Getting an effective Hamiltonian on a sublattice; effective hoppings

The idea is to start with a microscopic Hamiltonian describing the “actual” system and to find a reduced description on as subspace. For example, starting with a set of sites with various energies and hopping elements, we might want an effective description between a subset of them — which will make sense if they different energies — and to find the effective on-site and hopping elements for this subsystem. An example of the use would be for a metal oxide where one is interested in the bands and hoppings between the transition metal sites when they are mediated via the oxygen atoms in between (the direct hoppings between metal sites being negligible). This type of modeling is important for understanding superexchange and double-exchange (see below).