 Solution

For a unitary transformation $e^S$, take $S^{†}=-S$ and $S=O(t)$. Operator conjugation by an exponential givesSeparate the unperturbed energy $\mathcal{E}(\Delta )$ from $H_t=O(t)$. To first order,Therefore eliminating the first-order hopping requiresThis is a Schrieffer-Wolff transformation: virtual transitions are removed from the leading Hamiltonian and reappear as higher-order effects within a retained occupation sector. The same canonical-elimination strategy relates localized-moment and effective-exchange descriptions in the original construction.^[ref]

Such a choice requires the hopping to connect separated unperturbed energies. If it connects exactly degenerate states, those states must remain in the retained degenerate block; a singular perturbative generator is not an acceptable replacement.