Singular manifestation of square-planar coordination of a iridate Na$_4$IrO$_4$

Local environments and valence electron counts primarily determine the electronic states and physical properties of transition metal complexes. For example, square-planar surroundings found in transition oxometalates such as Curprates, Nickaltes are usually associated with the d$^8$ or d$^9$ electron configuration. In this work, we address an experimentally observed exotic square-planar mono-oxoanion [IrO$_4$]$^{4-}$ in Na$_4$IrO$_4$ with Ir(IV) in d$^5$ (S=$\frac{3}{2}$ state) configuration, using ab-initio calculations. On contrary, in its 3d counterpart, Na$_4$CoO$_4$, Co(IV) is in tetrahedral coordination with S=$\frac{5}{2}$ high spin state. Our ab-initio calculations reveal that the on-site Coulomb interaction $\textit{U}$ is the essential factor for determining the stability of the local coordination as well as spin state. We find that due to weak Coulomb repulsion of Ir-5d electrons, Na$_4$IrO$_4$ form in a square-planar coordination whereas for Na$_4$CoO$_4$, Co(IV) is in tetrahedral coordination, due to strong electron correlation at 3d Co site. Following the trend from 5d to 3d, we predict that the intermediate 4d material Na$_4$RhO$_4$, if synthesized, may favor tetrahedral coordination but with an S=$\frac{1}{2}$ low spin state.