Alpha characterizes the shape of the mass distribution. In the case of the dumb-bell:
For alpha>0: the lowest energy orientation is when the two masses are aligned. This is when theta=0.
For alpha=0: it is at any orientation as it is isotropic. It only depends on the distance of the rings and the masses.
For alpha<0: The radius of the ring is larger than their distance, so the dumb-bell will align with the xy axis. This is when theta=90.