Molecular orbitals vs. relativistic orbitals in t$_{\mathrm{2g}}$ honeycomb lattices: SrRu$_{\mathrm{2}}$O$_{\mathrm{6}}$ as compared to Na$_{\mathrm{2}}$IrO$_{\mathrm{3}}$, RuCl$_{\mathrm{3}}$, and Li$_{\mathrm{2}}$RuO$_{\mathrm{3}}$.

$t_{2g} $states on a honeycomb lattice tend to form non-dispersive localized states even if large intersite hopping is present. In the nonrelativistic case, these are molecular orbitals (MO) localized on metal hexagons, if the ligand-assisted nearest and next nearest neighbor hoppings, ${t}'_{1} $ and ${t}'_{2} $, dominate, or dimers (DO), if the direct overlap, $t_{1} $, dominates. In the ultrarelativistic limit $t_{2g} $ form effective relativistic orbitals (RO), $j_{eff} =\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}$, which are atomically localized if ${t}'_{1} $is the dominant hopping. On the first glance, the three regimes are defined by the conditions ${t}'_{1} \gg t_{1} ,\lambda $ or $t_{1} \gg {t}'_{1} ,\lambda $ or $\lambda \gg t_{1} ,{t}'_{1} $. In reality, the latter condition is never fulfilled, especially in ruthenates, yet not only Na$_{\mathrm{2}}$IrO$_{\mathrm{3}}$, but also RuCl$_{\mathrm{3}}$ appear to be in a regime dominated by RO, even though the residual effect of MO critically influences magnetic interactions, while Li$_{\mathrm{2}}$RuO$_{\mathrm{3}}$, not far removed from RuCl$_{\mathrm{3}}$ in the parameter space, is firmly in the DO regime. Most surprisingly, SrRu$_{\mathrm{2}}$O$_{\mathrm{6}}$, which is even closer to RuCl$_{\mathrm{3}}$, happens to be fully in the MO regime, with negligible spin-orbit effects. In this talk, we will show that an additional, decisive factor is the doping level per site. The principal difference between Na$_{\mathrm{2}}$IrO$_{\mathrm{3}}$ or RuCl$_{\mathrm{3}}$, Li$_{\mathrm{2}}$RuO$_{\mathrm{3}}$, and SrRu$_{\mathrm{2}}$O$_{\mathrm{6}}$ is that the first two have one $t_{2g} $hole per site, the second one two holes, and the last three electrons. In particular, the total dominance of MO in the latter compound fully explains its unique and unexpected magnetic properties.