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Undergraduate textbooks say that a molecule mustn't posses a reflection plane and an inversion center to be considered achiral. As mentioned by Ron in his answer to this question, molecules that don't possess a plane nor center of symmetry, but do possess an $S_4$ improper axis are considered to be achiral. Are there any other symmetry elements that need to be taken into consideration when determining whether a molecule is chiral or not?

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    $\begingroup$ Short answer: Anything that can be expressed in terms of $\mathrm{S}_n$ axes makes a molecule achiral (Schoenflies definition). Or alternatively, anything that can be expressed as $\bar n$ makes a molecule achiral (Hermann-Mauguin definition). $\endgroup$ – Jan Oct 27 '15 at 13:45
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    $\begingroup$ Actually, any reflection in a plane is just an $S_1$ operation, and an inversion through a point is just an $S_2$ operation. So, @Jan's comment is indeed the most concise yet complete definition: any $S_n$ axis present will make the molecule achiral. $\endgroup$ – orthocresol Oct 27 '15 at 13:49
  • $\begingroup$ Okay. I would like to see some exotic example :). $\endgroup$ – RBW Oct 27 '15 at 14:13
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    $\begingroup$ @Marko that's hard to find. Your best bet is looking for a molecule in the $S_4$ point group, which has the operations $E$, $S_4$, $C_2$, and $S_4^3$ - i.e. no mirror plane and no centre of inversion, but is still achiral because of the improper rotation axis. You can find some examples and pictures here csi.chemie.tu-darmstadt.de/ak/immel/tutorials/symmetry/… $\endgroup$ – orthocresol Oct 27 '15 at 14:29
  • $\begingroup$ Nice examples, especially the coronane. $\endgroup$ – RBW Oct 27 '15 at 14:44

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