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We develop an unrestricted Hartree-Fock mean-field method including Coulombrepulsion U, V and spin-orbital coupling $\lambda$ self$-$consistently toinvestigate the mechanism of structural instability and magnetic ordering inBa$_2$MgReO$_6$. A comprehensive quadrupole phase diagram versus U and V with$\lambda$=0.28eV is calculated. Our results demonstrate that, while U and$\lambda$ mainly lead to the onsite quadrupole $Q_{x^2-y^2}$ and $Q_{3z^2-r^2}$with orbital frustration, the easy-plane anisotropy or the intersite Coulombrepulsion V can remove the frustration. Finally, the V$>$10meV would arrange$Q_{x^2-y^2}$ antiparallelly, accompanied with small parallel $Q_{3z^2-r^2}$,and stabilize Ba$_2$MgReO$_6$ into the body centered tetragonal structure. Suchantiparallel $Q_{x^2-y^2}$ provides a new mechanism of Dzyaloshinskii-Moriyainteraction, and gives rise to the canted antiferromagnetic (CAF) state along[110] axis. Moreover, sizable octupoles such as $O_{21}^{31}$, $O_{21}^{33}$,$O_{21}^{34}$ and $O_{21}^{36}$ are discovered for the first time in CAFstates. Our study not only provides a comprehensive understanding of the natureand exotic properties in Ba$_2$MgReO$_6$, but also reveals some commonality of5d compounds.

Intersite Coulomb repulsion driven quadrupole instability and magnetic ordering in the orbital frustrated Ba$_2$MgReO$_6$