Use of Cyclic Matrices to Obtain Analytic Expressions for Crystals

Abstract

Whereas the recursion method [1,2] allows to calculate the electronic structure of systems without any symmetry, a profitable use of cyclic matrix functions (CMF) requires the presence of some translational symmetry. Actually, the latter can be often more fully exploited using CMF rather than Bloch waves. The distinct advantages offered by CMF include in particular: first the possibility to deal directly with the physical properties expressible in terms of some element or the trace of a CMF, so bypassing the band structure calculation in situations where it is irrelevant. Second, it is applicable to functions of several cyclic matrices, not merely a single one (the Hamiltonian for instance), thereby offering an analytic solution for the difficulty associated with the nonorthogonality of the localized orbitals. Third, the assumption of an infinite lattice, an integral step in band structure calculations, appears to be superfluous here, as CMF can be defined and analytically expressed for lattices of finite size. Fourth, CMF exhibit a good flexibility to be perturbed by the presence of defects, in a way that is well referenced to bulk properties.