A perturbative unitary conjugation eliminating couplings between chosen separated energy sectors. For and an anti-Hermitian satisfying at leading order, the transformed Hamiltonian contains at second order. Exactly resonant coupled states must be retained in a common block. The method was used for the Anderson-to-Kondo relation by John Robert Schrieffer and P. A. Wolff.
A perturbative generator dividing by an unperturbed energy difference cannot eliminate a nonzero coupling between exactly degenerate states. Those states must remain in the retained degenerate block. Small denominators also invalidate an expansion when the actual hopping matrix element, including occupation enhancement, is not small in comparison.
If site-transfer energies are , the left-ordered raising-hop generator has coefficient . Nonlinear occupation energies generally make the effective second-order shifts and spin-exchange amplitudes depend on the occupation sector.
With , and a real coefficient , the fixed-occupancy second-order operator is . Normal ordering gives and , where exchanges internal states without changing site totals.
