Fig. 3. Calculations of BMRO.

a The potential energy surface of a single ReO₆ octahedron in BMRO as a function of $d_{x^2-y^2}$ and $d_{z^2}$ distortions, compared to the experimental octahedral distortions (denoted with the red dots)³. b Total energy E_TOT versus the temperature T calculated within the mean-field approximation from the ab-initio IEI Hamiltonian (Eq. (1)). The solid and dashed curves are the total energies of the t_2g ferro-$q_{xy}^{\left[000\right]}$ and e_g antiferro-$q_{x^2-y^2}^{\left[001\right]}$ orders, respectively. Inset: the quadrupole-quadrupole block of the IEI matrix V for the nearest-neighbor xy Re–Re bond, blue (red) colors correspond to ferro (antiferro) couplings. c The electronic susceptibility χ_e over the temperature T associated with the ferro xy and antiferroic k = [001] x²−y² quadrupolar orders display Curie-Weiss behavior with a T_q ~ 50 K. On the other hand, the z² quadrupoles do not diverge at a temperature above zero and thus do not ferroically order spontaneously.