Direct and Indirect LCAO Recursion Methods for Surfaces

Abstract

In the LCAO representation our direct recursion (transfer matrix, T) method (DRM) applies strictly valid recurrence relations straight to the solution of the Hamiltonian matrix eigenvalue problem for the electronic system in bounded finite or infinite crystals and polymers with perturbed boundaries. We present the different reduced forms of the auxiliary equations of the DRM and introduce the notion of duality for these. Their relations to the equations of cyclic and half-infinite systems are given. The indirect recursion method (IRM) of Fromm and Koutecky (FK) utilize recurrence relations (strictly valid only for half-infinite perfect systems) to derive the bordering blocks of the resolvent (Green) matrix R. We completed their expressions with those of the inner blocks of R. Contrary to their statement, the inversion of matrix block B which describes the interactions between the consecutive sub-systems has similar role in both methods. We proved that the B^-1-ree reduced forms of the auxiliary equations of the DRM for semisimple T are identical to the FK equation of the IRM for the corner diagonal block of R multiplied by B’. Algebraic methods are proposed to eliminate the “B^-1-difficulty” of the transfer matrix method. The IRM was found generally much more complicated than the DRM is.