# Lecture 03. Transformation Matrices.

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group case it's look at why orbitals some meaning that the missile that fast because it's are you don't need writing down it's already on the character table which is that the joy of this whole thing is all this information already tabulated for you and all you have to do is figure out how to read it you don't have to generate yourself every time OK so we already went through facts we know how that 1 behaves so if we look at the PC or all we can see that that 1 doesn't change and the identity 2 doesn't do anything to it and neither do these reflections so in this particular case there's the orbital the PC orbital belongs to that symmetry group that you know that doesn't change under any transformation which in this case is a 1 OK so now I know some people might be thinking but what about in that case we were we were looking at the O C O 2 we had these orbitals you pointing in different directions and when we got differences for these that's because in that case my basis was the set of the 2 orbitals together and so that means that you want what we have so we have our own orbital isolation like this it in this case they belonged to the irreducible representations created few start building up sets of love than that might transform differently and you can make a producible representations for the England had a deal with that OK so if we look at why orbital we can do the same thing you again I'm just going to go through it quickly you can do it yourself for extra practice of the need to end then now at the next thing is let's look at this thing that's called a to here so that doesn't have anything that belongs to the x y and z unit vectors but
there's there still some characters for it if we think about it as a DXY orbital we can see that that's going to behave like this so he and Yancy to organize do anything to it but it will change signed under those reflections and so we get that under the 8 2 operation and so sometimes if you look at at these different .period groups there might be irreducible representations that are included for completeness but you might notice that there's nothing listed under that year in terms of linear quadratic terms of the Cartesian coordinates or occasions the other might just be nothing in those categories that's OK it it just means that there's nothing of chemical interests belonging to that symmetry species there yes he would be on the right of the people as you can get all hold of money all there will be able to transform according to that cemetery species and yet again how this is going to work and where these things go totally depends on the pointer so if you if you look around at different ones don't belong to different representations and so that's why it's so important when you start doing these problems to assign a molecule that to the correct .period OK
so now we have made up that we have looked at this we know what irreducible representations are we have the least scene some practice examples of how to make up reducible representations let's look at how we get from 1 to the other and there's a formula that we can use that so here you have to keep track of what you're doing but it's conceptually not really hard OK so what we want to know is we have some reducible representation that we're going to make up because it has to do with the bonding or molecular motion or some property of the molecule that were interested in and we want to see how can we add up the various irreducible representations and sorry in and get that reduce or representation so what we're looking for is for example the number of a ones that appears in Europe representation so sometimes we can do this by inspection but often it's not really that easy particularly when we weakening the problems we need to use this formula OK so boring and do it is add the following things so 1st of all we do we have a problem 1 over each so what's H. remember that some of what you get when you add up the total number of operations it's also called the order of the group so the character table I gave you unfortunately doesn't give you HC half-dead up yourself but that's a fact and then what's in this song here priority is the character in the reducible representation kind Is the character in the irreducible representation so remember the reducible representation is whatever you generated that's going to help you solve the the molecular problem the character and irreducible representation is the 1 from the table under that particular operation and the end is the number of equivalent operations in the class so that means the coefficient in front of the the operation at the top the table so let's let's look at this late hour the
OK so we're looking forward the number of the cemetery space is and again we go through a news our reduction formula for each operation in the table and we get 1 and so we went through all of the possible irreducible representations in this .period group and we use this reduction formula and so we're gone and here's what the former the insertion of a place we should say Demel 1 equals 2 8 2 was so then it's not it's not hard it's just you just have a lot of stuff to keep track of and you know it's fine to get 0 for these somethin you'll find that some reducible representations don't contain a particular irreducible representation what you don't want to get his fractions you you really do need to get the whole numbers for these things that we don't protect the work yes so I'm sorry Saigon negative to below the 3 sigma B. that's a problem that's what I gave you the problem is here's this reducible representation demo 1 reduces and the idea that I want to point out we do real ones that's going to come from something that we do as part of setting up a problem another question this that's right and the reason that that we want to do that is because we have all sorts of information about that on the character table and we can use it to learn stuff OK let's try to bring this back to 1 discussion instead of many because it's really hard to hear what's going on that question yes the 3rd term is the number of operations in that class the equivalent so that's how efficient informed about operations it's on the character table so if you look at C 3 there's a 2 in front of it if you look at that said Sigma Beta 2 3 in front of it OK so everybody clear on our terminology and and what we're going to do here is what's so let's move on OK let's see another example something to give you another example were in the same .period group there were going to go through this again and we're going we're going to do it fast because I we've already gone through red power if you get confused I'm happy to answer questions about it OK so we need to go through the number of a 1 again and so now we have a different reducible representation which recalling to and so in this case when we go through this we find that the number of a ones equals 1 and I guess what I want to point out here is that in this reducible representation the same to the 1st 2 characters are the same and only the 3rd 1 is different we can get quite differences from so it's not always easy to do them by inspection sometimes you can if you can just see what stands out that's a fine way to do it but it's not always easy and the formula always work so we definitely want to do that OK so if we don't fearing that the number of a choose now again using this formula we get 1 and the same thing for the asymmetry species and now we get 1 for that 1 and so we can write Gamache to A-1 plus a 2 plus OK so now we know how to use this reduction formula to reduced rates for representations and get some arbitrary representation in terms of on the irreducible representations as you can imagine you could make up ones that don't work right see if you if you make up a reducible representation without a corresponding to some to something real In terms of chemical bonding or something like that you can imagine coming up with sets of numbers that don't give you images in terms of the pointer so you know if you're going to look for practice examples you don't just make up arbitrary ones it may not work OK so let's arms I had another 1 in
here but I don't think we need to to do it let's quickly go through how we're going to set this up In
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 Title Lecture 03. Transformation Matrices. Title of Series Chem 131B: Molecular Structure & Statistical Mechanics Part Number 3 Number of Parts 26 Author Martin, Rachel License CC Attribution - ShareAlike 3.0 Unported:You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. DOI 10.5446/18911 Publisher University of California Irvine (UCI) Release Date 2013 Language English

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