Three-site charge-transfer superexchange ($J\sim 4t_{pd}^4/({\Delta }^2{U}_d)$)

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 $\Delta$. Two hops generate an effective metal–metal hopping of order $t_{pd}^2/\Delta$; two more hops through metal double occupancy generate the antiferromagnetic scale in the title. Ligand double occupancy adds a channel with denominator $2\Delta +U_p$. Orbital geometry and intra-atomic exchange determine when the antiferromagnetic channel dominates.