Local density approximation method

The application of density functional theory to isolated, organic molecules is still in relative infancy compared with the use of Hartree-Fock methods. There continues to be a steady stream of publications designed to assess the performance of the various approaches to DFT. As we have discussed there is a plethora of ways in which density functional theory can be implemented with different functional forms for the basis set (Gaussians, Slater type orbitals, or numerical), different expressions for the exchange and correlation contributions within the local density approximation, different expressions for the gradient corrections and different ways to solve the Kohn-Sham equations to achieve self-consistency. This contrasts with the situation for Hartree-Fock calculations, wlrich mostly use one of a series of tried and tested Gaussian basis sets and where there is a substantial body of literature to help choose the most appropriate method for incorporating post-Hartree-Fock methods, should that be desired. 

Transition structures are more dihicult to describe than equilibrium geometries. As such, lower levels of theory such as semiempirical methods, DFT using a local density approximation (LDA), and ah initio methods with small basis sets do not generally describe transition structures as accurately as they describe equilibrium geometries. There are, of course, exceptions to this, but they must be identihed on a case-by-case basis. As a general rule of thumb, methods that are empirically dehned, such as semiempirical methods or the G1 and G2 methods, describe transition structures more poorly than completely ah initio methods do. 

LCAO (linear combination of atomic orbitals) refers to construction of a wave function from atomic basis functions LDA (local density approximation) approximation used in some of the more approximate DFT methods... 

The electronic properties of single-walled carbon nanotubes have been studied theoretically using different methods[4-12. It is found that if n — wr is a multiple of 3, the nanotube will be metallic otherwise, it wiU exhibit a semiconducting behavior. Calculations on a 2D array of identical armchair nanotubes with parallel tube axes within the local density approximation framework indicate that a crystal with a hexagonal packing of the tubes is most stable, and that intertubule interactions render the system semiconducting with a zero energy gap[35]. 

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. 

Our work demonstrates that EELS and in particular the combination of this technique with first principles electronic structure calculations are very powerful methods to study the bonding character in intermetallic alloys and study the alloying effects of ternary elements on the electronic structure. Our success in modelling spectra indicates the validity of our methodology of calculating spectra using the local density approximation and the single particle approach. 

Hamada, N. and Ohnishi, S. (1986) Self-interaction correction to the local-density approximation in the calculation of the energy band gaps of semiconductors based on the full-potential linearized augmented-plane-wave method, Phys. Rev., B34,9042-9044. 

Within this local-density approximation one can obtain exact numerical solutions for the electronic density profile [5], but they require a major computational effort. Therefore the variational method is an attractive alternative. For this purpose one needs a local approximation for the kinetic energy. For a one-dimensional model the first two terms of a gradient expansion are ... 

Some of the major areas of activity in this field have been the application of the method to more complex materials, molecular dynamics, [28] and the treatment of excited states. [29] We will deal with some of the new materials in the next section. Two major goals of the molecular dynamics calculations are to determine crystal structures from first principles and to include finite temperature effects. By combining molecular dynamics techniques and ah initio pseudopotentials within the local density approximation, it becomes possible to consider complex, large, and disordered solids. 

Below is a brief review of the published calculations of yttrium ceramics based on the ECM approach. In studies by Goodman et al. [20] and Kaplan et al. [25,26], the embedded quantum clusters, representing the YBa2Cu307 x ceramics (with different x), were calculated by the discrete variation method in the local density approximation (EDA). Although in these studies many interesting results were obtained, it is necessary to keep in mind that the EDA approach has a restricted applicability to cuprate oxides, e.g. it does not describe correctly the magnetic properties [41] and gives an inadequate description of anisotropic effects [42,43]. Therefore, comparative ab initio calculations in the frame of the Hartree-Fock approximation are desirable. 

The local density approximation (LDA) and GGA within a plane-wave pseudopotential method was used in Ishibashi and Kohyama (2000) while DFT within the linearized augmented plane wave (LAPW) approach was employed in Sing et al. (2003b). 

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. 

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