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The single-band Hubbard model balances electron hopping between localized spatial orbitals against local interaction. A common form isThe first sum is over neighbouring unordered site pairs. The parameter $t$ sets hopping, while $U_H>0$ penalizes double occupancy. At $U_H=0$ electrons occupy independent energy bands; strong repulsion suppresses local charge fluctuations and can favour local-moment behaviour. This competition was the motivation for the model's treatment of narrow-band electron correlation.^[ref]

At one electron per site and $U_H\gg |t|$, virtual double occupancy yields the antiferromagnetic scale $J_{\mathrm{ex}}=4t^2/U_H$. The two-site Hubbard dimer makes this concrete: the singlet state is lowered by $(\sqrt{U_H^2+16t^2}-U_H)/2$, while triplet states have no such shift. Its small-hopping limit gives Hubbard kinetic exchange.

Interaction conventions must match. For dimensionless spin-half operators, $-2U({\mathbf{S}}^2-3n/4)=3U{n}_{\uparrow }{n}_{\downarrow }$, so that spin-square convention has $U_H=3U$, not $U_H=U$.

The first term assigns energy ${ϵ}_i$ to occupying the localized orbital at site $i$. The hopping term moves an electron between neighbouring sites, producing kinetic delocalization and band dispersion; Hermitian conjugate hops make the Hamiltonian Hermitian. The final term costs $U>0$ only when both spin states occupy the same local orbital. This is a Hubbard model with an on-site Coulomb penalty, not a classical interaction between magnetic dipoles.

At one electron per site and large $U$, a real hop is energetically costly, but a virtual hop into a doubly occupied intermediate state is possible. Opposite-spin states can exploit these virtual processes and lower their energy, while parallel spins cannot occupy the same one-orbital site. The resulting singlet–triplet splitting produces antiferromagnetic Hubbard kinetic exchange.

There is a terminology qualification. The paper calls this overlap-mediated process “direct exchange” because the hopping is directly between neighbouring magnetic orbitals. In the stricter distinction between direct Coulomb exchange versus kinetic exchange, the displayed Hamiltonian omits an explicit intersite Coulomb exchange integral. The antiferromagnetic $t^2/U$ mechanism derived below is kinetic exchange; a genuine intersite direct exchange term is separate and can compete with it. This distinction also separates the model from ligand-mediated superexchange in part f.