 In the last class, so we discussed in detail an experiment called as double quantum filtered cosy and that was simply a pulse sequence which was like this and we had 390 degree pulses this is 95, 95 is the phase of 5 and this is 90x and the data was collected here and this is the T1 period and this is the T2 period and this is called as the double quantum filtered cosy and this is very similar to the cosy which was also discussed in the previous classes and that was just this you have here T1 period here and the 2 period here just one pulse less and this is 90x 90x but we could have other things as well. In doing so we actually demonstrated how double quantum filtered cosy produces some features which are very advantageous compared to the cosy especially in the diagonal of the two dimensional spectrum the double quantum filtered cosy had anti-phase character. So you had minus, minus, minus plus, plus, minus, minus, plus, plus, minus, minus, plus and this was in comparison with the cosy where it was all plus, plus, plus, plus here, plus, minus, minus, plus, plus, plus, plus, plus, plus, minus, minus, plus and these are dispersive line shapes here and these are absorptive line shapes. Here we had all absorptive line shapes for both the diagonal as well as the cross peaks and therefore it had a big improvement in the resolution in the spectrum. So that was particularly advantageous from the point of view of improving the quality of the spectrum. But notice also we demonstrated a new concept in that process and that is called as the selection of coherence transfer pathways. In other words different components of the coherences were selected in the 2 cases. For example if I were to take a point like this here at this point or at this point the density operator we had created had this sort of terms ikz then ikx ily and iky and ikz ily. So in the cosy experiment this were ignored this first 2 terms were ignored this is ikx and ikz ily. These terms were retained in the cosy experiment and in the double counter filtered cosy experiment we retained this portion of the density operator by suitably adjusting the phases of this pulses 95, 95. In fact this brings in a new concept a totally a new thought with regard to manipulation of the magnetization transfers and manipulation of the nature of the spectra. So that is something which is quite important and it becomes applicable in many other experimental sequences as well. So I am going to demonstrate you one more such application where we choose a particular kind of a magnetization namely the ikz we choose ikz here. So let us see how that can be done and that is what I am going to show you in today's class. So this is an experiment called as two-dimensional nuclear overhouser effect spectroscopy or 2D nosy. This actually has been discussed previously by Professor Ashutosh Kumar during his lectures on polarization transfer. So this is not coherence transfer in some sense that it is a polarization transfer but nevertheless it relies on choice of a particular components of the magnetization through the pulse sequence. This experiment has the following pulse sequence you have this 95, 95 you can actually have 90x, 90x here it does not matter but let us keep 95, 95 to illustrate the point of selection of coherence transfer pathways. So here we have 95, 95 and 90x here and the phase cycling that is used for this experiment is given in this manner. You change the phase of the excitation pulses xy minus x minus y and plus plus plus plus here. Notice in the double quantum filtered cosy we have the phase cycles with initial single quantum case to magnetization is converted into double quantum coherence between the spins K and L then transferred to single quantum spin magnetization and vice versa in the KL spin systems. And here the phase cycle was this was plus minus plus minus for double quantum filtered cosy but now we are going to use this remaining the same we will make this plus plus plus plus. How does that help us? So this we can see but through this product arbitrary calculation once more. I am putting the same 4 equations which were given earlier in the case of double quantum filtered cosy. The first experiment with phi is equal to x. Often we also refer to this x phase as 0 phase and y phase as 90 degree phase minus x as 180 degrees and minus y also as 270 degrees. So sometimes you will find references as 0 degree phase, 90 degree phase, 180 phase and 270 degree phase and conventionally these are used in the coding on the spectrometers you put it as 0, 90, 180 and 270. So when you have 0 degree phase for the phi or the x phase then the density operator row 4 is the same as it was in the cosy. We have here minus ikz cosine pi jkl t1 minus 2 ikx ily sine pi jkl t1 cosine omega kt1 ikx cosine pi jkl t1 minus 2 ikz ily sine pi jkl t1 sine omega kt1. When phi was changed to 90 degree phase then of course you had these changes here this remained the same this changed to plus 2 iky ilx and this changed to iky coefficient being the same and this changed to plus 2 ikz ilx sine pi jkl t1. Notice here there is a change lx here and ly there if you change the phi to minus x there is 180 degrees phase shift then we got this remained the same ikz cosine pi jkl t1 and this became again minus 2 ikx ily as in this particular case and the third term became minus ikx as in this first case here except that it becomes a minus sign and the fourth term became plus 2 ikz ily as compared to this one. So this is the change that happened because of the change in the phase of the acceleration here. Finally when you had the phase as 270 degrees or the minus y then this first term remained the same this change the sign became plus 2 iky ilx and then you have minus iky you compare with this this is the minus iky and you compare with this this is minus 2 ikz ilx sine pi jkl omega kt1. I put all of them in the same page here for ease of understanding how if you manipulate these additions and subtractions different terms will cancel out. For example all of them I now add instead of subtracting anywhere in the double counter filtered cosy what I did this was plus this was minus this was plus and this was minus. Now what I do I add all of them plus plus plus plus so now we can go back and see which are the terms which will survive this term has remained the same throughout so this will remain. Now look at this term minus 2 ikx ily and where does that appear again it appears again here is minus 2 ikx ily but appears with the same sign and therefore this does not go away. And similarly this 2 iky ilx appears here and it also appears here with the same sign therefore we add all of this this also will remain. This will remain as these 2 terms will 2 iky ilx and this is minus 2 ikx ily this will remain with the same coefficients. Now what happens here now this iky cancels with this minus iky and this ikx cancels with this minus ikx and what happens to this this 2 ikz ily cancels with this and this 2 ikz ilx cancels with this. Therefore I will have only limited number of terms from the addition of all of this and that is indicated here I will have minus 4 ikz cosine pi jkl t1 plus 2 into bracket 2 iky ilx minus 2 ikx ily sin pi jkl t1 and the whole thing is multiplied by cosine omega k t1. Now look at what these terms are this is clearly z magnetization only this is the z magnetization of the kspin. Now if you go back and look at the natures of the product operators this term here 2 iky ilx minus 2 ikx ily is pure zero quantum coherence. So which means by this the operation of the phase cycling we have retained the z magnetization and zero quantum coherences. Now both these are not observable right. So now what happens to make them observable we will need another pulse and that of course is another 90 degree pulse which is required. So that is there in the pulse sequence the last 90 degree pulse converts them into observable magnetization for us to measure what happens before that. Now but during the next period in the pulse sequence during this period tau we now have period called tau m which is the mixing time. We looked at the density operator at this point in time and we found that operation of this type we had only the z magnetization here and the zero quantum coherence and this tau m is of the order of few hundred milliseconds. Now what happens during this period okay the following things happens. The z magnetization recovers towards the equilibrium value of the kspin because it is a kspin magnetization it has to come back to its equilibrium magnetization along the z axis and this will be dictated by the T1 relaxation time of the kspin. Now the zero quantum coherence is a coherence therefore it is a phase coherence this will actually decay due to T2 relaxation time. During the few hundred milliseconds it will decay due to T2 relaxation time. Now what else we can do? So now what we are interested in the nosy experiment is the z magnetization okay we do not want the zero quantum coherence here we want to keep only the z magnetization that means we want to eliminate this okay partially it will go away because of T2 relaxation time during the long mixing time we might have. But then we can also do a trick to see that okay this gets cancelled out what is the trick we do so we can systematically vary this tau m the mixing time over a large number of steps. So suppose you are recording experiments at 400 milliseconds or 200 milliseconds you record experiments for 200 if you are doing record with 190 millisecond 195 milliseconds 200 milliseconds 205 milliseconds 210 milliseconds the average still remains 200 milliseconds but just around that number you vary the mixing time to a certain degree. So what happens then is so because that time is variant the zero quantum will acquire different phases so which means acquire different phases meaning now if you take an average a sum of all of these ones it is possible that the residual zero quantum coherence will cancel out because these phases are very systematically varying so they can pass over the 360 degrees and if you have sufficient number of steps there they can be cancelled or if you do randomly vary this over a larger number of scans and then of course this also will cancel out if you do a larger number of increments scans because you collect the data for number of scans you collect 4 scans 16 scans 24 scans or 32 scans or whatever so that you have a good signal to noise ratio and for every scan you which occurs in of course in multiples of 4 as indicated in the phase cycle you keep changing the tau m value then when it is randomly varied the phases will also be random and when you add all of those the zero quantum coherence can get cancelled out. So this way you remove the zero quantum coherence and you retain only the Z magnetization. So this is an illustration of how phase cycles can be used for selecting particular magnetization pathways coherence transfer pathways or magnetization pathways through the experimental pulse sequence. Now during the mixing time the recovery of the Z magnetization of the k spin causes transfer of magnetization to a dipolarly coupled spin say how does the relaxation occur? The relaxation occurs due to interaction between two or more spins. So if the k spin is dipolarly coupled to some other spin let us say L spin then of course there will be a transfer of this magnetization because it is in a non-equilibrium state the system will have to return to equilibrium when it has to return to equilibrium it will pass on this disturbance of the non-equilibrium to some other spin which is interacting with it through dipolar interaction. So some portion of the Z magnetization will get transferred to a L spin therefore we can schematically say it this way that ikz during the period tau m becomes lambda ikz of course lambda will be less than 1 and the residual of that will be on the L spin. If lambda remains on k spin 1 minus lambda appears on L spin of course here we consider two spins only but there can be more spins if there are more spins some portion will stay here and the other will get distributed to more than one spins depending upon how many spins are dipolarly coupled to the k spin. So this is a network of coupled spins and magnetization gets distributed through this network coupled spins. So now the density operator at rho 5 I mean we have now looked at at the end of the mixing time so I have written here the density operator which consists of the Z magnetization of the k spin plus the Z magnetization of the L spin this actually has come from the k spin. Notice therefore that it also has the modulation in the T1 as per the precision of the k spin. So this is although this is Lz here this is modulated by cosine omega k T1 and cosine pi jkl T1. So this along the f1 dimension it will appear at the frequency of omega k and therefore this is the k magnetization Z spin. Now the last 90 degree pulse both these are not observable right the Z magnetization is not observable. The final 90 degree expulse converts this Z magnetization into observable transverse magnetization. So kz goes to minus ky so therefore you take away the minus sign and put it plus here lambda ky and similarly this will be 1 minus lambda i Ly and this remains the same. Now clearly this is k this is k therefore after T2 evolution and things like that this will produce the diagonal peak and this will produce a cross peak. So during the f1 dimension it will be k magnetization and the f2 dimension this will be L magnetization whereas this term is k magnetization along both f1 and f2 dimensions. So the both the diagonal and cross peaks have the same fine structure that is an important fact you look here both are y so this is the ky and Ly. So therefore they have the same phase they have the same phase and both are multiplied by the same coefficient as indicated by this therefore they will both have the same shape you know the both peak shapes on the f f1 dimension and here it will be absorbed to line shape because we have this cosine omega k T1 cosine pi jkl T1 this produces two peaks at omega k plus pi jkl and omega k minus pi jkl. So two frequencies but this will be cosine and therefore they will be absorptive line shapes. Of course in case of multiple spin systems with many dipolar couplings as I mentioned to you before the magnetization gets distributed through the network of spins and we have already done this explicit calculations of polarization transfer during the mixing time through cross relaxation or chemical exchange in the previous classes. So this we will not repeat that here because you remember this Solomon equations relaxation matrix that appears which describes the transfer of magnetization from one spin to another spin through the process of cross relaxation or chemical exchange and therefore there we are not going to repeat that question here and we simply take the final results. There we provide the final result of the intensities of the diagonal and cross peaks. Now considering a symmetrical two side exchange to A B of course this exchange it can be chemical exchange or it can also be cross relaxation with equal populations at the two sides if you consider a two side exchange process the equal populations of the two sides the equal spin lattice relaxation rates and equal transfer relaxation rates of the two spins of the two sides. The intensities of the diagonal which we label here as A A A and this A B B and this is A A B and A B A are given by the following equations. So these are for the diagonal and this will obviously depend upon the mixing time. So what we get here this is the final result as I mentioned to you this is the diagonal peak intensity will be given by half to 1 plus e to the minus 2 k tau m whole multiplied by e to the minus tau m by t 1 and this is of course dictated by the t 1 relaxation time that is z magnetization is recovering along the longitudinal axis and this is the exchange rate exchange rate and in case of cross relaxation there can be cross relaxation rate coming up here as well. So where k is the exchange rate and t 1 is the spin lattice relaxation time the equilibrium magnetization at the two sides is assumed to be the same. Now you can see here when tau m is 0 so this term is 1 and this term is 1 so what do you get here 1. So therefore the intensity of the diagonal peak is maximum and that is 1 it is essentially normalized and what about the intensity of the cross peak if tau m is equal to 0 see this one is 1 therefore 1 minus 1 goes to 0 and this is of course 1 so intensity of the cross peak is 0. So if you now plot these two functions as a function of tau m along this axis you have the intensity of the diagonal or the cross peak you see the diagonal peak intensity decreases like this as per this equation and the cross peak intensity goes up like this and transfer and then of course after sometime this also decreases this is because of what the process called as spin diffusion which has been discussed earlier that is the relay of magnetization from one spin through the network of napolylic coupled spins. Now it will initially increase to the nearest enable exchange spin and then if there is been diffusion occurring through multiple spins then of course it can decrease. This is in the case of cross relaxation and this dependent on the inter proton distances and how many protons are close by in space that determines how this diffusion transfer is happening what is the efficacy of this transfer from one spin to another spin but of course in the initial times when the mixing time is short then of course it is linearly dependent on the mixing time and the transfer will be restricted to the one spin only whichever one is the closest it will go there only and that will have a specific application. Now if you took the ratio of the diagonal to the cross peak intensities and this will give you aaa by aab this is 1 plus e to the minus 2k tau m divided by 1 minus e to the minus 2k tau m. Now for short mixing times tau m compared to t1 this will this will get reduced because you can simply expand this and you will get here this will be 1 minus 2k tau m 1 1 will cancel then you will have only k tau tau m below and here you will have 1 minus k tau m divided by k tau m. So you can see that this is directly related to tau m and by doing experiments at different tau m values you will be able to measure the exchange rates. Thus by monitoring the intensity ratios as a function of tau m the exchange rates can be calculated. Now in case of cross relaxation mediated transfer the cross peak intensity is proportional to the cross relaxation rate as discussed in the previous classes and there here we will write that the aab the cross peak intensity is proportional to sigma ab times tau m where sigma ab is the cross relaxation rate and we have also seen the cross relaxation rate is inversely proportional to the inter proton distance. If you are talking about proton-proton NOE it is related to the inter proton distance and to the inverse 6th power of the distance. Therefore if you perform experiments at different values of tau m then you can actually estimate the cross relaxation rate here what you do is you measure this cross peak intensities as a function of tau m and then if you fit it to a linear equation you get the sigma ab. Once you get sigma ab you can calculate the inter proton distance r. So this allows estimation of inter proton distances in complex molecules from nosy spectra recorded the short mixing times. So this is a very important observation and it has found extreme value in majority of applications. In fact a larger amount of biological NMR is now based on this kind of a concept. Structural determination of large molecules has become possible because of this kind of a strategy. So here is an experimental example here. This is a molecule which larger number of protons here. See so many protons are there all labeled here with ab, cd, ef, gh and so on and so forth. And this is the two dimensional nosy spectrum. This is the diagonal you can see the diagonal here here the black and all your cross peaks here are in red. So this is the one dimensional spectrum and lord we monitor the intensities of all of these and you can actually establish correlations firstly with regard to which protons are in close proximity and they also have different intensities because they have different inter proton distances. By looking at the intensities of the peaks here you can actually calculate the various inter proton distances. So you have so much information here and all of these peaks which are present here are the carriers of structural information. You quantify all of these peaks here then you will be able to establish the network of inter proton distances here and it will allow you to determine the structure of the molecule. And this is an extremely important application for chemistry and biology. This picture has been taken from this book here the Timothy Kaltrich high resolution NMR techniques in organic chemistry. So therefore we have discussed here the important application of the nosy with regard to the structure determination of molecules. And while this is an important thing from the application point of view we also demonstrated how pathways can be selected by appropriately choosing the phases of your pulses and the phases of the receivers adjust them so that you select what you want in your spectrum. If you chose coherences then you get j correlated spectra. If you choose z magnetization then you get dipolarly correlated spectra which protons are correlated through dipolar interactions and therefore they carry distance information. In the other case the j coupling information is obtainable. So this 2D NMR therefore provides a variety of possibilities for obtaining the desired kind of information in your spectrum and that has been the therefore the biggest revolution in NMR spectroscopy. So we will stop here and continue with other pulse sequences in the next class.