Consistent low-energy reduction of the three-band model for copper oxides with O-O hopping to the effectivet-Jmodel

A full three-band model for the ${\mathrm{CuO}}_2$ plane with inclusion of all essential interactions—Cu-O and O-O hopping, repulsion at the copper and oxygen and between them—is considered. A general procedure of the low-energy reduction of the primary Hamiltonian to the Hamiltonian of the generalized t-t’-J model is developed. An important role of the direct O-O hopping is discussed. Parameters of the effective low-energy model (the hopping integral, the band position, and the superexchange constant J) are calculated. An analysis of the obtained data shows that the experimental value of J fixes the charge-transfer energy Δ=(${\mathrm{\epsilon }}_{\mathit{p}}$-${\mathrm{\epsilon }}_{\mathit{d}}$) in a narrow region of energies.