Using the last formula in the derivation, the second-order Schrieffer-Wolff-transformed Hamiltonian H ′ = e S H D e − − S has now an explicit form as a function of its elementary terms D 0 and V ′ : H i j ′ = d i δ δ i j + 1 2 ∑ ∑ k V i k ′ V k j ′ ( 1 d i − − d k + 1 d j − − d k ) + O ( V ′ 3 )