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Some websites, such as this textbook, mention how generally a plane of symmetry or inversion center in a molecule is enough to consider it achiral. Nonetheless, this source clarifies

but if these symmetry elements are absent the molecule should be checked carefully for an S$_n$ axis before it is assumed to be chiral.

Furthermore, Wikipedia notes the definition of chirality as "a molecule which has no S$_n$ axis for any value of n is a chiral molecule."

Thus, are there any molecules lacking plane of symmetry/center of inversion but that are achiral due to presence of an axis of improper rotation? Are there are also examples of vice-versa, where a molecule has a plane of symmetry/center of inversion, but lacks an axis of improper rotation and is thus chiral?

edit: This post does cover some definitions which is helpful, but the point of this question was to see an actual example.

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    $\begingroup$ You have it all backwards. A plane of symmetry means no chirality, dead sure. Ditto for an inversion center. $\endgroup$ – Ivan Neretin Nov 1 '20 at 22:39
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Part 1

Are there any molecules lacking plane of symmetry/center of inversion but that are achiral due to presence of an axis of improper rotation?

Yes, although such examples are very rare. In general, the point groups $S_{2n}$ $(n \geq 2)$ are where you should look. These molecules have a $S_{2n}$ rotation axis, but no plane of symmetry, and no inversion centre.

Molecules in the $S_6$ and $S_8$ point group are are exceedingly rare, and generally need to be constructed specifically to satisfy these criteria. The best examples, therefore, come from the $S_4$ point group. You can find several examples on these websites:

In particular, one of the nicest examples is 2,3,7,8-tetramethyl-spiro[4.4]nonane from the second link. In the graphic below, I've taken the liberty of using four different coloured atoms in place of the methyl groups. The reader should therefore bear in mind that the third image is the same thing as the first, even though the colours are different.

Illustration of S4 symmetry in 2,3,7,8-tetramethyl-spiro[4.4]nonane

The XYZ coordinates that I used for the pictures above are as follows. You can download these and view them in software such as Avogadro or ChemCraft, although I don't think you can perform a "reflection" per se (I manually swapped the colours between the second and third pictures to achieve this effect).

C         -2.32545        1.12231        0.91714
C         -2.19292        2.47894        0.24035
C         -0.83350        2.34003       -0.44571
C         -1.75438        0.10430       -0.11277
H         -3.31782        0.88874        1.35769
H         -1.62382        1.17900        1.77968
C         -0.98281        0.99078       -1.13371
H         -1.65567        1.19844       -1.99603
H         -0.05915        0.56089       -1.57585
C         -2.87426       -0.73718       -0.79187
C         -0.83536       -0.95866        0.55714
C         -2.33811       -2.16138       -0.80738
H         -3.27791       -0.35456       -1.75318
H         -3.74616       -0.78244       -0.10108
C         -1.65652       -2.23999        0.55910
H         -0.36215       -0.67792        1.52187
H          0.00882       -1.17785       -0.13479
B         -1.40058       -2.29939       -1.82101
H         -3.13035       -2.92594       -0.95624
O         -2.60331       -2.18962        1.57227
H         -1.04117       -3.15429        0.69961
H         -2.24954        3.33056        0.95160
F         -3.19594        2.63169       -0.70627
N          0.17857        2.27493        0.50135
H         -0.60168        3.16745       -1.14998

Part 2

Are there are also examples of vice-versa, where a molecule has a plane of symmetry/center of inversion, but lacks an axis of improper rotation and is thus chiral?

An improper rotation $S_n$ is defined by a rotation about $360/n$ degrees, followed by reflection in a plane that is perpendicular to that rotation axis.

A plane of symmetry ($S_1$) and an inversion centre ($S_2$) are special cases of an improper rotation ($S_n$). It is easier to convince yourself of the $S_1$ case: according to the definition above, $S_1$ means rotation through $360^\circ$ followed by reflection in a plane. Since rotation through $360^\circ$ obviously does nothing, this is the same as a reflection in a plane.

So, the answer to this question is no. if a compound has either a plane of symmetry or an inversion centre, that automatically means it has an improper rotation axis.

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  • $\begingroup$ Neat, thanks @Karsten! That is a lot better than the figure. I was hoping it would be on chemtube or something, but alas. $\endgroup$ – orthocresol Nov 3 '20 at 22:14
  • $\begingroup$ i.imgur.com/PZ7M8UB.png For this molecules, does this count as a plane of symmetry? I feel it can be symmetrical when viewed in the animation. $\endgroup$ – ebehr Nov 6 '20 at 19:47
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    $\begingroup$ Hmm, in that frame, you are right that it certainly looks like it could be a mirror plane. I am not sure now whether it is just the angle which conceals some details, or whether that is the wrong conformation to begin with. The first figure I attached was only one of many conformations, albeit the most stable. If you look at that one, you can see that the CH2's next to the oxygens are not accurately preserved by the mirror plane you're proposing. $\endgroup$ – orthocresol Nov 6 '20 at 20:01
  • $\begingroup$ @ebehr: I changed the example to something that you may find easier to visualise (well, at least, I found it easier...!). $\endgroup$ – orthocresol Jan 15 at 16:04
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    $\begingroup$ More on this example: chemistry.stackexchange.com/questions/95979/… $\endgroup$ – user55119 Jan 15 at 17:29

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