Electronic structure of the harmonic-honeycomb iridates $\alpha$, $\beta$, $\gamma$-Li$_2$IrO$_3$*

Using ab-initio density functional theory we investigate the electronic and magnetic properties of the harmonic-honeycomb iridates $\alpha$, $\beta$, $\gamma$-Li$_2$IrO$_3$ with honeycomb, hyperhoneycomb and stripyhoneycomb crystal structures, respectively. We discuss the distinct features of each class of systems in terms of possible Ir-based molecular-orbitals and the implications on the magnetism in these materials. We further relate the electronic structure to proposals of generalized Kitaev-Heisenberg models.