Augmented-spherical-wave method

The augmented spherical wave method (ASW) describes the valence electrons by a set of spherical waves centred round ionic cores described by atomic orbital-like functions. ... 

The augmented spherical-wave method of Williams et al. [1.20] appeared in 1979 and is an efficient computational scheme to calculate self-consistent electronic structures and ground-state properties of crystalline solids. According to its inventors it is a "direct descendant of the LMTO technique", and a comparison will show that the two methods are indeed very similar. 

Self-consistent ab initio band-structure calculations using the augmented-spherical-wave method have been carried out by Coehoom (1990) for hypothetical YFen and YFei2-jM t (M=Ti, V, Cr, Mo, and W). The calculated value of magnetic moment per Fe atom is 2.02 Ub, in good agreement with experiment, particularly if one takes into account that the experimental total moment also contains a small orbital contribution. 

Eriksson and J. Wills, in The augmented spherical wave method a comprehensive treatment, edited by H. Dreysse, Lecture Notes in Physics, vol. 535, pp. 247-285 (Springer, Berlin/Heidelberg, 1999). 

The APW (augmented plane wave) method was devised by Slater (1937,1965), and is based on the solution of the Schrodinger equation for a spherical periodic potential using an expansion of the wavefunction in terms of solutions of the atomic problem near the nucleus, and an expansion in plane waves outside a predetermined sphere in the crystal. 

This monograph is based almost entirely on the work of O.K. Andersen. It is therefore appropriate to reveal the sources of the material presented, and at the same time give a brief history of the development of linear methods. At present several types of such methods are used, e.g. the linear muffin-tin orbitals (LMTO) method [1.19], the linear augmented plane-wave (LAPW) method [1.19], the augmented spherical-wave (ASW) method [1.20], and the linear rigorous cellular (LRC) method [1.15]. Of these the LMTO method, which was the earliest, will be our main concern. 

Within third-generation LMTO theory, also called NMTO theory, the Taylor series is not truncated after the second term but extended to go up to Nth order, thereby increasing its accuracy [235]. The augmented spherical wave (ASW) method [236] closely resembles LMTO theory despite a few technical differences (chosen energy for linearization and exact shape of the analytical envelope function). 

The authors of [20] used the method of augmented spherical waves (ASW) for calculation of bulk moduH and cohesive energies of metals. Figure 8.7 (lower part) shows the augmented spherical waves that are applied for simulating of interatomic electron waves. 

Linearized band structure methods were developed in the 1970s the linearized augmented plane wave (LAPW) method (36), the linear combination of muffin-tin orbitals (LMTO) method (37), the augmented spherical wave (ASW) method (38), and some others. In the LAPW method a warped muffin tin potential is frequently used, in which the real shape of the crystal potential in the interstitial region between the atomic spheres is taken into account. In the LMTO and ASW approaches the atomic sphere approximation (ASA) is frequently applied, in which— contrary to the muffin-tin approximation—overlapping atomic spheres are used. The crystal potential in the spheres is again assumed to be spherically symmetric. The sum of the atomic sphere volumes must be equal to the total volume of the unit cell. No interstitial space remains. 

Linear methods originated in the augmented plane wave (APW) method of Slater and were developed initially by Andersen. " Like the pseudopotential method, the linear augmented plane wave (LAPW) approach uses density functional theory, but the ionic cores are not represented by pseudopotentials. Instead each core is modelled by a sphere inside which the wave function has the form of a linear combination of radial functions times spherical harmonics. 

This is the simplified, two-region potential, on which the augmented-plane wave (APW) method by Slater [226,227] is essentially based. It stands for the rigorous approach which correctly describes both the spherical, atomic-like region close to the nucleus and also the flat region between the nuclei, by treating the two regions differently, in the spirit of the above muffin-tin idea. 

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