Character Table Hell: Understanding Irreducible Representations

[gravenewworld]

Character Table Hell!

Can someone please explain to me how the irreducible representations of a character table are assigned their letters? I figured out how A,B,E etc. are assigned to the irreducible representation-you look under the identity operation and if its 1 then its A or B, 2 then E, etc. If you look under the principal rotation axis operation then if its 1 then its an A, -1 its a B. The thing I don't understand is how the subscripts 1 and 2 are assigned. I know how subscripts u and g are assigned, by just looking down the ineversion operation column you can figure it out. But I have no idea where the 1 and 2 come from. Also, I don't understand what the right hand part of the characeter table means with the x,y,z, Rx,Ry,Rz's and how they relate to the orbitals of molecules. Also how do you assingn z,y,x, Rx,...etc. to an irreducible representation. For example in the C2v character table why does z go with A1, Rx, go with A2, x,Ry with B1, etc. ? Can someone please help?

 

[movies]

I'm not much of an inorganic chemist, but this is what I recall from my inorganic classes:

I think the subscript 1 and 2 depend on whether the irreducible representation is symetric with respect to the principle sigma plane (sigma-v for C2v). If the representation is symmetric (i.e. the character is 1) then the subscript is 1, if it is anti-symmetric (i.e. the character is -1) then the subscript is 2.

The right hand part tells you what atomic orbitals have that symmetry. The s-orbitals will always have the symmetry of the top row in the column and are therefore usually omitted from the right hand portion. The x, y, and z refer to the symmetry of the p-orbitals in that point group relative to the standard orientation. For example in C2v the p(z) orbital has A1 symmetry. The next column over has the same information, but for the d-orbitals (hence, xy, xz, z^2, etc.). This is useful when thinking of metal complexes because you know what the atomic orbitals on the central metal atom look like and you can figure out the overlap with the ligand orbitals of the same symmetry.

I can't recall what the Rx and Ry mean though. My textbook says that it "denotes a rotation about the axis" but I don't know what significance that has.

Hopefully I haven't made any errors in this post...good luck!

 

[altered-gravity]

Hello,

I completely agree with movies. Just wanted to add the detail of Rx, Ry, and Rz. You´ve to use them the same way that "x", "y", "z", "xy", "xz"... They indicate you the class of simetry of the rotations around each axis, in other words, how do the rotations transformate with each simetry operation of the group.

For example, if Rx is in the A1 row, means that the rotation around X axis is totally simetric.

For example in the C2v character table why does z go with A1, Rx, go with A2, x,Ry with B1, etc

z) Take the z axis and apply all the group operations to it, you´ll obtain 1, 1, 1, 1 that´s A1.

Rx) Take water molecule for example, make it rotate around X axis, now apply all the operations to the rotation, you must obtain 1, -1, -1, 1 that´s B2

and so on. Good luck!

 

[gravenewworld]

Thanks for the replies. I see where the subscripts 1 and 2 come from in the C2v point group by looking at the sigma v plane, but this doesn't work in every point group. For example where do the subscripts come from in the D4 and D3 point groups? There are no planes of symmetry in those groups. I hate how the undergrad. inorganic texts dumb down the introduction to group theory, and don't explain rigorously through mathematics why character tables are set up the way they are. I think the concepts would be much clearer if the derivation of character tables were shown mathematically.

 

[altered-gravity]

I agree with you

I confess that I stutied basis of applied Group Theory time ago and I forgot it in a week, as I only used some tricks to get the info I needed from the tables for spectroscopic works.

Anyway I will take a look at D3 and D4 groups, I will post as soon as possible.

 

[movies]

I just looked in a different book and it says that you can consider the subscript 1 or 2 to be arbitrary labels. I don't know if there is really more to it than that though. It seems to me like it is just a convention so that you can differentiate the two A irreducible representations.

 

[altered-gravity]

Sorry, we were mistaken,

Subscripts 1 and 2 indicate symetry/asymetry respect to C2 axis (perp. to the Cn principal axis), not respect sigma v.

 

[gravenewworld]

Ok this just raises some more questions. what if the C2 axis is the only and principle axis of rotation, like in C2v, where do the subscripts come from then since there are no other C2 axes? How can you tell if the C2 symmetry operation listed is perpendicular to the principle rotation axis in the first place without having the molecule infront of you to look at? For example, C4v has principle rotation axis of C4 and a C2 rotation axis, but the subscripts 1 and 2 don't match up to the characters under C2. Also,something like D2, D6, and D4d all have more than 1 C2 rotation axis, so how can I tell which one I should look at? Thanks again for all you help. I just have a huge midterm on this material really soon.

 

[altered-gravity]

How can you tell if the C2 symmetry operation listed is perpendicular to the principle rotation axis in the first place without having the molecule infront of you to look at? For example, C4v has principle rotation axis of C4 and a C2 rotation axis, but the subscripts 1 and 2 don't match up to the characters under C2. Also,something like D2, D6, and D4d all have more than 1 C2 rotation axis, so how can I tell which one I should look at?

I´m sorry, i should have to explain it better, although i´m not sure if i´ll be able to help you much more, I´m confused too

Work at the inverse, start imaging a molecule.

1.- Is it linear?
YES: Then we have D(infinite)h and C(inf)v
NO: Then we continue
2.- Has it 2 or more Cn (n>2)?
YES: Groups Ih, Oh, Td
NO: continue
3.- Has it principal Cn?
YES: Has it n C2 axis perp. to Cn?
Yes: Dnh Dnd Dn groups
No: Cn Cnv Cnh S2n groups And so on...

In Dnh Dnd and Dn groups there are n C2 axis perp to the Cn principal axis. Those were what I was talking about. There are no more cases where n C2 are perp. to Cn. For example, In D4h there are 5 C2, 4 of them are perp. to the C4.

what if the C2 axis is the only and principle axis of rotation, like in C2v, where do the subscripts come from then since there are no other C2 axes?

Good question, I´m not sure. In this case I think it refers to the symetry/asymetry respect to the sigma-v plane as we were talking before.

Take into account that when a sigma-h exist (horizontal plane) as in the Dnh Dnd and Dn groups, a C2 (perp. to Cn) is completely equivalent to a sigma-v that contains it.

I just espect not to get you more confused.

 

[Me..Sandeep]

HI to all...
Hey guys please help me understand the character table for Oh point group. in octahedral point group y C3 axis ( 8C3 ) axis are given priority than C4's...

 

[Me..Sandeep]

Character Table Hell!

Hi guys..
In octahedral character table y 8C3's are given priority than the highest fold rotational axis C4.