Local density formalism

Yamashita, J., and S. Asano (1983a). Cohesive properties of alkali halides and simple oxides in the local-density formalism. J. Phys. Soc. Jpn. 52, 3506-13. 

Fourier Transform and Discrete Variational Method Approach to the Self-Consistent Solution of the Electronic Band Structure Problem within the Local Density Formalism. 

Although the effect of the spin-orbit interaction was included for the band peak, it was not for the supercell peak since the effect was thought to be too small to be worth the effort (the two f-electrons in the core used to construct the potential were treated as j = 5/2 states, though). Multiplet dfects were also not taken into account. In principle, this can be done in the local-density formalism (von Barth 1979), but has not been applied to solids to the authors knowledge. If multiplet effects turn out to be crucial for interpreting the BIS spectra of cerium pnictides, then an effort could be made to incorporate those effects into the calculation. 

Band structure calculations of fee crystalline Cgo using a pseudopotential local density formalism predict C(io to be a semiconductor with a band gap in the range 0.9-1.5 eV, and widths of the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) derived bands in the range 0.4-... 

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. 

The multipole formalism described by Stewart (1976) deviates from Eq. (3.35) in several respects. It is a deformation density formalism in which the deformation from the IAM density is described by multipole functions with Slater-type radial dependence, without the K-type expansion and contraction of the valence shell. While Eq. (3.35) is commonly applied using local atomic coordinate systems to facilitate the introduction of chemical constraints (chapter 4), Stewart s formalism has been encoded using a single crystal-coordinate system. 

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. 

To treat the stochastic Lotka and Lotka-Volterra models, we have now to extend the formalism presented in Section 2.2.2, where collective variables-numbers of particles iVA and Vg were used to describe reactions. The point is that this approach neglects local density fluctuations in small element volumes. To incorporate both these fluctuations and their correlations due to diffusive conjunction, we are in position now to reformulate these models in terms of the diffusion-controlled processes - in contrast to the rather primitive birth-death formalism used in Section 2.2.2. It permits also to demonstrate in the non-trivial way a role of diffusion in the autowave processes. The main results of this Chapter are published in [21, 25]. 

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... 

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