Hubbard kinetic exchange ($H_{\mathrm{eff}}=\frac{4t^2}{U}({\mathbf{S}}_1\cdot {\mathbf{S}}_2-1/4)$)

At one electron per site and $U\gg |t|$, the low-energy singlet of a repulsive two-site Hubbard dimer mixes virtually with doubly occupied configurations. Its energy is $-4t^2/U$ to leading order, while the three triplets have no such shift. With dimensionless spin-half operators this yields the displayed effective antiferromagnetic Hamiltonian. Restoring angular-momentum operators with units of $\hslash$ requires the corresponding $1/{\hslash }^2$ in the spin product.