LSDA

The simplest approximation to the complete problem is one based only on the electron density, called a local density approximation (LDA). For high-spin systems, this is called the local spin density approximation (LSDA). LDA calculations have been widely used for band structure calculations. Their performance is less impressive for molecular calculations, where both qualitative and quantitative errors are encountered. For example, bonds tend to be too short and too strong. In recent years, LDA, LSDA, and VWN (the Vosko, Wilks, and Nusair functional) have become synonymous in the literature. 

LORG (localized orbital-local origin) technique for removing dependence on the coordinate system when computing NMR chemical shifts LSDA (local spin-density approximation) approximation used in more approximate DFT methods for open-shell systems LSER (linear solvent energy relationships) method for computing solvation energy... 

In the more general case, where the a and p densities are not equal, LDA (where the sum of the a and p densities is raised to the 4/3 power) has been virtually abandoned and replaced by the Local Spin Density Approximation (LSDA) (which is given as the sum of die individual densities raised to the 4/3 power, eq. (6.17)). 

LSDA may also be written in terms of the total density and the spin polarization. 

For closed-shell systems LSDA is equal to LDA, and since this is the most common case, LDA is often used interchangeably with LSDA, although this is not true in the general case (eqs. (6.16) and (6.17)). The method proposed by Slater in 1951 can be considered as an LDA mediod where die correlation energy is neglected and the exchange term is given as... 

The LSDA approximation in general underestimates the exchange energy by 10%, thereby creating errors which are larger tlian the whole correlation energy. Electron correlation is furthermore overestimated, often by a factor close to 2, and bond strengths are as a consequence overestimated. Despite the simplicity of the fundamental assumptions, LSDA methods are often found to provide results with an accuracy similar to that obtained by wave mechanics HE methods. 

Perdew and Wang (PW86) " proposed modifying the LSDA exchange expression to tliat shown in eq. (6.23), where x is a dimensionless gradient variable, and a, b and c being suitable constants (summation over equivalent expressions for the a and P densities is implicitly assumed). 

Becke proposed a widely used correction (B or B88) to tire LSDA exchange energy, which has the correct — asymptotic behaviour for the energy density (but not for the exchange potential). ... 

Perdew proposed a gradient correction to the LSDA result. It appeared in 1986 and is known by the acronym P86. 

This functional was later modified to the following form (also a correction to the LSDA energy) by Perdew and Wang in 1991 (PW91 or P91) ... 

Gradient corrected methods usually perform much better than LSDA. For the G2-1 data set (see Section 5.5), omitting electron affinities, the mean absolute deviations shown in Table 6.1 are obtained. The improvement achieved by adding gradient terms is impressive, and hybrid methods (like B3PW91) perform almost as well as the elaborate G2 model for these test cases. For a somewhat larger set of reference data, called the G2-2 set, the data shown in Table 6.2 are obtained. 

We have used the multisublattice generalization of the coherent potential approximation (CPA) in conjunction with the Linear-MufRn-Tin-Orbital (LMTO) method in the atomic sphere approximation (ASA). The LMTO-ASA is based on the work of Andersen and co-workers and the combined technique allows us to treat all phases on equal footing. To treat itinerant magnetism we have employed for the local spin density approximation (LSDA) the Vosko-Wilk-Nusair parameterization". 

Figure 1. The energy of bcc and hep randoiri alloys and the ])ai tially ordered a phase relative to the energy of the fee phase (a), of the Fe-Co alloy as a function of Co concentration. The corresponding mean magnetic moments are shown in (h). The ASA-LSDA-CPA results are shown as a dashed line for the o ])hase, as a full line for the her ]>hase, as a dot-dashed line for the hep phase, and as a dotted line for the fee phase. The FP-GGA results for pure Fe and Co are shown in (a) by the filled circles (bcc-fcc) and triangles (hep-fee). In (b) experimental mean magnetic moments are shown as open circles (bcc), open scpiares (fee) and open triangles (hep).

Figure 2. The structural energy difference (a) and the magnetic moment (b) as a function of the occupation of the canonical d-band n corresponding to the Fe-Co alloy. The same lines as in Fig. 1 are used for the different structures. In (b) the concentration dependence of the Stoner exchange integral Id used for the spin-polarized canonical d-band model calculations is shown as a thin dashed line with the solid circles. The value of Id for pure Fe and Co, calculated from LSDA and scaled to canonical units, are also shown in (b) as solid squares.

Figure l.The CPA-LSDA-ASA results for the energy of bcc (fat lines) and fee (thin lines) random and ordered alloys. The fee random phase is the referenee energy and defines zero in the graph. The random phases are given in full drawn lines and the ordered phases are given in dashed lines. 

In order to perform the calculation., of the conductivity shown here we first performed a calculation of the electronic structure of the material using first-principles techniques. The problem of many electrons interacting with each other was treated in a mean field approximation using the Local Spin Density Approximation (LSDA) which has been shown to be quite accurate for determining electronic densities and interatomic distances and forces. It is also known to reliably describe the magnetic structure of transition metal systems. 

The LSDA approach requires simultaneous self-consistent solutions of the Schrbdinger and Poisson equations. This was accomplished using the Layer Korringa-Kohn-Rostoker technique which has many useful features for calculations of properties of layered systems. It is, for example, one of only a few electronic structure techniques that can treat non-periodic infinite systems. It also has the virtue that the computational time required for a calculation scales linearly with the number of different layers, not as the third power as most other techniques. 

Antiphase boundary (APB) conservative vacancy segregation at Arrhenius plot Asymmetrical mixtures Atomic-sphere approximation (ASA) ASA-LSDA... 

Local-density approxmiation (LDA) comparison with GGA Local spm-deasity approxmiation (LSDA) Lmear-muffin-tm orbital (LMTO)... 

FIG. 25 Lime soap dispersing ability of linear alkylbenzenesulfonate-a-olefmsulfonate (LABS-AOS) mixed surfactant system. LSDA, lime soap dispersion ability. (From Ref. 3.)... 

---