Rashba splittings in the $d_{xz/yz}$ bands than $d_{xy}$ as shown in Fig. \[fig:bandpdos\], given that the $d_{xz/yz}$ bands are closer in energy to the $d_{xy}$ band than the $e_g$ bands (albeit within an order of magnitude). The intra-$t_{2g}$ ISB hopping $\gamma_1$ remains relatively small due to the orbital symmetry of Ta $d$ and O $p$ states, which we can see from a quantitative analysis with MLWFs that includes not only the Ta $d$ states but also the neighboring O $p$ states. Examining the hopping parameters relevant to $\gamma_1$, the particularly small ISB hopping between O $p_y$ and Ta $d_{yz}$ along $x$ direction (Fig. \[fig:wanhopping\]) can be attributed to the relative positions and shapes of the two orbitals; the lobes of the two orbitals lie on the $yz$ plane that is perpendicular to the hopping direction ($\hat{x}$), and $p_y$ has maximum amplitude along $y$ direction whereas $d_{yz}$ has a node along it. Thus, we have the negligible Rashba splitting of the $d_{xy}$ band as shown in Fig. \[fig:bandpdos\], despite the smaller energy difference with the $d_{xz/yz}$ bands.

The other reason why the $e_g$ orbital contribution is crucial for the Rashba splitting in the $t_{2g}$ bands is the reduced $t_{2g}$-$e_g$ energy splitting. 