Starting from tight binding Hamiltonians with strong short range Coulomb interactions we show how to derive a description where the fermions with short range Coulomb repulsion are replaced by pure localised spins and the remaining fermionic degrees of freedom are modified to take into account the charge degrees of freedom that the spin operators treated as fermions initially possessed. To leading order we find the Kondo Lattice, a single particle fermion Hamiltonian together with an exchange term between the localised spin and the fermion spin operator. At the next order, if we assume that the localised spins are random, we find a many body attraction between singlet pairs of fermions and non-linear-hopping terms. We analyse the spin half Anderson Lattice and find an attraction for pairs of electrons which sit on the same site but competing with this is a loss in mobility due to the non-linear-hopping if the electrons do pair. We also analyse a ‘HighT c ’ model where we find attraction between pairs of Oxygenp-electrons both on the same site and on neighbouring Oxygen sites due to enhanced hopping between these sites.