LMTO matrices

Although the errors introduced by the atomic-sphere approximation are unimportant for many applications, e.g. self-consistency procedures, there are cases where energy bands of high accuracy are needed, and where one should include the perturbation (6.2) in some form. Below, we derive an expression which accounts to first order for the differences between the sphere, atomic or muffin tin, and the atomic polyhedron, re-establishes the correct kinetic energy in the region between the sphere and the polyhedron, and corrects for the neglect of higher partial waves. The extra terms added to the LMTO matrices which accomplish these corrections are called the combined correction terms [6.2]. 

The COR is constructed to calculate those extra structure constants which may be used to correct the ASA for approximate treatment of the region between the sphere and the atomic polyhedron, and for the neglect of higher components as described in Sect.6.9. The programme produces and stores on disk or tape a set of correction-term matrices distributed on a suitable grid in the irreducible wedge of the Brillouin zone. Whenever requested the correction matrices may be retrieved by LMTO and used together with the canonical structure constants to set up the corrected LMTO matrices. 

In 1973 Andersen and Woolley [1.25] extended the LCMTO method to molecular calculations. At the end of their paper they introduced that choice of MTO tail, i.e. proportional to J = 9i/j/9E, which in a natural fashion ensured orthogonality to the core states and at the same time led to an accurate and elegant formulation of linear methods. The resulting, technique was immediately developed in a paper by Andersen [1.26] which, in a condensed form, contains most of what one need know about the simple concepts of linear band theory. Thus, we find here the KKR equation within the atomic-sphere approximation at this stage is called ASM the LCMTO secular matrix, latter called the LMTO matrix the energy-independent structure constants and the canonical bands and the Laurent expansion of the logarithmic-derivative function and the corresponding potential parameters. 

A typical example where such a procedure is needed is in the application of the LMTO method to molecules. Furthermore, in this situation Bloch s theorem does not apply and k is therefore not a good quantum number. Instead, the k dependence should be substituted by a Q , Q dependence, where Q is a site index. Formally, the LMTO matrix for molecules may be obtained by substituting... 

The contribution to the LMTO matrix (5.45) from the integral over the cell is obtained by adding... 

Theoretical calculations were performed with the linear muffin tin orbital (LMTO) method and the local density approximation for exchange and correlation. This method was used in combination with supercell models containing up to 16 atoms to calculate the DOS. The LMTO calculations are run self consistently and the DOS obtained are combined with the matrix elements for the transitions from initial to final states as described in detail elsewhere (Botton et al., 1996a) according to the method described by Vvedensky (1992). A comparison is also made between spectra calculated for some of the B2 compounds using the Korringa-Kohn-Rostoker (KKR) method. 

ANG AO ATA BF CB CF CNDO CPA DBA DOS FL GF HFA LDOS LMTO MO NN TBA VB VCA WSL Anderson-Newns-Grimley atomic orbital average t-matrix approximation Bessel function conduction band continued fraction complete neglect of differential overlap coherent-potential approximation disordered binary alloy density of states Fermi level Green function Flartree-Fock approximation local density of states linear muffin-tin orbital molecular orbital nearest neighbour tight-binding approximation valence band virtual crystal approximation Wannier-Stark ladder... 

The products (T I4 Tfp) are calculated from LMTO eigenstates by using a numerical derivation of the hamiltonian in the reciprocal space as explained in last section, diagonalization provides the components of eigentstates on the basis set Xthi k ). In this basis the velocity operator, Vx = [X,H], has the following matrix elements ... 

Not until the so called third generation of LMTOs [58], was there a way to properly include the interstitial part of the muffin-tin potential and perform calculations without the ASA, in fact it was possible to perform calculations for exact muffin-tins using the Exact Muffin-Tin Orbitals (EMTO) method. Since the structural dependent part is called the slope matrix in the EMTO method, this is the name I will use for the rest of this thesis when discussing the EMTO method. 

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. 

Expressions (2.32,33) are clearly the matrix generalisation of the unhybridised scaling relations (2.28,29), and they are much simpler than the conventional LMTO equations. They are, however, also slightly less accurate, and therefore we consider here the more versatile, conventional LMTO formalism. 

The LMTO secular matrix may now be written in the form H - EQ, which corresponds to the generalised eigenvalue problem... 

As shown in [6.6] the LMTO-ASA Hamiltonian matrix may be transformed into the two-centre form [6.7] where the hopping integrals are products of potential parameters and the canonical structure constants. This result was already stated in Sect.2.5. A less accurate two-centre approximation based upon the KKR-ASA equations will be presented in Sect.8.1.2. The canonical structure constants which, after multipiication by the appropriate potential parameters, form the two-centre hopping integrals are 1 isted in Table 6.1. The... 

Hence, the KKR-ASA matrix 4 is a factor of the LMTO-ASA matrix, and the LMTO and KKR methods are equivalent in the neighbourhood of E, as we wished to prove. 

With this definition, xL is equal to the proper orbital in the interstitial region only, while inside the spheres it is derived from a constant (pseudo) potential v(r) = E - k. For that reason the integral over the sphere appearing in (6.44) is simply the LMTO overlap matrix (5.47) evaluated for the free-electron potential parameters from Sect.4.4 corresponding to = D j (kS). Hence, the contribution from the second term in (6.44) may be included in the LMTO equations (5.45) by subtracting... 

As a final and obvious point we mention that the data for STR and COR for the same crystal structure should be consistent. If this is not the case LMTO will usually (fortunately) fail in the Cholesky decomposition of the overlap matrix. 

Within the ASA, the LMTO method not only acquires incredible speed -comparable to a semiempirical method - but both the Hamiltonian and the overlap matrix automatically decompose into solely structure-dependent... 

The electronic states of the phases in Ti-(A1, Ge)-(C, N) systems were studied by Ivanovsky et al (1987b, 1988b), using quantum-chemical calculations. The authors considered the initial stage of formation of these compounds and carried out LMTO-GF calculations of isolated Al impurities replacing Ti or C (N) atoms in the cubic TiC or TiN lattice. As can be seen from Fig. 5.15 for the TiC + Al system, Al is located at a site in the metal sublattice, Als states are hybridised mainly with the p-d-like subband of the matrix, while the Alp states are concentrated near the upper edge of the TiC valence band. On the Al - C substitution, s and p states of the impurity are shifted to the lower part of the occupied TiC band, and form nonbonding atom-like peaks at the DOS minima. A... 

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