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Lecture 14. Electronic Spectroscopy (Pt. III)


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Title Lecture 14. Electronic Spectroscopy (Pt. III)
Title of Series Chem 131B: Molecular Structure & Statistical Mechanics
Part Number 14
Number of Parts 26
Author Martin, Rachel
License CC Attribution - ShareAlike 3.0 Unported:
You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal and non-commercial purpose as long as the work is attributed to the author in the manner specified by the author or licensor and the work or content is shared also in adapted form only under the conditions of this license.
DOI 10.5446/18922
Publisher University of California Irvine (UCI)
Release Date 2013
Language English

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Subject Area Chemistry
Abstract UCI Chem 131B Molecular Structure & Statistical Mechanics (Winter 2013) Lec 14. Molecular Structure & Statistical Mechanics -- Electronic Spectroscopy -- Part 3. 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: 0:00:28 Term Symbols for Linear Molecules 0:04:29 Alternative Empirical Notation 0:06:48 Electronic Spectroscopy - Electrons are Moved Between Orbitals 0:14:11 Franck-Condon Principle 0:34:02 Transition Dipole 0:36:32 Weakly Allowed Transitions 0:38:22 l2 Energy Levels
1 he talking about electronics the cost of last only 1 over 100 you term symbols again some people have said that they don't like the way I do it they like doing the book doesn't matter that's fine it doesn't matter where you know you get the same answer either way just please do make sure you review it and and that you know how to do it it's
almost like any worried about term symbols for linear molecules form for this quarter we talked about this a little bit at the end last time that there are just some things there that I want to point out so 1 thing that people had questions about was the Harry G. and you know so whether whether it's even or odd when he reflected through inversion center a couple people asked well it's a linear molecule so you know most of them will have parity right said to things 1 as it could be a linear molecule were the 2 ends are different so could be see infinity the Infiniti age that's 1 issue the other 1 is that we're talking about the particular molecular orbital or the particular wave function of that state not the molecule itself and so the same thing for the plus or minus reflection symmetry so that along the plane containing into nuclear access and again the thing that we're looking at the symmetry of is the actual electronic wave function not not the molecules and so of course for talking about excited states the molecule they can have different cemeteries are it's so 1 consequence of this as we as we said last time is that you can't have transitions between 2 even states or to on states because the dipole moment operator is God and so Nissan which that between your state's you end up with a nonfunctional and selective as the origin of the selection rules electronic spectroscopy and let's look at some real examples so here is what we have for I to be looked at and potential land grant for this before but so we have we have a couple of other states here were not showing all but uses the ground state and an excited state so now we know how to read his terms we have a single state down here and we've got to hear it said signal state and then we can see that it has even symmetry and also it's got reflection symmetry with respect and in the access and then as so excited state up here Is it report stated and it is all so here we've got a molecule that is it's a diatonic were both Adams are the same but we can states with different symmetry that's just because the wave functions look different and also keep in mind that this x-axis is the bond lengths so you know we were talking about dissociation verses promoting electron to higher excited state what happens here depends on the bond like so as the molecules vibrating remember that the motion of the electrons is much faster than the motion of the nuclear so worried that molecule happens to be when the electronic when the electrons get excited that depends that affects whether we jump up to a different electronic state or maybe dissociate the molecule and different things can happen depending on word that the nuclear and we talked about that in a little bit more details OK so remember the rules for for naming these things are basically the same for as for atomic term symbols the best symbol is telling you about the total oral angular momentum and we're just using Greek letters instead of English letters and put the spring and will be more OK so there's another notation that you'll see In the literature in the peak can summon and I wanted to point out just that you know what it is so in this In this convention X is the ground state a and a B and C. we are excited states that have the same spin quantum numbers the ground state and small a B and C. are excited states with different quantum numbers from the ground state so again you know you don't happen to know how to generate this notation right now I just want I just want to know it's there because you will see in literature and you'll see it in seminars OK so
here's an experimental example of something like this from a paper that's a few years old but you can see how these things are are measured so in this case there are controlling which vibrational state gets excited by tuning laser pulses in this In this molecule so it's just a each enough in the gas phase and you can experiment with control which by regional states and an electronic states get excited and so then we have this terminology where we have the world had access the ground state and then a is an excited state with the same "quotation mark members against and so here the the collision between these molecules is controlled by laser pulses and you can get really fine control what happens in these processes this is something that's that's always really interesting about these old-school like the beam experiments it it's amazing that the final level of detail that we can learn about very simple molecules things things were you would think everything is known about it but actually there are a lot of interesting things you can do with that would really simple
things OK so let's get into the details of the mechanism of the school that more OK so the basic description of electronic spectroscopy is that electrons a route between different roles in the molecule as opposed to being ionized Mirasol little bit how that depends on where the nuclei are located of course you can always blasting a huge amount of energy in an ionized electrons over there OK so let's look at an example so here's of the electron configuration for nitrogen and you know this is our are diatonic molecule so we can draw molecular orbital diagram which is familiar from the General Chemistry and filling the electrons here and from in order from lowest energy highest energy and in this ground state we don't have anything but here In these high orbitals and if we put in the photo of photons of exactly the right energy so here that would be the 1 that has a wavelength of 145 that would cause 1 electron to get excited into 1 of these Hi orbitals so in that case you know we can see that there are various different ways it could go in right we have to work rules it can be the upper down so we we already know that we need to use the term symbols to describe the selectmen configuration more if we look at oxygen but in this case these a photon with a particular energy could give us 3 different states that are described by different terms symbols and so this gives us another example where we get something where the overall electron configuration is the same so this electron configuration is ambiguous because it doesn't really tell us about the difference the term symbols that we can get so we get treats with different than most multiple cities and with the different reflections cemeteries and so we
have to take into account that the whole election cloud changes state it's not just those upper electrons they're they're moving around and we really need the term symbols to describe what's going on so let's look at this in terms of the whole picture with the electronic and vibrational states so In this this should read the ironic states appear so rome vibrational states of our what we're talking about a vibrational spectroscopy we have the rotations and vibrations excited by 1 transitions are what we're talking about what we're looking at electronic spectroscopy were also exciting vibrational transitions of course rotational transitions are going on to what we don't usually have via the resolution to see them in the kind spectroscopy the RE so if we take our nuclear coordinator here so that we have a ground state and excited state and the x-axis is the nuclear ordnance a separation between nuclei which again is slow compared to the movements of the electrons each of the states is described by the total electronic wave function for the molecule and so state In just the 1 that has particular quantum numbers for the electronic and vibrational states so each of these guys is a away function and so we can take out some particular ones so here's 1 that has a vibrational quantum number 5 and elect the quantum number for its electronic stated 0 so this is called epsilon we can also College and Milicent is certainly in the case of Adams you'll see that so this is just 1 of these particular states that we can pick out and of course if we're going to have to look at this in greater detail we will need to know its rotational quantum number as well here were ignoring that were rescinding don't have the resolution to to see it and then a we pick another 1 in the associated with the excited states here's 1 words electronic quantum number is 1 and it's so irrational .period members I think we can write these and bracket notation to by just writing down the the quantum numbers of course you do that you have to be sure that everybody knows we're talking about the context OK so now if we look at the of the transitions you are subject to the same probabilities densities as any other kind of way functions we I have some ideas about where the electrons are going to be in there they had you in the excited states have nodes in them here just like any other way function but if we look at the dependence of the need of the hour excitation on the inner nuclear distance we can see that if the nuclei this position that electron is going to hop up to a different place than if it starts over here so for we don't have a lot of that electron density In the ground vibrational state at this in a nuclear distance that make sense right before starting to make the molecule by read a lot then it's going to spend more time being stretched apart then it is if it's if it's in the air and ground vibrational states and so that effects what kinds of transitions molecule can make so the nuclei are not affecting this process it in terms of moving on the same time scale but what it does mean is that the position of the nuclei at the beginning of the transition effects where the transitions can go and so that is so new that you can see that in the fact that we draw these expectations as vertical lines so weird that you there some assumptions in there somewhere so since were going excitation is vertical lines that's telling us that were you were using the born Oppenheimer approximation and were assuming that the nuclei are not moving around on the time scale of electronic transition and so
this can be stated as as franc common principle and conceptually it's pretty easy to understand but we need to go through and learn how to use it mathematically so that's just me you know the more formal statement of what I just said the nuclei a lot more massive than the electrons and so on the time scale the electronic transitions we can assume their stationary that really makes our lives like easier because it means that we don't have to 2 deal with all these wave functions in terms of the nuclear coordinates and the electronic ordnance at the same time we can separate the mn and so that also means that we're not going to have things like jumping up from this ground vibrational state in the lowest electronic state and coming over here In 2 the ground where racial state of this higher excited state there's a lot more density in some of these upper states here and so that's that's what makes them more likely to happen and so as you can imagine we need a way of dealing with this quantitatively so we already know a bunch of selection rules and and ways to look for whether enables them to 0 or not but that doesn't really tell us anything about the intensities of these things so we can see with the the electronic transitions now it matters a lot weather something has more less intensity so so far we've just been looking at you know does the cynical go exactly 2 0 but looking at these pictures we can see a lot of examples where something isn't actually forbidden by the selection role but it's not likely to have very much intensity just because there isn't a whole lot of overlap and so we do need to figure out how to to deal with that OK here just some more examples of the Franklin in principle this is for theoretical case but it just shows that these lines have different intensities and unlike the case of rotational vibrational spectroscopy where the intensities just depend on some combination of the 2 generously the states and their energy that's not really the case in these electronic transitions What's more important here is the overlap between the particular states involved so again here's a here's a real example for H 2 if we look at the transition between the ground electronic and vibrational state up to be excited electronic stating a bunch of associated racial states we can look at the transition intensities and we see that which lines or have stronger intensities depends a lot on the the between those particular states in terms of unit clear separation OK so what
kind this all up as far as what the French 100 principal means in practical terms so the relative intensity Of nearby ironic transition it depends on the relative horizontal placement of states and of course the horizontal distances the Inter nuclear course and so again here's what we're talking about when we're looking at that transition probability we have the transition dipole sandwiched between the final initial states and formerly we have to to sum up the effect of the transition dipole overall the electronic nuclear coordinates so Little R is the coordinator for the electrons and big is the 1 for the nuclei and these vectors here are the distances from the center of charge of the whole thing and again fortunately these things separable so we don't have to deal with that all at once the intensity of a transition is proportional to the magnitude of that transition which again as the is what we get we evaluate that integral and this is also a proportional to just the overlap between the states so the overlap b the square the overlap between the states is known as the Franklin factor I then again here's here's how we write that down you or we can react the full integral as well there :colon and so this remind you it S equals 1 there's complete overlap these things why right on top of each other in space and if as equals 0 there's no overlap and you'll be able to figure that out by usually using some sort of symmetry argument some and tourists some variant and even on Wall OK so if we
look at this transition dipole and a little bit more detail you we can write this out the 2nd step is I separated out things having to do with the electronic state and the vibrational state which again and what do because of the year 1 Oppenheimer approximation and we see that we get some some overlapping roles in here and of course if we have a set of of states that you know the makeup and working on the overlap between different functions it's 0 we just get he overlap for the Euphrates electronic states and so the overall transition moment has to do with it the overlap between vibrational wave functions in the upper and lower states involving nuclear coordinator so we're still talking about electronic transitions in the sense that we're moving from you know the the lower to the upper electronic transitions but Our overlap integral here is the 1 that's taking over these vibrational states and so we're going to assume that the ah dependence of our electronic dipole Smalls we can make some assumptions and that's going to give us the result that the relative intensity of a transition is the square of the vibrational overlap and so here I'm just writing this and in different ways so you should be able do and recognition but I've also written up in the role to and so that's what the actual Frank common factor it's this overlapping and role of the between initial and final vibrational states square thank you yet so think this year what that is the the average value so we're just we're just thinking that as the average OK so we're going to go back to talking about some symmetry arguments and switch gears a little bit before we gives anyone having more questions about Frank common factors or what were what we're doing here were agreeable to make these approximations yes it yeah exactly so that the Francona factors just telling us what these relative intensities are amino again they are relative like it's not giving us an exact number of Fortis if orders during the static population we need to use some more hardcore computational methods to 2 get that but we can at least compared them and say which ones are the more likely than others and in a relative sense those numbers will be accurate but also say about this I Oh yes don't get confused about you know we we canceled out the overlap integral the electronic states because they're all orthogonal to each other or normal to each other so if it's not the same statement that those 2 0 why can't we do that for the vibrational states to and say nothing of happens anybody know it's because they belong to different electronic states so we're talking about a vibrational states you know the starting electronic state and another regional state in the final electronic state OK so this is really important because it does enable us to get some relative intensities have so far only talked about selection rules and differences in whether transitions happen or not we've only worried about sort of yes-or-no answers you can can we say that the cynical goes to 0 or no it doesn't know now we have a method for looking at relative values lease for electronic transitions and it is kind of interesting that it depends on the corresponding vibrational states OK so much talk about our symmetry arguments a little bit more now that we have some more tools at our disposal for for looking at these kind of functions OK so selection rules in any kind of spectroscopy depend on a transition Bible and so what that means is we just we sandwich the transition but the dipole operator in between are final initial states N then we might look at this and see duties intervals go away by a symmetry of course if they don't know we have a method for finding out what they actually are but I do want to go through this again because I think that you know something that the people have trouble with looking at it 1st and now we've we've seen some more examples of different states in different cemeteries we can use so again we want to look at these kind of minerals and see whether something goes to 0 the 1st thing we have to do is find a cemetery species of each function and there are different ways to do that so we've seen examples where you just look at the the object that while working with and compare it to some things on the character table on CEO pay is there anything that we know about like say a PX P wire Keesee orbital that matches the symmetry of or another way to do it is to look at how your function transforms when you perform the various cemetery operations and whatever .period appearance and greater reusable representation for it and then reduce it that's another possibility and if we want to do this for 3 functions here and see if if the managers symmetry we defined species in each 1 and then we have to multiply the whole thing together and if that gives us a reducible red representation we need to reduce it or at least figure out the number of of anyone symmetry species in and if the answer that you get when you do that doesn't contain and 1 that means it has no overlap and again what's special about 81 that's the species where it's invariant all transformation so we get 1 as the character for everything and it's not always call a wide and some pointers it's called a here there are there different names for but it's the 1 word you get ones for every operation and if you think about that for a minute that makes a lot of sense talking about you know do we have overlap because what that means is that it's invariant to all transformations so for thinking about say a chemical bond of we wanted over to orbitals can overlap to make a bond that want better be invariant to all transformations that you can do in space so if you have a bond that vanishes when you uterine molecule over the other way that doesn't actually have any overlaps even remember it that way OK
so now what we need to do is take the size symmetry stuff that we've we've seen before and used for some things and apply it to Our electronic spectroscopy problems so the same thing we've seen before August arsenal that different ways so can export as radiation-induced transition between 3 easy orbital and the 3 DXY orbital and to the molecule so there's a lot of information here so the molecule belongs to you c to so that tells us what what .period to to look at and the character table to determine how you various things behave under the cemetery operations and that's important information another important piece of information is that the radiation is exported that means that we only have to worry about the the X component of the dipole operator if I didn't tell you that if I just said you have some photons although that would mean that we have components in all 3 directions and you would have to check each component of the vocal moment operator x y and z but in this case since we know it's acts we can write down the transition people like that so here's our integral inappropriate got the X dipole moment operator sandwiched between the 2 states and we need to see if that goes to 0 and so in this case finding the house of the various functions transform under the cemetery operations is easy because they're all things that are listed on the character of seeking does look at your character table and read this right off so we need to figure out you know which symmetry species DXY X's and Y's eastward along to but again you can just read the suffering of character table so DXY it belongs to the 8 2 so tree species and you can write on the characters for that and X belongs to be 1 and easy squared is a 1 and so now we can multiply these things together and remember that the the rules for that part you just multiply the coefficients in each column and so we multiply the whole thing together in this case we get the to and so we know that that doesn't have any overlap because we didn't get we didn't get any of the 8 1 component that had that's invariant all transformations so again in this case it was a simple example because we're just looking at things that are there already was on the character table we ended up with and irreducible representation at the end of we just look and see which 1 it is we know it's not anyone the careful because 1st of other examples you might have a case where you get a reducible representation at the end and that doesn't mean you can quit you have to reduce it and make sure that it doesn't have any it once before saying it so it was oral question the state of preservation it was a good thing so if you're if you're actually shooting electrons at your sample that's so that a different process so that you are who you might have some interaction but it's a it's probably not going to be exactly the same thing going on yes you have to look at all the of of the you recently wrote yeah that's a good question so if we if I just said we're shooting radiation that this thing can this transition happens Mennonite case you have to try this with x y and z separately but there's there's no way around you have to check what happens in all 3 dimensions and if you're in a point group were say 2 of them are degenerate for all 3 of them were degenerate that's a good way to get none review in order to get non-reusable representations at the Braves are multiplying these things together that have different coefficients yet the Hi how do you tell that you got a reducible representation at the end will also if you do if you don't get something that's already in the character table then need to reduce so like in this case you know we have 1 minus 1 minus 1 1 we can look at that and say OK that's already in the character tabled the 2 if you got something that is not already there then you need to reduce it and if you've got something that you can't reduce and it's not on the character table that means mistake well you know it sounds obvious but I always like to point these things out because you know things and things like that often happen on the exam were the up sometimes I think people running time but there definitely clues that it's time to go back and check your work OK so the transition is allowed if the integral for the transition dipole is non-zero and so then we've already seen this would just bring it back in a different context now that we know more about these different states and the symmetries thereof and hopefully it's a little bit more concrete because we're talking about actual warbles you know here we've been using examples of atomic orbitals just because we all know what the shapes are in their easy to visualize In the practice problems you'll see some things where there are some of the molecular orbitals that have different shapes and maybe it's not so easy to visualize and have to calculate some stuff OK so again if the polarization of the radiation isn't specified you have to consider all these components separately the guy took it fortunately there are only 3 choices OK so all of these things have you cemetery so if you just think about you know X acquires eh these are all on functions and what will see Is that segment Sigma transitions are related to these things have lost symmetry if we're going from plus 2 minus that's usually forbidden just plus is allowed and again remember all these rules only pertain to Center symmetric environments so things that happen in version center and that's a year that works pretty well it mostly in considering near molecules were talking about actually calculating his transition also just because stuff gets a lot more complex for larger molecules and that doesn't mean you
can't do it it just means you can't do do it easily in class with paper and pencil and that's where you need on you need Mathematica the or more sophisticated software for calculating an electronic transitions OK so I should point out that there is also such a thing as weekly allowed transitions so you can have something that is center symmetric services this molecule is opting has an aversion center and so you would think that this electronic transition is not allowed but you can have a case where vibrational modes break cemetery and again remember all these things like vibrational modes include the nuclei around our slow relative to the electrons so the excitation of the electronic transition can't catch the molecule in a state where it's bent in a way that the cemetery's broken we noticed that happens we've we've already talked about you know vibrational modes that change the dipole moment and and we know that these exist so we can have a cemetery broken temporarily but long enough for an electronic transition happen and we'll see some weekly allowed transitions so here's what that looks like in terms of an energy level diagram so we can have states where if if he did not have coupling than to this operational transition of breaks cemetery we wouldn't see any transitions but sense of vibrational state does break it a little bit that gives us enough time for the transition to happen OK so the last few minutes I wanna get back this picture of the army the eye to energy levels and some experimental data and look at what we can actually do with it OK so again here's our our picture of the the ground state and an excited state a 2 and you know notice we've got a and B. drive in here AT is another excited state but all of irrational levels on Otto Braun and so we're looking at the transition between exit the ground state and vibrational states and be so 1 thing you notice now that we know a little bit more about this is that these things are offset quite a bit in terms of horizontal displacement so we're not getting up to this excited state was the molecules is displaced from its site equilibrium position we know some things about this potential that we're going to see when we start calculating some some things from the data so 1 of the things that we needed to know how to do using electronic spectra is to look at the Spectrum and be able to infer some some things from the data about what this potential structural looks like and we've already done this for vibrational rotational spectroscopy so from the vibrational rotational spectra variable look at the Spectrum and you can see where does the main transition happened and then we can look at the spacing and get the rotational constants and use that to calculate things like the 1 length on the force constant and things like that from the we can induce some similar things electronic spectroscopy so things that we can get out of it include the vibrational frequency the dissociation energy of the molecule we can also get the you know the relative energies of the electronic states we can learn something about the potential so this on XE here wave numbers is the and Herman is the constant of about potential and notice that these potentials have different shapes so like particularly if you look at a hearing compare to ground state of the the potentials look very very different and so you're going to have a different and more ominously constant for each of these electronic states mn new prime this fever is the convergence limit that's where things start to look classical so new Prime is what that's the vibrational state of the chopper electronic state states so we're going from new double prior to new prime so why do double crime and new Prime Minister of New new prime animal that's that's convention so that steps were these are displayed descriptions of the vibrational states OK so here's what we see when we look at the electronic spectrum of a molecule like us so we've got all these individual lines that correspond to transitions between the vibrational states so you know here were going from you double granted to new crimes and will have a whole bunch of on that we can see and we've got a bunch of them that came from the lowest vibrational state of the bottom electronic state so that's Nunavut prime 0 at all these lines correspond to that going up to different vibrational states of the upper state then we also have a whole bunch of them that came from the new equals 1 state about lower electronic state so we have a bunch of states that started here and jumped up to various excited states and then we have a bunch of them started here and did the same thing and it's a little bit complicated because they all overlap with each other but if you know what you're looking at you can see that's you know we've got abandoned them here and then we've got another day on growing in that other excited vibrational state and so if you actually measure this data which you're probably not going to have to have to because this is so you it's something that yeah they're basically there are better ways to analyze this if you are trying to do this for much more complex molecules but you can actually do this and so you can make up lot of your own "quotation mark number that that you observed versus the change in frequency that you see which is called a British Phone replied and that is going to tell you some interesting things about the potential so Europe y intercept is going to tell you I am new prime so that's so instead that tell you something about the characteristic frequency about electronic state and then the slope you get in terms of the In her ministry constant for that potential see you can do this for various so excited states and learn something about the shape of the potential and also the X intercept using the convergence limit so that tells you where the topic potential and so on here's that part written out it also tells you about the dissociation energy N you learn about the various energies of the states I think let's leave more detailed discussion of this for next time there is also a practice problem for you you can try doing this and other than that we're pretty much done with my colleagues across the street the visitors for the next time
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