Direct exchange

Direct exchange comes from Coulomb matrix elements between overlapping localized electronic orbitals together with the antisymmetry required by the Pauli exclusion principle. It produces a spin-dependent energy without being a classical dipole interaction. Its competition with kinetic processes and orbital occupation helps determine whether neighbouring moments prefer parallel or antiparallel alignment.

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.