= 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:
$$
H=\Delta\sum_\sigma n_{p\sigma}
+U_d\sum_{i=1,2}n_{di\uparrow}n_{di\downarrow}
+U_p n_{p\uparrow}n_{p\downarrow}
-t_{pd}\sum_{i=1,2;\sigma}(d_{i\sigma}^\dagger p_\sigma+p_\sigma^\dagger d_{i\sigma}).
$$
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 $t_{dd}^{\rm eff}\sim t_{pd}^2/\Delta$. Combining it with the large metal repulsion gives
$$
\boxed{J_{\rm AF}\sim\frac{4t_{pd}^4}{\Delta^2U_d}>0.}
$$
A 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.
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