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

摘要

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