Fate of multipolar physics in 5d2 double perovskites

In a cubic environment, the ground state of spin-orbit coupled $5{d}²$ ions is a non-Kramers ${E}_g$ doublet, which hosts quadrupole and octupole moments. A series of $5{d}²$ osmium double perovskites ${\mathrm{Ba}}₂M{\mathrm{OsO}}₆$ $(M=\text{Mg}, \mathrm{Ca}, \mathrm{Zn}, \mathrm{Cd})$ have recently been proposed to exhibit multipolar orders. We investigate the structural properties of these materials using $ab$ initio calculations and find that the cubic structure is unstable for the Cd compound while the Mg, Ca, and Zn materials retain $Fm\overline{3}m$ symmetry. We show that ${\mathrm{Ba}}₂{\mathrm{CdOsO}}₆$ favors a rhombohedral $R\overline{3}$ structure characterized by ${a}^{\ensuremath{-}}{a}^{\ensuremath{-}}{a}^{\ensuremath{-}}$ octahedral tiltings as indicated by unstable ${\mathcal{T}}_1g$ phonon modes. Trigonal distortions split the excited ${T}_2g$ triplet into an ${E}_g^{\ensuremath{'}}$ doublet and an ${A}_g$ singlet, which may cross energy levels with the ${E}_g$ doublet and suppress the multipolar physics. We find a window where ${E}_g$ remains the lowest energy state under trigonal distortion, enabling the emergence of multipole phases in noncubic crystal environments.