Principle Rotation Axis in D2d Group (allene)

[bennybp]

I'm kinda pulling my hair out with this one. Basically, in a character table, A represents symmetry about the principle rotation axis, defined to be the highest order Cn axis (among other things). B is for anti-symmetry about the principle rotation axis. In the character table, there should be +1 for A and -1 for B under the column for the principle rotation axis.

In the D₂d group, however, this pattern occurs under the S₄ operation! In reviewing the literature (particularly F.A. Cotton, Mulliken, and Herzberg), the closest I can come up with is that it is due to D₂d being related to the C₄v group. Other than that, I (and my professor) have no idea.

For a little more background, I posted this on the Wikipedia Reference Desk a week or so ago, and it's available HERE

I'll keep looking, but let me know if anyone has anything.

Thanks for at least reading
Benny P.

[Vitalii]

I'm kinda pulling my hair out with this one. Basically, in a character table, A represents symmetry about the principle rotation axis, defined to be the highest order Cn axis (among other things). B is for anti-symmetry about the principle rotation axis. In the character table, there should be +1 for A and -1 for B under the column for the principle rotation axis.

In the D₂d group, however, this pattern occurs under the S₄ operation! In reviewing the literature (particularly F.A. Cotton, Mulliken, and Herzberg), the closest I can come up with is that it is due to D₂d being related to the C₄v group. Other than that, I (and my professor) have no idea.

For a little more background, I posted this on the Wikipedia Reference Desk a week or so ago, and it's available HERE

I'll keep looking, but let me know if anyone has anything.

Thanks for at least reading
Benny P.

The symmetry tables in their present form came from the accepted conventions. My guess is that these conventions had exceptions from the rules you stated above. I'd refer you to the original article on symmetry conventions:
R. S. Mulliken,. J. Chem. Phys., 23, (1955) 1997

[bennybp]

Yes, I have a copy of that article, which more or less says use the conventions in "Infrared and Raman Spectra of Polyatomic Molecules" by Gerhard Herzberg. That is where it discusses D₂d being isomorphic with C₄v. I suppose there isn't much more to it than that, without getting really heavy into group theory.

Thanks for thinking about it at least.

BennyP