Tuning Hole Mobility, Distribution and Repulsion in High-$T_c$ Cuprates via Apical Atoms*

Using a newly developed first-principles Wannier-states approach that takes into account large on-site Coulomb repulsion, we derive the low-energy effective one-band interacting Hamiltonians for several prototypical cuprate superconductors. 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, via additional intra-sublattice hoppings, nearest-neighbor ``super-repulsion,'' and other microscopic many-body processes. Possible implications on modulation of local pairing gaps, charge distribution, hole mobility, electron-phonon interaction, and multilayer effects will be discussed.