 Today we move on to the last or maybe second last topic that we will discuss in this course. As we said in the last module, we want to discuss two dimensional IR spectroscopy and then we want to discuss a little bit of surface some frequency surface non-linear spectroscopy. Now looking at the number of lectures already delivered and the number of hours prescribed for an NPTEL course, I am not very sure whether we will have time to go into this surface non-linear spectroscopy bit. But let us see how far we get. So today what we essentially want to do is we want to start learning what happens when we add one more dimension to the pump probe spectroscopic technique that we have studied and our discussion will be limited to the infrared region. But then after this if you read papers on say 2D electronic spectroscopy, the principles are pretty much the same. So I hope that nobody will have any difficulty understanding 2D electronic spectroscopy as well after this course. But we will limit our discussions to 2D IR spectroscopy and today we will get introduced in this module we are going to get introduced to this topic. Before proceeding further, let me acknowledge the contribution of my friend Dr. Shukendu from Bhabha Atomic Research Center. Shukendu has set up a 2 dimensional IR spectroscopy spectrometer in BRC and he is a real expert. I have learned 2D IR spectroscopy from him and some of the material in the slides that I am going to show you are actually from a presentation that he had made in our department. The other 2D IR spectrometer that is there is in Isar Pune in the lab of Professor Bagchi. So here goes what is the meaning of 2D IR spectroscopy. Before that let me remind you of something that we had studied about 10-12 modules ago. We had talked about vibrational spectrum of liquid water and we had said that when we talk about liquid water the molecules are all associated to each other by hydrogen bond. So, first of all this vibrational mode comes up which is not there for isolated water molecule and we saw that these modes are all coupled meaning if you can excite one mode then the vibration energy gets redistributed in an ultrafast time scale into other modes as well. I hope we have we all remember this discussion that we had made. So, 2D IR is sort of the next step of what we have studied in the course of discussion of this topic. There if you remember we did an IR pump IR probe we learned an IR pump IR probe spectroscopic technique. Here what is the how do we add another dimension and what is the advantage? The advantage is we get an idea of coupled vibrations. In the example discussed earlier we had to look at the time constants and then from there we had to get an idea of what kind of coupling is there. Here we see we get another very useful and interesting feature in the spectrum that comes up if you had one more dimension. But let us take it slow let us go step by step. Let us say I have an IR spectrum where there are two peaks W omega A and omega B. Now there are two possibilities one is that omega A and omega B arise from two different isolated normal modes of vibration of the molecule. I am talking about non-associated isolated molecule now right as we know molecules have these normal modes of vibration and each normal mode can be modeled as harmonic or an harmonic oscillator. Now let us say we have oscillators that are not coupled with each other then we expect to get two bands like this due to two transitions like this. But from this spectrum can we say that the picture is what we have drawn already and not this one. How do we know that the vibrations are not coupled? In a coupled system when you excite one vibration energy can be transferred to the other. So actually two quantum numbers are required one for each oscillator. This is something we will elaborate upon later. So looking at this spectrum there is no way in which we can say whether we have a case of isolated oscillators or coupled oscillators. If you read this paper in the published in 2001 we will see that they had studied IR spectra of tri-alanine and they had done isotopic substitution. So the IR spectrum of tri-alanine turns out to be something like this and here they focused on a particular region of the spectrum 1550 to 750 centimeter inverse. This is where the so-called amide 1 stretch amide 1 vibration shows up. Now there are several amide bonds here. It is possible that the amide bonds vibrate by themselves or maybe they are coupled. How do you know from this spectrum can you tell what is happening here? You cannot. But if you do 2D IR spectroscopy and here I am jumping the gun a little bit and showing you a 2D IR spectrum already. What you see is that this 2D IR spectrum is actually a 3-dimensional plot. On one axis we have pump frequency, on the other axis we have probe frequency. There is a third axis pointing out of the projection towards you or towards me. That axis gives you the intensity or absorbance whatever you choose to plot. So whenever we have a 3-dimensional plot and we have to draw it on a 2-dimensional paper or 2-dimensional surface it is most convenient to show it as contour diagrams. These contour lines essentially join all points where absorbance or intensity is the same. And what these contour lines look like they represent is that they represent hills. So this contour line outside is has the lowest magnitude. The point inside has the highest magnitude. As you go from out to in you see you can get a hill or you can get a trough if the sign of the absorbance is negative. In this course at this stage we are familiar with negative absorbance what we are really talking about is delta A. As we know for ground state bleach and for stimulated emission you actually get negative delta A signals. So depending on that usually it is color coded to show whether it is plus or minus and then you get contour lines like this to represent the 3-dimensional surfaces. 3-dimensions remember are pump frequency, first dimension, probe frequency and delta A or delta T or whatever you choose to plot is a third dimension. So this is conventionally called a 2D spectrum. Those of us who have studied NMR spectroscopy might be familiar with 2D NMR spectroscopy. In fact that came first the idea of 2D IR and 2D electronic spectroscopy borrows heavily from the understanding developed already from 2D NMR spectroscopy. So if you studied 2D NMR you would know that by 2D NMR one can actually understand what kind of coupling is there between different nuclei and from there one can predict structure function and so on and so forth that is why 2D NMR is very useful in elucidation of structure of complex molecules like proteins. So in IR spectroscopy what one can do and what has been done in this paper that we are citing is that actually coupling between different normal modes have been worked out and from there it has been shown that one can talk about structure and not only structure the advantage is that since this vibrational coherence is all decay in ultrafast time scale one can talk about ultrafast dynamics of evolution of structure by looking at how this coupling changes as a function of time that is the appeal of 2D IR spectroscopy to the ultrafast community. Now let us go back to basics once again and start from something that we have discussed many times in this course and that is pump probe spectroscopy. Here remember we are using ultrafast pulses so spectrally there is a lot of width. Band width is significant for ultrafast pulses as we know so we are talking about broadband pump probe spectroscopy and for the purpose of the present discussion we are talking about broadband pump probe spectroscopy in the IR range. Let us say these are the energy levels of a particular normal mode in a polyatomic molecule. For our purpose we will only talk about V equal to 0, V equal to 1, V equal to 2. One can talk about V equal to 3, 4, 5 but at least to start this is enough. So as we know in ground state at room temperature only V equal to 0 is populated for all practical purposes. So suppose I pump this we call this process 1 and corresponding band that we are going to show will be called band 1 we have pumped V 0 to V equal to 0 to V equal to 1. If we do a pump probe experiment and if 1 is a pump then what I can do is I can probe different regions. So this is the broadband pump that I am using and the probe that I can do is first of all I can probe the same region. What is it that we will get if we probe 2? 2 is essentially the same spectral region as 1. So what we can get is that we get contribution from ground state bleach of V equal to 0 and we can also get stimulated emission from V equal to 1 and they add up to give you a negative signal as we know. So we expect a negative signal like this. What happens if we probe V equal to 1 to V equal to 2 region? Let us call that region region 3. There we expect a positive signal is not it? There is going to be a transient absorption. Now if we vary the delay between pump and probe then we expect that this signal is going to decrease is going to become smaller and smaller at sufficiently long time is going to become 0. This is something that we know already. But now the additional dimension that comes is that since it is a broadband pump let us say we have the capability of exciting using narrower band light spanning the range of the broadband absorption of 0 to 1. Then what happens? Then I can record these transient spectra for each of these pump wavelengths and I can plot for again any given delay let us say I can plot a 3D plot like what we have discussed already. Pump frequency on one axis, probe frequency on the other, intensity or absorbance on the third axis represented by contours. So what do we expect? What happens when we pump at 1? Pump at 1 let us say the frequency is nu 0 1. We expect a negative signal for probe frequency of nu 0 1 as discussed already. We expect a positive signal for probe frequency of nu 1 2 when nu 0 1 corresponds to the frequency matching the energy gap between V equal to 0 and V equal to 1. Nu 1 2 is the frequency corresponding to the energy gap between V equal to 1 to V equal to 2. Of course, if this is a harmonic oscillator then nu 0 1 will be equal to nu 1 2 but for an harmonic oscillators they are going to be different. So what do I see? Do I see a point here and a point here not 3D because the thing is this think of the pump axis. So when I scan from say lower frequency to higher frequency of pump whatever signal I get here it is going to go up from this side to that right pump frequency lower to higher, lower to higher absorbance is increasing and then going through a maximum. So whatever is the intensity that I get at nu 0 equal to 1 magnitude of it remember we get a negative signal at nu 0 1 magnitude of it is going to go up and go down along this axis not very difficult to understand. And then if you look at the probe axis this is the probe axis here also for any given pump the magnitude of signal goes up negative sense here and then goes down until it becomes 0. And I am saying go up and go down I am only talking about magnitude. So what do you expect? You get a distorted well you get a 3 dimensional not really Gaussian not necessarily Gaussian distorted Gaussian kind of shape. So at center wavelength of nu 0 1 we expect this kind of a shape the negative signal due to here blue means negative red means positive negative signal due to ground state bleach of 1 and transient well stimulated emission denoted by 2. So we get a 3 dimensional surface and at nu 1 2 at the intersection of nu 0 1 pump and nu 1 2 probe we get a similar signal but positive. What is the difference between the maximum point in the positive signal and the negative signal not very difficult to see from here to here the difference in frequencies. So that would give you the difference in frequencies of nu 1 2 and nu 0 1 modal frequencies that is not very difficult to understand. But that has already introduced us to 2DIR this is the simplest possible 2DIR spectrum that one could think of. Now let us make the situation a little more complex because as we said earlier the appeal of 2DIR lies in the understanding of coupling between normal modes of vibration. So if coupling has to happen then you should have 2 normal modes. So now let us see what kind of 2DIR spectrum we expect when we have not 1 but 2 normal modes of vibration. To start with let us talk about 2 isolated modes 2 isolated modes denoted by these 2 potential energy surfaces since they are quantum oscillators the energy is quantized and here we have shown v equal to 0 v equal to 1 v equal to 2 and just to ensure that we do not forget that this mode is different from the other one we have represented the vibrational quantum numbers in the second mode as 0 dashed 1 dashed 2 dashed. So the way I have drawn it here it might look like the modes are similar they do not have to be the shapes can be different the energy minima have to be different this is not is a not 2 scale diagram. But now if we zoom in forget about the parabolas for the moment look at only the vibrational energies and zoom into the first one we already know what kind of transitions we can expect for the first normal mode and it is not very difficult to figure that we expect very similar kind of transitions for the second normal mode as well. Here instead of 1 we have written 4 for the 01 pump the instead of 2 we have written 5 for this v dashed equal to 1 dashed v dashed equal to 0 dashed transition and we have been 6 instead of 3 for v dashed equal to 1 to v dashed equal to 2 dashed transition. So what kind of IR 2D IR spectrum do we expect when the modes are isolated there is no coupling whatsoever they do not talk to each other. This is what we are going to get right now you have not shown any peak here but for the sake of understanding I have shown these spectra where the mode 1 and mode 2 are. Now when we pump mode 1 which means we pump here then what do we expect to get we expect to get a negative signal for 2 which will occur in this region and we expect to get a positive 1 for 3 which will occur also in this region this will be at lower frequency because for an harmonic oscillator nu equal to 1 to nu equal to 2 gap is smaller than nu equal to 0 to nu 1 to nu equal to 1 energy gap. So we expect this kind of a feature that we have discussed already negative signal due to ground state breach of 1 stimulated emission of 2 positive signal for transient absorption pathway 3 fine but now say we have the capability of scanning the pump wavelength. So we do not have to pump necessarily at mode 1 we can pump mode 2 also which means we can pump here what do we expect to see we expect to see an exactly similar feature not here but here in the region of mode 2 frequency again we expect to see a negative signal and a positive signal not very difficult to understand. So in a no coupling case we can expect to see modes along the diagonal this diagonal here represents the pump frequency equal to probe frequency situation. So we expect as many positive and negative pairs as the number of degenerate sorry as the number of non-degenerate vibrational normal modes we expect along the diagonal as many positive and negative pairs as the number of non-degenerate vibrational normal modes in the molecule. This is what we expect when coupling is not there. Now let us say coupling is there which means if I pump one normal mode then it can transfer the energy to the second normal mode as well and cause transition of the second normal mode from v dash equal to 0 to v dash equal to 1 state. To discuss such a situation first of all this is one way in which we can show coupling by a dual energy minimum kind double well kind of potential with a potential barrier. Now what I have drawn here is symmetric double well but the most general case would be an asymmetric double well this would be lower or higher the way I have drawn it here it should be lower. Now what happens since the system is coupled you cannot really talk about the 0, 1, 2 vibrational quantum numbers and 0 dash 1 dash 2 dash vibrational quantum number separately. To designate any particular energy state we need to specify both the quantum numbers as is shown here. So the lowest energy quantum number will be 0, 0 dashed which means this is how it is populated this normal mode is in the 0th state this normal mode is the 0 dashed state. And then what one can do is using light of suitable frequency one can do a promotion to 1, 0 dashed state 1, 0 dashed state would mean that this normal mode has undergone a promotion this has not. So 1, 0 dashed means the second normal mode continues to be in the 0 dashed state but the first one goes to a higher energy one state or you could have the other way round 0, 1 dashed 0, 1 dashed simply means no transition in the first normal mode 0 dash to 1 dash transition in the second normal mode. So these states are basically the same as the isolated ones we have just had to re designate them so that we show both the quantum numbers together. Similarly one can understand what the meaning of 2, 0 dashed and 0, 2 dashed is but that is not all as a result of coupling one needs to think of some other states as well. One state that arises is 1, 1 dashed state where the where both the normal modes have been excited to 1 dashed. It is not necessary that only one normal mode is excited right one can have both the normal modes in the excited state that is 1, 1 dashed normal mode. It is important to understand that the 1, 1 dashed normal mode can be produced in 2 ways. You cannot go from 0, 0 dashed state to 1, 1 dashed state directly and once again for those who have studied NMR spectroscopy I would like you to think what happens when you talk about 2 nuclei. Suppose you have alpha, alpha you cannot go from alpha, alpha to beta, beta by itself because that would require in a single transition you cannot go from alpha, alpha to beta, beta right because one photon can only bring about one transition that is called one photon rule I think we have talked about this earlier in this course as well. So, in one photon transition you cannot go from 0, 0 dashed to 1, 1 dashed. So, only one normal mode can undergo excitation when one photon impinges on the molecule. So, you cannot go from 0, 0 dashed to 1, 1 dashed but one can go from 1, 0 dashed to 1, 1 dashed right because 1, 0 dashed to 1, 1 dashed essentially means the first oscillator is kept well sorry the second oscillator is kept what am I saying 1, 0 dashed to 1, 1 dashed means already the first oscillator is in the v equal to 1 state. Now one photon comes and all it has to do is to promote the second oscillator from 0 dash to 1 dashed state 1, 0 dashed to 1, 1 dashed essentially means promotion of the second oscillator from 0 dash to 1 dashed state 1 dash level when the first oscillator is already at v equal to 1. So, we will call this pathway 7 and it is important to understand that the energy of this is equal to the 0 dash to 1 dashed transition what else can we do we can do the other thing from 0, 1 dashed we can go to 1, 1 dashed also if you want to go from 0, 1 dashed to 1, 1 dashed then what we are doing essentially is that already the second oscillator is in the 1 dash level. Now the first oscillator is in 0 level one photon comes and promotes it from 1, 0 dashed to 1, 1 dashed state we call that pathway 8. Now let us think how this 2DIR spectrum is going to change if at all if we pump either mode 1 or mode 2 let us say we pump mode 1 in addition to pathway 2 and pathway 3 the other pathway that is available is this right because the what the pump has done is that the pump has populated the 1, 0 dashed state. Now the probe if the frequency is right can bring about a 1, 0 dashed to 1, 1 dashed transition that is a transient absorption. So, we expect a positive going signal and where do we expect the positive going signal remember we have pumped mode 1. So, this is the pump frequency and the positive going signal as we said earlier should appear in the same frequency as this 0 dash to 1 dash transition because that is the transition that is taking place here. So, we should get something that comes here where pump frequency is that of the of mode 1 but the probe frequency is the same as that of 6. The other thing that happens is that the moment you have this 1, 0 dashed to 1, 1 dashed transition the other thing that happens is that the second oscillator well not other thing the same thing the second oscillator gets promoted from 0 dash to 1 dash level yeah 1, 0 dash to 1, 1 dash as we said essentially is 0 dash to 1 dash promotion when the first oscillator is in v equal to 1 level. So, 0 dash to 1 dash promotion it is manifested in 2 things 1 is transient absorption that is 7 second thing is again ground state bleach. So, here we get another positive negative signal but this time it is off diagonal we have pumped mode 1 and we have got in the probe signature that we expect when we pumped mode 2 similarly if we now well and this is delta 1, 2 as we have discussed earlier. Now, if you pump mode 4 what will happen again the same thing will happen we will get another cross diagonal peak in this position because pumping mode 4 is going to bring about a transient absorption here in pathway 8 which is 0, 1 dash to 0, 1, 1 dashed remains 1 dashed the first oscillator goes from 0 to 1. So, pump wavelength will be for mode 2 probe wavelength we get the feature where we got it for pumping at mode 1. So, the significant new feature that we get if you perform 2 dimensional IR spectroscopy is off diagonal peaks. And as we have seen in the discussion so far off diagonal peaks do not arise if coupling is not there if you have off diagonal peaks that means the 2 modes have coupled very similar to 2D NMR spectroscopy and that gives us an idea about if we extrapolate further and interpret a little more that can give us an idea of the structure. So, take home message is that off diagonal peak in 2D IR spectrum is a signature of coupling. Now, after all this is an ultrafast dynamics course. So, it is logical to ask can we follow some kind of a dynamics using this of course we can we are doing pump probe remember. So, what we can do is in addition to scanning the pump frequency we can also vary the delay time between pump and probe like what we have discussed so many times earlier. Let us take this example where we have this situation let us say we have this molecule this C double bond O let us say or some molecule that can do hydrogen bonding let us say initially there is no hydrogen bonding new one is the wavelength of this vibration new 2 is the wavelength of this vibration. So, if there is no coupling no hydrogen bonding then we expect this kind of a spectrum only diagonal peaks. If hydrogen bonding is there then we expect off diagonal peaks this is something that we know already from our previous discussion. Now, see if the situation is such that the hydrogen bonding is not there in down state we excited and then in the excited state the hydrogen bond gets formed then what happens then with time we go from this non hydrogen bonded non coupled structure to hydrogen bonded coupled structure. So, time 0 we expect to have a 2D spectrum without off diagonal peaks the progress in time the off diagonal peaks slowly emerge and the dynamics of emergence of the off diagonal peaks gives us the dynamics of formation of hydrogen bond. So, this is an introduction to the 2D IR spectroscopy technique well this is an introduction and this tells us what we can do by 2D IR. Next day in well next module we are going to learn what happens rather how are we supposed to do it we understand that 2D IR spectroscopy gives us additional information but we have to do a frequency domain as well as time domain measurement how do we do it we will start with this simplest technique frequency frequency double resonance 2D IR and we will learn something about February pair filter but then we will move on to a little more complicated technique. So, in the next module we are going to discuss techniques of 2D IR spectroscopy.