Exchange energy free-electron approximation

We see that this spin polarization is a spherical oscillating function of distance. If the free electron approximation is good the kp can be expressed as (3ir n) and the effective electronic mass m = mo. The spin polarization interacts by exchange with a localized neighbour ion at distance Ra from the scattering centre. The corresponding energy is... 

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] ... 

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... 

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. 

Here, i CT(r) and (r) represent the ordinary nonrelativistic electron annihilation and creation operators. An LDA-type approximation has recently been derived for the exchange-correlation free energy Fxc[n, xl leading to explicit expressions for the effective potentials Veg(r) and Aeff (r, r ) (Kurth et al. 1999). 

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... 

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... 

Notice that in the simple approximation we are considering, the free-electron gas, there are no other terms involved in the total energy since we neglect the Coulomb interactions and there is no exchange interaction between spin-up and spin-down electrons. 

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