Dirac exchange formula

Note that the choice k = kp, regardless of the value of a, yields the LDA (Dirac) exchange formula (58) as the first term of x [p]- The remaining terms containing Vp, V p, and T, naturally arise as corrections to the LDA. 

Dirac 3 made an improvement to the TF model by introducing a formula for the exchange energy of a uniform electron gas. The Dirac exchange formula is... 

Unfortunately, adding the Dirac exchange formula to the TF model does not improve the quality of the calculated electron density. The TFD density suffers from the same undesirable characteristics as does the TF density. A major enhancement of these two overly simplified models was made through addition of an inhomogeneity the electron density correction to the kinetic energy density functional. This was first investigated by von Weizsacker, who derived a correction that depends upon the gradient of the density, namely. 

After the discovery of the relativistic wave equation for the electron by Dirac in 1928, it seems that all the problems in condensed-matter physics become a matter of mathematics. However, the theoretical calculations for surfaces were not practical until the discovery of the density-functional formalism by Hohenberg and Kohn (1964). Although it is already simpler than the Hartree-Fock formalism, the form of the exchange and correlation interactions in it is still too complicated for practical problems. Kohn and Sham (1965) then proposed the local density approximation, which assumes that the exchange and correlation interaction at a point is a universal function of the total electron density at the same point, and uses a semiempirical analytical formula to represent such universal interactions. The resulting equations, the Kohn-Sham equations, are much easier to handle, especially by using modern computers. This method has been the standard approach for first-principles calculations for solid surfaces. 

The first generation is the local density approximation (LDA). This estimation involves the Dirac functional for exchange, which is nothing else than the functional proposed by Dirac [15] in 1927 for the so-called Thomas-Fermi-Dirac model of the atoms. For the correlation energy, some parameterizations have been proposed, and the formula can be considered as the limit of what can be obtained at this level of approximation [16-18], The Xa approximation falls into this category, since a known proportion of the exchange energy approximates the correlation. 

In the Local Density Approximation (LDA) it is assumed that the density locally can be treated as a uniform electron gas, or equivalently that the density is a slowly varying function. The exchange energy for a uniform electron gas is given by the Dirac formula (eq. (6.2)). 

There is, however, one paper of Dirac [5] that keeps being cited, namely the one at which we want to have a look now. Most people who cite this paper hardly know that its title is Quantum mechanics of many-electron systems and are unaware of its scientific context. It deals mainly with the relation between permutation symmetry and spin and contains a formula which relates the expectation value of the operator of electron exchange to the total spin of the state. 

Use the Dirac spin-exchange identity (Problem 4.4) to obtain an alternative proof of the energy formula (7.1.9). Extend your result to the case of N electrons, for any perfect-pairing wavefhnction (p. 219), and express (7.3.7) as the expectation value of a spin Hamiltonian. 

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