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Corundum oxide Ti$_2$O$_3$ shows the metal-insulator transition around400-600 K accompanying the nearest Ti$^{3+}$-Ti$^{3+}$ bond ($a_{1g}a_{1g}$singlet state) formation along the $c$ axis. In order to clarify thehole-doping effect for the $a_{1g}a_{1g}$ singlet bond in Ti$_2$O$_3$, weinvestigated Ti $3d$ orbital anisotropy between corundum-type Ti$_2$O$_3$ andilmenite-type MgTiO$_3$ using linear dichroism of soft x-ray absorptionspectroscopy of the Ti $L_{2,3}$ edge. From the linear dichroic spectral weightin Mg$_y$Ti$_{2-y}$O$_3$, we confirmed that the $a_{1g}a_{1g}$ state isdominant not only in $y=0.01$ (almost Ti$_2$O$_3$), but also in $y = 0.29$,indicating that the Ti-Ti bond survives against a certain level of hole doping.In $y=0.63$ corresponding to 46% hole doping per Ti, the $3d$ orbital symmetrychanges from $a_{1g}$ to $e_g^{\pi}$.

Linear dichroic x-ray absorption response of Ti-Ti dimers along the $c$ axis in Ti$_2$O$_3$ upon Mg substitution