 Direct Coulomb exchange versus kinetic exchange

A direct intersite Coulomb exchange integral acts without an intermediate doubly occupied state. In a repulsive one-band Hubbard model, by contrast, virtual hopping into a doubly occupied state lowers the singlet energy by $4t^2/U$ relative to the triplet. The latter is antiferromagnetic kinetic exchange, even when hopping is directly between neighbouring magnetic orbitals. Ligand-mediated superexchange introduces an intermediate ligand orbital as well.