Slater augmented plane wave

The first accurate band structure calculations with inclusion of relativistic effects were published in the mid-sixties. Loucks published [64-67] his relativistic generalization of Slaters Augmented Plane Wave (APW) method. [68] Neither the first APW, nor its relativistic version (RAPW), were linearized, and calculations used ad hoc potentials based on Slaters s Xa scheme, [69] and were thus not strictly consistent with the density-functional theory. Nevertheless (or, maybe therefore ) good descriptions of the bands, Fermi surfaces etc. of heavy-element solids like W and Au were obtained.[3,65,70,71] With this background it was a rather simple matter to include [4,31,32,72] relativistic effects in the linear methods [30] when they (LMTO, LAPW) appeared in 1975. 

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. 

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 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. 

In the present paper we focus on crystals and surfaces. We choose DFT as the quantum mechanical treatment of exchange and correlation. This means that we must solve the Kohn-Sham (KS) equations by means of a proper basis set. For this purpose we use the augmented plane wave (APW) scheme, which originally was proposed by Slater [10]. The development of APW and its linearized version, which led to the WIEN code [11] and its present version WIEN2k [12], was described in detail in a recent review [1] and previous articles [13-15]. The main concepts are summarized below ... 

Freeman, A.J., J.O. Dimmock, and R.E. Watson, 1966, The augmented plane wave method and the electronic properties of rare earth metals, Ldwdin, P.O., ed.. Quantum Theory of Atoms, Molecules and the Solid State, a Tribute to John C. Slater (Academic Press, New York), pp. 361-380. 

Augmented Plane Waves (APW) This method, introduced by Slater [45], consists of expanding the wavefunctions in plane waves in the regions between the atomic spheres, and in functions with spherical symmetry within the spheres. Then the two expressions must be matched at the sphere boundary so that the wave-functions and their first and second derivatives are continuous. For core states, the wavefunctions are essentially unchanged within the spheres. It is only valence states that have significant weight in the regions outside the atomic spheres. 

The band structure of semiconducting SmSe shown in Fig. 57 is calculated with the augmented plane-wave method (APW). The approximation is used to obtain the muffin-tin potentials with a = 0.67 for the exchange, assuming an intermediate state for Sm with the configuration 4f d instead of pure 4f . The atomic wave functions are derived in the Hartree-Fock-Slater approximation, Farberovich [1]. The band structure model of [1] is qualitatively confirmed by an analysis of the reflection and electroreflection spectra of SmSe single crystals with the minimum direct gap located at the X point. However, the next direct gap is at r and no indications of reflection structures which are attributable to K point excitations are observed, Kurita et al. [2]. Earlier, the band structure for the T-X (i.e. (100)) direction was calculated by... 

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