A new bonding insight on Ba$_{2}$NaOsO$_{6}$ based on orbital-quenching-induced magnetism

Double perovskite Ba$_{2}$NaOsO$_{6}$ (BNOO) is an exotic example of a heptavalent osmium compound, and also uncommon by being a ferromagnetic insulator. Although the single 5d t$_{\mathrm{2g}}$ electron from Os orders magnetically, there is no evidence of orbital order that would destroy its cubic symmetry. Local density approximation with Hubbard U (LDA$+$U) calculation revealed very strong Os d -- O p hybridization into weakly overlapping cluster orbitals, but was unable to obtain the observed Mott insulating behavior. The gap can obtained using DFT$+$U$+$ spin-orbit coupling (SOC) with unreasonably high value of U. Building from the basic understanding from LDA$+$U calculations, we have performed hybrid DFT studies, including SOC, implemented in Wien2k. This method obtains a narrow gap, and an orbital moment of -0.42 muB that strongly compensates the $+$0.52 muB spin moment. The effects of SOC on the spin density will be presented and discussed, as will the change in the electronic and magnetic properties under pressure.