The Thomas-Fermi Model

We now switch on an external potential U(r) that is slowly varying over the dimensions of the metallic box. This makes the conduction electron density 

Our treatment so far has dealt with non-interacting electrons, yet we know for sure that electrons do interact with each other. Dirac (1930b) studied the effects of exchange interactions on the Thomas-Fermi model, and he soon discovered that this effect could be modelled by adding an extra term 

This result was rediscovered by Slater (1951) with a slightly different numerical coefficient of C. Authors often refer to a term Vx which is proportional to the one-third power of the electron density as a Slater-Dirac exchange potential. 

Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as 

By analogy with solid-state studies. Slater had the idea of writing the atomic Hartree-Fock eigenvalue equation 

Actually, the first attempts to use the electron density rather than the wave function for obtaining information about atomic and molecular systems are almost as old as is quantum mechanics itself and date back to the early work of Thomas, 1927 and Fermi, 1927. In the present context, their approach is of only historical interest. We therefore refrain from an in-depth discussion of the Thomas-Fermi model and restrict ourselves to a brief summary of the conclusions important to the general discussion of DFT. The reader interested in learning more about this approach is encouraged to consult the rich review literature on this subject, for example by March, 1975, 1992 or by Parr and Yang, 1989. 

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 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. 

In the Thomas-Fermi model,49 the kinetic energy density of the electron gas is written as... 

The calculations were subsequently extended to moderate surface charges and electrolyte concentrations.8 The compact-layer capacitance, in this approach, clearly depends on the nature of the solvent, the nature of the metal electrode, and the interaction between solvent and metal. The work8,109 describing the electrodesolvent system with the use of nonlocal dielectric functions e(x, x ) is reviewed and discussed by Vorotyntsev, Kornyshev, and coworkers.6,77 With several assumptions for e(x,x ), related to the Thomas-Fermi model, an explicit expression6 for the compact-layer capacitance could be derived ... 

For neutral atoms, N = Z, (23) requires x ixo) = 0> so that x vanishes at the same point as x- Since this condition cannot be satisfied for a finite value xo by non-trivial solutions, the point xo must be at infinity. The solution X(x) for a neutral atom must hence be asymptotic to the x-axis, x(°°) = 0. There is no boundary to the neutral atom in the Thomas-Fermi model. 

The universal function x(x) obtained by numerical integration and valid for all neutral atoms decreases monotonically. The electron density is similar for all atoms, except for a different length scale, which is determined by the quantity b and proportional to Z. The density is poorly determined at both small and large values of r. However, since most electrons in complex atoms are at intermediate distances from the nucleus the Thomas-Fermi model is useful for calculating quantities that depend on the average electron density, such as the total energy. The Thomas-Fermi model therefore cannot account for the periodic properties of atoms, but provides a good estimate of initial fields used in more elaborate calculations like those to be discussed in the next section. 

The most obvious defect of the Thomas-Fermi model is the neglect of interaction between electrons, but even in the most advanced modern methods this interaction still presents the most difficult problem. The most useful practical procedure to calculate the electronic structure of complex atoms is by means of the Hartree-Fock procedure, which is not by solution of the atomic wave equation, but by iterative numerical procedures, based on the hydrogen model. In this method the exact Hamiltonian is replaced by... 

The electronic wave function of an n-electron molecule is defined in 3n-dimensional configuration space, consistent with any conceivable molecular geometry. If the only aim is to characterize a molecule of fixed Born-Oppenheimer geometry the amount of information contained in the molecular wave function is therefore quite excessive. It turns out that the three-dimensional electron density function contains adequate information to uniquely determine the ground-state electronic properties of the molecule, as first demonstrated by Hohenberg and Kohn [104]. The approach is equivalent to the Thomas-Fermi model of an atom applied to molecules. 

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... 

Consider two different metals in contact and assume that both are well described by the Thomas-Fermi model (see Problem 3.3) with a decay length of Ltf 0.5 A. (a) Calculate the dipole potential drop at the contact if both metals carry equal and opposite charges of 0.1 C m 2. (b) If the work functions of the two metals differ by 0.5 eV, how large is the surface-charge density on each meted ... 

The important features of ael are represented by the Thomas-Fermi model of the atom which assumes that orbital electrons screen exponentially the nuclear charge 7. On this model we have,... 

Moreover, as the Thomas-Fermi model for atomic systems becomes more accurate when increasing Z (asymptotically exact in the large Z-limit) with respect to the non-relativistic solution of Schrodinger equation, but relativistic effects increases with Z does, the inclusion of these is demanded for its application. 

The p ) moments for all the neutral atoms can be fit to simple functions of the number of electrons [232,263]. These functions are heuristic extensions of expressions [264-266] derived on the basis of the Thomas-Fermi model [267-271], with the Scott-Schwinger correction [272,273] for strongly bound electrons. Thus, the Hartree-Fock (p ) for the neutral atoms from hydrogen (A = 1) through lawrencium (A = 103) can be fit [232] as follows ... 

Consideration of the valence region of a ground-state molecule leads to a formula for the energy of atomization in an approximation akin to the Thomas-Fermi model. The final result, from Eq. (10.33), is... 

During this period, accurate solutions for the electronic structure of helium (1) and the hydrogen molecule (2) were obtained in order to verify that the Schrodinger equation was useful. Most of the effort, however, was devoted to developing a simple quantum model of electronic structure. Hartree (3) and others developed the self-consistent-field model for the structure of light atoms. For heavier atoms, the Thomas-Fermi model (4) based on total charge density rather than individual orbitals was used. 

The electrical conductivities of several alkali metals dissolved in liquid ammonia are shown in Figure 1 (7, 119 15). The strong variation of the conductivity, a, with concentration has been most difficult to explain. This difficulty can be assessed by referring to a simple model of conductance, the Thomas-Fermi model of a screened Coulomb potential (19). This model has been used in describing semiconductors as well as in theories of metal-ammonia solutions (1). 

---