c- However, two very favourable aspects of this procedure are clearly operative. First, the Kohn-Sham orbitals control the physical class of density functions which are allowed (in contrast, for example, to simpler Thomas-Fermi theories). Second, local density approximations for are mild-mannered,... 

The origins of density functional theory (DFT) are to be found in the statistical theory of atoms proposed independently by Thomas in 1926 [1] and Fermi in 1928 [2]. The inclusion of exchange in this theory was proposed by Dirac in 1930 [3]. In his paper, Dirac introduced the idempotent first-order density matrix which now carries his name and is the result of a total wave function which is approximated by a single Slater determinant. The total energy underlying the Thomas-Fermi-Dirac (TFD) theory can be written (see, e.g. March [4], [5]) as... 

In the simplest form, the Thomas-Fermi-Dirac model, the functionals are those which are valid for an electronic gas with slow spatial variations (the nearly free electron gas ). In this approximation, the kinetic energy T is given by... 

Equations (3.20) and (3.21) represent an identity in Hartree-Fock theory. (The Hellmann-Feynman and virial theorems are satisfied by Hartree-Fock wavefunc-tions.) The particular interest offered by (3.21) lies in the fact that 7 = 1 appears to be the characteristic homogeneity of both Thomas-Fermi [62,75,76] and local density functional theory [77], in which case (3.20) gives the Ruedenberg approximation [78], E = v,e,-, while (3.21) gives the Politzer formula [79], E = Vne-... 

An especially useful approximation for the Thomas-Fermi potential has been developed by Lindhard and co-workers where the screening function is assumed to have the form ... 

---