= Solution
For a <unitary transformation> $e^S$, take $S^\dagger=-S$ and $S=O(t)$. <Operator conjugation by an exponential> gives
$$
H'=H+[S,H]+\frac12[S,[S,H]]+\cdots.
$$
Separate the unperturbed energy $\mathcal E(\Delta)$ from $H_t=O(t)$. To first order,
$$
H'=\mathcal E(\Delta)+H_t+[S,\mathcal E(\Delta)]+O(t^2).
$$
Therefore \b[eliminating the first-order hopping requires]
$$
\boxed{[S,\mathcal E(\Delta)]=-H_t.}
$$
This 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.<Relation between the Anderson and Kondo Hamiltonians by John Robert Schrieffer and P. A. Wolff>[Abstract, canonical transformation in the small-mixing limit]{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.
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