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When alpha < 0 we know that the minimun value of the quadrupole energy is -alpha. And thus the lowest-energy orientation of the dumb-bell is E – alpha
When alpha = 0 we know that there is no quadrupole energy contribution. And thus the lowest-energy orientation of the dumb-bell is E.
When alpha > 0 we know that the minimun value of the quadrupole energy is -2*alpha. And thus the lowest-energy orientation of the dumb-bell is E – 2*alpha
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