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Joseph Keane, Muhlenberg College
Last seen: 3 months 4 days ago
Joined: 03/29/2009 - 9:35pm

Symmetry Questions

The D5h character table lists 2 S53 operations.  My background in group theory is (obviously) not the strongest.  I would have told my students to list 2 S52 operations, as consistent with the listed 2 C52 operations.  Is there a short explanation of why S53 is the prefered notation for the improper rotations?  When I first introduce this notation, I tell them that we try to minimize the value of the superscripts, thus, for instance, C3 rather than C62 in D6h.  Calling these operations 2 S53 seems to disagree with that approach.


Joseph Keane, Muhlenberg College
Last seen: 3 months 4 days ago
Joined: 03/29/2009 - 9:35pm

Following from the glowing success of my D5h question above (Anyone? Bueller?) I'll try another.

Does anyone have a concise way to explain the sigma-v vs. sigma-d notation?  So, for instance, why in D3h molecules the (vertical) mirror planes are labeled sigma-v, in D3d molecules they are labeled sigma-d, and in benzene (D6h) those that follow the C-H bonds are sigma-v while those between the C-H bonds are sigma-d?

Be well.

- Joe

Kyle Grice, DePaul University
Last seen: 2 days 8 hours ago
Joined: 07/22/2010 - 5:29pm

Hi Joe

As for the D5h table, The reason there are two of the Sis just like there are two C5's. You can do the rotation in opposite directions and get a different product of the operation, the product of which should have an operation that results in the identity operation. 


As for sigma-d, that's because they're dihedral to the secondary rotational axes rather than containing them like sigma-v. Its a convention to help show which mirror planes are which. 



Anne Bentley, Lewis & Clark College
Last seen: 1 week 4 days ago
Joined: 04/01/2008 - 8:09pm

Hi Joe,

I'm responding back to your first question about why S53 is the second type of S rotation instead of S52. To see it better, I drew a pentagon and put a little "x" at the top corner.  I then carried out an S5 operation two and then three times.  I changed my "x" to an "o" to indicate that my point had gone to the bottom half of my pentagon.

When I did two S5 rotations, my point returns to the top of my pentagon, so an S52 is equivalent to doing a  C52.  So you need to do three S5 rotations to get something unique.

I hope this helps!