By using a new downfolding procedure based on the density functional theory and the muffin-tin-orbital method, we have derived a simple, generic conduction band Hamiltonian for these systems. We have shown that the essential material dependence is contained in only one parameter, namely r=t'/t, the range of the hopping integrals in the CuO2 layer. This range parameter is controlled by the energy of the so called axial orbital, a hybrid between Cu 4s, Cu 3d x^2-y^2, Cu 3d 3z^2-1, apical O 2pz, and farther orbitals such as La 5d 3z^2-1.
We observe that the materials with higher Tc max have larger hopping ranges and axial orbitals more
localized in the CuO2 layers. Possible reasons for this correlation will be mentioned.
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