Free-electron exchange approximation

Energy bands can be calculated from first principles, without any experimental input. The main approximation required is the one-electron approximation (see Appendix A), which we use throughout this text. Then the two remaining questions are what does one use for the potential and what representation does one use to describe the wave function At present the same essential view of the potential is taken by almost all workers, based upon free-electron exchange and little, if any, modification for correlation. (This is discussed in Appendixes A and... 

The details of the modified electron-gas (MEG) ionic model method have been fully described by Gordon and Kim (1972). The fundamental assumptions of the method are (1) the total electron density at each point is simply the sum of the free-ion densities, with no rearangements or distortion taking place (2) ion-ion interactions are calculated using Coulomb s law, and the free-electron gas approximation is employed to evaluate the electronic kinetic, exchange, and correlation energies (3) the free ions are described by wave functions of Hartree-Fock accuracy. Note that this method does not iterate to a self-consistent electron density. 

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) ... 

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] ... 

The so-called Hartree-Fock-Slater method is much more widely utilized, and is a hybrid of the Hartree and Thomas-Fermi-Dirac methods. In this method the direct part of the potential is calculated using the Hartree-Fock approach, whereas the exchange part is approximated by some statistical expression of the model of free electrons. The Slater potential is given by ... 

The electron exchange rate k (Eq. 10.5) is a function of the transmission coefficient k (approximately 1 for reactions with substantial electronic coupling (>4 kJ), i.e., for adiabatic reactions), the effective collision frequency in solution (Z 1011 M 1 s 1 Ar2) and the free energy term AG. ... 

An alternative to the perturbation theory approach is the approximate method of Gordon and Kim.60 In this method the electron density is first calculated as the sum of the densities of the separate atoms and the energy is then obtained as the sum of a Coulomb term calculated exactly, and kinetic energy, exchange, and correlation terms calculated from the free electron gas model. Though it worked well for larger... 

We may also compute the electrostatic energy of this electron distribution combined with the nuclear charge from each atom. This is the most intricate part of the calculation but is a straightforward hiachine calculation. Next we may add the exchange energy. It is given for a free-electron gas by, for example, Kittel (1963, p. 92), and leads to a total approximate... 

Kinetic energy of electrons, 3 free-electron gas, 348f local approximation, 351, 377f, 541 Kleinman s internal displacement parameter, 198f tables, 196, 208, 220 Kohn anomalies, 395f Kohn-Sham exchange, 540 Koster-Slaler tables, 481 Kramers-Kronig relations, 99 Krypton, properties of. See Inert gas solids... 

More recently, Lang and Kohn (1970) treated the jellium model with modern methods and modern computers. In particular, they included exchange terms in the free-electron approximation, as we have discussed. In this context it is inter-csiing that the potential well that holds the electrons in the metal comes predominantly from the exchange potential the electrostatic potential itself gives only... 

From the application of the free-electron approximation in the Hartree-Fock exchange potential... 

The exchange potential is further simplified by the Xa approximation of Slater (1951). For the free electron gas of density p one can show that the average exchange energy is... 

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