  By calculation and analysis of the bare conduction bands in a large number of
hole-doped high-temperature superconductors, we have identified the energy of
the so-called axial-orbital as the essential, material-dependent parameter. It
is uniquely related to the range of the intra-layer hopping. It controls the Cu
4s-character, influences the perpendicular hopping, and correlates with the
observed Tc at optimal doping. We explain its dependence on chemical
composition and structure, and present a generic tight-binding model.
