Direct exchange and superexchange interactions

Throughout the intersite spin models in this module, write $H_{12}=J\mathbf s_1\cdot\mathbf s_2$ for one unordered bond, with dimensionless spins. Positive J is antiferromagnetic and negative J is ferromagnetic. For two spin-half particles, the scalar product has eigenvalues minus three quarters in the singlet and plus one quarter in the triplet. Consequently J equals the triplet energy minus the singlet energy after a common energy shift has been removed.

For two electrons, a symmetric triplet spin wavefunction requires an antisymmetric spatial wavefunction; the singlet has the opposite pairing of symmetries. Their Coulomb matrix elements can therefore differ even when the microscopic potential contains no explicit spin operator. A positive exchange integral in a simple fixed-orbital treatment lowers the triplet and contributes a ferromagnetic term in our convention. Orbital relaxation, kinetic energy, and other channels also enter the actual energy difference, so this argument does not assign a universal sign to a material's net exchange.

Consider two sites of a one-orbital repulsive Hubbard model at half filling, with hopping amplitude t and on-site repulsion U. The low-energy subspace has one electron on each site when $U\gg|t|$. Hopping briefly accesses an empty site paired with a doubly occupied site, then returns. Such intermediate configurations cost approximately U. Second-order perturbation theory lowers energies by a scale quadratic in hopping and inverse in the charge-excitation energy.

The spin singlet can couple to virtual doubly occupied states, whereas a triplet cannot occupy one orbital twice without violating the Pauli principle. Accounting for the allowed paths gives a singlet lowering of four times hopping squared divided by U relative to the triplet. The projected Hamiltonian includes a convenient constant that leaves the triplet energy zero. Removing that constant yields the same spin-only singlet–triplet gap. Changing the phase or sign of a single hopping amplitude does not change this leading positive scale.

In an oxide, magnetic ions may communicate through a ligand rather than overlap directly. Eliminating ligand charge states can generate an effective hopping and then an exchange, but the resulting denominators involve charge-transfer energies and interactions on several orbitals. Orbital orientation, bond geometry, and Hund coupling can even favor ferromagnetic contributions. Calling every ligand-mediated path positive four t squared over U conceals precisely the information that determines its sign.

A spin-only description requires separated energy scales: the measurements of interest should probe energies well below the charge excitations that were eliminated. If doping introduces mobile carriers or the repulsion is too small, keeping only fixed local spins may fail. Adding phenomenological direct and kinetic contributions can be useful when their definitions avoid double counting, but the sum is a model assumption. Exchange constants inferred from spectroscopy or total-energy differences must therefore be accompanied by the Hamiltonian normalization and the fitting regime.