Self local density approximation

Stampfl C, van de Walle C G, Vogel D, Kruger P and Pollmann J 2000 Native defects and impurities in InN First-principles studies using the local-density approximation and self-interaction and relaxation-corrected pseudopotentials Phys. Rev. B 61 R7846-9... 

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. 

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. 

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. 

Aside from the well-known LDA (local density approximation (26, 27) BLYP (Becke-Lee-Yang-Parr (28, 29)) and B3LYP (29, 30) functionals, we considered the more recent B97-1 functional (which is a reparametrization (31) of Becke s 1997 hybrid functional), the B97-2 functional (32) (a variation of B97-1 which includes a kinetic energy density term), and the HCTH-407 (33) functional of Boese and Handy (arguably the best GGA functional in existence at the time of writing). The rationale behind the Wlc (Weizmann-1 cheap) approach is extensively discussed elsewhere. (34,35) For the sake of self-containedness of the paper, we briefly summarize the steps involved for the specific system discussed here ... 

Fig. 3.12 The binding energies, equilibrium internuclear separations and vibrational frequencies across the first-row diatomic molecules. Note the good agreement between the self-consistent local density approximation calculations and experiment for R and coe but the larger systematic error of up to 2 eV for the binding energy. (After Gunnarsson et aL (1977).)...

Fig. 3.13 Left-hand panel The electronic structure of an sp-valent diatomic molecule as a function of the internuclear separation. Labels Et and Ep mark the positions of the free-atomic valence s and p levels respectively and = ( s + p). The quantity Rx is the distance at which the and upper tr9 levels cross. The region between the upper and lower n levels has been shaded to emphasize the increase in their separation with decreasing distance that is responsible for this crossing. Right-hand panel The self-consistent local density approximation electronic structure for C2 and Si2 whose equilibrium internuclear separations are marked by RCz and RSa respectively. (After Harris 1984.)...

Fig. 6.8 The normalized heat of formation, AAZ/fZfAp1 3)2], as a function of the average cube root of the electron density, p1/3, for sp-valent AB compounds. The solid curve is the electron-gas contribution, eqn (6.88). The open circles are the self-consistent local density approximation predictions for the CsCI lattice. (From Pettifor and Gelatt (1983).)...

The relaxation of the structure in the KMC-DR method was done using an approach based on the density functional theory and linear combination of atomic orbitals implemented in the Siesta code [97]. The minimum basis set of localized numerical orbitals of Sankey type [98] was used for all atoms except silicon atoms near the interface, for which polarization functions were added to improve the description of the SiOx layer. The core electrons were replaced with norm-conserving Troullier-Martins pseudopotentials [99] (Zr atoms also include 4p electrons in the valence shell). Calculations were done in the local density approximation (LDA) of DFT. The grid in the real space for the calculation of matrix elements has an equivalent cutoff energy of 60 Ry. The standard diagonalization scheme with Pulay mixing was used to get a self-consistent solution. In the framework of the KMC-DR method, it is not necessary to perform an accurate optimization of the structure, since structure relaxation is performed many times. 

Here Ho is the kinetic energy operator of valence electrons Vps is the pseudopotential [40,41] which defines the atomic core. V = eUn(r) is the Hartree energy which satisfies the Poisson equation ArUn(r) = —4nep(r) with proper boundary conditions as discussed in the previous subsection. The last term is the exchange-correlation potential Vxc [p which is a functional of the density. Many forms of 14c exist and we use the simplest one which is the local density approximation [42] (LDA). One may also consider the generalized gradient approximation (GGA) [43,44] which can be implemented for transport calculations without too much difficulty [45]. Importantly a self-consistent solution of Eq. (2) is necessary because Hks is a functional of the charge density p. One constructs p from the KS states Ts, p(r) = (r p r) = ns Fs(r) 2, where p is the density matrix,... 

Fig. 1 Asymptotic structure coefficients as(j8), c ks,xG ) crw03), a CS), and aKS gOS) as fimction of barrier height parameter )S =VW/eF, where W is the barrier height and eF the Fermi energy. Corresponding values of the Wigner-Seitz radius rs for jellium and structureless-pseudopotential models over the metallic range of densities are also given. The relationship between rs and ff is via self-consistent calculations in the local density approximation for exchange-correlation.

A plot of wfPp(z) for rs = 3.24 employing the orbitals of the finite-linear-potential moder is given in Fig. 6. The corresponding local density approximation (LDA) potential is also plotted. In the interior and about the surface of the metal the two potentials are equivalent. But outside the surface vxapp(z) improves upon the LDA significantly and approaches the exact structure asymptotically. We thus expect that properties such as the surface energy and work function obtained with Eapp[p] and v pp(r) to be superior to those of the LDA. Such self-consistent calculations are in progress. 

For perfectly ordered crystals at absolute zero, solutions to the Schrodinger equation can be calculated on fast computers using density functional theory (DFT) based on the self-consistent local density approximation (LDA) simplifying procedures using different basis functions include augmented... 

An important new development within solid-state theory is the combination of self-consistent band structure, structure determination, and molecular dynamics within the local-density approximation as developed by Car and Parrinello (1985). Our discussion follows that of Srivastava and Weaire (1987). 

LDA SCF VWN (Local-Density Approximation, Self-Consistent Field)... 

Huang and Ching [129 131] followed the scheme elaborated by Sipe and coworkers while using the self-consistent orthogonalized-LCAO method in the local density approximation (LDA) to determine the band structures and... 

Several authors " have attempted to use density functional type approaches for only the correlation energy. If the Hartree-Fock expression for exchange is kept, this of course ensures that the self-Coulomb integrals are properly cancelled by self-exchange one goes back (for better or worse) to the HF level as the point of reference. The computational demands are of the same order as those of the HF calculation itself Results of early attempts of this nature have been summarized by Stoll et al If the local density approximation to correlation. 

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