 Occupation-dependent virtual hopping

If site-transfer energies are $\mathcal{E}(\Delta )$, the left-ordered raising-hop generator has coefficient $f(\Delta )=t/[\mathcal{E}(\Delta )-\mathcal{E}(\Delta -2)]$. Nonlinear occupation energies generally make the effective second-order shifts and spin-exchange amplitudes depend on the occupation sector.