[bunker_man]

The long and short of it is that I'm in advanced inorganic chemistry, and my teacher wants me to know this, but not only didn't give out a handout that was in any way coherent, but it doesn't appear to show up in my book either. (And blind internet searches have led to things which were less than helpful.)

So my first thing I need, is *** how to make reducible representations. (I know how to use the flowchart for symmetry, but not what to do after that.) I kind of know how to convert them to irreducible once I have them, but actually putting them together I just have a hard time understanding at all where the numbers come from. I'm also not sure what decides whether the representation gets one or two Γ on the left, or how to make an irreducible representation if there's two.

***

Here's an example of my other problems. Note, I have no clue how to solve these at all, or what the answer would even look like. I'm not looking for answers, since this is from a test I finished, I'm looking for information on how to determine and do these types of problems in general.

-IV. Determine the number of IR-active and Raman active CO stretching modes for the mar- and fac- Mo(Co)3(NCCH3)3 isomers.

-V. Determine how you can distinguish between cis- and trans- isomers for FE(CO)4(Cl)2 using IR spectroscopy.

(Note for that problem: use point groups to determine reproducible & irreducible representation for C---O stretching. {triple bond})

VI: The Ion [AuBr4]- could be expected to adopt either of two common geometries: tetrahedral or square planar. In the infrared spectrum of this compound one stretching vibration is observed at 196 cm-1. In the raman spectrum of this compound, two stretching vibrations are observed one at 212 cm-1 and one at 102 cm-1 . From this information provided, assign the structure of the [AuBr4]- ion to its appropriate geometry.

(Note: use point groups to differentiate between them)

[Corribus]

Unfortunately a rather difficult topic to try to explain without visual aids, but I can try to help.  However, let's try some easier examples than those you've listed in your post.

First, there's no formulaic way to build a general reducible symmetry representation of something because it depends to some extent on what you're trying to represent. Nevertheless, generally a reducible representation is built by determining what the characters are for the various transformation matrices corresponding to the symmetry operations applied to the system of interest. Real straightforward, right? LOL. Ok, an example will help.

Let's do something simple: vibrational modes of water.  How can we use symmetry and character tables to determine what the IR- and Raman-active modes of water are? You need to build a reducible symmetry representation of the possible motions of all the atoms in a water molecule, then reduce it to irreducible representations, which can tell you about what vibrational modes are IR and raman active. 

Do you have any idea how to do this? I'll walk you through it, but first I want to see what your level of knowledge is.

[bunker_man]

I know how to determine it's point group. Not anything after that.

[Corribus]

Ok, well that's something at least. How's your matrix math?  You know how to do simple stuff, like calculate a trace, multiply matrices together, that kind of thing?

[bunker_man]

Um... I don't think so?

[Corribus]

Well, either you do or you don't. For instance, do you know how to solve this equation?

[tex]\begin{pmatrix} a\\b\\c \end{pmatrix}\begin{pmatrix} 0&-1&0\\1&0&0\\0&0&1 \end{pmatrix}=?[/tex]

And what is the trace (character) of the 3x3 matrix in the middle? What is its determinant?

If you don't know how to find this information, this is probably 90% of your problem.  In which case I would suggest finding some linear algebra text and reading through the first chapter or two about basic matrix mechanics.  Most physical chemistry texts, such as McQuarrie, also have short math sections that provide an introduction to this kind of thing.

It'll be hard for me to help you learn about irreducible representations without some basic knowledge of how to work with matrices.  If you do a little reading and satisfy yourself that you can answer the above question, please come back and I'll walk you through the chemical applications of reducible and irreducible representations.

[Corribus]

LOL, I realize the equation I gave you is not solvable as it's written. Try this one instead:

[tex]\begin{pmatrix} 0&-1&0\\1&0&0\\0&0&1 \end{pmatrix} \begin{pmatrix} a\\b\\c \end{pmatrix}=?[/tex]

[bunker_man]

Nope. -.-'' But we moved past this area. I'll have to put learning this on hold til the final if we re-go 0ver it.