Theory of projections with non-orthogonal basis sets: Partitioning techniques and effective Hamiltonians

Here we present a detailed account of the fundamental problems one encounters in projection theory when non-orthogonal basis sets are used for representation of the operators. In particular, we re-examine the use of projection operators in connection with the calculation of projected (or reduced) Green's functions and associated physical quantities such as the local density of states (LDOS), local charge, and conductance. The unavoidable ambiguity in the evaluation of the LDOS and charge is made explicit with the help of simple examples of metallic nanocontacts while the conductance, within certain obvious limits, remains invariant against the type of projection. We also examine the procedure to obtain effective Hamiltonians from reduced Green's functions. For completeness we include a comparison with results obtained with block-orthogonal basis sets where both direct and dual spaces are used.