Figure 1. Crystal structure of Na₃Ir₃O₈.

(a) The unit cell of Na₃Ir₃O₈ with a space group of P4₁32. The orange, yellow, silver and blue spheres represent Na1, Na2, Ir and O atoms, respectively¹⁹. While Na1 atoms are enclosed in distorted O₆ octahedra, Na2 atoms are coordinated tetrahedrally by oxygen atoms. The iridium atoms comprise corner-sharing triangles in three-dimensions called hyper-kagome lattice. (b) Schematic illustration of the hyper-kagome lattice showing formation of molecular orbitals on Ir₃ triangles seen in the scalar relativistic calculation. d_zx on Ir(1) has hopping paths to Ir(2) and Ir(3) via O2 p-orbitals, forming a localized molecular orbitals. Likewise, d_yz on Ir(1) forms another molecular orbital with the like orbitals on Ir(4) and Ir(5). Note that the t_2g orbitals are defined in a local frame (sketched in the inset) which is unique for each Ir site. The hopping via O1, which is likely much smaller than the one via O2¹⁷, is neglected here.