The LCMTO Secular Matrix

Instead of applying tail cancellation as in Sect.2.1 where we derived the KKR-ASA equations, one may use the linear combination of muffin-tin orbitals (5.27) directly in a variational procedure. This has the advantages that it leads to an eigenvalue problem and that it is possible to include non-muffin-tin perturbations to the potential. According to the Rayleigh-Ritz variational principle, one varies y to make the energy functional stationary, i.e. 

Since the muffin-tin orbital is everywhere continuous and differentiable we may evaluate the integral over all space in (5.36) as a sum of integrals over all atomic polyhedra. After repeated use of the Bloch condition (5.28) and rearrangement of the lattice sums, we obtain the well-known result 

The LCMTO secular matrix is now simply obtained by inserting the one-centre expansion (5.29) into the matrix (5.37). We find 

If the cellular potential, which has not been specified so far in this section, is spherically symmetric and the cells approximated by spheres (or 2 

Then the L 1 summations vanish and the LCMTO secular matrix reduces to 

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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. 

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