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Lecture 10. Anharmonic Potential.
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00:06
this morning "quotation mark floods it started on time we have a busy day today OK 1st of all of those who turn in your your sheets from the summer thanks on if anybody else has done you can turn them in to me as you're the case were really happened so that the rules on when you have to turn those in is you can turn into near the TVA's is anytime you can find us office hours discussion is fine but it turns into a pumpkin 1 week after the seminar attended so you can save enough for the rest of the quarter you have to to turn then promptly on usually albeit the seminar yesterday I had to skip because I was working on a proposal that was submitted today but usually only their details will probably be OK so I think the last time I said that the material from today's lecture was going to be in the exam I changed my mind when I started making that the exam would cover a lot of stuff so I said this stuff from from today's lecture I think will actually help you understand some of the material that's going to be on the exam but it itself is not going to be there the anybody have any questions about the examiner extra credit that kind of stuff very good looks continue talking about diatonic molecules and selection rules of this the last time I briefly showed Indian harmonic potential and kind of running out of time so I didn't talk about every much few questions about it the right so there is a problem 111 harmonic potentials of looking at the correction for the fact that you're diatonic molecule not a perfect harmonic oscillator that's homework it's good to do it's good practice what I want you to know about it right now is really conceptual OK so we have a harmonic oscillator potential that's just a parabola and so that means as you got to higher and higher energy in others this potential extends forever and so what that would mean in a physical sense is that your molecules by reading of course the vibrational states are quantized but we've put in more energy connect next aided by rates faster and on and on and on and if it were perfect harmonic oscillator no matter how much energy you put into it it would just do that forever will vibrate faster and faster but nothing whatever happened to it of course in reality we know that's not true if you put enough energy there's some threshold which is the dissociation energy above which the molecule just falls apart has so much vibrational energy that the nuclei just fly apart and go off into space and so that's what the unwanted potential is telling you so 1st of all you know we have this this funny shape to the potential and that energy where you go over the wall threshold is the dissociation energy and it's another thing to notice is that the levels get closer and closer together as we approach that association and so that's all I want to know about it right now and it's just at a conceptual model what's what's going on with it and again there is a homework problem about it where you have to look at it more quantitatively good practice but right now we understand the stranded ship OK so as promised Now what we're going to do is we're going to go through that Gross and specific selection rules for vibrational spectroscopy spectroscopy and for rotational Rahman and this is going to tie together a bunch of things hopefully so it's going to tie together the threads between looking at Spectrum atomic molecules that the vibrations invitations and also talking about symmetry and in particular the things that we've talked about with the even and odd rules and intervals going 0 because of symmetry and the plan is for this to give you a little bit better idea of how use this OK so we know how to state the gross selection rule for Irish spectroscopy inwards the molecule has to change its dipole moment when its buy rating in order for for that migration to the fire active refers to observe spectrum so let's translate that into mass so what that means is that it's trade we we have this value the transition dipole that's and that is to be not 0 in order for there to be and I are active mode so remember your direct notation here we have in the cat I knew I that the initial state and then the broad is new efforts the final state and you know we're looking at a transition probability between them but here were saying that those states are connected by the dipole moment operator which is near and of course we have to remember our terminology In context New is sometimes reduced mass here it's not about Paloma operator OK so if we think about what our dipole moment looks like in a little bit more formal way it comes from having to partial
05:51
charges and I'm calling impostor dealt a you on opposite sides of a molecule were just assume it's diatonic and Paul work for the sake of making the problem easy and it separated by some distance are so are is your equilibrium radius between the the 2 molecules at the and at some point in time it gives it also has some displacement exits to the the vibrations so RE is the equilibrium distance between nuclei and then X is your displacement as it's moving around OK so the dipole that we have at any point is the radius that yet the distance and the charge and so that's all I'm sorry times Delta Q plus extra indulgent so we have we have a term that's going to stay with us that that has to do with the equilibrium distance and then if we also have a soul displacement and the term that's always going to be there that equilibrium is called New not so that's that the time independent sort of standard that dipole moment volatility seen before and that's going to be a constant it's just the value of that operator evaluated at that distance OK so now let's look at what happens when these states are not equal so we're making a transition to new new initial is not equal to new final and here we can plug look that take a moment operator represents where we were just talking about 80 linear diatonic molecule all although atomic molecules linear so we only have a onedimensional foreign all I can do is vibrate along this axis that were calling acts if we had a more complicated system we would have to break this out and doing all 3 directions arena see that in the case of rotational spectroscopy we look at the whole ability but for this case we can simplify a lot because they were constrained to onedimensional system just because of the the set up of the problem it's just a vibration in 1 direction OK so my dipole moment operator just gets replaced by X that's the the only component the Bible overlooking and so I can evaluate that and so we get that Hugh is just the media access and if we just look at that we can see that that's going to be unless the dipole moment is varying with displacement yeah yes yesterday it is that you would think that if you don't do this don't do yes sorry hopefully that's the only when there's a lot of math today OK so so we can work this through and you know it takes a little bit of getting used to the notation but hopefully you can see a relatively straightforward we this argument comes from words were translating this this statement that we can already see how it works into the mass OK so that's the gross selection rule that tells us that the molecule has to have a vital moments and that their words sorry I should be more careful with that the dipole moment should change as a result of that vibration and that's what gives us the the higher stock so now let's talk about the specifics selection will service specific selection will remember is the statement that we can only change vibrational states by increments of customers 1 alright so to start to think about how this works remember what the harmonic oscillator functions on the polynomials which have these kind of functional forms there during your textbook immediately chapters I think maybe it's chapter 3 if you need to go to review what those look like it might be good to go check that out in here's what they look like sort of plotted out and 1 thing that that we need to remember when we start talking about the selection rules and how we know whether we can get a transition between 1 or the other is leaving on so if we look at these just you know without even seeing the mathematically we can see that the cemetery changes every other state so this 1 is even the next 1 is on the next 1 is even 1 after that is odd etc. And remember if we agreed and on function over a symmetric interval that enables them to go to 0 and we also know that if you multiply to even functions are 2 on functions together you get an even function and that doesn't 0 but if you multiply even and odd to get on and that's OK so what does this have to do with our selection rules we have to look at what happens when we in dipole moment operator in between those 2 states Oregon and integrating over a pair states connected by that dipole moment operator and we're going to look at whether the resultant is even or odd enough controls whether there's a signal that OK so
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before we do that I just want to point out here's what that looks like 4 the harmonic potential the picture I found only the even states are drawn ending just for clarity but you get the idea so here's yeah just what that looks like in a more realistic potential but the argument is the same OK
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so we're back to having a transition dipole again we know what that is we've got an initial state in the final state and they're connected by the dipole moment operator and so now In order to figure out what we're doing it actually write down the wave functions and so we go look them up in broker or look on the wall from site and find out what the air Her meet polynomials are and they're written in terms of this term Alpha which is not polarize ability again were we have to pay attention to context so here or Her polynomial functions and again we replaced our transition dipole with just acts because we're still talking about the harmonic oscillator which is we just had displacement in 1 dimension that's all there is going on OK so just to remind you how this works I wrote this out in the long version so not reputation and you can start to understand why people who do quantum mechanics for a living like to use direct attention it's really compact it's nice to not have to write down of all these animals all the time of course the tradeoff is you have to make sure that everybody knows exactly what you're talking about because there's a lot of ambiguity that could be hidden in the notation otherwise but so here we're talking about the welldefined case of a harmonic oscillator we all know the wave functions we can look them up so that's OK right so if we work out this animal we can you know we can pull out the constants and we need to know and identity relating to the polynomials and again we don't have time to go through this but you can check it out on the wall from mathematical website for your resources like that but so here's here's a nice nicer identity involving her meet polynomials that's going to help us evaluate this integral or least simplifying the former we can see what it does OK so having rewritten the soccer in the form that this identity gives us I swear that's going to do something useful for me in the 2nd we need to use another identity In order to get there this is rule equals 0 if new primed as equals New again we're did I get this I looked it up and 1 of the things about you know going through these kinds of arguments when your 1st learning how to do it you really wonder you know how does anybody think of the staff when it you know when you know when to do what it takes a lot of trial and error and so I'm I'm running through here hearing lecture you know we don't have bring much time it's all nice and neat if you actually want workout stuff like this for yourself starting from from scratch and if you don't know the answer to it it's a really long time and there's a lot of trial and error so if if it doesn't you know immediately seem trivial that's completely fine because it's not OK so this identity below 0 IV new prime doesn't equal New and does it equals this function of new if they are article and so that means the 1st term in this whole thing equals 0 unless the final stage the initial statements 1 the 2nd 1 equals 0 unless you jumped up quite lost 1 and so we get the fact that the transition dipole equals 0 loss for a change in newest Costa minus 1 so it's called the specific selection rule for reason it's very very specific you have to plug any actual functions for the be the wave functions here and look at these animals in great detail to see what's going on so anyway now you know where that comes from and you see how symmetry arguments and being able to use the even on Roland and decide which animals go to 0 by symmetry can help us out in looking at further complicated things so here it's nice because we happen know this identity so we can actually get the value but even without having a value you could say here from cemetery argument this is either 0 or it's not traders 1
17:00
last thing that I am not going to spend a lot of time on greatest man it will come back to it later I want to point out that there are some spectra for which you do see if you can the middle the spectrum so I just spent all this time proving you that you don't see a peak in the middle the spectrum ever for a reason but it turns out your molecule has orbital angular momentum so that means not only electrons repaired In other words it's a radical then you can have a transition Delta JPAL's 0 and so remember we have seen no undecided spectra were going from 0 1 in the vibrational state and your daytoday minus 1 in the rotational state the opposite on the other side and we just proved that for a normal diatonic molecule it's illegal to go from 0 to 1 in the vibrational state and stay the same in the rotational states however if you have a really good momentum this is allowed and were and talk about why later on when we talk more about electronic structure electronic spectroscopy but I just want to point out that that this is their 1 of the things that that a lot of people find harder frustrating about chemistry is that there are lots of rules that are given and then later you find out all the exceptions to the rules while we told you this but it's not entirely true but I want to point out those things of front that were you were operating under a particular set of assumptions and were using some rules that apply to specific cases they're not universal there or there are other cases we'll talk about that later OK so now we have gone over our selection rules Furuya spectroscopy and we're going to talk about Roman question 1st year the 2 sides but when he uses for That's a good question so designed the IRA instrument is another hole on area of research that actually have some some are really advance researchers at UCI during the summer professor go out designs new IRA expert ,comma hopefully will get a chance to talk about some applications at the end but beginning of why do you need to know the selection rules so imagine that you made and new molecule and you don't know its properties and you get its spectrum we need to be able to interpret that in terms of which levels overlooking at server we see a peak that represents a transition between 1 little another and if we don't know like Can we only go up by 1 can we go up by 2 or 3 or 4 then we wouldn't know what that spacing represents we wouldn't be able to interpret the spectrum and so that's that's why we need to know these things in order to to look at the specter and so if you're if your physical chemist working in this area there's a really tight connection between theory and experiment so you be the experiments in loud and then you have to work out a theory to explain why you results look like they do and sometimes that's done by the same groups sometimes that's done in collaboration so I'm an experimentalist but I have collaborations with people like Professor Tobias who does theory you know and and some other things will work out ourselves but it's always really important to me to understand the physical basis for this strategy you measure OK so that desire and again I want to emphasize remember Our whole lecture about the differences between absorption and scattering don't get confused about the difference between buyer and vibrational Ronald spectroscopy they have a different physical basis and Iowa talking about absorbing a photon and going up and energy levels in Rahman people were talking about a photon being scattered and either giving up or getting a quantum of energy from the molecule it is a case so right now we're going to talk about rotational so we just talked about fire spectroscopy which is direct vibrational spectroscopy is now we're happy about Roman scattering and were also talking about workstations OK so let's think about a molecule in an electric field and again remembering words are words selection rule for rotational Ramos when the molecule rotates around the polarize ability has to change with time otherwise constantly think and and so again news by that's induced by the electric field but now we have a more complicated situation it's not a onedimensional problem they still have a diatonic molecule but it has this threedimensional electron of that's shaped like a football and it has components in all 3 dimensions and so it's a secondranked cancer so we have to break down our new in 2 x y and z components and so here's what these are and again if if you're not quite comfortable with her where that came from going review no conversions tween competition and spherical coordinates systems again so and here I mean little bit careful to try to call it new induced because this molecule may or may not have a viable on its own it's got 1 induced by the electric field and that's what we're interested in Invitational rollin OK so this thing has annexed component the electric field also has an x y and z component so applying electric fields in whatever direction and it's it's got components each dimension OK so then when it makes sense to do is to Split up my induced by Paul In terms of the polarize ability contributions from the parallel and perpendicular orientation of the molecule so remember we sat here to a 1st approximation it doesn't matter where the molecules pointing like this like this it just matters whether it's Carol perpendicular and so we can rewrite that in terms of Alpha parallel and Alpha perpendicular to hammer out here is our polarize ability and it's time dependent it's going from heralded perpendicular twice petition and so we can simplify this expression for induced by Paul right so again remember the point of of wire working through this is that we won a seat you know what happens to the transition Bible so that is the transition is allowed or for the view that the molecule shows us something Invitational Roman spectrum only if the transition dipole nonzero so here is a transition dipole we have quantum numbers for rotational states and we have the x y and z component of the induced dipole sandwiched in between so again we've got our final state and our initial stage and it's connected by the operator In the operator in this case is this transition dipole that is an interplay between the plausibility of the molecule in the applied electric field that's coming from a spare ,comma last year did that's right John M. Geraghty the quantum numbers for the for the angular momentum and I I put both of them in here nearnaked general yes it was yes thank you sir the director of the and history of the only thing I can say is that it is but that's a really good question so you know that the the question is do you have to have a you have to have a component in x y and z for it to be allowed now attending Obama's allowed the allowed unions you'll see a spectrum OK so here's our
26:02
dipole again and you know see we again hinted wave functions with the Brok notation which looks nice and clean now let's go back to what the wave functions for region where are they are spherical harmonics and they're kind of of big and ugly but you can you can see the cemetery easily enough if you plot so again this is something to look up if you need to do is go check it out in the early chapters of the book or on the ball from website OK so here's some spherical harmonics will and we think we can use a trade identity To simplify expression for the new Stifel In this case and so if we're looking at the transition between 2 rotational states it has 2 components 1 has to do with the polarize ability and the other 1 has to do with that the electric field and so this power base is 0 and last Delta 0 so this is the 1 that contributes to the really line so this is the you know transition in rotational energy that photon hits it just bounces off elastically and it doesn't change frequency and so if you're doing this experiment to find out something about the molecule this is really boring so he shined a laser at it in the same color laser light comes back off and that's all that's all you get the other 1 this is 0 and last you have a change Invitational state of course minus 2 and you have to have an isotropic forays ability so that Delta Alpha has not equal 0 so between between parallel and perpendicular you have to have a difference enough otherwise this whole thing is going 0 yes there would be no way to know what the JI People's Daily Post reminders to an Delta Delta alpha does not equal 0 right because if that is the change in employers ability equals 0 then that term is going to go 0 right away just making sure everybody's away thank you very much for protection OK so we can put this in terms of an integral involving spherical harmonics but you get most of the takehome message just by looking at best OK
29:05
so just for completeness here's what you'd get if you right outside the X and Y components this is not so the details are not the most important thing is to know right now but I just wanted right enough for completeness OK so let's now that we have that we got through this election rules and we understand where they come from looks talk about how this actually gets used in a practical sense it's it can be hard to make the connection between the examples that are simple enough that we can work out in class and what you would actually doing research contracts a pace of 1 of the things that I R & Roman Specter argues for most frequently is identifying functional groups in organic molecules and as you can see here so here's a specter here's a spectra of styrene and heated dying lover so this is a real material that somebody made and they're interested in knowing what some you know what's going on in its structure and I want you to notice a couple things about it so what there are these these peaks in the EIA and Rahman the plotted on the same scale so here we have this in wave numbers it's given as Roman shift but you know the IRA 1 is on the same scale and so a couple things to notice some of the I enrollment bands coincide so you'll see some vibrations that have a transition in both the I R the Roman spectrum just like you know when we do our cemetery analysis and we come up with you know some of the vibrations of molecules active and run an active and some them 1 or the other so here we see for instance this set of transitions is really strong in both but if we look at this 1 that has a big peak the Roman spectrum then only a tiny 1 in the eye are so what that means is you know we have some vibrational mode that really changes the electrical ability a lot of it means there's there's a lot of overlap between those 2 states when they're connected by that polarize ability operator but in the vibrational In the transition between vibrational states it doesn't change the dipole moment very much so again the symmetry arguments alone don't tell us anything about magnitude it just tells you whether there's people there but you have to actually plugin what the values are and calculate the intensities if you want that information Stephen have something like like this where there is a huge value in the Roman and a tiny little 1 In the I and symmetry will just tell you that they both exist that that's it on other things that that useful to know here for a really complex molecules of water times on if you're like so for instance if synthetic chemists it's not really worth the the effort to fit every single band in the spectrum and really understand everything about all the details physical chemist like to do that and that's you know that's an important part of the peak camp researchers trying to understand why things are the way they are for synthetic chemists this is still incredibly useful even if you don't go through and fit everything because people have really well documented tables of where bonds belonging to certain functional groups show up in the spectrum so you've probably use this in Labastida some of them some of your drinks and better research you can use it to just see what kind of functional groups you having a molecule it's very powerful here's another 1 that's even more complicated where somebody has assigned you know what types of bonds are likely to to vibrate in these regions and so you again you see things like the CEO stretch shows up in both but it's a lot more intense and the Roman which kind of makes sense if you think about it Kerviel that's stretching out that is going to change the dipole moment a lot more than changes the Flores abilities to some memories he visualized and rationalize some them myself if you actually want to get into all the details and c how this works you have to do a computationally and people do have pretty good success predicting these things and an understanding all the year the modes computationally on a good example of somebody who does this for research is Philip 1st at UCI so that's not something that his group because there are many others in different places OK so what else can we do With vibrational spectroscopy in a more advanced contacts were really interested in physical chemistry this is an example from the website of Charles Harris and his group at Berkeley and they're illustrating how we can use vibrational spectroscopy in a timeresolved way so instead of just taking a spectrum at a particular point in time and and measuring the bond vibration they're interested in looking at some chemical process that's happening and using fast laser pulses to capture snapshots of the system as it's undergoing some kind of a change and so here's the schematic picture of what that's what that looks like there's a chemical reaction and it's going between different States here through throughout the course of the reaction and they're able to approach this thing with by our calls and look at it at different times .period and actually
34:55
watch bonds being broken informed so they have their ire spectrum and and as as they watch it as a function of time they can see vibrational modes "quotation mark corresponding to 1 set of bonds disappearing and another wondering in so you can actually use these kinds of things in a timeresolved way to learn something about the processes that are happening in real time so that was an example from
35:22
Berkeley here is an example of another buyer the application from UCI services from our professor grows louder 1 of the things that she does is twodimensional air spectroscopy so in this case we have all Ireland spectrum the news it's basically each of these 2 dimensions is correlated with higher spectrum with that gives us is something looks like a topological maps were looking down on it and we're going to talk about this in a lot more detail as far as how it actually works when we get into twodimensional and mark but for now just treated as it's 2 spectra their correlated to each other if you took a slice through vertically or horizontally you would see the normal onedimensional infrared spectrum and what you get by doing this in a correlated fashion looking at it this way is you see connections between different vibrational states and so what that is telling us is therefore work for a large complicated molecule like this this peptide in solution if you have 1 bond vibration somewhere in the molecule it's not completely independent from everything else if you will respond over here that's going to affect house 1 on a different part of the molecule by rates that's what the speaks telling you their correlations between pairs of vibrations and some I'm looking at that you can learn interesting things about the molecular structure you know how it interacts with with the solvent through its environment and also how it changes over time OK so
36:57
here's our Grand summary for vibrational rotational spectroscopy were for the most part on talking about it this able this enables us to learned things about molecular geometry for simple molecules we can identify functional groups and figure out what bonds are being destroyed a form for more complicated ones and of course we can look at things that are as complex as we always have to proteins but in that case we need a lot more sophisticated computational methods to be able to fit the data and understand what we're seeing and for those of you who are interested in doing research these are the groups that you are doing vibrational spectroscopy research at UCI and I'm sure that I forgot at least 1 group but here are some some examples yeah I forgot to professor whose career off that's that's 1 and as I mentioned Professor version looks theoretical modeling of different kinds of spectroscopy including but not limited to vibrational spectroscopy so the have I couldn't do that not forgetting someone right so we are done a little bit early today that's because the minister Brown and answer questions and good luck on the exam await the there is 1 thing I want announced for those who didn't see this job market is having a review session this evening it's 8 to 9 20 PM and it's Roland Hall 104 is right drama OK cool on TV me basic stick around 4 minutes after half past right Rodionov don't forget to check the seating chart tomorrow nite and make sure that you already know and set out for the exam
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Formal Metadata
Title  Lecture 10. Anharmonic Potential. 
Title of Series  Chem 131B: Molecular Structure & Statistical Mechanics 
Part Number  10 
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  10.5446/18918 
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 10. Molecular Structure & Statistical Mechanics  Anharmonic Potential. 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: 0:00:08 Anharmonic Potential 0:03:35 IR Selection Rules: Gross Selection 0:09:33 IR Selection Rules: Specific Selection 0:11:41 IR Selection Rules: Anharmonic Potential 0:16:59 IR Spectrum of NO 0:18:40 Raman Spectroscopy 0:26:56 Rotational Transition 0:29:20 Spectra of Styrene/Butadiene Rubber 0:32:50 2,5Dichloroacetophenone 0:34:02 TimeResolved Spectroscopy 0:35:19 2D IR 0:36:55 Summary: Spectroscopy 