Lecture 01. Symmetry and Spectroscopy Pt. 1.
This is a modal window.
Das Video konnte nicht geladen werden, da entweder ein Server- oder Netzwerkfehler auftrat oder das Format nicht unterstützt wird.
Formale Metadaten
| Titel |
| |
| Serientitel | ||
| Teil | 1 | |
| Anzahl der Teile | 26 | |
| Autor | ||
| Lizenz | CC-Namensnennung - Weitergabe unter gleichen Bedingungen 3.0 Unported: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen und nicht-kommerziellen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen und das Werk bzw. diesen Inhalt auch in veränderter Form nur unter den Bedingungen dieser Lizenz weitergeben. | |
| Identifikatoren | 10.5446/18909 (DOI) | |
| Herausgeber | ||
| Erscheinungsjahr | ||
| Sprache |
Inhaltliche Metadaten
| Fachgebiet | ||
| Genre | ||
| Abstract |
|
17
25
26
00:00
MolekülstrukturMonomolekulare ReaktionPhysikalische ChemieNMR-SpektroskopieBetäubungsmittelWerkzeugstahlChemieRepetitive DNSChemische StrukturTerminations-CodonGletscherzungeChemieingenieurinMolekülSchönenFülle <Speise>Gangart <Erzlagerstätte>Organische ChemieGradingBewegungThermoformenLactitolWassertropfenBukett <Wein>WerkstoffkundeReaktionsgleichungNahrungsergänzungsmittelFormaldehydSpektroskopieConsensus-SequenzTopizitätElektronische ZigaretteHalbedelsteinVorlesung/Konferenz
09:30
StickstoffatomDipol <1,3->MannoseBiologisches MaterialElementanalyseDipol <1,3->AzokupplungAtombindungBaseWasserstoffbrückenbindungIonenbindungBukett <Wein>MolekülChemische EigenschaftWasserChemieChemische BindungAlkoholische LösungOperonVerhungernOrganische ChemieCobaltoxideFülle <Speise>WursthülleDeprotonierungAnorganische ChemieElektronische ZigaretteSulfurMolekülstrukturCHARGE-AssoziationErdrutschFluor <Medizin>BewegungAlignment <Biochemie>Hybridisierung <Chemie>Physikalische ChemieBleierzChemische ForschungEinsames ElektronenpaarAtomorbitalComputeranimation
16:07
MannoseIonenpumpeBohriumNatriumcarbonatOptische AnalyseOptische AktivitätReflexionsspektrumChiralität <Chemie>OperonProteineOrganische ChemieChemisches ElementChemische StrukturOptische AktivitätTiermodellBraunes FettgewebeBindegewebeVSEPR-ModellWursthülleLactitolAzokupplungFülle <Speise>KristallkörperSymptomatologieMonomolekulare ReaktionErdrutschMineralElektronische ZigaretteMolekülVorlesung/Konferenz
22:45
Optische AktivitätPlasmamembranPolyethylenterephthalateSerotonin-Reuptake-HemmerMannoseOptische AnalyseEnoleStickstofffixierungOptische AktivitätElektronische ZigaretteMethanisierungMolekülEthanChemisches ElementChemische ForschungWasserstoffIonenbindungPeriodateOperonTiermodellPharmazieSetzen <Verfahrenstechnik>Quarz <alpha->Chemische StrukturChemieingenieurinVersetzung <Kristallographie>Hope <Diamant>Translation <Genetik>AbschreckenChiralität <Chemie>FormaldehydComputeranimationVorlesung/Konferenz
29:21
Aspartat-AminotransferasenOperonHydroxybuttersäure <gamma->InsulinGraukäsePolyurethaneMannoseSeltenerdmetallBohriumKorkenColaminEnoleTransformation <Genetik>MolekülSchmerzChemische EigenschaftMetallmatrix-VerbundwerkstoffOperonSenseOptische AktivitätScreeningElektronische ZigaretteGesundheitsstörungTabletteSchönenLactitolSingle electron transferSpektroskopieReaktionsmechanismusAktives ZentrumAlterungSetzen <Verfahrenstechnik>PeriodateWursthülleCHARGE-AssoziationMähdrescherComputeranimationVorlesung/Konferenz
35:58
KorkenMannoseEnoleZeitverschiebungBohriumAlterungReflexionsspektrumBleitetraethylVSEPR-ModellScreeningKorkenWursthülleChemischer ProzessChemieanlageOktanzahlChlorOptische AktivitätElektronische ZigaretteProteineMolekülPeriodateKohlenstoff-14WasserstoffGangart <Erzlagerstätte>Chemische EigenschaftChemische BindungVerzweigte VerbindungenSchwingungsspektroskopieErdrutschSekretSirupChemische StrukturStickstofffixierungCobaltoxideWasserCadmiumsulfidVersetzung <Kristallographie>DeprotonierungLot <Werkstoff>Monomolekulare ReaktionChemieChemische ForschungSubstituentVorlesung/Konferenz
Transkript: Englisch(automatisch erzeugt)
00:06
Good morning everyone. Welcome to second quarter PChem. I'm Professor Rachel Martin and I'm going to be teaching this second quarter that has to do with spectroscopy which is broadly defined the interaction
00:23
of light with matter. So we're going to talk about a lot of things that really get to the heart of how do we know that molecules will look like, look the way they do. So a lot of the beginning of this class is going to look like a repeat of general chemistry.
00:41
And I hope it will be really fun because we get to see a lot of these molecular shapes and structures that we know about from general chemistry where it's presented as here's a bunch of stuff to memorize. And we're going to learn to understand why they look the way they do and how we know. So a lot of what this quarter of PChem gets
01:02
at is taking the knowledge that you've learned last quarter in quantum mechanics and applying it to practical problems in molecular structure, molecular motion and things like that. Another really important part of second quarter PChem is really learning how to use mathematical tools and mathematical formalisms
01:21
to describe nature. So we're going to be learning a lot of problem solving methods that might be different from things that you've done before and that's going to require a lot of practice. But hopefully you'll find it worthwhile. I do. I think it's a lot of fun. So this is what I do.
01:41
I'm an NMR spectroscopist. So you've probably all seen NMR in the context of organic chemistry, you know learning how molecules look the way they do. In my group, we work on NMR methods development. So being able to use the NMR experiment in new ways
02:01
to learn different things about molecules. And I'll get to share a little bit of that with you later on when we get to the NMR part of the class. OK. So basically, I'm handing out the packet for you. There are a few things in there. So one is the syllabus and that has a schedule
02:20
with reading assignments from the chapter. The book is the same book as last quarter. And if you have slightly different editions, that isn't going to matter. It'll be close enough, just some of the page numbers might be a little bit off. Some notes about the schedule.
02:41
It's all subject to change except the exam dates. The exam dates are set in stone. That's definitely when it will be. If we get a little bit ahead or a little bit behind on the topics, that might change what's on the exam. But the exam dates are when they are. My office hours, I understand maybe I should change
03:01
because I understand that the chemical engineers have classes that conflict with those times. Is that correct? OK. So, let's give this a try. If there's not consensus right away, I'll have to regroup and look at my schedule. So, what if we move the Tuesday one to an hour earlier
03:24
if we do one to two, how many people could make that? All right, let's try this a different way. How many people absolutely couldn't make it if it's one to two on Tuesday? OK. That's not very many. So, one to two on Tuesday it is.
03:47
All right. And then on, if we move the Wednesday one an hour later and do it three to four, how is that for people? Who couldn't make it if it's three to four? OK. Who can't, who absolutely can't make it
04:00
to either of those times? All right. That's few enough people that we're just going to have to work something out. So, that's when it will be. One to two on Tuesday and three to four on Wednesday. And I will update the syllabus and put that on the website. OK. So, there are a number of resources online for this class. So, one is the course website which a lot of you have discovered.
04:24
It's pretty basic. It's not fancy but all the information will be there. I will hand out things like this in class once in a while but for the most part, course materials that I want you to have supplemental readings and things like that will just be uploaded on the website and you can download it and look at it in whatever form you prefer.
04:43
There's also a group Facebook page which I encourage you to use. The group ID is listed on the class website. And that is, it's, I found it to be a really neat supplemental tool for interacting with the class. You don't have to use it. If you hate Facebook, that's fine. It doesn't matter.
05:00
But, I found that it's actually a pretty useful tool. So, you can get on there, ask questions. The TAs and I will check it really regularly. Another advantage that it has is you can discuss things with your classmates and a lot of times people can answer each other's questions, you know, before we have a chance to get on there. So, it's a lot more efficient than 50 people all emailing me the same question
05:22
and waiting a long time for me to answer. So, I do try to answer email. You can email me. I get a lot of emails. So, the Facebook page is definitely more efficient. Other resources that are available. You might notice that this class is being videotaped. That means that the videos will be available online later
05:43
on. You can watch them. If I make mistakes in class, you can laugh at them. That's fine. Hopefully, that will be a resource for when I talk really fast and it's hard to write everything down which does occasionally happen. Another good strategy for dealing with that is to ask lots of questions.
06:00
That's a good thing to do anyway. It's good to have a lot of class participation. It makes things more interesting. I mean, I really like to hear myself talk and that's why I became a professor, but, you know, it's better if we get everybody participating. And also, that does help slow things down. So, anyway, I never mind answering questions.
06:21
If stuff isn't clear, I would rather clear it up right away. OK. Yes? Sure. Let's see. All right. First time in this room. Let's see how that, how this goes.
06:45
OK. How's that? All right. Good. OK. Other logistical things. Wow, that's annoying.
07:03
OK. Maybe that, is that a better balance between too much feedback and being able to hear me? OK. Does anybody have any more questions about logistical stuff and how the course is going to go, things like that? I have one more thing that I need to say before I forget.
07:22
And that is discussion sections are canceled for the first week just to give everybody a chance to settle into the new quarter. If you need help, you can use the Facebook page, you can email me, email the TAs, stop by after class. Every Monday, Wednesday, and Friday after class,
07:41
you know, unless I have some specific thing that I have to run off to, I will try to hang out outside the lecture hall. We can't all hang out in the front of the room because the next class has to come in and get ready, but we'll step right outside and I'll answer questions for a while after each class. OK. So, no more questions about logistical stuff.
08:02
There's one. The first homework assignment, I'm so glad you asked. It's, I'm going to give it to you today. So, here's how homework is going to work in this course. You're going to have a lot of homework. But, I'm not going to grade it.
08:21
So, there's going to be homework problems. There are things that you really need to do to understand the material. But, I'm going to measure what you actually know rather than whether you did all the homework. So that means we're going to have quizzes. Sometimes they're going to be in lecture, sometimes they're going to be in discussion. If they're in discussion, don't worry, there will be a different one for every day of the week
08:41
so that there's no unfairness there. You don't have to go to a particular discussion. So, if you can't make it to the one you're registered for, you know, don't drop the class and re-register. It will make your life difficult. Just go to a different one. You know, nobody is going to keep track of it. Just as long as you go to one every week, that's fine.
09:01
You can go to more than one if you like it so much you just can't resist. So, anyway, the way we're going to do it is there will be a lot of practice problems along the way. And there are things that I think you should know how to do. And it's just really for your own benefit and these are things that are going to come back and show up on the exams
09:20
and if they'll be on the quizzes. In general, the quizzes will be very similar to the homework problems. And then the things on the exam might have a little twist. It will be a little bit beyond what we did in class or what we looked at in the homework. But it will be basically related. And so the quizzes serve a couple of purposes.
09:41
One is it helps me and the TAs to know what people are getting, what they're not getting, maybe what we need to spend a little bit more time on. And it's also useful for you because you know what you need to understand better. OK, so now that we're, yes, another question.
10:02
I haven't decided yet. Let me think about it. So, the, let's see how it goes. So my inclination is to just let people get help and discussion and work it out and, you know, do things
10:20
in office hours and things like that. But if it turns out that we just have too much, then I might get into posting solutions on the website or maybe I'll do it for selected ones. Let's see how it goes. I'm not against doing it in general but I don't want people to get into a mode where you just don't do it and then wait
10:40
for the solutions to be posted because it is seductive to do that. But then you don't know what you don't understand. One thing about PChem is that for a lot of people there's a way of doing well in classes that involves learning a lot of information
11:02
and memorizing specific things. And that doesn't really work with these kind of problems. You really have to learn a new way of working things out. And for some of you, you know, maybe you've already learned that in some of your other classes but for some people it's going to be new. So, it's really important to spend time working them out. OK. So, let's go ahead and move on to the actual good stuff.
11:28
We are going to start with our review of general chemistry. So, we're going back to the time in Chem 1A when we talk
11:42
about molecular geometries and what different shapes molecules are and the consequences that this has. And we're going to learn some more quantitative ways to describe that. So, one of the first things that we learn about molecular shapes is that if you have polar covalent bonds,
12:05
you know, particularly depending on the geometry of the molecule, this could make your molecule have a dipole moment. You have an unequal distribution of electron density. Yes? Yes? I will eventually, so probably at the end of every week
12:20
but the whole lecture is being videotaped. So, there's going to be a video of the whole thing. So, I'm not, you know, I don't think the slides are going, themselves are going to add very much to that. OK. So, we have an unequal electron density in this molecule. So, we know that fluorine is really electronegative. So, in this case of HF, there's going
12:42
to be more electron density around the fluorine. That's going to give it a partial negative charge. There's a partial positive charge on the side with the proton. And that's going to make this molecule tend to have some alignment in the electric field. And of course, this is drawn in a really exaggerated way but you get the idea.
13:00
These things are going to be aligned in the electric field. And of course, we know that many molecules that have polar covalent bonds have this property. If we look at water, same kind of thing. We've got those lone pairs on the oxygen. There's a partial negative charge in that direction. There's a partial positive charge on the protons.
13:21
And all of these leads to hydrogen bonding and the interesting properties of water. But, we also know that not every molecule that has polar covalent bonds has a dipole moment. So, if we look at HCL, that's another case of a linear moment of a linear molecule that has a dipole.
13:44
H2S, same basic shape as water. But if we look at SO3, that's something that does have polar bonds. Here our oxygen has a partial negative charge and then the sulfur has partial positive charge.
14:02
But it has no net dipole because of the orientations of the bonds. And so, in general chemistry, we talk about this in kind of a hand wavy way and say, all right, the electron density is being pulled toward those oxygens but they're all equivalent to each other and so the molecule overall doesn't have a dipole moment.
14:22
And there's nothing wrong with that. It's fine, but we're going to learn a more quantitative way to describe that. And we're going to use that in our description of how chemical bonding works and also how spectroscopy can tell us about molecular motions. So we're going to start with just the mathematical descriptions of symmetry
14:43
that we're going to use in this course. And then we're going to build up to how can we use that to determine hybridization of orbitals, which orbitals have the right symmetry to be involved in particular bonds. And if you're taking inorganic chemistry, this will overlap very well with that. How many people are in inorganic chemistry right now?
15:04
A few. How many have already taken it? OK. That's good. So some of that will be familiar to you. And if you haven't, don't worry. We'll definitely cover what you need to know. OK. So the idea here is that some objects have more symmetry than others.
15:22
And this is something that we know intuitively. We can look at things and say this object is more symmetrical than that one. But there are ways to quantify that. And what we're really interested in is quantifying these things in terms of symmetry operators and then assigning each object
15:43
that we're interested in. In our case, it's usually going to be a molecule. We want to assign it to a point group. And the point group is a descriptor of symmetry that contains a lot of mathematical information. So in general chemistry,
16:00
we talked about the molecular shapes. You know, we could say something is trigonal by pyramidal or T-shaped or seesaw. There are all these names for the shapes. We're not going to do that anymore. We're going to talk about these things in a more mathematical way by assigning it to a point group. And that is more general and it gives us a lot of information.
16:21
OK. So symmetry elements are related to symmetry operations. So for example, a symmetry operation could be something like a rotation. So if I rotate something, I'm rotating it about an axis. So the symmetry element is the axis and the operation is the rotation.
16:42
We could also have reflections. So again, you know, we have a reflection axis and that's the reflection plane is the element and the reflection is the operation.
17:01
And I have a couple of links here to pages that have good examples. I will go ahead and reproduce those on the class website so you can use them. OK. So let's just look at some abstract examples. I'm going to be using some of the works of M.C. Escher because he drew these really beautiful things
17:22
that illustrate the symmetry operations very well. So if we have a rotation axis, that means that if you rotate the object by 360 degrees over N and it looks unchanged, then you have an N-fold rotation axis.
17:43
And that's called CN. So for this particular object, it has a C3 axis because you can rotate it 120 degrees and it looks the same. So it's got a C3 axis, that's its highest axis of symmetry.
18:03
There are things that have higher order axes. Don't get fooled by this one because it's got these little stars in the corners. So I think the highest axis it has is C4. There are all kinds of examples in nature including protein structures.
18:22
So if we look at protein crystal structures, these rotation axes are really important parts of the symmetry that we see. So to recap, the definition of a symmetry element is,
18:45
it can be an axis, it can be a plane or a point about which we perform the operation. And the operation is the action that you perform to the object that leaves it looking unchanged at the end. So in the case of rotations, we can describe this
19:01
by the rotation 360 over N. OK. So we can also think about reflections. So a reflection, you know, the symmetry stuff is really easy. Even golden retrievers understand it. You know, they're not the most intelligent creatures
19:22
but they get it. The reflection can be vertical which is defined as it contains the principal axis. So if we look, if we think about the Escher print with the bats that had a C3 axis, that's the principal axis and that's just defined as the highest order CN symmetry axis
19:43
that the thing has. It can have all kinds of different axes. We'll see examples of that but the highest order one is the principal axis. And a vertical plane contains the principal axis and a horizontal one is perpendicular to it. Of course, there are all sorts of examples here
20:02
that have reflection planes. This one is related to within a sign change. It's not, you know, it doesn't exactly have a reflection plane along here. It changes sign. But otherwise, these are pretty familiar operations.
20:24
We're just getting new ways to describe them. I should also point out, I didn't put it on the slide, the symbol that's traditionally used for planes is sigma. So, we'll see those. OK. The next one is inversion. That's a little bit harder
20:42
because you can't physically do it to a model. But if you have an inversion center, that means you can basically turn your molecule inside out. So you take every point and it's like you're pulling it through the center and having it come out the other side. So everything, all the coordinates of the positions go
21:03
from x to minus x, y to minus y, z to minus z. So, things like this cubic lattice, this organic molecule, if you look at it really carefully, that has an inversion center. Here's a virus capsid.
21:23
All of these things have inversion centers. And the inversion operation is usually called I. Yes?
21:51
So, it has a chiral center. Yeah, then it can't have inversion symmetry, right? Because it's always going to be different
22:01
when you take the mirror image of it. So a chiral molecule doesn't have inversion symmetry. And that's, you know, it's really, you know, it's good to make that connection. That's one of the really important practical consequences of molecular symmetry. We'll see some other ones later on.
22:24
OK. So now we get to the hardest one of these things to visualize, which is an improper rotation. So these things are called S axes or improper rotation. And for all of these symmetry things, I really recommend
22:43
if you still have your models from organic chemistry, get them out and look at some of these things and convince yourself how the symmetry operations work. If you don't have them, come to office hours. You can play with mine. It really helps when you're first learning to visualize these things.
23:02
OK. So if we take something like methane, it has a fourfold improper rotation axis. And that seems a little bit counterintuitive at first because methane definitely does not have a fourfold
23:20
rotation axis. It doesn't have a C4 but it does have an S4. There was a question in the middle here. It is supposed to be 360 over N. Sorry about that.
23:42
OK. So in order to perform the improper rotation, we need to rotate by 360 over N and reflect perpendicular to the rotation axis. So for methane, you know, imagine my fingers are three of the hydrogens and then the other one is sticking out. So that means we have to rotate it 90 degrees and reflect it this way perpendicular to that axis.
24:03
And if you play around with the models, you can hopefully convince yourself that that actually does work. It gives you the same type of thing. We'll see more examples of this later. Probably next time in class, I'll bring in some models and use them on the document camera and we can see how these things work.
24:20
I guess what I'm saying is if that one is hard to visualize, it might take a little practice. Yes. How do you know N? That's a good question. You just have to get started looking at these things. So the improper rotation is the hardest one to visualize. And so, fortunately, there's a really easy way to cheat
24:41
and that is you have all of these things listed in the point group table. So there's a point group table in the back of your book. It's not as complete as I would like so I'm going to be giving you a better one that you can use and you know, you'll get these things on the exam. Hopefully next time, I'll have that printed out for you. But the point group table contains all kinds
25:03
of information about the molecule. And one thing it has is it lists all the symmetry elements that that molecule has. And so one of the things in that packet I gave you is a flowchart that's useful for assigning molecules to a point group. So what I would recommend if you have trouble visualizing some of these symmetry elements is when you have a problem that has
25:21
to do with molecular symmetry, the first thing you should do is just go down that flowchart and assign it to a point group and then that will tell you all of the operations that there are. And then after you know that, you can go back and try to visualize them and it's a little bit easier.
25:40
OK. So as we saw in the methane example, you can have an improper rotation axis. You can have an N-fold improper rotation axis even if you don't have the N-fold rotation axis. Another example is if you have staggered ethane, that has an S6 axis. So staggered ethane is like I take my ethane molecule
26:03
have the hydrogen set up in a staggered confirmation and then cool it down to zero kelvin so that it can't freely rotate around that single bond. And then that has an S6 axis. And so I kind of drew this out for you so you can see how it works. So we rotate by 360 over 6 and then reflect and you do end
26:26
up with the same thing. OK. So now we get to some consequences of symmetry. So chirality has to do with, as you know,
26:47
whether molecules are left-handed and right-handed. And it turns out that only molecules from some specific point groups can be chiral. And we'll talk about this some more later as we get into talking about specific molecules.
27:02
So basically only molecules that don't have an S axis can be chiral. And that's also implied by having an inversion center. So as we already talked about, if you can invert it, it's not chiral. And if it has both CN axis and a horizontal plane,
27:21
that also means it can't be chiral. And again, this is just a different way of saying things that you already know. So, you know, chemists and chemical engineers and biochemists, you know, at this stage, you definitely know how to tell which molecules are chiral. This is just a different formalism for talking about it.
27:43
OK. So we have one symmetry operation left that's not going to be important for talking about individual molecules, but it will come up when we get into looking at crystal structures, which we'll do briefly. And that's translation. So that means that we can move things up and down,
28:00
side to side in space, and maintain the overall structure. And again, this is really important for studying crystals. A lot of what we're going to talk about during the first part of the course is going to be isolated molecules. So we're imagining that we just have one molecule in the gas phase and that's all or in the liquid phase
28:22
for that matter and that's all we're thinking about. But in crystals, we can see translational symmetry as well. OK. So that's the introduction to what the symmetry elements and symmetry operations are. Those are all of them that we need to worry about. Now, let's start getting into how to use them and how to deal
28:44
with the formal representation of these things. Before we do that, anybody have any more questions? OK. Good. OK. Sorry.
29:00
That you would rotate it by 90 degrees, meaning N would be 4. That's right. So where would the principal rotation axis be? Well, so it doesn't have a C4 rotation axis. It's principal rotation axis is C3.
29:24
Well, that's if I wanted to do a normal rotation. So if I wanted to rotate about the principal axis, I would have to do 120 degrees. You're totally right. But I'm talking about the improper rotation axis. And so that's a really important point. It can have an S3 axis or sorry, an S4 axis
29:41
without having a C4 axis. And that is the hardest one to visualize. It's going to take some practice. OK. So let's talk about group theory. I really like group theory because it gives you a really simple problem solving method.
30:04
In other words, once we assign these molecules to the point group, the point group table contains all sorts of useful information about everything of that symmetry class and we can just read it off and learn all kinds of important things. It also helps build intuition about things like how
30:23
to make matrix representations of operations which we're definitely going to do. So last quarter in quantum mechanics, it's all, that's all about the properties of linear operators. Did you get into the matrix representation of quantum mechanical operators at all or no?
30:42
Raise your hand if you have no idea what I'm talking about. OK. That's fine. So there are different ways of introducing that in the beginning quantum mechanics. You know, one is not better than the other. It's just a matter of preference. For the type of spectroscopy that I do,
31:02
it's really important to be able to look at these things as matrix representations. And so I want to give you a flavor for that. And group theory is a really good way to do that in a painless way because it's clear what the operations are. So that's something that we're going to spend some time on.
31:21
OK. So, group theory pertains to this notion of a mathematical group of operations. Basically, that means that for something of a particular symmetry, it describes all the operations that you can do to this object. And each one is like its own little kingdom.
31:41
It has its own rules and defines what you can do to the object. All right. So in order to have a mathematical group, it has to follow all of these rules. So one of the transformations that you have is the identity operator. And the identity operator is easy. It just means don't do anything.
32:02
And that is usually represented as capital E on the tables that we'll use. You'll also see capital I in some other books. You know, we'll stick with E but don't get confused if you see this in other places.
32:21
OK. So another condition that has to be met for something to be a group is that for every transformation which we can call R, the inverse transformation is also in the group. And if you apply a transformation and its inverse, that gives you the identity. So that just means, you know, if I have something
32:42
and I reflect it and then I reflect it back, it's like I didn't do anything. OK. So the, this is something that's very intuitive. If you do something and then do the inverse, you didn't get anything. OK. So then, if we have any two transformations,
33:01
column R and R prime, the combination, so you do one and then the other, that is going to be equivalent to some other transformation in the group. That doesn't mean it has to be the identity. That doesn't mean it has to be either of the original transformations. It's just that if you do one and then the other, that has to be equivalent
33:20
to some other transformation that's in the group. And the last property that we need to know is that these things have associative properties.
33:45
OK. So this is sort of highly abstract at this point. And we'll kind of do some practice problems where we have to set up some principles of groups. But really understanding the mathematical underpinnings
34:01
of this is neat, but without getting that all right away, we can still use it to solve practical problems. And so the first thing that we need to be able to do in a practical sense is assign molecules to a point group. And so this is one of the things that I have given you.
34:20
It's also on the website if you want to look at it on your laptop screen or tablet or something. This is just a flowchart for looking at molecular symmetries. There are a million different versions of this flowchart on the web.
34:41
I made this one because it makes sense to me. If you find other ones that you like to use, that's absolutely fine. With the caveat that this is the one I'm going to give you on the exam. So you should get to be familiar with it. So you definitely don't want to memorize all these rules.
35:02
Just, you know, learn how to use the chart. Okay, so we're going to have a bunch of practice problems involving doing this, but let's just work through what the various things on this chart mean. Okay, so the first set of questions that you have to ask yourself about your molecule is does it belong
35:23
to various special groups? So the first question is, is it linear? And if so, does it, are the ends the same or different? So if the ends are the same,
35:43
it belongs to the D infinity H group. So something like CO2 belongs there. And if the ends are different, so if it's like HCL it belongs to C infinity V. We're going to talk
36:00
about the low symmetry groups a little bit later because they're hard to explain, but we'll get there. Question? I think I'm going to defer that one until maybe next time.
36:22
Yeah, they do have a meaning, but and I think it'll be easier when you do some examples and you'll get a feel for it. And we'll talk about it next time. It's not that I don't want to answer it. I just, I think it'll come more naturally at a future point. Yes? In the back.
36:44
I'm sorry, could you repeat your question? The same as each other. So like CO2 where you have a linear molecule and you have an oxygen on both sides, then that's D infinity H. Okay,
37:04
so the low symmetry groups are basically C1 is it has no symmetry. So like if I have a carbon atom that has four different substituents on it and they're all different, that's C1.
37:22
CI is it just has an inversion center. CS is just has an improper rotation. The high symmetry groups are pretty easy to remember. They are tetrahedral, TD, octahedral is OH, and IH is icosahedral. So that's something
37:40
like a virus capsid would be icosahedral symmetry if you ignore the fact that the proteins that make it up are chiral. Again, if you read the chapter and look at some of the examples that we'll go through, you'll get a little bit of a feel for these.
38:02
Okay, so if you look at your molecule and determine that it doesn't belong to any of these special groups, then the next thing to do is find the principal rotation axis. So what's your CN? And then we go down to the next step which is does it have C2 axis,
38:24
so 180-degree rotations perpendicular to CN? And if it does, then it belongs to the D groups. So I guess this is a fine first answer to your question
38:44
about what do C and D mean. The D groups mean it has C2 axis perpendicular to the CN axis. And we'll see a little bit more about what that implies for the shapes of these things. Okay, so then we go through these D groups and we can look
39:03
at does it have a horizontal reflection plane? So remember that a horizontal reflection plane is defined as perpendicular to the principal axis. And if it does, then it belongs to a DNH group. If it doesn't, then you look
39:20
at whether it has a dihedral reflection plane which also contains the principal axis and classify it accordingly. And then if it doesn't have C2 axis perpendicular to the principal axis, then we go down the other branch of this tree.
39:42
And I think rather than going through all of this on the chart because it's kind of abstract, let's just do some examples. So let me switch over to this document camera thing. Yes?
40:07
You're right, I do. I need to fix that. So yeah, that is supposed to be the no arrow. Please mark that on your paper copy that you have. I will fix it online. The one going down.
40:22
Yeah. And this is why I do not post my slides online before class because no matter how hard you try, there's always at least one.
41:14
OK. So here's the other way in which we're going back to general chemistry. You have to be really good at Lewis structures.
41:23
So if you need a little bit of review on that, now is a good time to go do it. OK. So let's look at some examples. So if we have CH2Cl2, and of course, which one it is depends
41:47
on how you draw it, but I'm going to draw it like this.
42:07
OK. Well, we could also draw. That's a perfectly fine one too.
42:24
So we can decide whether these things have the right, have the same symmetry or not. And then of course, remember that, you know, they're three dimensional. OK. So do I have a brave volunteer to assign one
42:43
of these things to a point group with using the flowchart? You want to come do it?
43:03
Yes. Don't worry. Everyone will help you because of course, when you're up here, you want help, right?
43:22
OK. So what do we think? So for C2H2Cl2, does it belong to one of these special groups, like it's not linear, it's not one of the low symmetry ones, it's not one of the high symmetry ones, so no, right?
43:42
OK. So that means it must have a principal axis. So what do you think its principal axis is? That's right, it's the C2 axis. Does it have perpendicular C2 axis?
44:00
So let's see. It's got this thing. Yeah, what do you think? No, right? That's right. It has to be either C2H or C2V. So which is it?
44:20
C2V, the mirror plane. So then it's mirror plane. That's right. All right, great job. Thanks for being the brave first volunteer.
44:42
OK. Now, does everybody understand how she got that? Question. Does it have a C2 axis in the plane coming out of the paper? Well, so it's principal axis is in the paper, right?
45:03
And so, it's got different substituents on these things, so no, right? Yes? So you want to draw an axis like that?
45:23
Well, so if I flip that over then the chlorines are going to be on top and the hydrogens will be on the bottom, so it's not the same. So, I can flip it over this way. I can't flip it over that way.
45:41
Yes. OK, absolutely. All right, so first of all, does everyone understand how we got it's not one of the special groups? OK, good. So then, we went to find the principal axis.
46:04
So, this you just kind of have to look at and say, all right, what's the highest order rotation axis that this molecule has? And in this case, it looks like I can flip it over this way and do a 180-degree rotation. But I can't do anything like you could think about, OK,
46:24
I could try to turn it 90 degrees but that wouldn't work, right, because I'd have a proton and a chlorine swapping places and it wouldn't be the same. And it doesn't have any C3 symmetry. So the highest order axis that I can get is that C2. Do you agree with me?
46:42
OK. So now, the next question is does it have a C2 axis that's perpendicular to this C2 axis? So we've narrowed it down to either, you know,
47:08
the C groups or the D groups. And so, we already decided, all right, here's our C2 axis this way. We can't flip it over like that.
47:21
So, that looks like it puts us in the C groups, right? Everybody agree with me? OK, good. OK. So then, we have narrowed it down to C2H or C2V.
47:41
So, which is it? So the question is, it comes down to, do we have a horizontal reflection plane perpendicular to the principal axis? So what that means is if I take this molecule and stand it up like this and cut it through the middle and do a reflection, what will happen? If I do that, the chlorines will swap places
48:00
with the protons and it won't be symmetric. So it doesn't have a horizontal reflection plane. And I think what people were asking about is it has two vertical planes, right? Because you can do it this way and you can do it this way and that is why it ends up being C2V because you do have those vertical reflection planes.
48:20
And so, the neat thing about this is that when we talk about all kinds of properties like chemical bonding, molecular vibrations, all sorts of things that you would think would be very specific to a particular molecule, it turns out that a lot of them just depend on the symmetry that we have. And so, this molecule and things like water that all belong
48:46
to the C2V point group, we can learn a lot of things about, for example, their vibrational spectra. And I think we're out of time so we're going to leave it at that for right now and don't run away yet because I have one more thing to say.
49:02
And that is for those who asked about homework, your homework to be done before next class is to assign all these molecules to a point group which, of course, this is not my screen is showing the document camera, but they're online anyway.
49:25
So, go check out the website, get all your examples and also be sure to bring your packet to the next class because we're going to need it to do examples. All right, thanks. Have a good day.