Tin Orbitals and the Atomic Sphere Approximation

Energy bands and a number of properties have been described here in terms of L( A() theory and matrix elements given by formulae such as Eq. (20-6). We turn now to the origin of those formulae and to a description of the electronic structure that proves useful for other properties. The formulae for the matrix elements will in fact be obtained from transition-metal pseudopotential theory, but the principal results can be obtained from the theory of Miiflin-Tin Orbitals, which we discu.ss first. Moreover, one of the central concepts of Muffin-Tin Orbital theory is necessary for using transition metal pseudopotential theory to obtain the formulae for the interatomic matrix elements. The analysis in this section and the next is somewhat analogous to the use of free-clectron theory to obtain the form and estimates of the magnitudes of the matrix elements used in the LCAO theory, and here the consequences arc just as rich. 

Muffin-Tin Orbital theory is in the spirit of the very early treatment of alkali metals by Wigner and Seitz (1934), who focused on a single atomic cell (those points nearer the atom being studied than any other atom) in which the potential is nearly spherically symmetric. They then replaced the cell by a sphere of equal volume, the sphere of radius /q that we introduced in the discussion of simple metal.s. This is illustrated in Fig. 20-12 for a face-centered cubic lattice. Wigner 

Two very important points can be made about this. First, the approximation makes sense for a range of values of r, and so our formal results for physical 

I-IGUKH 20-12 A (100) section of a face-centered cubic lattice with atomic cells constructed around each atom. Also shown are atomic spheres, of equal volume, constructed around each atom. 

Andersen considered atomiclike orbitals calculated with a potential equal to the atomic potential for r r, , but constant for r these might have more relevance to the metal than do true atomic orbitals. The orbital at any given 

---

The investigations of Asada et al. and Christensen - were carried out with linear-muffin-tin orbitals within the atomic sphere approximation (LMTO- AS A) Within the muffin-tin model suitable s, p and d basis functions (muffin-tin orbitals, MTO) are chosen. In contrast to the APW procedure the radial wave functions chosen in the linear MTO approach are not exact solutions of the radial Schrodinger (or Dirac) equation. Furthermore, in the atomic sphere approximation (ASA) the radii of the atomic spheres are of the Wigner-Seitz type (for metals the spheres have the volume of the Wigner-Seitz cell) and therefore the atomic spheres overlap. The ASA procedure is less accurate than the APW method. However, the advantage of the ASA-LMTO method is the drastic reduction of computer time compared to the APW procedure. 

Norman et al. performed additional calculations to test this last result. They used a supercell structure with one Ce site in the center that had a 4f hole. The ena-gy differences between the poorly (5d)- and fully (4f)-screened 4f holes were 1.9 and 2.4 eV for a- and y-Ce, respedtively - close to the experimental results but in reverse order. They tentatively attributed this reversal to the use of the linear mulfin-tin-orbital approximation and the atomic-sphere approximation. 

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". 

TABLES 1 and 2 show the calculated and measured results of splitting energies in WZ and ZB structures, respectively. Suzuki et al derived the values of A and Ar for WZ and ZN GaN and AIN from a full-potential linearised augmented plane wave (FLAPW) and band calculation [3,4], Another result with LAPW calculation was given by Wei and Zunger [5], Kim et al [6] determined them by the linear muffm-tin orbital (LMTO) method within the atomic sphere approximation (ASA). Majewski... 

In the LDA, Adolph and Bechstedt [157,158] adopted the approach of Aspnes [116] with a plane-wave-pseudopotential method to determine the dynamic x of the usual IB V semiconductors as well as of SiC polytypes. They emphasized (i) the difficulty to obtain converged Brillouin zone integration and (ii) the relatively good quality of the scissors operator for including quasiparticle effects (from a comparison with the GW approximation, which takes into account wave-vector- and band-dependent shifts). Another implementation of the SOS x —2 ffi, ffi) expressions at the independent-particle level was carried out by Raskheev et al. [159] by using the linearized muffin-tin orbital (LMTO) method in the atomic sphere approximation. They considered... 

With the later introduction of the Linear Muffin-Tin Orbital (LMTO) method [46, 47, 48, 49, 50], a formulation of the multiple scattering problem in terms of Hamiltonians was introduced. This provided another way to gain more knowledge about the KKR method, which, although elegant, was not so easily understood. In the LMTO method one had to use energy linearizations of the MTOs to be able to put it into a Hamiltonian formalism. The two methods (KKR and LMTO) were shown [51] to be very closely related within the Atomic Sphere Approximation (ASA) [46, 52], which was used in conjunction with the LMTO method to provide an accurate and computationally efficient technique. 

In Chap.5 we derive the LCMTO equations in a form not restricted to the atomic-sphere approximation, and use the , technique introduced in Chap.3 to turn these equations into the linear muffin-tin orbital method. Here we also give a description of the partial waves and the muffin-tin orbitals for a single muffin-tin sphere, define the energy-independent muffin-tin orbitals and present the LMTO secular matrix in the form used in the actual programming, Sect.9.3. 

In Chap.6 the atomic-sphere approximation is introduced and discussed, canonical structure constants are presented, and it is shown that the LMTO-ASA and KKR-ASA equations are mathematically equivalent in the sense that the KKR-ASA matrix is a factor of the LMTO-ASA secular matrix. In addition, we treat muffin-tin orbitals in the ASA, project out the i character of the eigenvectors, derive expressions for the spherically averaged electron density, and develop a correction to the ASA. 

To turn the LCMTO method into an efficient calculational technique, in the following we introduce the atomic-sphere approximation and parametrise the energy dependence of the one-, two-, and three-centre or overlap integrals appearing in (5.40) by means of the results in Sect.3.5. The resulting procedure constitutes the so-called linear muffin-tin orbital (LMTO) method. 

In the k = 0 limit used in the atomic-sphere approximation, the wave equation (5.7) used to construct the tail of the partial wave (5.10) turns into the Laplace equation. Hence, in the definition of the muffin-tin orbitals (5.13,25) the spherical Bessel and Neumann functions should be substituted by the harmonic functions (r/S)z and (r/S)" "1, respectively. By means of the small kr limits (5.8) of the spherical Bessel and Neumann functions, the expansion theorem (5.14) becomes... 

The LMTO method has the computational speed and flexibility needed to perform calculations of electron states in molecules and compounds. Therefore in the present chapter we shall generalise the LMTO formalism purely within the atomic-sphere approximation to include the case of many inequivalent atoms per cell. The LMTO method is based on the variational principle in conjunction with energy-independent muffin-tin orbitals but, in addition to this approach, we have also considered the tail-cancellation principle which led to the KKR-ASA condition (2.8). Since the latter has conceptual advantages, we apply the tail-cancellation principle to the simplest possible case of more than one atom, namely the diatomic molecule. After that, we turn to crystalline solids and generalise or sometimes rederive the important equations of LMTO formalism. Hence, in addition to giving the LMTO equations for many atoms per cell, the present chapter may also serve as a short and compact presentation of the crystal-structure-dependent part of LMTO formalism. The potential-dependent part is treated in Chap.3. In the final sections are listed the modifications needed to calculate ground-state properties for materials with several atoms per cell. 

Shape approximations limit the precision of the calculations whether they be the muifin-tin or the atomic-sphere approximations. The minority f character induced at the Fermi energy is not spherically symmetric and will be influenced by these shape approximations, as discussed by Harmon and Freeman (1974). The non-spherical potential terms, not incorporated by Temmerman and Sterne, could significantly modify the interaction of the 5p and 4f orbitals. 

The electronic structure is determined using the ab initio all-electron scalar-relativistic tight-binding linecir muffin-tin orbital (TB-LMTO) method in the atomic-sphere approximation (ASA). The nnderlying lattice, zincblende structure, refers to an fee Bravais lattice with a basis which contains a cation site (at a(0,0,0)), an anion site (at o(j,, )), and two interstitial sites occupied by empty spheres (at a(, 5, h) and a(, , )) which in turn are necessary for a correct description of open lattices . ... 

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. 

The results are conveniently and clearly expressed in a thermodynamic formalism this is why they find their place in this chapter. They depend however on parameters which are drawn from band-theory, especially from the LMTO-ASA (Linear Muffin-Tin Orbitals-Atomic Sphere Approximation) method. 

---