Lecture 26. Final Exam Review
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Martin, Rachel
 Principles of quantum mechanics with application to the elements of atomic structure and energy levels, diatomic molecular spectroscopy and structure determination, and chemical bonding in simple molecules. 
 Good morning. Welcome to our last PCAM lecture. So I know I'm going to miss you guys. This has been a really great class. I've been really happy about, you know, how much everybody participates and is really excited about learning PCAM and that's really cool. It's been a lot of fun. So thanks for being such a great class. Today we're just going to do a review of what's going to be on the final. It is completely cumulative. It covers everything that we've talked about in the course, which is really a lot of stuff. So we're just going to go back through and, you know, not go into anything in too much depth, but talk about everything that's going to be on there. And if we run out of time, which we might, the slides are posted and I'm going to have a lot of office hours next week. I still don't have it posted when I'm going to do it, but I'm planning to have, you know, at least one every day, Monday through Thursday, possibly more. It just depends how I can work out the schedule. Just a quick poll. Who has a final Monday morning? Who has a final Monday afternoon? Tuesday morning. Tuesday afternoon. Okay, looks like Tuesday is a good day. How about Wednesday morning? Wednesday afternoon? Thursday morning? Thursday afternoon. Okay, that's unfortunate. That's a bad schedule. Okay. Yeah, well, I figure, you know, people are mostly taking the same classes, so I'm trying to avoid scheduling the office hours when a lot of people won't be able to make it. Okay, so it looks like Tuesday is a pretty good day if I'm going to do extra ones. And then I'll still do, you know, a bunch of last-minute stuff Thursday, but it's too bad that a lot of people will have a final. When is it over? Six. Ouch. Okay, we'll see what we can do. I have to check the schedule. Okay, another thing I want to mention before we get started is that a lot of people have sent me emails about the seminar extra credit sheets and exam regrades and people are getting anxious because I haven't worked on them yet. So I was out of town the last three days, and I did not have a lot of internet access. So I could see emails on my phone sort of when the plane landed and whatever, but, you know, I've been traveling a lot and or I was in an eight-hour meeting reviewing grant proposals. So I just haven't had a lot of internet access to upload stuff. So I totally get it. I know that it's anxiety-provoking that you turned in your stuff and you don't know whether you got credit for it or not. The issue is I'm just behind. So when I started doing this extra credit thing with the seminars, I didn't know everybody was going to do it. And that's awesome. I'm glad that you're doing it. But it means that, you know, whenever you give me a piece of paper, it turns into a Russian novel when everybody does and I am just behind. So one of my plans for the weekend is to get caught up on the stuff. When I get all the seminar extra credit things done, I'm going to post something. I'll post it on the Facebook page and the class website. And, you know, I'll say, okay, I think I have all of them done now. And at that point, if I still didn't get yours, then please do send me an email and I'll go look through the stack again. Same thing with the exam regrades. The deadline to ask me about it is tonight at midnight. I'm going to wait until I have all of them and then just do it all at once. That'll make sure that it's, you know, that I'm definitely doing it consistently and also hopefully get it done quickly. And again, I will post when I think I have them all done and then if I missed yours, go ahead and let me know at that time. I know it's tough not to know what's going on and I'll try to get caught up as soon as I can. All right. Does anybody have any more questions about general stuff before we start reviewing for the exam? Okay. Let's do it. All right. So as I said, this exam is really cumulative. And before we get into the older material, I want to just talk about the canonical ensemble a little bit. So we got about this far, you know, talking about the, the connections between the canonical ensemble and just the standard partition function that we've looked at. But we didn't quite get to finish up and then in between, we had example day which, by the way, I heard John Mark's lecture debut was awesome. So that's great. I'm not surprised but I'm glad it went really well. Okay. So we have, we learned about the canonical distribution and the canonical partition function. So this is for our canonical ensemble which, remember, is a collection of little individual ensembles that are all at the same temperature. And the thing that we like to use this for, or its key feature, is the fact that the canonical partition function is more general than our normal partition function. And that's because it doesn't assume that all the particles are independent. And so that is really useful when we want to use it for studying condensed phases, so liquids and solids, or even gases that don't behave ideally. So it's a lot more general and it can be used for more things. And, you know, obviously we're about at a time this quarter, but so we're not going to do too much with this, but I want to make sure that we cover it to set you up for next quarter. So next quarter with Dr. Gerber, you're going to do a lot of working with the canonical ensemble, statistical mechanics, and get into thermodynamic properties. So the last thing that we need to talk about is the fact that you can get bulk properties of the system from the partition function. So the average energy of one of our little member ensembles is, it's given by, you know, we've got the, it's just the average energy for our individual ones. And we can write that down in terms of the relative populations, which, again, we remember what that is. And what would be really nice is to have this in terms of just Q, because then it's a lot more useful. And so we can substitute using the derivative of Q with respect to beta. I remember beta is 1 over kT. And we know that that equals dLn Q d beta. And so the result we get is for distinguishable molecules, big Q is little Q to the n. So distinguishable molecules could be, it could be, you know, molecules that are in a crystal lattice so that they occupy a particular position all the time. So it could be that they're all the same, but they're in a particular point in space. And so you can distinguish them that way. It could be that they're all the same, that they're not all the same type of molecule. So you might have a solution of, say, ethanol and water. So they're moving around, but some of the molecules are distinguishable because they're different molecules. And in that case, Q is little Q to the n over n factorial. And so this is stuff that I basically just want you to hold in your mind for next quarter, for when you work on thermodynamics with Dr. Gerber. It would have been ideal if we had time to get to it in the last lecture, but we didn't quite. So that is what we're going to say about statistical mechanics. And with that, let's move on to the review for the final. Okay, so what do you need to know? So the first thing is being able to assign molecules to a point group. This is really important because there are lots of types of problems where you have to assign stuff to the right point group in order to get the right answer. And that will be the key to the whole thing. There might be, you know, maybe one problem that's not worth very much that says, you know, just assign something to a point group, but most of what's going to be going on is just having to use this information to learn something about bonding or molecular motion or, you know, whether certain kinds of orbitals can overlap or whether certain kinds of wave functions can overlap. And this is something that you definitely need to review if you had trouble with it or even if you didn't have trouble with it and you just haven't looked at it in a little while. It's an important skill. And so remember things like, you know, we've talked about different objects and how they transform under the operations in a point group. And so, you know, I bring up this OCL2 example again that we saw before in class where depending on whether the phase of the orbitals is in phase or out of phase, they have different behavior with respect to the operations and so you get different matrices for those operations. This is something that you definitely need to be able to do. And hopefully you see a little bit more why it's important now that we've talked about the rotation statistics of things like molecules that have fermions versus bosons where you might have them transforming differently under this rotation operation. So this is something that you should definitely be able to do. So with a basis set that's described in words, you should be able to figure out, you know, how to draw it and write down appropriate matrices for these operations. And so again, depending on whether the orbitals are out of phase or in phase, in this case, you get different answers. Another thing that's important to point out is that in this particular case, we talked about the basis set being the p orbitals. So you treat them separately. We've also seen other examples where the basis set is a molecular orbital consisting of a linear combination of those p orbitals. That's a little bit of a subtle difference, but you definitely have to pay attention to it. So if our basis set is the linear combination of the orbitals, then you treat it as just one thing and, you know, how you count what happens with the operations is a little different. So keep that in mind. We also need to do things like looking at the molecular motion and determining which vibrational modes are IR and Raman active. So remember the general procedure for doing this. So we have, you know, some molecule we want to learn about its IR and Raman active vibrational modes. The first thing to do is set up your basis, which is going to be X, Y, and Z unit vectors on each atom. Mistakes that I saw people make on the first couple of exams included, you know, not putting a basis set on the central atom and just doing the outer ones. That's, you know, make sure you don't make that mistake again. The issue is we're interested in the relative displacements of all of the atoms, and so we have to include all of them. So of course getting the molecule into the right point group is an important part of this. You need to be able to set up your basis and then look at this and determine whether you can use the shortcut or not to just get the character. I'm probably not mean enough to give you one where you can't in the context of a molecular vibration problem. Maybe in some other context I might, but in this case it's probably too long. And then, you know, be able to write down your reducible representation representing the molecular motion and reduce it to get the modes. And so again, we should have nine, we have nine elements in the basis because we have three unit vectors and that's going to be three times the number of atoms in the molecule. So that's a good way to check yourself. You should get nine symmetry species in the final answer then. And then what we have to do is go through and take out the translations and rotations because those are something that we don't see in vibrational spectroscopy. But they're accounting for some of those symmetry species. And so you do this by looking at the character table and finding the symmetry species that correspond to X, Y, and Z and then RX, Ry, and RZ. And then the vibrational modes are whatever is left over. So here's another note about this. So in this case we only have symmetry species that belong to A and B type representations. That means they're non-degenerate. When you have things like E which is doubly degenerate and T which is triply degenerate. If you have say a T something symmetry species and X, Y, and Z all belong to it, that takes out the T one time from your representation of what's left. So if you have three T in your reducible representation and then you see that T is X, Y, and Z, you have two T left for the vibrations. Does that make sense if it's not clear? Ask me about it. That's something that people got a little bit confused on last time. Like if you have an E representation and you're removing X and Y, that only removes one E from whatever you have left, not two because they are doubly degenerate. Okay. Yes. Oh, okay. There wasn't a question. All right. Okay. So what's left are the vibrations. And you figure out whether they're IR or Raman active by looking at the character table and you see whether there's a component of, if it belongs to the same symmetry species as a component of the dipole moment, that being X, Y, or Z. And if it does, then it's IR active. And if it's Raman active, that means it belongs to the same symmetry species as a component of the polarizability. So something like X, Y, Y, Z, X squared minus Y squared, Z squared, something like that. And of course, the symmetry species can be both or neither or one or the other. So I think we've done plenty of examples of these in class and they've showed up on the other exam. So just make sure you go back and review. And, you know, if you made mistakes, be sure that you understand how to do it. Okay. So that's what we have to say about group theory. We also need to talk about different kinds of spectroscopy. So you should know sort of the big picture of spectroscopy. What are you measuring when we talk about different kinds of spectroscopy? And we've talked about quite a few. So we have electronic spectroscopy. We've got vibrational spectroscopy, IR and Raman. We also have rotational spectroscopy, which could either be, you know, direct rotational spectroscopy or rotational Raman. And you need to know how the mechanisms for these things are different from each other. You know, how you're physically measuring a signal. And we've also talked about NMR, which is different from all of these. And you should also know the relative energies that are involved. So bless you. So if you only have enough energy to excite rotational states, you should know that everything vibrational is in the ground state. But the opposite isn't true, right? If you're exciting vibrational states, then you get the rotational excitations along with it. Okay, again, you should be able to look at the Raman spectrum and know what's different about it than the absorption spectrum. And again, there are two kinds of Raman spectroscopy, rotational and vibrational. If it's not mentioned which kind it is, then it's vibrational. That's the one that's most commonly used. But rotational Raman spectroscopy does come up as well. Okay, so once you have these either vibrational or rotational spectra, we should be able to analyze them. And so remember that if you have a, this one's an IR spectrum. If you have the IR spectrum, you should be able to make some, the correspondence between that spectrum and the energy level diagram. So remember that the peaks in the spectrum correspond to transitions between the levels, you know, not the levels themselves. So given one of these things, like the, either the potential energy diagram or the spectrum, you should be able to draw the other one and say, you know, which levels correspond to what? Also, you should be able to explain sort of general features of the spectrum. You should know about the selection rules, both the gross and specific selection rules for all the kinds of spectroscopy that we've talked about. This is just another picture of what these things look like. So remember IR spectra are sometimes plotted like this with the peaks going down. They're sometimes plotted with the peaks going up. It doesn't matter. It gives you the same information. Okay, so given these kinds of spectra, we should be able to calculate different things from them. So for a simple molecule, we should be able to get a bond length from this. And so things that you need to know include the spacing between the lines for the different rotational states is 2B or it's 4B if you take it across the middle where there's no peak in the center because the J equals 0 to J equals 0 transition is forbidden. So if you do that, it's 4B. And so based on the rotational constant, you should be able to get the bond length using these equations that we've used before. You should also be able to estimate the force constant which is something that we did on the last exam. And for the force constant, you just need the fundamental frequency of the whole thing which of course is the point in the center where there would be a line if the J equals 0 to J equals 0 transition were allowed. And I also want to point out that these things could be in all kinds of crazy units. They could be in wave numbers or frequency or energy or, you know, some combinations of these things. And no matter what it is, you should be able to convert back and forth and use all of these things. It's an important skill because when you read the literature, you'll actually see these things written down in different ways and it's important to be able to convert among them fluently. It would be nice if everything were consistent and in units that make sense but alas, it's not like that. Okay, so from, this is from the practice exam. There are questions like why isn't there a peak in the middle? You know, again, that's because the specific selection rule for rotational transitions is that J has to equal, delta J has to equal plus or minus 1. And getting the energy for the new equals 0 to new equals 1 transition is just, you read off the point in the center of the spectrum where there's a line missing. You know, or if you had a molecule that had a peak there because it has an unpaired electron say, then that would be where that is. Other questions include, you know, is our molecule a perfect rigid rotor and how do you tell? So again, this was on the last exam. If the spacings are really exactly equivalent, then you can say this thing behaves as a perfect rigid rotor and if they're not, if it's stretched on one side and squished on the other, then you know that you have centripetal distortion and it's not a perfect rigid rotor. So what that means is as the molecule rotates really, really quickly, then it starts to stretch out and it doesn't behave as an ideal case. So in these particular set of examples, I would say that CO2 actually looks like a pretty good rigid rotor. The spacings are quite even and N2O really doesn't. Question? Yes? What in the intensity matter also? Do we have to make that the intensity on one side? Well, the intensities, yeah, that's a good point as to whether they're the same on both sides, but the intensities mostly come from the Boltzmann distribution of the populations and we're going to talk about that a little bit. Yes? So can we say that CO2 is a perfect rigid rotor? Or in this case it's still not a perfect rigid rotor? If you said that CO2 is a perfect rigid rotor, for this one, I mean it can be hard to tell, but for this one, it does look like the line spacings are really even and particularly, you know, maybe a better question would be compare them. Which one is a better rigid rotor? And then, you know, then it's clearly CO2. Okay, so back to the question of the line intensities. If we say these spectra were collected at room temperature and then we lower the temperature to 10 Kelvin, how would they change? How would they look different? And in that case, we would see more intensity in the lower transitions and also the distribution would sharpen up because we're filling fewer states. There are just fewer states available at lower energy. And so we've also seen this picture before. Here's the, how the populations look different at low temperature versus at high temperature. So notice that the scales on these are different. It's a little bit hard to see, but it goes from, you know, 10 to 100 gigahertz on the top and 0 to 1,000 on the bottom. So that's just showing you that at higher temperature, there are many, many, many more states that can be populated than at low temperature. And also, we see, you know, so at low temperature, we see everything piling up in just a few states. So that's something that you should be able to explain and, you know, be able to sketch what it looks like sort of qualitatively. And so that's most of the effect that accounts for the distribution of the levels. Okay. And we've already talked about Raman spectroscopy. Okay. So we went through that pretty fast, but there's a lot of information in there about vibrational and rotational spectroscopy. So that's something to spend some time reviewing. Look at your exams from this corner and make sure that you know how to do the problems that were there. You know, also look at the practice exams from a couple of years ago that I posted and make sure that you know how to do those problems. And then we get to electronic spectroscopy. So make sure that you know how to write term symbols. So here are the rules for term symbols for atoms again. Probably not going to have a question directly about term symbols for atoms because I know that you covered it last quarter. What we're more going to be concerned about is the term symbols for diatomic molecules. But of course you have to understand how to do the ones for atoms in order to be able to do that. It's also important to remember Hun's rules in determining which of these states are lower energy. So a lot of times for a particular electron configuration, the electron configuration itself will be ambiguous. You can get different arrangements of electrons for the same electron configuration, which is of course why we need term symbols in the first place. They're a lot more specific than just the electron configuration. And so you should be able to use this to figure out which one is the ground state. And then the part that you're actually going to have to do is figuring this out for diatomic molecules. And so this will be pretty similar to what we did on the last exam. You get some diatomic molecule. You have to draw the molecular orbital diagram and figure out the properties of these electrons and see, you know, how many electron configurations you can get out of a particular, well, how many arrangements you can get out of a particular electron configuration and then write down the term symbol. So again, make sure that you know how to do the example from the last exam. I think the TAs did a really excellent job of this in the review session last time. So look over your notes from that. And also remember things like the even odd rule and, you know, how to determine whether particular transitions are allowed or not by symmetry. And so for electronic spectroscopy, there are going to be two considerations for transitions. So one is just are the transitions allowed or not by symmetry and that's something that you get by looking at the symmetry of the wave function. You have to take into account, you know, both G and U and if it's a sigma term plus and minus. And then the other thing that we have to remember is Frank Condon factors. So you should know how to write down an expression for the Frank Condon factor between pairs of states. Again, you're probably not going to need to evaluate that because you don't have time to do hard integrals during the time of the exam and honestly you're not going to have a lot of extra time to do much of anything because it's going to be long. So just like the midterms have been. It won't be twice as long and you have twice as much time. So that's a little bit better. But, you know, just I want to take this point to say make sure that you read the directions really carefully because there will be things where I'm trying to save you time by either giving you an intermediate step or saying only do this part of it. Just make sure you read them really carefully and if you're confused about it ask. I, you know, I do stay here for the exam and run around and answer questions and, you know, if you ask something that is information you should already know I'll just say sorry you already need to know that. There's no harm in asking though. Question? Is it possible to provide us with like a practice final so we can just know the format? Well, you kind of do know the format. So I've already given you two practice midterms and then we just have the stat mech questions from, you know, that the quiz kind of serves as that and I gave you a bunch of practice homework problems. So I think by now you do pretty much know what the format is going to be like and you know sort of how I ask questions. So I'm not going to give you a separate practice final but I do think you have a lot of practice problems that you can work with. So, yeah, read the directions very carefully. Ask if you don't understand. And the other thing is when you get the exam, you know, take a deep breath and read the whole thing and make sure that you do the easiest problems first because what's easy and what's hard is a matter of opinion. Some people understand some concepts more readily than others and so I want everybody to do their very best. So make sure that you do the ones that you think you can do really quickly first and then go back and work on the things where maybe you need more time. Because I don't want people to get into a situation where you spend all your time on something that's really hard and then you find that, you know, oh, there was an easy one that you could have done quickly. So just some general exam strategy. Okay, so we definitely need to know about selection rules and this slide on selection rules is really general. This could be for just about anything. So it depends on the, the selection rules depend on a transition dipole and there are different ways to look at this. Sometimes you can just do it by inspection basically like if you have say the harmonic oscillator wave functions and you can just look at whether they're even and odd and remember that the transition dipole, you know, the dipole moment operator is always odd. You know, it either goes as X, Y or Z. And then if you can't just do it by looking at it and say, okay, they're even and odd, then you need to do it by looking at the character table. And so in that case, you need to find the symmetry species of each function and then multiply the characters for all three of them together and then you'll get some reducible representation in general. Sometimes you're lucky and you get something that looks like a reducible, an irreducible representation that's already on the character table and you can just look at it. But sometimes you'll get something that's a reducible representation. And then what you need to do with that reducible representation is see if there's a component of A1 in it or, you know, if it's not called A1 in that point group, whatever the symmetry species is that has 1s under every operation. So again, that's the one that's invariant to all transformations. And so what we want to see there is that if you have things that actually overlap and are coupled by that dipole moment operator, then they're going to overlap no matter how you move the thing around in space. So we just have to make sure that there's a component of the symmetry species that's invariant to all transformations in order to say whether that exists or not. Question? So if it has A1 then the integral does not vanish? That's right. That means that it has a component that's invariant to all transformations and the integral does not vanish and you get an answer. Another mistake that I've seen people make with us on the previous exams is that that doesn't mean that the overlap is 1. So you can tell whether it's 0 really easily from this treatment. But you don't know what the value is. So all you can say is it's not 0. So, you know, if people put it's equal to 1 for these things, you got some partial credit but be careful. You know, you don't actually know what the value is just from the symmetry treatment. It could be really tiny. It could be 1. You don't know. You have to do some more sophisticated computation to be able to get that. Okay. So continuing with electronic spectroscopy, we looked at birds sponer plots and how these compare to the potentials associated with electronic spectroscopy. We've seen a couple of different examples of this. There was an example on the practice exam that I gave you. The exam on the actual test was maybe a little bit harder because it was a weird example where the thing had a break in the slope. So, you know, again, just make sure you read the directions and look at what the question is actually asking and know your equations for how this plot pertains to the potential energy diagram of the molecule. Okay. So that's what we have done with electronic spectroscopy. You need to know how to write your term symbols and be able to figure out which transitions are allowed. You need to figure out the Frank Condon factors and also be able to use some of these plots to find out some properties of the molecules. We've also talked about NMR and that's a little bit different from these other types of spectroscopy but what you need to know about it is kind of similar. So there are both sort of theoretical things that we need to understand about it and then also in a practical sense being able to look at the spectrum and learn something about the molecule or look at the molecule and figure out what its spectrum is going to look like. So important things to know here is how the Zeeman effect works. We have our nuclear spins. They're in all kinds of different states. If there's no magnetic field, they're all equivalent in energy. If we put the magnetic field on, that degeneracy is broken and we've got a couple of states. For a spin one-half, we can have plus or minus a half that corresponds to spin up or spin down. And in the spin one-half case, these states have nicknames. You can call them alpha and beta. If it's not a spin one-half, then you have to use their full names. You have to write a ket that has, you know, so for instance, for spin one, you'd have 0, 1, sorry, 1, 1, 1, 0, 1 minus 1. And, you know, you can do this for any kind of nucleus. It's just, you have to remember that whatever the value of I is, you go from plus I to minus I in increments of 1. So here that is written down. Our z component of the angular momentum goes from plus I to minus I in increments of 1. And there are a few spin operators that we learned how to use in the context of NMR. And we talked about how you can use these things to generate pulse sequences and flip the spins. We're not going to get into, you know, I'm not going to expect you to know too much of that, but as far as how you use it at this point, I just wanted to introduce you to it. But you do need to know how to use a couple of these spin operators that we've talked about. So one is IZ. So when we, these states that we're looking at in the Zamen basis are the eigenstates of IZ. And so you should know that when you operate IZ on a state, you get its L value back. That's the eigenvalue. And then the eigenstate is that same ket. So for a spin 1 half case, if you operate IZ on alpha, you get 1 half alpha. And if you operate IZ on beta, you get minus 1 half beta. But again, the equation at the top here is the general definition of IZ and you should be able to apply that to any I value for your spin. And you should also be able to write a matrix representation of something like IZ in the Zamen basis. And so you do that by generating each of the matrix elements. And so the important things to know here is first of all, how to operate IZ on the states. And you do these things from right to left. So you operate IZ on the ket first. And then you take the overlap integral of whatever is left. And these things make up an orthonormal basis. So if the states are the same, the value is 1. If they're different, it's 0. And so you should be able to use that to generate a matrix representation for something like IZ. We also learned how to use the raising and lowering operators. So here this is just written down for the spin 1 half case. So if you operate I plus on alpha, you can't raise alpha anymore so you get 0. If you operate I plus on beta, you get alpha. I would recommend checking out, you know, on your exam, look at the general definition of the raising and lowering operators. So we mostly talked about this in the spin 1 half case. But on your exam, there's a general definition of them. And you should check that out and make sure that you know how to use it. And so again, we can write down the matrix representations of these things because we know how to operate the operators on the states and then we can take the overlap integrals of the states with each other. And so you should know how to write down these matrix representations. And when you do this on the exam, you should definitely show your work. If you don't want to write out all the matrix elements, at least write out a couple of them so that you show you know how to do it. So if you just write the answer, I'm going to assume that you put it on your cheat sheet and wrote it down and you won't get full credit. You get some credit, but you have to show some work or explain your rationale to get all of it. Okay, IX and IY can also be written in terms of the raising and lowering operators. This is something that you should know about the actual derivation of how you get this. We did in the homework and it's important. It's probably too long to do on the exam, but you should remember what these are. Okay, so now we get to what to the spectra actually look like. So that's something that you have a lot of experience here from organic chemistry to build on. The rules are the same as far as what the spectra look like and you already know a lot of that or you already did know a lot of it before starting this class. And the difference is now you know how it works or why the spectra look like that. So this is something that you should be able to do. If you have a molecule, you should be able to generate its NMR spectrum. And you know that should be true for any kind of nucleus that we want to talk about. The same principles apply. So whether it's proton C13, you know 31P, you know anything like this you should be able to generate what the spectra kind of look like. Same as last time, I'll give you a chemical shift table. So you just have to figure out what functional group is what and put things in the right general places. I'm not really worried about people memorizing the chemical shifts of different things. If you either become a synthetic chemist or an NMR spectroscopist and you really need to work with this all the time, you'll definitely remember it then. But for now you just need to be able to use the table. You should also be able to generate the coupling patterns for J couplings and you should be able to explain where these come from in a physical sense. And you know you should be able to draw spectra for different kinds of molecules. Basically putting everything in the right general location and figuring out the J coupling pattern. And you know again you should know what those things look like with and without decoupling a various nuclei. And then the last thing we talked about is statistical mechanics. And so we don't have time to review all of that because you know we're almost out of time and also we just went over it. But important things that you should be able to know is how to set up your partition function for an ensemble of molecules. You should be able to think about the most probable configurations of various states. So and for instance you should know that you don't pile everything into the ground state necessarily because it often doesn't have any degeneracy whereas higher states do have multiple ways to get the same configuration. You should be comfortable with how these configurations are written down and with Boltzmann distributions and how we get the relative populations of the states. These are all kind of good things to write down on your cheat sheet. You should know how to find the relative populations of two states or the population of a particular state relative to the whole ensemble. And you should also be able to write partition functions for various things. So here's a general case of a partition function and also how you write that down, how you write the relative populations in terms of that. And you should be able to do this. We've talked about some specific examples in class. So we've talked about rotational states a lot. So you probably want to know how to do that. We've also talked about vibrational states a bit so that's a good thing to know. We've also talked about the NMR case of this so that's another one where it would be good to know the specifics about that one. There are also these questions where you're given a description of the system in words and you have to write an energy level diagram and write the partition function. And so here's a case where people seem to get confused about this a lot where people look at this and say, well I don't know the value of J and the degeneracy is 2J plus 1 so how do I do that? Remember that's for the case of a rotational spectrum. And so you should know that the degeneracy of a rotational level is 2J plus 1. That's an important thing to know. But don't try to apply it to other cases. So in the case of the vibrational partition function, if we just have a harmonic oscillator which we're going to for anything that we talked about in this class, those states are all non-degenerate, right? You just have your parabolic potential and then you have all these states. So those are non-degenerate. For electronic states it's more complicated. You don't know what it is. But for this kind of a general system, we already told you the degeneracy in the problem. And so that's, you know, so don't get hung up there on, you know, wanting to use these other rules that you know for different systems. If it just says the degeneracy of state 1 is x and it's for state 2 it's y, then just directly write that down and use it and it doesn't matter where it comes from in that case. But you should know how to write these partition functions in a general sense for a system like this if it's described in words. And you should know how that changes with respect to temperature. So if you make the temperature really low or really high, you should know how that affects the relative populations and how it affects the partition function. And I think we are about done. That's what's going to be on the final. So thanks again for a really great class. I really enjoyed it. Thank you. Thank you.
 Good morning.  Welcome to our last PCAM lecture.  So I know I'm going to miss you guys.  This has been a really great class.  I've been really happy about, you know,  how much everybody participates and is really excited  about learning PCAM and that's really cool.  It's been a lot of fun.  So thanks for being such a great class.  Today we're just going to do a review of what's going  to be on the final.  It is completely cumulative.  It covers everything that we've talked about in the course,  which is really a lot of stuff.  So we're just going to go back through and, you know,  not go into anything in too much depth, but talk  about everything that's going to be on there.  And if we run out of time, which we might,  the slides are posted and I'm going to have a lot  of office hours next week.  I still don't have it posted when I'm going to do it,  but I'm planning to have, you know, at least one every day,  Monday through Thursday, possibly more.  It just depends how I can work out the schedule.  Just a quick poll.  Who has a final Monday morning?  Who has a final Monday afternoon?  Tuesday morning.  Tuesday afternoon.  Okay, looks like Tuesday is a good day.  How about Wednesday morning?  Wednesday afternoon?  Thursday morning?  Thursday afternoon.  Okay, that's unfortunate.  That's a bad schedule.  Okay. Yeah, well, I figure, you know,  people are mostly taking the same classes, so I'm trying  to avoid scheduling the office hours  when a lot of people won't be able to make it.  Okay, so it looks like Tuesday is a pretty good day  if I'm going to do extra ones.  And then I'll still do, you know, a bunch  of last-minute stuff Thursday, but it's too bad  that a lot of people will have a final.  When is it over?  Six. Ouch.  Okay, we'll see what we can do.  I have to check the schedule.  Okay, another thing I want to mention before we get started is  that a lot of people have sent me emails  about the seminar extra credit sheets and exam regrades  and people are getting anxious  because I haven't worked on them yet.  So I was out of town the last three days, and I did not have a  lot of internet access.  So I could see emails on my phone sort of when the plane landed  and whatever, but, you know, I've been traveling a lot and  or I was in an eight-hour meeting reviewing grant proposals.  So I just haven't had a lot of internet access  to upload stuff.  So I totally get it.  I know that it's anxiety-provoking that you turned  in your stuff and you don't know whether you got credit  for it or not.  The issue is I'm just behind.  So when I started doing this extra credit thing  with the seminars, I didn't know everybody was going to do it.  And that's awesome.  I'm glad that you're doing it.  But it means that, you know, whenever you give me a piece  of paper, it turns into a Russian novel when everybody does  and I am just behind.  So one of my plans for the weekend is to get caught  up on the stuff.  When I get all the seminar extra credit things done,  I'm going to post something.  I'll post it on the Facebook page and the class website.  And, you know, I'll say, okay, I think I have all of them done now.  And at that point, if I still didn't get yours,  then please do send me an email and I'll go look  through the stack again.  Same thing with the exam regrades.  The deadline to ask me about it is tonight at midnight.  I'm going to wait until I have all of them and then just do it  all at once.  That'll make sure that it's, you know,  that I'm definitely doing it consistently  and also hopefully get it done quickly.  And again, I will post when I think I have them all done  and then if I missed yours, go ahead and let me know at that time.  I know it's tough not to know what's going on and I'll try  to get caught up as soon as I can.  All right.  Does anybody have any more questions about general stuff  before we start reviewing for the exam?  Okay. Let's do it.  All right.  So as I said, this exam is really cumulative.  And before we get into the older material,  I want to just talk  about the canonical ensemble a little bit.  So we got about this far, you know, talking about the,  the connections between the canonical ensemble  and just the standard partition function that we've looked at.  But we didn't quite get to finish up and then in between,  we had example day which, by the way,  I heard John Mark's lecture debut was awesome.  So that's great.  I'm not surprised but I'm glad it went really well.  Okay. So we have, we learned about the canonical distribution  and the canonical partition function.  So this is for our canonical ensemble which,  remember, is a collection of little individual ensembles  that are all at the same temperature.  And the thing that we like to use this for, or its key feature,  is the fact that the canonical partition function is more  general than our normal partition function.  And that's because it doesn't assume  that all the particles are independent.  And so that is really useful when we want to use it  for studying condensed phases, so liquids and solids,  or even gases that don't behave ideally.  So it's a lot more general and it can be used for more things.  And, you know, obviously we're about at a time this quarter,  but so we're not going to do too much with this,  but I want to make sure that we cover it to set you  up for next quarter.  So next quarter with Dr. Gerber, you're going to do a lot  of working with the canonical ensemble, statistical mechanics,  and get into thermodynamic properties.  So the last thing that we need to talk about is the fact  that you can get bulk properties of the system  from the partition function.  So the average energy of one  of our little member ensembles is, it's given by, you know,  we've got the, it's just the average energy  for our individual ones.  And we can write that down in terms  of the relative populations, which,  again, we remember what that is.  And what would be really nice is to have this in terms  of just Q, because then it's a lot more useful.  And so we can substitute using the derivative of Q  with respect to beta.  I remember beta is 1 over kT.  And we know that that equals dLn Q d beta.  And so the result we get is for distinguishable molecules,  big Q is little Q to the n.  So distinguishable molecules could be, it could be, you know,  molecules that are in a crystal lattice  so that they occupy a particular position all the time.  So it could be that they're all the same,  but they're in a particular point in space.  And so you can distinguish them that way.  It could be that they're all the same,  that they're not all the same type of molecule.  So you might have a solution of, say, ethanol and water.  So they're moving around, but some  of the molecules are distinguishable  because they're different molecules.  And in that case, Q is little Q to the n over n factorial.  And so this is stuff that I basically just want you to hold  in your mind for next quarter,  for when you work on thermodynamics with Dr. Gerber.  It would have been ideal if we had time to get to it  in the last lecture, but we didn't quite.  So that is what we're going to say about statistical mechanics.  And with that, let's move on to the review for the final.  Okay, so what do you need to know?  So the first thing is being able to assign molecules  to a point group.  This is really important because there are lots of types  of problems where you have to assign stuff  to the right point group in order to get the right answer.  And that will be the key to the whole thing.  There might be, you know, maybe one problem that's not worth  very much that says, you know, just assign something  to a point group, but most of what's going to be going  on is just having to use this information to learn something  about bonding or molecular motion or, you know,  whether certain kinds of orbitals can overlap  or whether certain kinds of wave functions can overlap.  And this is something that you definitely need to review  if you had trouble with it or even if you didn't have trouble  with it and you just haven't looked at it in a little while.  It's an important skill.  And so remember things like, you know, we've talked  about different objects and how they transform under the  operations in a point group.  And so, you know, I bring up this OCL2 example again  that we saw before in class where depending on whether the phase  of the orbitals is in phase or out of phase,  they have different behavior with respect to the operations  and so you get different matrices for those operations.  This is something that you definitely need to be able to do.  And hopefully you see a little bit more why it's important now  that we've talked about the rotation statistics  of things like molecules that have fermions versus bosons  where you might have them transforming differently  under this rotation operation.  So this is something that you should definitely be able to do.  So with a basis set that's described in words, you should be  able to figure out, you know, how to draw it  and write down appropriate matrices for these operations.  And so again, depending on whether the orbitals are out of phase  or in phase, in this case, you get different answers.  Another thing that's important to point out is  that in this particular case, we talked about the basis set being  the p orbitals.  So you treat them separately.  We've also seen other examples where the basis set is a molecular  orbital consisting of a linear combination of those p orbitals.  That's a little bit of a subtle difference,  but you definitely have to pay attention to it.  So if our basis set is the linear combination of the orbitals,  then you treat it as just one thing and, you know,  how you count what happens with the operations is a little different.  So keep that in mind.  We also need to do things like looking at the molecular motion  and determining which vibrational modes are IR and Raman active.  So remember the general procedure for doing this.  So we have, you know, some molecule we want to learn  about its IR and Raman active vibrational modes.  The first thing to do is set up your basis, which is going  to be X, Y, and Z unit vectors on each atom.  Mistakes that I saw people make on the first couple of exams included,  you know, not putting a basis set on the central atom  and just doing the outer ones.  That's, you know, make sure you don't make that mistake again.  The issue is we're interested in the relative displacements  of all of the atoms, and so we have to include all of them.  So of course getting the molecule  into the right point group is an important part of this.  You need to be able to set up your basis and then look at this  and determine whether you can use the shortcut or not  to just get the character.  I'm probably not mean enough to give you one where you can't  in the context of a molecular vibration problem.  Maybe in some other context I might, but in this case it's  probably too long.  And then, you know, be able to write  down your reducible representation representing the  molecular motion and reduce it to get the modes.  And so again, we should have nine, we have nine elements  in the basis because we have three unit vectors  and that's going to be three times the number of atoms  in the molecule.  So that's a good way to check yourself.  You should get nine symmetry species  in the final answer then.  And then what we have to do is go through  and take out the translations and rotations  because those are something that we don't see  in vibrational spectroscopy.  But they're accounting for some of those symmetry species.  And so you do this by looking at the character table  and finding the symmetry species that correspond to X, Y,  and Z and then RX, Ry, and RZ.  And then the vibrational modes are whatever is left over.  So here's another note about this.  So in this case we only have symmetry species that belong  to A and B type representations.  That means they're non-degenerate.  When you have things like E which is doubly degenerate  and T which is triply degenerate.  If you have say a T something symmetry species and X, Y,  and Z all belong to it, that takes out the T one time  from your representation of what's left.  So if you have three T in your reducible representation  and then you see that T is X, Y, and Z,  you have two T left for the vibrations.  Does that make sense if it's not clear?  Ask me about it.  That's something that people got a little bit confused  on last time.  Like if you have an E representation  and you're removing X and Y, that only removes one E  from whatever you have left, not two  because they are doubly degenerate.  Okay. Yes.  Oh, okay.  There wasn't a question.  All right.  Okay. So what's left are the vibrations.  And you figure out whether they're IR or Raman active  by looking at the character table and you see whether there's  a component of, if it belongs to the same symmetry species  as a component of the dipole moment, that being X, Y, or Z.  And if it does, then it's IR active.  And if it's Raman active, that means it belongs  to the same symmetry species as a component  of the polarizability.  So something like X, Y, Y, Z, X squared minus Y squared,  Z squared, something like that.  And of course, the symmetry species can be both  or neither or one or the other.  So I think we've done plenty of examples of these in class  and they've showed up on the other exam.  So just make sure you go back and review.  And, you know, if you made mistakes, be sure  that you understand how to do it.  Okay. So that's what we have to say about group theory.  We also need to talk about different kinds of spectroscopy.  So you should know sort of the big picture of spectroscopy.  What are you measuring when we talk about different kinds  of spectroscopy?  And we've talked about quite a few.  So we have electronic spectroscopy.  We've got vibrational spectroscopy, IR and Raman.  We also have rotational spectroscopy,  which could either be, you know,  direct rotational spectroscopy or rotational Raman.  And you need to know how the mechanisms  for these things are different from each other.  You know, how you're physically measuring a signal.  And we've also talked about NMR, which is different  from all of these.  And you should also know the relative energies  that are involved.  So bless you.  So if you only have enough energy  to excite rotational states, you should know  that everything vibrational is in the ground state.  But the opposite isn't true, right?  If you're exciting vibrational states,  then you get the rotational excitations along with it.  Okay, again, you should be able to look  at the Raman spectrum and know what's different  about it than the absorption spectrum.  And again, there are two kinds  of Raman spectroscopy, rotational and vibrational.  If it's not mentioned which kind it is,  then it's vibrational.  That's the one that's most commonly used.  But rotational Raman spectroscopy does come up as well.  Okay, so once you have these either vibrational  or rotational spectra, we should be able to analyze them.  And so remember that if you have a,  this one's an IR spectrum.  If you have the IR spectrum, you should be able  to make some, the correspondence between that spectrum  and the energy level diagram.  So remember that the peaks in the spectrum correspond  to transitions between the levels, you know,  not the levels themselves.  So given one of these things, like the,  either the potential energy diagram or the spectrum,  you should be able to draw the other one and say, you know,  which levels correspond to what?  Also, you should be able to explain sort of general features  of the spectrum.  You should know about the selection rules,  both the gross and specific selection rules for all the  kinds of spectroscopy that we've talked about.  This is just another picture of what these things look like.  So remember IR spectra are sometimes plotted like this  with the peaks going down.  They're sometimes plotted with the peaks going up.  It doesn't matter.  It gives you the same information.  Okay, so given these kinds of spectra, we should be able  to calculate different things from them.  So for a simple molecule, we should be able to get a bond  length from this.  And so things that you need to know include the spacing  between the lines for the different rotational states is  2B or it's 4B if you take it across the middle  where there's no peak in the center because the J equals 0  to J equals 0 transition is forbidden.  So if you do that, it's 4B.  And so based on the rotational constant, you should be able  to get the bond length using these equations  that we've used before.  You should also be able to estimate the force constant  which is something that we did on the last exam.  And for the force constant, you just need the fundamental  frequency of the whole thing which of course is the point  in the center where there would be a line if the J equals 0  to J equals 0 transition were allowed.  And I also want to point out that these things could be  in all kinds of crazy units.  They could be in wave numbers or frequency or energy or,  you know, some combinations of these things.  And no matter what it is, you should be able to convert back  and forth and use all of these things.  It's an important skill because when you read the literature,  you'll actually see these things written down in different ways  and it's important to be able to convert among them fluently.  It would be nice if everything were consistent  and in units that make sense but alas, it's not like that.  Okay, so from, this is from the practice exam.  There are questions like why isn't there a peak in the middle?  You know, again, that's because the specific selection rule  for rotational transitions is that J has to equal,  delta J has to equal plus or minus 1.  And getting the energy for the new equals 0  to new equals 1 transition is just, you read off the point  in the center of the spectrum where there's a line missing.  You know, or if you had a molecule that had a peak there  because it has an unpaired electron say,  then that would be where that is.  Other questions include, you know,  is our molecule a perfect rigid rotor and how do you tell?  So again, this was on the last exam.  If the spacings are really exactly equivalent,  then you can say this thing behaves as a perfect rigid rotor  and if they're not, if it's stretched on one side  and squished on the other, then you know  that you have centripetal distortion  and it's not a perfect rigid rotor.  So what that means is as the molecule rotates really,  really quickly, then it starts to stretch out  and it doesn't behave as an ideal case.  So in these particular set of examples, I would say  that CO2 actually looks like a pretty good rigid rotor.  The spacings are quite even and N2O really doesn't.  Question?  Yes?  What in the intensity matter also?  Do we have to make that the intensity on one side?  Well, the intensities, yeah, that's a good point  as to whether they're the same on both sides,  but the intensities mostly come  from the Boltzmann distribution of the populations  and we're going to talk about that a little bit.  Yes?  So can we say that CO2 is a perfect rigid rotor?  Or in this case it's still not a perfect rigid rotor?  If you said that CO2 is a perfect rigid rotor,  for this one, I mean it can be hard to tell, but for this one,  it does look like the line spacings are really even  and particularly, you know,  maybe a better question would be compare them.  Which one is a better rigid rotor?  And then, you know, then it's clearly CO2.  Okay, so back to the question of the line intensities.  If we say these spectra were collected at room temperature  and then we lower the temperature to 10 Kelvin,  how would they change?  How would they look different?  And in that case, we would see more intensity  in the lower transitions and also the distribution would  sharpen up because we're filling fewer states.  There are just fewer states available at lower energy.  And so we've also seen this picture before.  Here's the, how the populations look different  at low temperature versus at high temperature.  So notice that the scales on these are different.  It's a little bit hard to see, but it goes from, you know,  10 to 100 gigahertz on the top and 0 to 1,000 on the bottom.  So that's just showing you that at higher temperature,  there are many, many, many more states that can be populated  than at low temperature.  And also, we see, you know, so at low temperature,  we see everything piling up in just a few states.  So that's something that you should be able to explain  and, you know, be able to sketch what it looks  like sort of qualitatively.  And so that's most of the effect that accounts  for the distribution of the levels.  Okay. And we've already talked about Raman spectroscopy.  Okay. So we went through that pretty fast, but there's a lot  of information in there about vibrational  and rotational spectroscopy.  So that's something to spend some time reviewing.  Look at your exams from this corner and make sure  that you know how to do the problems that were there.  You know, also look at the practice exams from a couple  of years ago that I posted and make sure  that you know how to do those problems.  And then we get to electronic spectroscopy.  So make sure that you know how to write term symbols.  So here are the rules for term symbols for atoms again.  Probably not going to have a question directly  about term symbols for atoms because I know  that you covered it last quarter.  What we're more going to be concerned  about is the term symbols for diatomic molecules.  But of course you have to understand how to do the ones  for atoms in order to be able to do that.  It's also important to remember Hun's rules  in determining which of these states are lower energy.  So a lot of times for a particular electron configuration,  the electron configuration itself will be ambiguous.  You can get different arrangements of electrons  for the same electron configuration,  which is of course why we need term symbols  in the first place.  They're a lot more specific  than just the electron configuration.  And so you should be able to use this to figure  out which one is the ground state.  And then the part that you're actually going to have  to do is figuring this out for diatomic molecules.  And so this will be pretty similar to what we did  on the last exam.  You get some diatomic molecule.  You have to draw the molecular orbital diagram  and figure out the properties of these electrons  and see, you know, how many electron configurations you can  get out of a particular, well, how many arrangements you can get  out of a particular electron configuration  and then write down the term symbol.  So again, make sure that you know how to do the example  from the last exam.  I think the TAs did a really excellent job of this  in the review session last time.  So look over your notes from that.  And also remember things like the even odd rule and, you know,  how to determine whether particular transitions are allowed  or not by symmetry.  And so for electronic spectroscopy,  there are going to be two considerations for transitions.  So one is just are the transitions allowed or not  by symmetry and that's something that you get by looking  at the symmetry of the wave function.  You have to take into account, you know, both G and U  and if it's a sigma term plus and minus.  And then the other thing that we have  to remember is Frank Condon factors.  So you should know how to write down an expression  for the Frank Condon factor between pairs of states.  Again, you're probably not going to need to evaluate that  because you don't have time to do hard integrals during the time  of the exam and honestly you're not going to have a lot  of extra time to do much of anything  because it's going to be long.  So just like the midterms have been.  It won't be twice as long and you have twice as much time.  So that's a little bit better.  But, you know, just I want to take this point to say make sure  that you read the directions really carefully  because there will be things where I'm trying to save you time  by either giving you an intermediate step  or saying only do this part of it.  Just make sure you read them really carefully  and if you're confused about it ask.  I, you know, I do stay here for the exam and run around  and answer questions and, you know,  if you ask something that is information you should already  know I'll just say sorry you already need to know that.  There's no harm in asking though.  Question?  Is it possible to provide us with like a practice final  so we can just know the format?  Well, you kind of do know the format.  So I've already given you two practice midterms  and then we just have the stat mech questions from, you know,  that the quiz kind of serves as that and I gave you a bunch  of practice homework problems.  So I think by now you do pretty much know what the format is  going to be like and you know sort of how I ask questions.  So I'm not going to give you a separate practice final  but I do think you have a lot of practice problems  that you can work with.  So, yeah, read the directions very carefully.  Ask if you don't understand.  And the other thing is when you get the exam, you know,  take a deep breath and read the whole thing and make sure  that you do the easiest problems first because what's easy  and what's hard is a matter of opinion.  Some people understand some concepts more readily than others  and so I want everybody to do their very best.  So make sure that you do the ones that you think you can do  really quickly first and then go back and work  on the things where maybe you need more time.  Because I don't want people to get into a situation  where you spend all your time on something that's really hard  and then you find that, you know, oh, there was an easy one  that you could have done quickly.  So just some general exam strategy.  Okay, so we definitely need to know about selection rules  and this slide on selection rules is really general.  This could be for just about anything.  So it depends on the, the selection rules depend  on a transition dipole and there are different ways  to look at this.  Sometimes you can just do it by inspection basically  like if you have say the harmonic oscillator wave functions  and you can just look at whether they're even and odd  and remember that the transition dipole,  you know, the dipole moment operator is always odd.  You know, it either goes as X, Y or Z. And then if you can't just  do it by looking at it and say, okay, they're even and odd,  then you need to do it by looking at the character table.  And so in that case, you need to find the symmetry species  of each function and then multiply the characters  for all three of them together  and then you'll get some reducible representation in general.  Sometimes you're lucky and you get something that looks  like a reducible, an irreducible representation that's already  on the character table and you can just look at it.  But sometimes you'll get something  that's a reducible representation.  And then what you need to do  with that reducible representation is see if there's a  component of A1 in it or, you know, if it's not called A1  in that point group, whatever the symmetry species is  that has 1s under every operation.  So again, that's the one that's invariant  to all transformations.  And so what we want to see there is that if you have things  that actually overlap and are coupled  by that dipole moment operator, then they're going  to overlap no matter how you move the thing around in space.  So we just have to make sure that there's a component  of the symmetry species that's invariant to all transformations  in order to say whether that exists or not.  Question?  So if it has A1 then the integral does not vanish?  That's right.  That means that it has a component that's invariant  to all transformations and the integral does not vanish  and you get an answer.  Another mistake that I've seen people make with us  on the previous exams is that that doesn't mean  that the overlap is 1.  So you can tell whether it's 0 really easily  from this treatment.  But you don't know what the value is.  So all you can say is it's not 0.  So, you know, if people put it's equal to 1 for these things,  you got some partial credit but be careful.  You know, you don't actually know what the value is just  from the symmetry treatment.  It could be really tiny.  It could be 1.  You don't know.  You have to do some more sophisticated computation  to be able to get that.  Okay. So continuing with electronic spectroscopy,  we looked at birds sponer plots and how these compare  to the potentials associated with electronic spectroscopy.  We've seen a couple of different examples of this.  There was an example on the practice exam that I gave you.  The exam on the actual test was maybe a little bit harder  because it was a weird example where the thing had a break  in the slope.  So, you know, again, just make sure you read the directions  and look at what the question is actually asking  and know your equations for how this plot pertains  to the potential energy diagram of the molecule.  Okay. So that's what we have done  with electronic spectroscopy.  You need to know how to write your term symbols and be able  to figure out which transitions are allowed.  You need to figure out the Frank Condon factors and also be able  to use some of these plots to find out some properties  of the molecules.  We've also talked about NMR and that's a little bit different  from these other types of spectroscopy  but what you need to know about it is kind of similar.  So there are both sort of theoretical things  that we need to understand about it and then also  in a practical sense being able to look at the spectrum  and learn something about the molecule or look at the molecule  and figure out what its spectrum is going to look like.  So important things to know here is how the Zeeman effect works.  We have our nuclear spins.  They're in all kinds of different states.  If there's no magnetic field, they're all equivalent in energy.  If we put the magnetic field on, that degeneracy is broken  and we've got a couple of states.  For a spin one-half, we can have plus or minus a half  that corresponds to spin up or spin down.  And in the spin one-half case, these states have nicknames.  You can call them alpha and beta.  If it's not a spin one-half, then you have  to use their full names.  You have to write a ket that has, you know, so for instance,  for spin one, you'd have 0, 1, sorry, 1, 1, 1, 0, 1 minus 1.  And, you know, you can do this for any kind of nucleus.  It's just, you have to remember that whatever the value of I is,  you go from plus I to minus I in increments of 1.  So here that is written down.  Our z component of the angular momentum goes  from plus I to minus I in increments of 1.  And there are a few spin operators that we learned how  to use in the context of NMR.  And we talked about how you can use these things  to generate pulse sequences and flip the spins.  We're not going to get into, you know, I'm not going to expect you  to know too much of that, but as far as how you use it  at this point, I just wanted to introduce you to it.  But you do need to know how to use a couple  of these spin operators that we've talked about.  So one is IZ.  So when we, these states that we're looking at in the Zamen basis  are the eigenstates of IZ.  And so you should know that when you operate IZ on a state,  you get its L value back.  That's the eigenvalue.  And then the eigenstate is that same ket.  So for a spin 1 half case, if you operate IZ on alpha,  you get 1 half alpha.  And if you operate IZ on beta, you get minus 1 half beta.  But again, the equation at the top here is the general definition  of IZ and you should be able to apply that to any I value  for your spin.  And you should also be able to write a matrix representation  of something like IZ in the Zamen basis.  And so you do that by generating each of the matrix elements.  And so the important things to know here is first of all,  how to operate IZ on the states.  And you do these things from right to left.  So you operate IZ on the ket first.  And then you take the overlap integral of whatever is left.  And these things make up an orthonormal basis.  So if the states are the same, the value is 1.  If they're different, it's 0.  And so you should be able to use that to generate a matrix  representation for something like IZ.  We also learned how to use the raising and lowering operators.  So here this is just written down for the spin 1 half case.  So if you operate I plus on alpha, you can't raise alpha anymore  so you get 0.  If you operate I plus on beta, you get alpha.  I would recommend checking out, you know, on your exam,  look at the general definition of the raising  and lowering operators.  So we mostly talked about this in the spin 1 half case.  But on your exam, there's a general definition of them.  And you should check that out and make sure  that you know how to use it.  And so again, we can write down the matrix representations  of these things because we know how to operate the operators  on the states and then we can take the overlap integrals  of the states with each other.  And so you should know how  to write down these matrix representations.  And when you do this on the exam,  you should definitely show your work.  If you don't want to write out all the matrix elements,  at least write out a couple of them  so that you show you know how to do it.  So if you just write the answer, I'm going to assume  that you put it on your cheat sheet and wrote it down  and you won't get full credit.  You get some credit, but you have to show some work  or explain your rationale to get all of it.  Okay, IX and IY can also be written in terms of the raising  and lowering operators.  This is something that you should know  about the actual derivation of how you get this.  We did in the homework and it's important.  It's probably too long to do on the exam,  but you should remember what these are.  Okay, so now we get to what  to the spectra actually look like.  So that's something that you have a lot of experience here  from organic chemistry to build on.  The rules are the same as far as what the spectra look like  and you already know a lot of that or you already did know a  lot of it before starting this class.  And the difference is now you know how it works  or why the spectra look like that.  So this is something that you should be able to do.  If you have a molecule, you should be able  to generate its NMR spectrum.  And you know that should be true for any kind of nucleus  that we want to talk about.  The same principles apply.  So whether it's proton C13, you know 31P, you know anything  like this you should be able  to generate what the spectra kind of look like.  Same as last time, I'll give you a chemical shift table.  So you just have to figure  out what functional group is what and put things  in the right general places.  I'm not really worried about people memorizing the chemical  shifts of different things.  If you either become a synthetic chemist  or an NMR spectroscopist and you really need to work  with this all the time, you'll definitely remember it then.  But for now you just need to be able to use the table.  You should also be able to generate the coupling patterns  for J couplings and you should be able to explain  where these come from in a physical sense.  And you know you should be able to draw spectra  for different kinds of molecules.  Basically putting everything in the right general location  and figuring out the J coupling pattern.  And you know again you should know what those things look  like with and without decoupling a various nuclei.  And then the last thing we talked  about is statistical mechanics.  And so we don't have time to review all of that because you  know we're almost out of time and also we just went over it.  But important things that you should be able to know is how  to set up your partition function for an ensemble  of molecules.  You should be able to think  about the most probable configurations of various states.  So and for instance you should know  that you don't pile everything  into the ground state necessarily  because it often doesn't have any degeneracy  whereas higher states do have multiple ways  to get the same configuration.  You should be comfortable with how these configurations are  written down and with Boltzmann distributions  and how we get the relative populations of the states.  These are all kind of good things to write  down on your cheat sheet.  You should know how to find the relative populations  of two states or the population  of a particular state relative to the whole ensemble.  And you should also be able  to write partition functions for various things.  So here's a general case of a partition function  and also how you write that down,  how you write the relative populations in terms of that.  And you should be able to do this.  We've talked about some specific examples in class.  So we've talked about rotational states a lot.  So you probably want to know how to do that.  We've also talked about vibrational states a bit  so that's a good thing to know.  We've also talked about the NMR case of this  so that's another one where it would be good  to know the specifics about that one.  There are also these questions where you're given a description  of the system in words and you have to write an energy level  diagram and write the partition function.  And so here's a case where people seem to get confused  about this a lot where people look at this and say,  well I don't know the value of J  and the degeneracy is 2J plus 1 so how do I do that?  Remember that's for the case of a rotational spectrum.  And so you should know that the degeneracy  of a rotational level is 2J plus 1.  That's an important thing to know.  But don't try to apply it to other cases.  So in the case of the vibrational partition function,  if we just have a harmonic oscillator which we're going  to for anything that we talked about in this class,  those states are all non-degenerate, right?  You just have your parabolic potential  and then you have all these states.  So those are non-degenerate.  For electronic states it's more complicated.  You don't know what it is.  But for this kind of a general system,  we already told you the degeneracy in the problem.  And so that's, you know, so don't get hung up there on,  you know, wanting to use these other rules  that you know for different systems.  If it just says the degeneracy of state 1 is x  and it's for state 2 it's y, then just directly write that  down and use it and it doesn't matter  where it comes from in that case.  But you should know how to write these partition functions  in a general sense for a system like this  if it's described in words.  And you should know how that changes with respect to temperature.  So if you make the temperature really low or really high,  you should know how that affects the relative populations  and how it affects the partition function.  And I think we are about done.  That's what's going to be on the final.  So thanks again for a really great class.  I really enjoyed it.  Thank you.  Thank you.
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