Bridge-mediated electronic interactions: Differences between Hamiltonian and Green function partitioning in a non-orthogonal basis

An analysis of the partitioning (projection) technique is given with emphasis on non-orthogonal basis sets. The general expression for the effective Hamiltonian obtained via Löwdin partitioning of the Schrödinger equation is discussed in the context of semi-empirical theories and electron transfer matrix elements. Numerous pitfalls in calculations of matrix elements are pointed out. More importantly, it is shown that contrary to the case of an orthogonal basis, for a non-orthogonal basis Löwdin partitioning of the Schrödinger equation and partitioning of the Green function equation are not equivalent. The latter method provides a more general prescription for deriving effective Hamiltonians. Such Hamiltonians reproduce the full propagation in the partitioned subspace.