The D_{5}h character table lists 2 S_{5}^{3} operations. My background in group theory is (obviously) not the strongest. I would have told my students to list 2 S_{5}^{2} operations, as consistent with the listed 2 C_{5}^{2} operations. Is there a short explanation of why S_{5}^{3} 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, C_{3} rather than C_{6}^{2} in D_{6}h. Calling these operations 2 S_{5}^{3} seems to disagree with that approach.

Thoughts?

Following from the glowing success of my D

_{5}h 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

Hi Joe

As for the D5h table, The reason there are two of the S

_{5 }is 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.

Kyle

Hi Joe,

I'm responding back to your first question about why S

_{5}^{3}is the second type of S rotation instead of S_{5}^{2}. To see it better, I drew a pentagon and put a little "x" at the top corner. I then carried out an S_{5}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 S

_{5}rotations, my point returns to the top of my pentagon, so an S_{5}^{2 }is equivalent to doing a C_{5}^{2}. So you need to do three S_{5}rotations to get something unique.I hope this helps!

-Anne