The Fermi-Thomas Theory

Instead of using V = 0 as in the simple electron gas theory, or a constant, as in the jellium theory, a pseudo-potential V(R, z) can be used to calculate the electronic wavefunction 

The concept of screening, i.e., the ability of an electron gas to screen the positive ionic charge or background, plays a role in simplistic treatments of atom/molecule-surface interaction. The simplest way of treating this at a semi-quantitative level is through the Fermi-Thomas theory. 

If the electron moves in a potential V(r), the total energy is a sum of the kinetic and potential energy 

A fundamental equation in electrostatic theory is the Poisson equation. The Poisson equation can be derived from the Gauss theorem [245] and relates the potential to the charge density or distribution by the equation 

Numerical techniques for solving the Poisson equation for an arbitrary charge distribution exist (see, e.g., [246]). Here we shall just consider the simple case where the distribution is given by the expression (9.32). Using the fact that p(r) = —en(r) we can obtain the Fermi-Thomas equation [248] 

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So far, the electrons have been treated as a free-electron gas. In the Hartree and Hartree-Fock theory the electrons are allowed to interact. We shall therefore briefly consider the electron gas in these theories and then introduce the approximate solution in a potential according to the Fermi-Thomas theory [217, 218]. 

In examining how changes in the electron states caused by the pseiidopotential change the total energy of the electron gas, it is best not to use the Fermi-Thomas approximation, used in Section 16-P but to compute the energy of the electrons directly by applying perturbation theory. For a particular electron of wave number k, Eq, (1-14) directly gives... 

A similar general approach was given by Slater (1951) and made variational by Kohn and Sham (1965). In fact, the same theory had been developed earlier as an extension of Fermi-Thomas theory by Gombas (1949). It is the basis of the one-electron approximation contemplated throughout this text, and the resulting equation,... 

The equations must be solved iteratively since the density enters the expressions for Veff. The free electron gas, the Fermi-Thomas, and the DFT theory will be treated in more detail in Chapters 9 and 10. The DFT theory has been used recently in a number of calculations of molecule-surface interaction (sec Table 4.7). Some studies just involves the chemisorption to a small cluster of metal atoms [189], and it is questionable how well this represents the behavior of a bulk metal. 

For liquid metals, one has to set up density functionals for the electrons and for the particles making up the positive background (ion cores). Since the electrons are to be treated quantum mechanically, their density functional will not be the same as that used for the ions. The simplest quantum statistical theories of electrons, such as the Thomas-Fermi and Thomas-Fermi-Dirac theories, write the electronic energy as the integral of an energy density e(n), a function of the local density n. Then, the actual density is found by minimizing e(n) + vn, where v is the potential energy. Such... 

The Thomas-Fermi model of a metal is similar to the Gouy-Chapman theory for electrolytes. In this model the surface-charge density o is... 

Beginning way back in the 20s, Thomas and Fermi had put forward a theory using just the diagonal element of the first-order density matrix, the electron density itself. This so-called statistical theory totally failed for chemistry because it could not account for the existence of molecules. Nevertheless, in 1968, after years of doing wonders with various free-electron-like descriptions of molecular electron distributions, the physicist John Platt wrote [2] We must find an equation for, or a way of computing directly, total electron density. [This was very soon after Hohenberg and Kohn, but Platt certainly was not aware of HK by that time he had left physics.]... 

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 relativistic formulation of Thomas-Fermi theory started at the same time as the original non-relativistic one, the first work being of Vallarta and Rosen [9] in 1932. The result they arrived at can be found by replacing the kinetic energy fimctional by the result of the integration of the relativistic kinetic energy in terms of the momentum p times the number of electrons with a given momentum p from /i = 0 to the Fermi momentum p = Pp. ... 

The failure is not limited to metal-ammonia solutions nor to the linear Thomas-Fermi theory (19). The metals physicist has known for 30 years that the theory of electron interactions is unsatisfactory. E. Wigner showed in 1934 that a dilute electron gas (in the presence of a uniform positive charge density) would condense into an electron crystal wherein the electrons occupy the fixed positions of a lattice. Weaker correlations doubtless exist in the present case and have not been properly treated as yet. Studies on metal-ammonia solutions may help resolve this problem. But one or another form of this problem—the inadequate understanding of electron correlations—precludes any conclusive theoretical treatment of the conductivity in terms of, say, effective mass at present. The effective mass may be introduced to account for errors in the density of states—not in the electron correlations. 

The idea of calculating atomic and molecular properties from electron density appears to have arisen from calculations made independently by Enrico Fermi and P.A.M. Dirac in the 1920s on an ideal electron gas, work now well-known as the Fermi-Dirac statistics [19]. In independent work by Fermi [20] and Thomas [21], atoms were modelled as systems with a positive potential (the nucleus) located in a uniform (homogeneous) electron gas. This obviously unrealistic idealization, the Thomas-Fermi model [22], or with embellishments by Dirac the Thomas-Fermi-Dirac model [22], gave surprisingly good results for atoms, but failed completely for molecules it predicted all molecules to be unstable toward dissociation into their atoms (indeed, this is a theorem in Thomas-Fermi theory). 

The Hohenberg-Kohn theory of /V-clcctron ground states is based on consideration of the spin-indexed density function. Much earlier in the development of quantum mechanics, Thomas-Fermi theory [402, 108] (TFT) was formulated as exactly such a density-dependent formalism, justified as a semiclassical statistical theory [231, 232], Since Hohenberg-Kohn theory establishes the existence of an exact universal functional Fs [p] for ground states, it apparently implies the existence of an exact ground-state Thomas-Fermi theory. The variational theory that might support such a conclusion is considered here. 

As already remarked, the idea underlying the Thomas-Fermi (TF) statistical theory is to treat the electrons around a point r in the electron cloud as though they were a completely degenerate electron gas. Then the lowest states in momentum space are all doubly occupied by electrons with opposed spins, out to the Fermi sphere radius corresponding to a maximum or Fermi momentum pt(r) at this position r. Therefore if we consider a volume dr of configuration space around r, the volume of occupied phase space is simply the product dr 47ipf(r)/3. However, we know that two electrons can occupy each cell of phase space of volume h3 and hence we may write for the number of electrons per unit volume at r,... 

Wang et al.s0 have calculated T0, T2, and Tt using good wave-mechanical densities for closed-shell atoms and a selection of their results is recorded in Table 5. The inequality (80) is seen to be fulfilled. Furthermore, since the Thomas-Fermi statistical theory becomes correct for sufficiently large numbers of electrons, it follows that the importance of T% diminishes continually for heavier atoms. 

Fermi energy in a-Si H cannot be brought closer than about 0.1 eV to Ef., so that the conductivity can hardly be measured below about 100 K and there is only limited information about the sharpness of the mobility edge. Most of the detailed tests of the mobility edge theories are made on disordered crystals and in metals in which the Fermi energy can be made to cross the mobility edge, giving measurable conductivity at low temperatures (Thomas 1985). 

The energy functional Etf[p] = Ttf[p] + ne />] + T[p] is known as Thomas-Fermi (TF) theory, inHiiHinv the Atp[p] exchange part (first derived by Block but commonly associated with the name of Dirac (constitutes the Thomas-Fermi-Dirac (TFD)... 

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. 

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