Tuning effective interactions in high-$T_c$ cuprates via apical oxygen atoms: New realization from the first-principles Wannier function approach

Based on a novel first-principles Wannier function approach, the low-energy effective Hamiltonian for high-$T_c$ cuprates is derived. The apical oxygen atoms are found to significantly modify the mobility and distribution of the Zhang-Rice singlets in the CuO$_2$ plane, by tuning the effective hopping parameters $t^\prime$ and $t^{\prime\prime}$, and local chemical potential, $\mu_{\mathrm{eff}}$. Most remarkably, $V_{\mathrm{eff}}$, an additional effective repulsion (de- pairing) between neighboring doped holes, is found to be significantly tuned by a ``vacuum fluctuation'' mechanism inherited from the correlated multiband nature of the cuprates. Our results identify the apical oxygen states as the main material dependence of these systems and provide a microscopic avenue to the understanding of recent spectroscopic imaging STM data [K. McElroy \textsl{et al.}, Science \textbf{309}, 1048 (2005)].