I still don't see it. If the nuclei are distinct (say, oxygen and hydrogen), and they are initially at (0,1) and (0,-1), respectively, then only rotation through 360[itex]^\circ[/itex] will leave the system unchanged. If the nuclei are not distinct, however, then I see how rotation through 180[itex]^\circ[/itex] will leave the system unchanged. But I'm not assuming that the nuclei are identical.Keep reading; especially the section titled "n-fold rotational symmetry".
Sorry; maybe I should be more specific. I'm reading a paper, and the argument is made that, because of the "rotational symmetry" of a diatomic molecule, the energy levels of the electrons depend only on the magnitude of the distance between the two nuclei. Does the fact that the system is invariant under 360-degree rotation mean it has rotational symmetry? If so, why should that be sufficient for us to conclude that the electron energy levels are only dependent on the distance between the nuclei and not their orientation?OK, then I don't get your point. Where does it claim that such a system has anything other than trivial rotational symmetry (except about its central axis, of course)?
The rotational symmetry referred to is with respect to the central axis (the line between the two nuclei).I'm reading a paper, and the argument is made that, because of the "rotational symmetry" of a diatomic molecule, the energy levels of the electrons depend only on the magnitude of the distance between the two nuclei.