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\subsection{Hg-based Cuprates}  \label{sec:cuprates2}  With the hindsight of the experience of our first attempt in designing new  cuprates, we revisited the problem question of designing novel cuprates  from a slightly different broader  viewpoint. The We view the  cuprates are as  a stack of  functional stack: layers, with  the composition of each layeris  chosen to play a specific role. The central copper oxide (CuO$_2$) plane supports superconductivity and roughly constrains the in-plane lattice constant. The  remaining layers must tune the chemical potential of the CuO$_2$ layer without  rumpling the plane or introducing disorder, and isolate each CuO$_2$ plane to