along the BZ boundary is even more pronounced, with a giant splitting ($\approx 180$ meV), which is nearly twice the maximum reported value [-@Santander2014] in the perovskite oxide 2DEG, occurring near $\mathrm{X}$ along $k_{x/y} = \pi$.

  ----------- ------------------- --------------------------------------------------------------------------------------------------------- -- -- --
   $\vec{k}$       Manifold                                    Splitting terms in $\mathcal{H}_{\mathrm{eff}}$                                    

               upper $d_{xz/yz}$   $\left[\frac{-2\gamma_3 \xi}{\Delta_{uxz/yz,x^2-y^2}}+\frac{-2\gamma_1 \xi}{\Delta_{uxz/yz,xy}}\right]         
                                                                 (\vec{\sigma}\times \vec{k})\cdot \hat{z}$                                       

               lower $d_{xz/yz}$        $\frac{2\sqrt{3}\gamma_2 \xi}{\Delta_{lxz/yz,z^2}} (\vec{\sigma}\times \vec{k})\cdot \hat{z}$             

                   $d_{xy}$                 $\frac{-2\gamma_1 \xi}{\Delta_{xy,uxz/yz}} (\vec{\sigma}\times \vec{k})\cdot \hat{z}$                 

                   $d_{yz}$                             $\left[\frac{-2\sqrt{3}\gamma_2 \xi}{\tilde{\Delta}_{yz,z^2}}                             
                                                         +\frac{2\gamma_3 \xi}{\tilde{\Delta}_{yz,x^2-y^2}}\right]                                
                                                   \sigma_x k_y -\frac{2\gamma_1 \xi}{\tilde{\Delta}_{yz,xy}}\sigma_y k_x$                        
  ----------- ------------------- --------------------------------------------------------------------------------------------------------- -- -- --

  : \[tab:heff\]Splitting terms of the effective Hamiltonian $\mathcal{H}_{eff}$ for Ta $t_{2g}$ manifolds. $\vec{k}$ denotes the reference point of the effective Hamiltonian with $\Gamma=(0,0)$ and $\mathrm{X}=(\pi,0)$. $\Delta (\tilde{\Delta})$ represents the energy difference between the states in the subscript at $\Gamma (\mathrm{X})$, where $uxz/yz$ and $lxz/yz$ mean the upper $xz/yz$ and the lower $xz/yz$, respectively.

An analysis that includes all Ta $d$-orbitals – not only the $t_{2g}$ orbitals but also the $e_g$ orbitals – is required in order to understand the two conspicuous features of Fig. \[fig:bandpdos\], the discrepancy between the Rashba splitting of the $d_{xy}$ and the $d_{xz/yz}$ bands, and the giant band splitting along $k_{x/y} = \pi$. We employ an analytic TB model for a 