Criterion for the validity of thet-Jmodel based upon a two-band model for copper oxides

The second-order effective Hamiltonian in the representation of a set of orthogonal operators is derived from a two-band model for Cu oxides. The contribution from the local ${\mathrm{Cu}}^{2+}$ and surrounding ${\mathrm{O}}^{\mathrm{-}}$ spin triplet state to various hopping processes has been included. From the fourth-order perturbation we study doping effects on the background spin-spin interaction and obtain the expression for the ferromagnetic superexchange interaction between two neighboring ${\mathrm{Cu}}^{2+}$ spins with an ${\mathrm{O}}^{\mathrm{-}}$ hole in the middle. The condition for the equivalence between the t-J model and the two-band model for Cu oxides is discussed. In terms of nonorthogonal operators for a single ${\mathrm{O}}^{\mathrm{-}}$ hole, the condition for the exact mapping of a two-band model to a single-band model has also been derived.