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Lecture 06. Rotational Spectroscopy Pt. II.
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Title  Lecture 06. Rotational Spectroscopy Pt. II. 
Title of Series  Chem 131B: Molecular Structure & Statistical Mechanics 
Part Number  6 
Number of Parts  26 
Author 
Martin, Rachel

License 
CC Attribution  ShareAlike 3.0 Unported: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and noncommercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license. 
DOI  10.5446/18914 
Publisher  University of California Irvine (UCI) 
Release Date  2013 
Language  English 
Content Metadata
Subject Area  Chemistry 
Abstract  UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 06. Molecular Structure & Statistical Mechanics  Rotational Spectroscopy  Part 2. Instructor: Rachel Martin, Ph.D. Description: 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. Index of Topics: 0:01:53 Angular Momentum 0:08:13 Spherical Harmonics 0:13:25 Rotational Spectroscopy 0:26:00 Energies and Frequencies 0:31:55 Types of Rigid Motors 0:36:28 Symmetric Rotor 0:39:40 Degeneracy 
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00:07
come on what is the experience that at time it is it's time for and so it's really cool that everyone comes to Obasanjo's office hours it's finally getting record everyone on a more personal basis and thinking about having more of them may be on Thursday so if you have an opinion about when it should be on Thursday please post them on Facebook page all look at it and see what I see in the circumstances my scandal isn't completely flexible but I will take people's preferences into account so you go ahead and if you if you have anything in his bid on a Facebook page I'm so we're tight at this point we're talking about recreational spectroscopy were going through a different kinds of excited states that that molecules can be put into as a result of interacting with electromagnetic radiation and
01:13
last time we looked at this picture that's that's the big picture of spectroscopy In the end we see if I can fix the street because that's going to be annoying much good the peso right now and we are awaiting the bottom of the ground electronic state and were all the way to the bottom of that well in the excited and the ground vibrational state and we're just talking about exciting rotational transitions on their own and last time we got around to talking about a rigid rotor in a plane so like a linear died ,comma molecule that's in a plane it's rotating about the access now we need to talk about the more general case of something that is working on a sphere and it has more degrees of freedom and again you've seen this last quarter this was described as a political atmosphere and or the hydrogen atoms wave functions which of course familiar from General Chemistry trip also so we can write down our shorting equation for this and now in our cats we have to quantum members to keep track of elements said those of quantum numbers for the spherical harmonics and we remember what they values need to be and if we write down the Hambletonian in spherical coordinates With are fixed because of course we're talking about a molecule that worsening universal that Adams art by laying around here's what we get for the Hambletonian and actually look really familiar from last quarter and I definitely recommend if it's a little rusty go back and remind yourself how to convert it into a spherical and cylindrical coordinates and check out the Hambletonian for these things because I'm not going to go through and for equations for the systems I'm I'm assuming that you were done with industries the results I am however writing them down and direct rotation just so we can get used to making that transition between the effects of a severe delays also can write Hambletonian in terms of the angular momentum operator elsewhere and here's what we get and so From that we can pull out our energy eigenvalues so things that should look familiar this Hambletonian the solution for region in this kind of of system and these energy titan values I also want to briefly talk about because the commutation relationships of the angular rental operators in the important for things that we're doing and so on again is totally review from last quarter if 2 operators commute then it doesn't matter where ordering the men and an example that I know you've all seen is position and momentum and we talked about this last time that in terms of a agreement the equivalent of a pair of complementary observables are angle and angular momentum we notice things with precision and so on I'm sure this is familiar enough but now wants look at it in terms of the angular momentum operators so a lamented come up over and over again in the camp so this is kind of the the most liberal version of were actually talking about a molecule rotating around we're looking at its rotational states but it's worth spending a little bit of extra time thinking about England momentum because were also need to deal with it in terms of spin and things like electrons on proton see 13 nuclei habits transit property called spin that this kind mysterious actually but it behaves In the same way as angular about Mac mathematically we can treated using the same formalism as something moving around this is going to come up with it over and over again and it's worth looking at these things OK so if we look at our angular momentum operators India x y and z directions here's how the defined and again this should be familiar from from last quarter but maybe for looking at all the different contexts the main thing that I want to point out here is will 1st of all this remind you what they are and also I want to point out that they don't you with each other and in fact they have a special commutation relations you can prove this yourself it's kind of tedious but if you look at ease that the commentators of these operators and working out here's what you get for the country as a commentator Aleksandr Y is I Elzy the ones were Elzy Alexis I O Y and etc. call that a psychic commutation relationship so we have a set of 3 operators and their commentators are related to each other in site where questions that the capital D is the arm it's the partial derivatives With respect to whatever the subscript so it's so it's defined down here on the right just done it's a useful shorthand for later on when I can have so much space OK so that's just some position useful properties of the angular momentum operators I'm going to have to prove that the homework another 1 of their properties but so now let's look at the actual spherical harmonics so we've been talking about a particle atmosphere or the general case of a molecule that is free to move around in any way in space and we have to deal with its angular momentum about each of the 3 accidents so we have no written arbitration around z for wire X and we have to to be able to do no that's OK to me
08:14
with each other let's look at the Eigen functions of all square so squared is the total angular momentum operator operators is a pretty fundamental property in quantum mechanics and just remind you here's what the spherical harmonics look like and I see people writing stuff down don't you don't want secured creditor all these things you know the views of the post online you can also Google's spherical harmonics or hydrogen atoms wave functions and you'll see what there are lots of unique 3 D representations of these that you can play with but I want to remind you what they are and draw a connection to what the functions look like mathematically because 1 of the things that going have to do In the practice problems is we're going to look at selection rules of organization OK can you have a transition from a particular state particularly rotational state to another 1 and you're going to have to do that recent symmetry which means that you don't have to take intervals with respect to these functions and safe right to these things overlap by symmetry and wasted that 1 is if you really really good visualizing stuff head you can look at these things imagine where they overlap I'm unless you really great drawing stuff really fast that's my work in the context of an exam or something like that so you need do you remind yourself of the symmetry properties of these things and so it's important to know what the functions actually look like so again Morgan represent our state's yeah you've seen them as Weiss of Bell and Sevele that means the describing these DuPont numbers we're going to look at that and direct notation by just sticking those 2 quantum numbers that we have to keep track of the cat so the state 0 0 looks like this and then as we go through we can put in the rest of the Lozada on wheels again if you're frantically writing this down don't you can now you can look it up I just want to remind you that these these familiar shapes from the hydrogen Adam wave functions have mathematical forms that are easy to write down we know what they are and we can take animals with respect to and do things like figuratively whether they overlap and get derived selection rules for different rotational states so another thing that I want to mention here is that in my mechanics we call you know all these states that we're talking about wave functions and you get really used to thinking about the terms of an electronic wave function so don't get confused about that you were talking about different rotational states little the type of vibrational states there are all kinds of different things that we have with functions for OK so that brings us to practice problems which means economy posted so wet so things that I would like to be able to do I want show that l squared commutes with Elsy so weird he said that elects Hawaiian don't me with each other you can prove that to yourself if you want takes a long time but I do want to show that elsewhere all the new and I would also like to take this Elzy operator that we have in Cartesian coordinates and converted to spherical coordinates have you done this before is that something that came out last quarter OK see you've seen it but I haven't necessarily done as of yesterday practice to go ahead and do it and there are also a bunch of extra practice problems many of them were from the book some of them are not some of them I made out there not posting yet but it but all do and I get back from lunch so that the 2 practice problems of every website 2 things the need to know about them 1 is there's a lot of them on the other 1 is you don't have enough information to them all right now so they practice problems for for occasional spectroscopy and vibrational spectroscopy so if you don't know how to do all them yet don't worry well you'll see it as we go along in class or if really hadn't booked if you if you want to do something had questioned yes I am not going to post the answers however you know Fiesta is nicely will probably help you with that discussion I will definitely help you with that in office hours if you wanna come on anything like that but I'm not going dispose of the key OK so we talked about were rotational spectroscopy fits in the grand scheme of spectroscopy it's a very humble modest little spectroscopy doesn't take very much and you do it and we talked about some properties angular momentum which would be important for a lot of different things will to get into the details of rotational spectroscopy OK
13:28
so 1 of the things that we really need to know To get started learning about us is the rotational constants so it's called B it has until told they over which indicates that it's even stranger units and here's how it's defined it's just age 1st over 2 times the moment of inertia for the particular molecule now as we see when we look at the pictures of different molecules that have different shapes in the book some have more than 1 moment the show and that has some implications for what the spectral light that this rotational constant tells you something fundamental about the molecule because the the moment of inertia is in there and it comes up inspectors so let's talk about what that telling means that means that its units are in we've numbers and and in the context of rotational and vibrational spectroscopy whenever you see something has to over that's what it means and that's the way numbers and part of that I think 1 of the hardest things about merging spectroscopy in general is that it is the land of messedup units and sloppy notation and we just have to deal with it if you wanna reading literature it's an old field a lot of stuff is historical it's not necessarily consistent among different parts of it and there are certain different units so far exceed the weight number unit in and of itself isn't sloppy that's just it's defined as reciprocal centimeters and why do we use it historically it's because no more talking about rotational vibrational spectra This is a unit that gives us reasonable values you know we don't have values of you know gigantic numbers so a typical rotational constant for her Little molecules of the type that we're talking about is something like a 10th of a wave number to attend with numbers and for vibrational ones will be doubly larger so now I I say it's sloppy well when you get into the sloppy notation is when people start expressing energies and where numbers so that doesn't make sense right we have a reciprocal wavelength and people referring to it as an energy and you'll hear this seminars people say they're skipping some steps so if we have something at him with numbers we can get the frequency of that electromagnetic radiation and they know that the frequency is related to the energy you have multiplied by planks constant so there is a really straightforward relationship between this and that the annual you'll see people use that as a shorthand OK something customer here yet the speed of light on the day .period this year yeah that should be the focus the thanks for pointing that out the so yeah that's sir that's relevant we start talking about the energy ventilation in the recreational constant RE so if we're talking about a molecule that's true that's rotate about 3 different accidents now we need to consider different moments of inertia so we look at our classical rotational and kinetic energy we've got these 3 moments of inertia their labeled a B and C just to emphasize the top free space you know we could call xyz but what however we wanted to find Coronets so this is the general case so this is a molecule this is the case of a molecule that doesn't have any cemetery has 3 separate moments of inertia and so it's classical angular momentum around any 1 of these axioms is related to its frequency and so here's its overall honors and what we're going to be dealing with is the quantum analog of the situation and we're going to have to look at what that looks like for the cases of different shaped molecules again we're not going to get into huge levels of detail about how you calculate the different moments of inertia for molecules of different shapes that's a good thing that without OK so here is the general case we have 3 different moments of inertia during spend a little bit of time talking about simplifying
18:41
cases so in the region were approximation were were making the approximation of the bonds are rated and they're not moving around so the industry in Internet nuclear distance stays the same and we also have to worry about the selection rules so selection roller just telling us you know by symmetry which transitions can be observed and the current selection wall sort of that you can think of is the answer to the largescale course rule is that a molecule you can only have a pure rotational spectrum if it has a dipole moment so let's think about why areas so we're talking about doing rotational spectroscopy we have some electromagnetic radiation it's exciting rotational transitions and interacting with the field of that electromagnetic radiation and so on the only gonna see anything if it has about also molecules rotating around and if there's no change in the the electron density of the molecule as that happens like say you have and to as its rotating around there's nothing for you observe it's like a tree falling in the forest you can't see it doesn't interact that radiation so yes what it this about so here again I'm not I'm not going to go into twoyear too much I calculate these things there's a really nice table in the book that shows you know how to get at the moment of inertia whereas most nothing focus on it OK so were only able to observe these transitions if the molecule actually has a permanent dipole moment and of course it's chemistry so you can have a role without exceptions to the rule will talk about what they are up as we get further on but the gross selection rule is yesterday ,comma some change in electron density of the molecules retaining around in order to observe and we get the transitions when the molecule absorbs the photons and it that it that residents it's a bit like energy too excited to higher rotational state and then it changes from Danish shoulder J. 5 and so here's how we write that down In a more formal way so we have transition dipole therefore that rotational transitions and we can write down it's matrix element in this form were Jason Binder said after the initial and final states and a formal way of expressing their gross selection rule is that that transition that will have to be nonzero and it turns out that the answer you get 4 what it has to be or is it you can have Delta J. being 0 the molecule can just stay in the same state or it can be customized 1 and I'm not going to prove that you at this point I will for other types of spectroscopy later for this 1 was going to leave it at that the duration is in your book Atlantic about but for now let's just use the resultant if everybody understands how already is down I'm happy with that OK
22:33
so let's look at what the energy levels look like so this is for a diatribe molecules so it's really simple so tonight ,comma molecule we know that has a when they will see anything and here's what the spectrum of flight so notation is you'll see you know you'll see J. plus 1 and Jay or J. prime NJ and the arrow is going in the direction of the transition still sees these things written down great so are rotational constant again is is a spot squared over 2 eyes and the energy for a particular level J can also be expressed in terms of the rotational constants it's just that rotational constant times J. Township was 1 here again just from the the item values of the LC operator which know again we we have the result here of what it what it is important coordinates with showing of homework OK so as a result of this we see we have these equally spaced levels and the occasional spectrum has these wines that come in increments of 2 B remember that whenever you see a line in the spectrum that represents a transition so we have the levels and and it's tempting to look at all those lines in the spectrum and think that those correspond to the levels but remember that a spectral line is where you have a transition from 1 state to another OK so if we look at the separation between adjacent wines and this f the till they is lower your energy of a particular state In wave numbers and we can write down our separation between adjacent lines and get out relationship between that and the frequency so this is a fancy way of saying that we can look at the Spectrum and we know that Flinders space increments of 2 be and from that we can calculate rotational constants and we can get the moment of inertia of the molecule and so we can figure out this year something fundamental about the molecule from this that kind of spectroscopy OK so as spectroscopic method goes this one's a little lame doesn't actually contain much information means a lot a lot of times you're going to know the moment of inertia that molecule anywhere there are better ways to get it this is not you know the most useful method in allowed said there are some situations where it is useful which in talk about a later the the main thing is in space it's really cold out there and you don't have the luxury of aiming a giant laser at some galaxy and seeing what molecules that have to to deal with the ambient radiation OK so before we talk about applications was just go through here again the notation is a little bit it was just go :colon recapped what everything as a base of the someday as the energy of some rotational level J and that's in normal good old energy units after a day With the tell over it is the energy of the world today In which number so again we can convert readily between real energy units and wave numbers because we know the relationship there and if we have to be if we have a new of j of the transition J. today plus 1 so again knew with the telly over it is year spectral frequency but it's and wave numbers and that is for a particular transition JJ plus 1 and that corresponds to the position of the line that you see on the spectrum when something changes From plus 1 and that sounds a little bit convoluted in terms of you know thinking about the interview diagrams but it's important because that's what we actually measure if we take a rotational spectrum that's what we're going to see and so we have to get to know how to look at that and back out all this other stuff tells us about the states and then I have 1 more confusing occasional issue too remind everyone about which is that you is the reduced mask and that's a constant and there's also an operator told you which is the dipole moment and of course that's an operator How do you know the difference context and if you get confused please ask so I know there are a lot of nutritional things that are confusing and hard to get used to it we just have to deal with them it's an old feel that's been around for a long time itself it's something that that we just have to learn her reading OK so let's talk about a occasional energy levels and all
28:38
that more details so we're back to talking about the diatonic molecule and these are things that can already seen here we were attacked but that's 2 2 and where he tell about it OK so what I want to point out now is that real molecules might not always follow rigid liver approximation and that's something that that we should be aware of OK so I'm going to make this point by showing real data so it's just a table data don't right up down all these numbers but I think it really makes the point if you see what's going on a case of Frasier all we can measure real numbers for these rotational transitions and so I have some of the the actual measured numbers were these said states so the radius that we measure for it's Cl if we look at him again we can just look at the spacing between the states and get the rotational constant and measure these things and if we do not for different transitions in the spectrum is what we get for the rate in Netanya so we take the 1 from going from 3 to 4 that's the frequency and here's the bond length that we get we calculated and now we take the higher energy states the bond lengths starts to increase so that that we measure and again if we keep going increases a little bit more and a little bit more and as we go up to higher energy states are bond is actually starting stretched so what's happening is we have a HCl molecule putting some energy in minutes rotating around and at low energies the bond as the region but at higher energies it's rotating faster and faster and there's some centripetal distortion there N we can compensate for that there is a correction term for diatonic molecules I'm not going to make you use it for anything in particular right now I just want you to know that exists so it's important to Be aware that a lot of times were using approximations because that makes things easy to treat and you know we can understand the basics of how something works but we should always know about the assumptions behind the approximations that we're making and understand when the appropriate when they're not so if we're looking at really high energy states in this region were approximation it might not be the best it also depends on the particular chemical bond that you're looking at so if it's a really rigid bond if the idea of it's very it's a very stiff kind of bond then this is going to happen so much higher energy then it will offer I thought you find that they can move around more OK so let's of talk about the types of rigid rivers so again there's a nice table in your book of world moments of inertia are but I just want talk conceptually about what what they look like OK so we have diatonic and and the other 1 year molecules and the moment of inertia defined differently here and In the case of diatonic molecules we really only have 1 axis of rotation don't worry about so for women automata linear molecule you were looking at the Musee access is here and we're talking about rotation in this place we don't have to worry about the larger picture of what's happening it's working atmosphere in that case and so on they do generosity of those states Gee isn't too generously of stage is just to date plus 1 and so all it means is that as we go 2 higher and higher energy the states get more generous there are more ways to generated that state then if you're a lower and so if you have you 0 and alignment of our particular access there's only 1 way to do that but then as we had more energy the higher states become more populated and part of the reason for that is that they have hired to generously OK so we we can also look at an asymmetric molecule that has 3 different moments of inertia and what that means is so water molecule if I rotated around z axis of pirated around X or Y each of those things different doesn't it doesn't have any cemetery in that sense so In this case were not talking about you know we've spent all this time popular rotations as symmetry operations here we're not talking about in that sense were just talking about like characters a water molecule in the gas phase minding its own business and it can rotate about the x y and z axis In the X and Y cases that's another cemetery operations but it's it's still doing that in terms of additional spectroscopy we have to worry about it and I remember it on the picture here the 2nd molecule here is C O 2 1 of these things is not like the others remember that you have adverse selection rule is that you have have moment to see that Europe recreational Specter cited appeared because it's an example of a linear molecule for this particular type of across used
34:36
by comport OK so the other types of rigid rivers that we have our symmetric orders and spherical workers and again he said these names are a little confusing so the symmetric work is something like ammonia where we have 2 different types of rotation that we have to worry about so we can rotated around the access so that surround its principal axis of rotation and then it has 2 other equal moments of inertia and that's because it's a metric in the sense that if we rotated around acts a irritated around y those look the same so that's a sense in which it's it's a it's a matter the it's not the same as the cemetery operations we talked about it .period and so 1 consequence of that is it has 2 different rotational constants and they're called a and B. and there are defined as parallel perpendicular the unparalleled perpendicular to what the principal axis of rotation and on the bottom is an example of part of the rotational spectrum of cf 3 i which of course is a symmetric so there are lots of transitions going there a patient for the spherical all the moments of inertia the the same and that's all that's all is meant by a spherical so something like methane Nikos neutral molecule like 6 of buckyball anything like that it's going to be a spherical and we can simplify things by noting that all of these moments of inertia the the same focus also because for them the class in the case of symmetrical and I just want to draw the parallel between the classical and quantum cases suffer the classical case here's the angular momentum we got to moments of inertia the equal were calling Indian see and you so that has to do with perpendicular and we've also got 5 parallel which is that the unique 1 that's about the principal axis and so we can write down the total angular momentum we can write down energy in terms of and then we can also look at this in the quantum case by just making the analogy that we know what the item values of the total angular momentum tomorrow and we can relate it to the expression for the iron the position of the lines on the spot so here's where you're going to get in terms of where the lines show up in the spectrum with respect to the 2 were occasional Constance the area so other things that we need to think about we have different rotational quantum numbers here because rotation is quantized around each axis that worried about so we've got a rotation about the principal axis and we've also got these other 2 recreational use of the 2 sets of recreational motion and we have quantum numbers for all things and so what that means is that you know if we have if key equals 0 that means there's no rotation of the principal blacks so the molecule is in space and its rotating your purely around X or Y or somewhere in between there and if case equals plus minus J that means although rotations about the principal axis so it's just rotating like this so that we can think about the relationship between those quantum numbers it's just were just talking about what direction is quantized and again those are always quantise increments of H so it said it's written as H there should be at far and so 4 symmetric the specific selection role is that we can have worshiped rotational transitions where 10 that the year change in K is 0 and we can have Delta Jr being plus or minus 1 and then also has to take these values up to and including it was a nice day question it is a spot they are excellent that means is that for spherical writers we have a lot more generous because there are more taxis there were things are quantized to be worried about so in general this work has 80 To date plus 1 folder generous because of its orientation in space and it has another 1 With respect to its orientation but in the molecular friend so we've got access the molecule without access because its orientation in space and the generous for this thing gets larger really quickly so if we have to equals 10 so were only in the you 10 state the 441 ways to get there and this has some important consequences for what the spectral light so this
40:33
is a simulated spectrum for FCL 0 3 and it's said 1 Calvin so it's really really cold this molecule it's not rotating very much so when it's really cold you were used to thinking about if we don't have much energy everything must be piled on the ground state trade well In these kinds of experiments that's not true and the reason for that is that the littered the ground state is the lowest energy sherbet there's only 1 way to get it that state is not degenerate everything in there the there are entropy considerations if you well too to getting that there's only 1 way to do it so it's rare whereas a little bit higher energy states just have more ways to get that value and so we see that the maximum population it is not piled into the lowest energy state even at 3 cold temperatures if we look at something at more like room temperature we see a couple of things 1 is that the distribution is shifted and also it's brought about a lot sold so some really high energy states can be populated because there's a lot of the generously there's a lot of different ways to to get that N this is the introduction to kind of 1st statement which will see at the very end of the class but will try to bring in at least conceptual representations of us all 1 because it have a feel for how it works so to use another analogy it's like saying you know the most likely state for the 1st midterm is that everyone gets 100 because everybody's really smart and that's true that's the that's lowest energy state right but it's really really unlikely because there's only 1 way to get everything right and there were are lots of ways to To make the mistakes so those those states there are populated so the last
42:40
thing I want to mention is an actual application of rotational spectroscopy mentioned that this is the main 1 it's really useful for looking at interstellar molecules so there's a picture of this a cloud of gas that has a bunch of molecules that's it that's out of space and many of the molecules that are known to be out there were discovered near this feature and you know how do you know what molecules in space again you can't shine giant laser out there and do laser spectroscopy you have to deal with the ambient radiation it's there and it's really low energy cold and so the way people do that it
43:22
bye In measuring the specter of using a radio telescope and then they make mixtures of molecules in their laps you get the specter that Irving masses of all but different rotational states and then they can kind guess based on pattern recognition and knowing what the specter different molecules look like and makeup mixtures of molecules and allowed that can match the the spectra so this is from these status from Lavrov professor Lucy's Rees who is at the University of Arizona I visited alive a couple years ago it's pretty interesting so she has to face she has these giant telescopes which is in charge of 1 of these facilities in Hawaii where you can you know she can widen from her computer in Arizona and run these giant telescopes and she gets factor from space that have a bunch of invitational features of different molecules and then in order to figure out what's there she goes and makes mixtures of molecules that she thinks that she thinks might match in vacuum chambers of a really cold rollout and compares it to and so there's a there's a lot of effort there in your 1st of all instrumentation as far as being of a measure these things and also in data analysis because you have to do a lot of pattern recognition and sift through a lot of Spector and compare with the same not so this is what this stuff is actually used for In real life 10
44:48
years the instrument that you need to do it so it's kind of exotic doesn't come up much it's me but it's not use all over the place next time were talk about vibrational spectroscopy which is used all the time in research labs you and you probably all you see yourself in the context of a or maybe Parliament as well OK happy Martin Luther King Day on Monday and the rebellion Wednesday
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