[emissivity]

Hi! I have been really really struggled for this problem. But I couldn't make it..
So I just decided to ask you this..
please help me..



First, I will write some problems. I'm not asking you to solve these problemsC. It's about the concept that I want to get.
==========================
1. One of the first metal carbonyls to be characterized was the tetrahedral molecule Ni(CO)₄. The vibrational modes of the molecule that arise from stretching motions of the CO groups are four combinations of the four CO displacement vectors. Which modes are IR or Raman active?

2. Show that the four CO displacements in the square-planar [Pt(CO)₄]²⁺ cation transform as A_1g+B_1g+E_u.
How many bands would you expect in the IR and Raman spectra for the [Pt(CO)₄]²⁺ cation?

3. Consider the molecule cis-[PdCl₂(NH₃)₂], which, if we ignore the hydrogen atoms, belongs to the point group C_2v. What are the symmetry species spanned by the displacements of the atoms?

4. Determine the symmetries of all the vibration modes of [PdCl₄]^2-, a D_4h molecule.
==========================

Well... Actually the biggest thing that I'm really confusing is this. Problem 1, 2 and 4. You can see the picture..






OK...
Maybe it is hard for you to read my question because my English is not good..
I hope you understand..


Thank you
======
Summary
1. count or not count central atom
2. x,y,z coordinates or not
3. why should I subtract rotation and translation?(despite the problem actually didn't mention it.)

[Corribus]

I'm not going to go through and do all of these time consuming problems, but here are two thoughts with respect to problem 1.

First: In your own attempt at doing the problem, your 15-dimensional irreducible representation is wrong.  Actually very wrong.  You will never get the right answer with it.  Aside from the character of the identity operation, none of your characters are right. (In your defense, Td is tricky because the angles of rotation aren't 90 degrees. You have to use some trigonometry to get the right characters.)

Second: If you had a CORRECT 15D reducible representation for the 3N coordinate transformations, and you reduced this to a sum of the irreducible represendtations for the Td point group, and then you subtracted out the irreps corresponding to translational and rotational motion, you would be left with a total of 9 vibrational modes (3N-6) - some of them combined into degenerate irreps  - corresponding to the normal modes of the molecule. This approach to a problem like this will ALWAYS give you the right answer about what all the vibrational modes are - although it can be quite a bit of work to get there.

Third: However the problem is not asking you to do quite this much work.  Some of the irreps you would find if you followed the previous process correspond to vibrational normal modes that involve the central atom moving and some of them do not.  The question asks you only to identify those involving vibrational stretching motion of the ligands.  This is why you don't need to consider the central atom in this problem and it's also why you're not considering a full 3D range of motion of each ligand - only an outward displacement of the ligand from the metal.  This simplifies the problem quite a bit and leaves you with a 4-dimensional problem that results in 4 normal modes (3 of them degenerate). These normal modes are a subset of those identified in the more complex problem, and are ONLY the ones that do not involve changing the position of the central metal.

I do have to admit that the intended approach of the problem not obvious to me until I solved the problem the long way.  I think it's written in a rather confusing way. Then again, it's been a while since I solved for vibrational modes of polyatomics using group theory, so . . .

Anyway, looking at the other problems: problem 2 also asks you only to consider the four vectoral displacements of the ligands, so this should be solved in a similar way.  Problem 3 wants to know what all the symmmetry species are by "displacement of the atoms".  Here the ligands are NOT specifically (and exclusively) highlighted so a full treatment is necessary (though again, whoever wrote the problem should have done a better job of making this subtle distinction clear).  C2v is an easier point group to work with, so the full treatment isn't quite as tricky as it is in the Td case.  Problem four you identify as being treated in a similar way as problem 1. But according to my interpretation of the wording, a full 3D treatment seems to be required.  After all it asks for "all symmetry modes".  Restriction to displacement of the ligands only is not specified.

So in short, I sympathize. The questions aren't particularly well written and unless you had a lot of experience with symmetry treatments I can see what you were confused.  You just need to read carefully what the problems want - if they tell you only to consider displacements of ligands, then you know you don't need to do a 3D treatment of each atom's position.  If they want you to solve for every vibrational symmetry mode of a molecule, you know you've got to do the full deal. 

Of course, you do have solve for your reducible representations correctly - fail to do this and you won't get it right now matter what approach you take.

EDIT: By the way, the nice thing about these kinds of problems is that it is very easy to check if your reducible representation is correct: simply try to reduce it to a sum or irreducible representations. If you don't get integer values of the irreducible representations, you've done something wrong.  Take yours for problem 1 for example (15 2 0 0 2). If you try to reduce this you get 1.79 as the number of A₁ irreps.  At this point you know something ain't right.

[emissivity]

*MOD Edit* Remove massive quote

Thank you Corribus. I really appreciate your comment. I didn't think that someone would read my post and reply in detail.
Anyways Your last tip was quite helpful for me.

Well, after I read your comment, I've just decided to study more..
I am really confused.. some problems are treated in this way and the other problems are in that way..
one is just thinking about ligands and the other is thinking about whole atoms of the molecule..

Actually, I don't know how to get all of this over.. I don't even know where to start again..
Anyway, Thank you. Maybe to keep going is the only way to get over this situation....

[Corribus]

Well let me see if I can't raise your morale a bit.

I'm not sure if those problems were from a textbook or written by a professor. Looking over them, I suspect the latter (professors are notorious for writing needlessly complicated, ambiguously worded problems to test comprehension), but even textbooks can have duds. Most mainstream textbooks, however do have problems that are vetted and tested for clarity.

Anyway, the point is that solving any problem in physical chemistry correctly, like pretty much any other subject, requires two critical components: setting up the problem (finding the right strategy to solve) and then executing the solution in a technically correct way.  In my opinion, while it is chronologically the second step, the latter portion is more critical as a first step toward comprehension of a subject area.  Moreover, it's the part that more under your control as a student.  Setting up a problem is largely dependent on how the problem is written.  Executing the solution is completely dependent on your skill and confidence in the subject matter, and doesn't really depend on whether you've set the problem up correctly.  That is to say, even if you set up a problem wrong, you should still be able to execute the solution correctly.  This may not give you a correct answer to the question being asked, but it will give you a correct answer to the question you thought was being asked.  When it comes to grades that may seem like a meaningless distinction, but it's not a meaningless distinction when it comes to learning the material.  More importantly, if you are confident in your ability to execute a solution, this will eventually start to help you recognize when you've interpreted a poorly worded question wrong.  Remember, in "real life", questions aren't usually worded like textbook problems.  Sometimes solving a real world problem requires several different strategies until you find the one that gives you an answer that makes sense.  If you aren't executing solutions properly, then NONE of your attempts will make sense, so you'll never hone in on the right solution.  Make sense?

What I'm getting at is to not get caught up at this stage in being able to answer complicated questions like those posed in your opening post.  My advice is to find yourself a mainstream textbook and do the normal difficulty problems, which usually involve simple, straightforward (albeit boring) systems with simple, straightforward solutions.  Solving for the vibrational modes of water with a symmetry treatment may not sound interesting, but doing so will teach you the TECHNIQUE.  Once you go through all these types of problems, you will become confident in the METHOD, and won't make errors like you did in your solution to problem 1.  Remember that if a problem is confusing because of the way it's worded, a good portion of your classmates will also find it confusing.  Many of them, like you, will set up the problem wrong.  What is important at that point is to execute the solution to your wrong setup correctly.  If I'm a professor, that's what I'm looking for when it comes to grading.  Most likely you will also start to solve the problem and realize the math is more complicated that what would be expected of you (something you'll only recognize if you're confident in your technique) and this will cue you into the fact that you didn't set up the problem as the professor probably intended. Which may make you search for a different interpretation of the problem.

[emissivity]

More importantly, if you are confident in your ability to execute a solution, this will eventually start to help you recognize when you've interpreted a poorly worded question wrong.  Remember, in "real life", questions aren't usually worded like textbook problems.  Sometimes solving a real world problem requires several different strategies until you find the one that gives you an answer that makes sense.  If you aren't executing solutions properly, then NONE of your attempts will make sense, so you'll never hone in on the right solution.  Make sense?

Wow.. I think I have learned something important about life though I've come to study chemistry. You gave me really helpful advice. It is quite impressive. After getting the technique and skills, I can figure it out whether it is wrong or not.. Maybe I've thought that the comprehension of the problems goes to only my comprehension skill(It's sort of like language skill). But after I read your comment I am very very impressed. I think I have just learned the direction to my study.
 Thank you anyway. Your advice is very helpful.