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The original resource

"Tables for Group Theory" By Peter Atkins, Mark Child and Courtenay Phillips (1970). I have been using this 2008 version, in which Pg. 37 has the following table:

enter image description here

Some minor inconsistencies

These inconsistencies don't bother me as much as the ones in the next section, but I'm mentioning them because the existence of these inconsistencies could possibly offer a simple explanation for the ones that I describe in the next section (i.e. maybe they simply didn't put much effort into perfecting the diagrams).

  • For C4 one of the lines is black instead of purple
  • For C6 there's extra purple lines which seem to be spurious
  • For C4, C5 and C6 some lines are different sizes when compared to the other lines in the same diagram, or when comparing the diagrams of C5 and C5h.
  • If the Cnv series approaches a cone as nā†’āˆž, then the Dnh series would approach a circle as nā†’āˆž, but this is not depicted.

More uncomfortable inconsistencies

For the inconsistencies that were listed in the previous section, I'm comfortable concluding that the first three were just issues with aesthetics (poor work on the graphics), and that the other inconsistency was simply an oversight.

For the following two, I'm a bit more uncomfortable because they seem much less accidental:

  • The purple lines in C4-C6 are perfectly parallel with beige lines, but for C2 and C3 they are not. This seems much less accidental than coloring a line black when all other such lines are purple.
  • The purple lines in the Cn and Cnh series are just thin straight lines, but the Dn series has triangles. Perhaps in the Dn series, since there's two different types of appendages (beige and purple), they felt the need to use thicker triangles rather than just thin lines, so that the difference between the two types is made more obvious than it would be if they were just thin lines with different colors (rather than thick triangles with different colors). I'll ask a question about this inconsistency in a separate post, in line with the one question per post policy.

Question

Are the C2 and C3 diagrams simply in error? For example, is there actually a molecule with a structure like depicted in the C2 diagram, or is it meant to be just a straight horizontal line with two appendages, like the O-O bond with two H appendages in H2O2, rather than a 3D diamond with two appendages?

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  • $\begingroup$ Also it would probably be more correct for the Dnd diagram to converge to two circles on different planes, rather than just one circle! $\endgroup$ Commented Sep 10, 2023 at 20:39

1 Answer 1

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After hours of conversation with three participants including me, I can make the following conclusions.

Answer to the question

It's likely an unintentional "error" in the publication. In the C2 and C3 figures, the purple lines can indeed be aligned with the beige lines, as in the diagrams for C4, C5, C6, C2h, C3h, C4h, C5h, C6h. The authors likely did not intend to draw the purple lines with non-zero angles with respect to the beige lines, especially in the case of the C3 diagram, which seems to be unintentionally divergent from the C3h diagram (although they may have had specific molecules in mind when choosing the diagrams for C2 and C3).

Further conclusions

It is not technically incorrect for the angles between the purple lines and the beige lines to be non-zero in C2 and C3 molecules (as long as the angles are not chosen in such a way to promote the point groups to a higher one, like D3h in the case of C3), but if drawn that way, it would have been more consistent for the angles to be non-zero in the C4, C5 and C6 cases as well.

Also, the diagram for C2 is indeed a valid structure with that point group, but it may not be as easy to find a real-world molecule that has that shape, compared to the H2O2 example that I suggested in the question post. The diagram for C2v is essentially the one in the top-right corner of the screenshot below from the Landolt-Boernstein series, so it's a pyramid whose 2D diagram looks like a diamond:

enter image description here

By adding one atom along the H-Si line, and another identical atom in the H-Si line directly opposite to the first H-Si line, we can maintain the C2 axis while breaking the vertical planes, therefore demoting the point group from C2v to C2. The new atoms that are added along the H-Si lines could be added in between the H and Si atoms, or outside of the original pyramid, as in the diagram in the original question post: in both cases we would get a C2 molecule, but in both cases it's harder to find a real molecule that would have this shape, compared to the ease with which we found H2O2 as a simpler example of a C2 molecule. Likewise, H2O would have been a more obvious choice for the C2v example, compared to the above (non-standard) isomer of H2Si2.

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