 Solution

Use an oxygen-mediated superexchange path between two magnetic metal orbitals. A minimal one-active-orbital three-site charge-transfer superexchange Hamiltonian is clearest in a hole representation:Here $d_i$ describes a metal hole, $p$ a ligand hole, and $\Delta >0$ is the energy needed to create the virtual ligand hole. The low-energy configuration has one hole on each metal orbital and no ligand hole, corresponding to a filled bridging ligand orbital in the electron picture. $U_d$ and $U_p$ penalize double hole occupancy on the corresponding sites.

Virtual metal–ligand hopping transfers charge through oxygen without producing a real mobile carrier. Eliminating the ligand gives an effective metal–metal hopping $t_{dd}^{\mathrm{eff}}\sim t_{pd}^2/\Delta$. Combining it with the large metal repulsion givesA ligand double-occupancy channel adds a contribution of the same sign for the simple half-filled straight-bond model, often organized as $J\sim (4t_{pd}^4/{\Delta }^2)[1/U_d+2/(2\Delta +U_p)]$. The precise coefficients depend on the orbitals and hopping conventions; the essential mechanism is virtual charge transfer plus spin-selective access to intermediate states. The effective spin interaction favors antiparallel alignment while the material can remain insulating.

For MnO's rock salt structure, the strong approximately $180^{\circ }$ Mn–O–Mn path joins next-nearest Mn sites. It is not the nearest fcc Mn–Mn separation used in the direct-overlap estimate of part e. The oxygen-mediated path avoids relying on the tiny direct overlap of the two distant 3d tails. A realistic model includes several Mn 3d orbitals, Hund coupling and the high-spin Mn ion; the single-orbital Hamiltonian above isolates the antiferromagnetic exchange channel. Approximately $90^{\circ }$ paths and different orbital occupancies need not have the same sign.