The free-electron gas

The sp-valent metals such as sodium, magnesium and aluminium constitute the simplest form of condensed matter. They are archetypal of the textbook metallic bond in which the outer shell of electrons form a gas of free particles that are only very weakly perturbed by the underlying ionic lattice. The classical free-electron gas model of Drude accounted very well for the electrical and thermal conductivities of metals, linking their ratio in the very simple form of the Wiedemann-Franz law. However, we shall now see that a proper quantum mechanical treatment is required in order to explain not only the binding properties of a free-electron gas at zero temperature but also the observed linear temperature dependence of its heat capacity. According to classical mechanics the heat capacity should be temperature-independent, taking the constant value of kB per free particle. 

The Schrodinger equation for a free-electron gas takes the form 

If the electrons are confined within a box of side L, then the normalized eigenfunctions are the plane waves 

For a free electron this energy is purely kinetic, so that E = p2/2m. Hence p = hk = h/X, as we have already found experimentally for free particles, namely eqs (18) and (2.14). 

Following Pauli s exclusion principle, each state corresponding to a given can contain at most two electrons of opposite spin. Therefore, at the absolute zero of temperature aU the states, k, will be occupied within a sphere of radius, kF, the so-called Fermi sphere because these correspond to the states of lowest energy, as can be seen from Fig. 2.9(a). The magnitude of the Fermi wave vector, kF, may be related to the total number of valence electrons N by 

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The free-electron gas was first applied to a metal by A. Sommerfeld (1928) and this application is also known as the Sommerfeld model. Although the model does not give results that are in quantitative agreement with experiments, it does predict the qualitative behavior of the electronic contribution to the heat capacity, electrical and thermal conductivity, and thermionic emission. The reason for the success of this model is that the quantum effects due to the antisymmetric character of the electronic wave function are very large and dominate the effects of the Coulombic interactions. 

The free-electron gas exerts a pressure on the walls of the infinite potential well in which it is contained. If the volume v of the gas is increased slightly by an amount dv, then the energy levels in equation (8.56) decrease... 

Space-coordinate density transformations have been used by a number of authors in various contexts related to density functional theory [26,27, 53-64, 85-87]. As the free-electron gas wavefunction is expressed in terms of plane waves associated with a constant density, these transformations were introduced by Macke in 1955 for the purpose of producing modified plane waves that incorporate the density as a variable. In this manner, the density could be then be regarded as the variational object [53, 54]. Thus, explicitly a set of plane waves (defined in the volume V in and having uniform density po = N/V) ... 

An extension of Macke s formulation to the 1-matrix was given by March and Young in 1958 [55], These authors applied these generalized transformations to the free-electron gas 1-matrix Do r,r ) which is carried into the new 1-matrix D r,r ) ... 

The general Jacobian problem associated with the transformation of a density Pi(r) into a density p2(r) (where these densities differ from that of the free-electron gas) was discussed by Moser in 1965 [58]. This work was not performed in the framework of orbital transformations - which might have interested chemists, nor was it done in the context of density functional theory - which might have attracted the attention of physicists. It was a paper written for mathematicians and, as such, it remained unknown to the quantum chemistry community. In the discussion that follows, we use the more accessible reformulation of Bokanowski and Grebert (1995) [65] which relies heavily on the work of Zumbach and Maschke (1983) [61]. Let us define as ifjy = the space of... 

Note that the exchange term is of the form / y(r,r ) h(r )dr instead of the y (r) (r) type. Equation (1.12), known as the Hartree-Fock equation, is intractable except for the free-electron gas case. Hence the interest in sticking to the conceptually simple free-electron case as the basis for solving the more realistic case of electrons in periodic potentials. The question is how far can this approximation be driven. Landau s approach, known as the Fermi liquid theory, establishes that the electron-electron interactions do not appear to invalidate the one-electron picture, even when such interactions are strong, provided that the levels involved are located within kBT of Ep. For metals, electrons are distributed close to Ep according to the Fermi function f E) ... 

A theoretically well-founded theory of two-photon absorption using the free electron-gas model of dye molecules 50> is to be found in 49>. 

The free-electron gas model is a good starting point for the sp-valent metals where the loosely bound valence electrons are stripped off from their ion cores as the atoms are brought together to form the solid. However, bonding in the majority of elements and compounds takes place through saturated... 

Thus, we have come a long way from the exactly soluble problems of quantum mechanics, the free-electron gas and the hydrogen atom. The concept of the exchange-correlation hole linked with the LDA has allowed... 

The jellium model of the free-electron gas can account for the increased abundance of alkali metal clusters of a certain size which are observed in mass spectroscopy experiments. This occurrence of so-called magic numbers is related directly to the electronic shell structure of the atomic clusters. Rather than solving the Schrodinger equation self-consistently for jellium clusters, we first consider the two simpler problems of a free-electron gas that is confined either within a sphere of radius, R, or within a cubic box of edge length, L (cf. problem 28 of Sutton (1993)). This corresponds to imposing hard-wall boundary conditions on the electrons, namely... 

The eigenspectrum of the free-electron gas confined within a sphere of... 

We can understand the behaviour of the binding energy curves of monovalent sodium and other polyvalent metals by considering the metallic bond as arising from the immersion of an ionic lattice of empty core pseudopotentials into a free-electron gas as illustrated schematically in Fig. 5.15. We have seen that the pseudopotentials will only perturb the free-electron gas weakly so that, as a first approximation, we may assume that the free-electron gas remains uniformly distributed throughout the metal. Thus, the total binding energy per atom may be written as... 

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. 

The energy-wave-number characteristic, ( )> depends only on the density of the free-electron gas and the nature of the pseudopotential core but not on the structural arrangement of the atoms. Its behaviour as a function of the wave vector, q, is illustrated in Fig. 6.6, where we see that it vanishes at q0 as expected. It also has a weak logarithmic singularity in its slope at q = 2kF. 

In 1974 Finnis showed that the q - 0 limit could be evaluated directly by using the compressibility sum rule of the free-electron gas, which relates the long wavelength behaviour of the dielectric constant to its compressibility. He found that... 

The wave vector, k , and the screening length, 1/ , depend only on the density of the free-electron gas through the poles of the approximated inverse dielectric response function, whereas the amplitude, A , and the phase shift, a , depend also on the nature of the ion-core pseudopotential through eqs (6.96) and (6.97). For the particular case of the Ashcroft empty-core pseudopotential, where tfj fa) = cos qRc, the modulus and phase are given explicitly by... 

There exists a whole number of approximate expressions for Vl(r) (see, for example [139]). The simplest, called the Thomas-Fermi potential, follows from the statistical model of an atom. Unfortunately, it leads to results of very low accuracy. More accurate is the Thomas-Fermi-Dirac model, in which an attempt is made to account for the exchange part of the potential energy of an electron in the framework of the free electron gas approach. Various forms of the parametric potential method are fairly widely utilized, particularly for multiply charged ions. Such potentials may look as follows [16] ... 

In the previous two chapters we have described basic structural properties of the components of an interphase. In Chapter 2 we have shown that water molecules form clusters and that ions in a water solution are hydrated. Each ion in an ionic solution is surrounded predominantly by ions of opposite charge. In Chapter 3 we have shown that a metal is composed of positive ions distributed on crystal lattice points and surrounded by a free-electron gas. The free-electron gas extends outside the ionic lattice to form a surface dipole layer. 

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