Superexchange couples localized moments through virtual hopping via intermediate, often nonmagnetic, ligand orbitals. Half-filled orbitals sharing an effective hopping amplitude give an antiferromagnetic scale in the repulsive one-band limit. Ligand geometry, orbital occupation and exchange within intermediate states can change the balance and allow ferromagnetic coupling. Real charge motion can remain gapped while these virtual processes couple an insulating magnet's spins.
In a minimal hole description of a metal–ligand–metal path, two holes initially occupy separate metal orbitals and the ligand has no hole. Creating a virtual ligand hole costs a charge-transfer energy . Two hops generate an effective metal–metal hopping of order ; two more hops through metal double occupancy generate the antiferromagnetic scale in the title. Ligand double occupancy adds a channel with denominator . Orbital geometry and intra-atomic exchange determine when the antiferromagnetic channel dominates.
