Inhomogeneity effects in oxygen doped HgBa$_2$CuO$_{4}$
  We theoretically investigate inhomogeneity effects on the charges, electric field gradients and site-projected densities of states in HgBa$_2$CuO$_{4+\delta}$. We find pronounced differences in the doping-induced number of holes at different atomic sites. The contributions of these sites to the density of states in the vicinity of the Fermi level are peaked at the same energy, but vary in magnitude by up to 70 percent and have different energy dependence. Due to this energy dependence the role of the intrinsic inhomogeneities for superconductivity strongly depends on the energy and character of the quasiparticle mediating the Cooper pairing. Our results can explain the origin of doping-induced effects observed either by local or macroscopic experimental probes. 
C. Ambrosch-Draxl and E. Ya. Sherman