  Using a newly developed first-principles Wannier-states approach that takes
into account large on-site Coulomb repulsion, we derive the effective
low-energy interacting Hamiltonians for several prototypical high-$T_c$
superconducting cuprates. The material dependence is found to originate
primarily from the different energy of the apical atom $p_z$ state.
Specifically, the general properties of the low-energy hole state, namely the
Zhang-Rice singlet, are significantly modified by a triplet state associated
with this $p_z$ state, via additional intra-sublattice hoppings,
nearest-neighbor "super-repulsion", and other microscopic many-body processes.
Possible implications on modulation of $T_c$, local superconducting gaps,
charge distribution, hole mobility, electron-phonon interaction, and
multi-layer effects are discussed.
