Thomas-Fermi approximation

Another model for the metal in the interface, also employing the Thomas-Fermi approximation for the electrons, was presented by Kuklin.63 It posited, in addition, a sharp boundary between metal and solution, and made other assumptions which have been criticized.6 In spite of its errors, it was one of the first attempts since Rice5 at a model for the interface which treated the metal as... 

We will solve eqn (6.2) within the Thomas-Fermi approximation by linking the change in electron density, <5p(r), to the local potential, K(r). At equilibrium the chemical potential or Fermi energy must be constant everywhere as illustrated in Fig. 6.1, so that... 

The Thomas-Fermi approximation assumes the variation in the potential, K(r), to be sufficiently slow that the local kinetic energy, 7T(r), is equal to that of an homogeneous free electron gas with the same density p(r) as seen locally, that is... 

The Thomas-Fermi approximation is, unfortunately, a poor approximation for the sp-valent metals. It is based on the assumption that the potential varies much more slowly than the screening length of the electrons themselves, so that the local approximation for the kinetic energy, eqn (6.6), is valid. In practice, however, the variation in the ionic potential is measured by the core radius, Rc (cf Fig. 5.11), which is not large but of the same size as the screening length, XTF. Thus, we do not satisfy the criterion for the validity... 

We will find that the Thomas-Fermi approximation totally fails to distinguish correctly between the different competing close-packed structure types such as fee, bcc or hep. We must, therefore, go beyond the Thomas-Fermi approximation and evaluate the proper screening behaviour of the free-electron gas at equilibrium metallic densities. 

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

If a positive charge ze is immersed in a degenerate electron gas, the Coulomb field is screened by the electrons. The screening was first estimated by the present author using the Thomas-Fermi approximation (Mott 1936, Mott and Jones 1936, p. 86), and by this method one finds for the potential energy of an electron... 

March, N.H. (1957). The Thomas-Fermi approximation in quantum mechanics, Adv. Phys. 6, 1-101. 

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... 

Physical properties of atoms and ions in intense magnetic fields are hence obtained in the statistical limit of Thomas-Fermi theory. This discussion is then supplemented by the hyperstrong limit, considered especially by Lieb and co-workers. Chemistry in intense magnetic fields is thereby compared and contrasted with terrestrial chemistry. Some emphasis is then placed on a model of confined atoms in intense electric fields the statistical Thomas-Fermi approximation again being the central tool employed. 

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. 

A major defect of the Thomas-Fermi approximation is that within it molecules are unstable the energy of a set of isolated atoms is lower than that of the bound molecule. This fundamental deficiency, and the lack of accuracy resulting from neglect of correlations in (32) and from using the local approximation (34) for the kinetic energy, limit the practical use of the Thomas-Fermi... 

The Thomas-Fermi approximation for screening, discussed in many books on solid-state physics, is obtained by minimizing ETF[n] with respect to n and linearizing the resulting relation between v(r) and n(r). It thus involves one more approximation (the linearization) compared to what is called the Thomas-Fermi approximation in DFT [44]. In two dimensions no linearization is required and both become equivalent [44]. 

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... 

Historically (and in many applications also practically) the most important type of approximation is the local-density approximation (LDA). To understand the concept of an LDA recall first how the noninteracting kinetic energy Ts [n] is treated in the Thomas-Fermi approximation In a homogeneous system one knows that, per volume45... 

Figure 1. Evolution of the radial electron density for argon from each of the three limits described in Table 1. The limits (top row) are those given by the Thomas-Fermi approximation, the non-interacting electron model, and the large-dimension limit. The abscissae are Hn-ear in y/r, where r is given in units of (IS/JV) for the first column, (18/Z) Co for the second, and (D/Zy a<, for the third. All curves except the Thomas-Fermi limit were obtained by adding Slater distributions whose exponents were determined from the subhamiltonian minima described in Sec. 4.

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]... 

Hodges, C. H. Quantum corrections to the Thomas-Fermi approximation the Kirzhnits method. Can. J. Phys. 1973, 51, 1428-1437. 

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