Approximations for exchange and correlation

The exchange and correlation polarizability in the local-density approximation therefore takes the form (see Eq. 54-55)  

The meaning of this approximation can be elucidated by considering the uniform electron gas as the equilibrium system. In this case, the equilibrium density is independent of the position, and only the 5 — 0 term in Eq. 67 contributes. As a result, one obtains  

Note that the exchange contribution is thus given by (see Eq. 23)  

Instead of assuming the LDA-form (Eq. 69), one could introduce a function G(q) which immitates this exchange and correlation hole, and which is to be determined later on from many-body theory  

The determination of 0(5) from the dielectric theory, as discussed in the next chapter, is quite essential in order to satisfy some known sum rules. Indeed, from Ekj. 69, GiDAig) is obviously divergent at large wave vectors. 

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Theoretical calculations were performed with the linear muffin tin orbital (LMTO) method and the local density approximation for exchange and correlation. This method was used in combination with supercell models containing up to 16 atoms to calculate the DOS. The LMTO calculations are run self consistently and the DOS obtained are combined with the matrix elements for the transitions from initial to final states as described in detail elsewhere (Botton et al., 1996a) according to the method described by Vvedensky (1992). A comparison is also made between spectra calculated for some of the B2 compounds using the Korringa-Kohn-Rostoker (KKR) method. 

Fiolhais, Phys. Rev. B, 46, 6671 (1992). Atoms, Molecules, Solids, and Surfaces Application of the Generalized Gradient Approximation for Exchange and Correlation. 

Perdew JP (1991a) Generalized gradient approximations for exchange and correlation - A look backward and forward, Physica B, 172 1-6... 

Perdew JP, Chevary JA, Vosko SH, Jackson KA, Pederson MR, Singh DJ, Fiolhais C (1992) Atoms molecules, solids, and surfaces applications of the generalized gradient approximation for exchange and correlation, Phys Rev B, 46 6671—6687... 

There is, in principle, nothing which limits the self-consistent field method to any particular form of the exchange-correlation potential, and the procedure outlined above has been used in connection with several approximations for exchange and correlation. Most notable in this respect is SLATER S Xa method [1.4] which has been applied to all atoms in the periodic table, to some molecules, and in the majority of the existing electronic-structure calculations for crystalline solids. 

Causa, M., Dovesi, R., Pisani, C., and Roetti, C. (1986) Electronic structure and stability of different crystal phases of magnesium oxide, Phys. Rev. B 33, 1308-1316. Perdew, J.P., Chevary, J.A., Vosko, S.H., Jackson, K.A., Pederson, M.R., Singh, D.J., and Fiolhais, C. (1992) Atoms, molecules, solids, and surfaces applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B 46, 6671-6687. 

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