Local density approximation formalism

TIME-DEPENDENT LOCAL DENSITY APPROXIMATION Formalism... 

The inherent problems associated with the computation of the properties of solids have been reduced by a computational technique called Density Functional Theory. This approach to the calculation of the properties of solids again stems from solid-state physics. In Hartree-Fock equations the N electrons need to be specified by 3/V variables, indicating the position of each electron in space. The density functional theory replaces these with just the electron density at a point, specified by just three variables. In the commonest formalism of the theory, due to Kohn and Sham, called the local density approximation (LDA), noninteracting electrons move in an effective potential that is described in terms of a uniform electron gas. Density functional theory is now widely used for many chemical calculations, including the stabilities and bulk properties of solids, as well as defect formation energies and configurations in materials such as silicon, GaN, and Agl. At present, the excited states of solids are not well treated in this way. 

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. 

We also discuss the generalization of density-functional theory to n-partical states, nDFT, and the possible extension of the local density approximation , nLDA. We will see there that the difficulty of describing the state of a system properly in terms of n-particle states presents no formal difficultie since DFT is directed only at the determination of the particle density rather than individual-particle wave functions. The extent to which practical applications of nDFT within a generalized Kohn-Sham scheme will provide a viable procedure is commented upon below. 

At this stage, the formalism can be implemented in a computer program. The applications described below [15-21] rely on the expansion of the electronic wavefunctions in terms of a large number of plane waves, as well as on the replacement of nuclear bare potentials by accurate norm-conserving pseudopotentials. The Local Density Approximation was used, with the Ceperley and Alder data for the exchange-correlation energy of the homogeneous electron gas. 

The results of our band structure calculations for GaN crystals are based on the local-density approximation (LDA) treatment of electronic exchange and correlation [17-19] and on the augmented spherical wave (ASW) formalism [20] for the solution of the effective single-particle equations. For the calculations, the atomic sphere approximation (ASA) with a correction term is adopted. For valence electrons, we employ outermost s and p orbitals for each atom. The Madelung energy, which reflects the long-range electrostatic interactions in the system, is assumed to be restricted to a sum over monopoles. 

In the Local Density Approximation (LDA), if the charge density p0 varies only slowly with position, then a formal expression for Exc is... 

Until now we only considered the formal framework of density functional theory. However, the theory would be of little use if we would not be able to construct good approximate functionals for the exchange-correlation energy and exchange-correlation functional. Historically the first approximation for the exchange-correlation functional to be used was the local density approximation. In this approximation the exchange-correlation functional is taken to be... 

At present the most satisfactory foundation of the one-electron picture for metals is provided by the local approximation to the density-functional formalism of Hohenberg and Kohn [1.5] and Kohn and Sham [1.6]. This one-electron theory is presented in Chap.7 when ground-state properties of crystalline solids are discussed. Here we note that the local-density (LD) formalism, like the Xa method, leads to an effective one-electron potential which is a function of the local electron density as expressed by (1.2). 

In Chaps.7 and 8 it is shown how the LMTO method and the physically simple concepts contained in linear theory may be used in self-consistent calculations to estimate ground-state properties of metals and compounds. Here we treat the local-density approximation to the functional formalism of Hohenberg3 Kohn, and Sham, and the force relation derived by Andersen together with an accurate and a first-order pressure relation. In addition, the LMTO-ASA and KKR-ASA methods are generalised to the case of many atoms per cell. 

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