# Irreducible representation of C3v

Hello to everyone I'm a newcomer to the blog.
I'm studying group theory and I'm dealing with irreducible representations (irreps) of C3v. Now since C3v is invariant after 6 simmetry operations I expect, as a consequence of the well known relationship g=Sum_i n^2_i (where g is the order of the group and n_i is the size of the matrices corresponding the i-th irrep), to have 2 irreps of dimension 1 and 1 irrep of dimension 2. This 3 irreps are found to be named A1, A2 (1-dimensional) and E (2-dimensional). There are examples of functions transforming as the A1 and E irreps (for example the pz orbital in the ammonia molecule transforms as A1, while px and py mix up following the rules of E). Similarly s-orbitals and d-orbitals transform as A1 and E, but I have not found so far an example of a function transforming as the 1-dim A2.
It therefore seems to me that there are irreps that have a profound physical meaning such as A1 and E irreps for c3v, being closely associated with the properties of transformation of orbitals, and others, like A2, having less physical meaning.
Do you agree? Could you please comment?
Thanks a lot,
R.Gaspari

PS is it possible to use latex formulae?

Gokul43201
Staff Emeritus
Gold Member
Hello Gaspari. Welcome to PF!

Yes, LaTeX is enabled on this site. Use [tex ] latex here [/tex ] tags (without the spaces). For inline TeX, use [itex ] instead.

$$g=\sum_in_i^2$$

Click on LaTeX to see code, or look here: https://www.physicsforums.com/showthread.php?t=8997

I think the answer to your question is yes and no. While there seem to be more functions that transform as A1 or E among common groups, there are some functions that transform as A2, for example, $d_{xy}$ in a C2v molecule or $p_z$ in D3 and D4 molecules.

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Similarly s-orbitals and d-orbitals transform as A1 and E, but I have not found so far an example of a function transforming as the 1-dim A2.
It therefore seems to me that there are irreps that have a profound physical meaning such as A1 and E irreps for c3v, being closely associated with the properties of transformation of orbitals, and others, like A2, having less physical meaning.
Do you agree? Could you please comment?
I often don't remember the irreps, so I had to look them up again.
I find the tables always make the physical meaning more clear anyway.
http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/C3v.html
http://newton.ex.ac.uk/research/qsystems/people/goss/symmetry/C3v_cor.html

It is clear that A2 involves a sign change when performing the mirror operation. (which can also be seen in the correlation table if we break c3v -> cs)

So the reason you are having difficulty finding a physical example here is that you only considered systems with a central atom on the axis and all dangling bonds terminated with hydrogen.

http://en.wikipedia.org/wiki/Cyclopropane

Thanks a lot for the latex suggestions, first.
I lacked the examples you provided me, thank you Gokul and Justin. Browsing on internet I actually see that $C_{2v}$ has an irrep labelled by $A_2$. Similarly the cyclopropane molecule belongs to the point group $D_{3h}$ and has an irrep called $A_2^'$. In these latter cases you can find functions that form a basis for the irreducible representation, such as those mentioned by Gokul. I'm nevertheless surprised a little bit by the fact that the same labels can be used for irreps of different point groups. Actually they represent a set of different operations: $(E,2C_3,3\sigma_v )$ for the point group $C_{3v}$ and $(E,C_2,2\sigma_v )$ for the point group $C_{2v}$. I wonder if the description of an irrep would be more complete if the name of the label came with the particular point group to which it is referred. If we refer to the $A_2$ representation corresponding to the $C_{3v}$ point group I keep failing to find orbital combinations that transform accordingly.
Anyway I'm pheraps worrying too much about that..

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Similarly the cyclopropane molecule belongs to the point group $D_{3h}$ and has an irrep called $A_2^'$.
Sorry, I couldn't think of a better example off the top of my head.
Yes, it has D3h symmetry. But the point is that it also has C3v symmetry (a subgroup of D3h). So analyze it with C3v, not D3h symmetry.

If we refer to the $A_2$ representation corresponding to the $C_{3v}$ point group I keep failing to find orbital combinations that transform accordingly.
The reason I suggested cyclopropane was so you could consider p orbitals on atoms not centered on the axis. Try playing with combinations of the carbon p-orbitals.

I looked with more patience to the links you have provided me. Actually no orbitals are reported as basis functions for the irrep $A_2$, so I think there are really none. This situation stands similarly for the $D_{3h}$ point group (I looked on a textbook). But as it is reported, the $z$-component of the angular momentum of a particle transforms as $A_2$.
This comes from the fact that the angle between the projection of vectors $\vec{r}$ and $\vec{p}$ on $xy$ plane does not change when a reflection is made with respect to a plane containing the main axis of rotation, $e_z$ here (this is the case of the $\sigma$ operations in $C_{3v}$ point groups ). Instead the orientation of the angle is inverted. Thus we could say
$\sigma_v(\vec{r} \wedge \vec{p}) = - \vec{r} \wedge{p}$ or $\sigma_v(\vec{l_z}) = -\vec{l_z}$. One sees soon that $C(l_z) =l_z$ and $E(l_z) =l_z$ then $l_z$ actually transforms as $A_2$.
I think I found an example. I'm quite satisfied.
Thanks a lot to Gokul and JustinLevy

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Actually no orbitals are reported as basis functions for the irrep $A_2$, so I think there are really none. This situation stands similarly for the $D_{3h}$ point group (I looked on a textbook).
You are correct for orbital functions centered on the origin.

Instead consider the p orbitals of the carbons in cyclopropane. Since A2 would require odd symmetry across the mirror plane, you could see it as an anti-bonding molecular orbital for the C-C bonds. Does that makes sense?