FLAPW

The AFC method does not have the same demanding convergence criteria as the FLAPW method but yields physically equivalent results [69]. 

The projector augmented-wave (PAW) DFT method was invented by Blochl to generalize both the pseudopotential and the LAPW DFT teclmiques [M]- PAW, however, provides all-electron one-particle wavefiinctions not accessible with the pseudopotential approach. The central idea of the PAW is to express the all-electron quantities in tenns of a pseudo-wavefiinction (easily expanded in plane waves) tenn that describes mterstitial contributions well, and one-centre corrections expanded in tenns of atom-centred fiinctions, that allow for the recovery of the all-electron quantities. The LAPW method is a special case of the PAW method and the pseudopotential fonnalism is obtained by an approximation. Comparisons of the PAW method to other all-electron methods show an accuracy similar to the FLAPW results and an efficiency comparable to plane wave pseudopotential calculations [, ]. PAW is also fonnulated to carry out DFT dynamics, where the forces on nuclei and wavefiinctions are calculated from the PAW wavefiinctions. (Another all-electron DFT molecular dynamics teclmique using a mixed-basis approach is applied in [84].)... 

As an indication of the types of infonnation gleaned from all-electron methods, we focus on one recent approach, the FLAPW method. It has been used to detennine the band stmcture and optical properties over a wide energy range for a variety of crystal stmctures and chemical compositions ranging from elementary metals [ ] to complex oxides [M], layered dichalcogenides [, and nanoporous semiconductors The k p fonnulation has also enabled calculation of the complex band stmcture of the A1 (100) surface... 

Table 2 Lattice parameter ao (in A) and elastic constants B, Cn, C12, and C44 for CoSi2 (in GPa), calculated using all-electron (FLAPW) and pseudopotential (VASP) techniques in the LDA and using GGC corrections.

We have investigated ground state properties on a first principles basis. Total energy as well as magnetic moment (for FeaNi) were determined with the FLAPW method and the GGA introduced by Perdew and Wang in 1992 by employing the WIEN95 code developed by Blaha et al. 

The calculated and experimental values of the equilibrium lattice constant, bulk modulus and elastic stiffness constants across the M3X series are listed in Table I. With the exception of NiaGa, the calculated values of the elastic constants agree with the experimental values to within 30 %. The calculated elastic constants of NiaGa show a large discrepancy with the experimental values. Our calculated value of 2.49 for the bulk modulus for NiaGa, which agrees well with the FLAPW result of 2.24 differs substantially from experiment. The error in C44 of NiaGe is... 

One of the most accurate approaches to solve the LDF equations for the single slab geometry is the full-potential linearized augmented plane wave (FLAPW) method (10). Here, we highlight only the essential characteristics of this approach for further details the reader is referred to a recent review article (11). 

One obvious drawback of the LDA-based band theory is that the self-interaction term in the Coulomb interaction is not completely canceled out by the approximate self-exchange term, particularly in the case of a tightly bound electron system. Next, the discrepancy is believed to be due to the DFT which is a ground-state theory, because we have to treat quasi-particle states in the calculation of CPs. To correct these drawbacks the so-called self-interaction correction (SIC) [6] and GW-approximation (GWA) [7] are introduced in the calculations of CPs and the full-potential linearized APW (FLAPW) method [8] is employed to find out the effects. No established formula is known to take into account the SIC. 

Equation (15) is solved self-consistently employing the FLAPW method. Using the solutions, wave functions and energies, momentum densities in Equation (8) are calculated. In this step, one more drastic approximation we are going to make is that the occupation number in Equation (10) is replaced by the step function... 

Typical CPs calculated by the FLAPW-LDA are shown compared with experiments measured by Sakurai [14] in Figures 1 and 2, for Li and Cu, respectively. As seen in both figures, there are serious discrepancies between the experiments and the calculations. That is, the calculated profiles are higher than the experimental profiles at small momenta and lower at large momenta, as observed consistently in studies of other solids. Therefore, I take this as an indication that we have to go beyond the LDA. 

Figure 1. The valence-electron CPs of Li along the three principal symmetry directions. The solid curves represent the FLAPW-LDA calculations. The dots represent the experimental results measured by Sakurai etal. [33],... 

This FLAPW-SIC scheme has been applied to the CP calculations of Cu, Si and diamond. The semiconductor Si and the insulator diamond have energy gaps and the most upper valence electrons are regarded as being a slightly bound state. The noble metal Cu has tightly bound d-electrons. 

Table 1. Energy band gaps of diamond and silicon calculated by FLAPW-LDA and FLAPW-SIC schemes. The experimental values [34] are also shown. Units are in eV.

Figure 3. Measured (dotted) and calculated CPs of Si by the FLAPW-LDA (dashed) and the FLAPW-SIC (solid) schemes, The theoretical core profile is represented by a dash-dotted curve (after Kubo et al. [10]). 

Table 2. The dHvA frequencies of some symmetry orbits in Cu calculated by the FLAPW-LDA (denoted as LDA) and FLAPW-SIC (SIC), respectively. The experimental values measured by Shoenberg [22], and Coleridge and Templeton [23], respectively. 

Figure 7. The occupation number densities as functions of wave vector for Na. The thick curves labeled (100), (110) and (111) represent the three principal directions within the first Brillouin zone, obtained by the FLAPW-GWA. The thin solid curve is obtained from an interacting electron-gas model [27]. The dash-dotted line represents the Fermi momentum. 

We have studied the effects of the SIC for the filled and tightly bound bands for Si, diamond and Cu , respectively, by utilizing the FLAPW method. In the case of Si,... 

Figure 10. The valence-electron CPs of Li along the three principal symmetry directions. The solid and dotted curves represent the FLAPW-GWA and FLAPW-LDA calculations, respectively. The dashed curves represent the FLAPW-LDA calculations including correlation effects according to Lundqvist and Lyden [27], The EXPI and EXPII represent the experimental results measured by Sakurai etal. [33] and Schillke etal. [32], respectively (after Kubo [13]).

FLAPW Full potential linearised augmented plane wave Jansen and Freeman (1984)... 

More recently, Zhao et al. (2001a) employed the FLAPW method within the generalized gradient approximation (GGA) to investigate further electronic and magnetic properties of ordered Gai xMnxAs alloys with low and high Mn concentrations, 0.031 x 0.5. 

The theoretical description of the electronic structure has been obtained by means of the LAPW method on an ab-initio basis1101. The electronic potential is determined self-consistently for the elementary cell of the bare host structure, which consists of 44 atoms. More complicated systems, where the tubes are filled with water molecules are also taken into account. Recent self-consistent full-potential calculations (FLAPW) are performed to refine the results 11 . 

As mentioned earlier, the existence of surface shifted core levels has been questioned.6 Calculated results for TiC(lOO) using the full potential linearized augmented plane wave method (FLAPW) predicted6 no surface core level shift in the C Is level but a surface shift of about +0.05 eV for the Tis levels. The absence of a shift in the C Is level was attributed to a similar electrostatic potential for the surface and bulk atoms in TiC. The same result was predicted for TiN because its ionicity is close to that of TiC. This cast doubts on earlier interpretations of the surface states observed on the (100) surface of TiN and ZrN which were thought to be Tamm states (see references given in Reference 4), i.e. states pulled out of the bulk band by a shift in the surface layer potential. High resolution core level studies could possibly resolve this issue, since the presence of surface shifted C Is and N Is levels could imply an overall electrostatic shift in the surface potential, as suggested for the formation of the surface states. 

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