Electronic structure of finite or infinite systems in the tight-binding model with overlap

Much interest is devoted to the study of the electronic structure of defects in bulk and near surfaces and interfaces, with the tight-binding method. Recent progress in this field, namely the concepts of ‘‘adspace and subspace,’’ allow a better description of the defects. However, the nonorthogonality of the atomic-orbital basis can no longer be neglected; the overlap matrix has to be considered on the same footing as the Hamiltonian matrix. This is even more crucial in reactive systems, where the overlap and Hamiltonian matrices vary in time, according to the positions of constituent ions. Different representations of specific variance may be built up for an operator; some are more adequate for a given physical property. The recursion method is one of the most powerful methods for computing a finite number of matrix elements of the Green’s operator, as continued fractions. We discuss its relationship to variance and we propose a new accurate method for generating the continued fraction which avoids any explicit or implicit usage of the inverse overlap matrix in the molecular case. It provides a new way for doing ‘‘quantum-chemistry’’ calculations even for reactive or diffusive systems. It extends, in principle, to infinite systems, irrespective of the dimension of space. However, numerical problems associated with truncation and convergence remain open in the application made to a nitrogen overlayer adsorbed on chromium.