= 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.
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