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In the case of alpha > 0, as the slide shows, the case for the lowest energy is the one with the dumbbell parallel to the z-axis.
In the case of alpha < 0, since the sign is inverted, the case for the lowest energy must be the one where the dumbbell is perpendicular to the z-axis.
In the case of alpha = 0, since there is no quadrupole correction regardless of orientation, all orientations are the lowest-energy case.
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