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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.