KKR matrix

If the p labels refer to lattice sites j, this matrix reduces to 6(k) in the KKR matrix M(k) and Eq. (15) can be shown to reduce to Eq. (14). The evaluation of is hindered by the free-electron poles in the b matrices. This has formed a barrier for electronic structure calculations of interstitial impurities, but in some cases this problem was bypassed by using an extended lattice in which interstitial atoms occupy a lattice site. For the calculation of Dingle temperatures [1.3] and interstitial electromigration [14] the accuracy was just sufficient. Recently this accuracy problem has been solved [15, 16]. 

For a finite cluster of atoms, Equation (5.28) can be solved by inverting the corresponding real-space KKR matrix (Ebert et al. 1999),... 

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. 

In a matrix notation omitting L indices and summations, the KKR/MST equation is... 

If the KKR functional A were treated as a functional of the coefficient matrix co, the derived variational equations would be a set of linear equations of the form S1 J]v[- ] = 0, where the bracketed term is the same as in Eqs. (7.10). The solution of these simplified equations for a given value of X,L and all values of pt, L is a column vector of the o -matrix. These simplified equations were tested by empty-lattice calculations on an fee space-lattice [280]. 

ZnA(r, E) and JnA (r, E) (see Equation (5.31)), while the scheme to set up the singlesite r-matrix (Equation (5.27)) and the various ways for dealing with the multiple scattering problem (Equations (5.29) and (5.30)) remain unchanged. This important feature of the KKR formalism also applies to the use of more complex Hamiltonians, as demonstrated by the inclusion of the Breit interaction (Ebert 1995) and the OP term (Battocletti and Ebert 1996), as well as for the use of CDFT (Ebert et al. 1997a). 

In actual calculations one does not solve this equation, but instead calculates the poles of the scattering path operator to the kink matrix. Nevertheless, it is clear that we need to find an expression for the slope matrix. This can be derived from the bare (or canonical) KKR structure constant matrix Sr,l,rl(k2), and this will be shown below, as well as how to compute the first energy derivative of the slope matrix. 

The EMTO slope matrix can be obtained from the bare KKR structure constant matrix Sr,l,rl(k2) as [58]... 

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. 

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. 

The LCMTO matrix (5.40), which forms the basis of the band-structure technique we are about to develop, is in a form closely related to the LCAO method. One clearly recognises the one-centre term, zeroth order in B, the two-centre terms, first order in B, and the three-centre or crystal-field term, second order in B. The convergence of the LCMTO method is similar to that of the KKR method and occurs when the phase shifts vanish for l > . ... 

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. 

The calculation of the diffracted intensities usually proceeds in two steps. The first step is the construction of the crystal potential and the calculation of the scattering amplitudes from a single atom, and the second step is the calculation of scattering processes within a single atomic layer and the calculation of scattering between different atomic layers. In the second step the multiple scattering processes are based on the condition that the scattered wave from one atom is an incident wave on all other atoms. This leads to a set of linear equations that is solved by matrix inversion. The formulation of the theory is based on the KKR (Korringa-Kohn-Rostocker) method used for band structure calculations. 

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