Cubic interaction parameters for t2g Wannier orbitals

Many-body calculations for multi-orbital systems at present typically employ Slater or Kanamori interactions which implicitly assume a full rotational invariance of the orbitals, whereas the real crystal has a lower symmetry. In cubic symmetry, the low-energy $t_{2g}$ orbitals have an on-site Kanamori interaction, albeit without the constraint $U = U' + 2J$ implied by spherical symmetry ($U$: intra-orbital interaction, $U'$: inter-orbital interaction, $J$: Hund's exchange). Using maximally localized Wannier functions we show that deviations from the standard, spherically symmetric interactions are indeed significant for $5d$ orbitals ($\sim25$ for BaOsO$_3$ ; $\sim12$ if screening is included), but less important for $3d$ orbitals ($\sim6\%$ for SrVO$_3$; $\sim1\%$ if screened).