Eigenvalues calculations using canonical

The STR may be used to calculate the canonical structure constants defined by (6.7-9) or (8.23,24). In a typical application the programme is executed once for a given crystal structure. It produces and stores on disk or tape a set of structure-constant matrices distributed on a suitable grid in the irreducible wedge of the Brillouin zone. Whenever that particular crystal structure is encountered, the structure constant matrices may be retrieved and used to set up the LMTO eigenvalue problem which, in turn, leads to the energy bands of the material considered. 

Let us now turn to the problem switching on a model potential V(r) to the Hamiltonian used above. Denoting the canonical density matrix calculated there by = C(V =0), the simplest approximation is to follow the ideas of the Thomas-Fermi (TF) method. Then, with slowly varying V(r) for which the assumptions of this approximation are valid, one can return to the definition at Eq. (2.2), and simply move all eigenvalues a,- by the same (almost constant— ) amount F(r), the wavefunctions ( i(r) being unaffected to the same order of approximation. Hence one can write for the diagonal form of the canonical density matrix... 

In Chap.2 we introduced the concept of canonical bands based upon the KKR-ASA equations and used it to interpret energy bands calculated by the LMTO method. We did this because the KKR-ASA and LMTO-ASA methods are mathematically equivalent, as proven below. Specifically, we show that in a range around so narrow that the small parameter may be neglected the LMTO-ASA and KKR-ASA equations will lead to the same eigenvalues. 

Figure 4.9 Application of the canonical orthonormalization procedure of Section 3.6 to the calculation of the 1 s and 2s eigenfunctions and eigenvalues approximations for the Is and 2s orbitals in hydrogen over Slater functions. Note the exact fit of the Is Slater, which is an eigenfunction of the Fock matrix for the hydrogen atom and the relatively close agreement of the ls/2s linear combinations based on simple canonical orthogonalization and also direct orthonormalization using the matrix procedure of Section 3.7.

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