 We have been discussing two-dimensional correlation experiments. This was the Cauchy experiment was the first experiment to be developed and that was the most fundamental of all of these correlation experiments. Let us do a quick recap of what we did in the last class. This is the pulse sequence of the Cauchy experiment which is very simple consists of two pulses 90X and then the T1 period, then 90X, then the T2 period. So T1 is one independent time variable which is systematically incremented from one experiment to another experiment and T2 is the time period which is the acquisition period where you actually collect the data. Two-dimensional Fourier transformation of such a data results in a spectrum which looks like this. So we have this peaks along the diagonal which are called as the diagonal peaks. You have one frequency axis, this is called the F2 axis or this is the F1 axis and on the diagonal F1 is equal to F2 so therefore it is called the diagonal and we have cross peaks here arising between two spins which are J coupled. So this is the extremely useful part of the Cauchy experiment. So it displays correlations between J coupled spins. So if you have a molecule which has such a kind of a network that this spin is coupled to this spin then it will produce a cross peak here and this spin is coupled to this spin it will produce a cross peak here and this will appear symmetrically and if there are singlets that means which they are not coupled to anybody there they lie as singlets here on the diagonal. Notice here that this period which is given as a small gap here this gap is really 0 it is not really a gap. This has been given here in this figure simply to accommodate this number 4 which is the time point we have indicated here this various time points 1, 2, 3, 4 for the purpose of evaluation of the product operators at different time points through the pulse sequence. So it is just for that purpose otherwise this gap is actually 0 it is just minimum required to see that there are no effects of the transmitter on the receiver directly. Here the transmitter is put off and the receiver is on there should be no direct interference between the transmitter and the receiver only that much gap is given which is of the order of few microseconds sometimes. So therefore this is not really a long time gap and started looking at the theory of this how these diagonal peaks arise, how the cross peaks arise, what are their structures and that is we did part of that in the last class. So we calculated the density operator at different time points and then we arrived at this point 4 and that density operator is called as row 4 it had 2 parts one was non-observable other one was an observable part non-observable part we have just ignored here and only the observable part is written here observable non-observable non-observable part is because of what because the trace of the IKX or the IKY operators this has to be non-zero with the density operator. If you measure the X magnetization you take the IKX operator if you take the Y magnetization you take the IKY operator so each way so this was the calculation which was arrived at starting from the K spin magnetization to begin with similar things will appear when you start from the L magnetization also but it is enough to demonstrate taking the K spin alone. So therefore and we calculated the density operator at point 4 and that is row 4 it has 2 terms here. So it has IKX cosine pi Jkl t1 minus 2 IKZIL Y sine pi Jkl t1 and sine omega k t1. So this is the X magnetization of the K spin and you see here is the Y magnetization of the L spin which is anti-phase with respect to K whereas here it is the in phase magnetization of the K spin and this is the anti-phase magnetization of the L spin and this is responsible for the cross peak because in the evolution period the t1 period the K spin was evolving with this characteristic frequency omega k and that appears as sine omega k t1 here and eventually this during the t2 period this will evolve with the frequency of L and this will evolve with the frequency of K. Therefore, this one will produce the diagonal peak in the end and this term will produce the cross peak in the end and we started actually calculating the evolution in t2 taking the first term which will produce the diagonal peak. So what is the fine structure in the diagonal peak does it have a fine structure? So we started calculating that so if a way to represent the part of the density operator which belongs to the diagonal peak so I call it as rho 5d prime and this is during the acquisition period and that one comes as a result of evolution of this part of the density operator here. So this is IKX cosine pi Jkl t2 plus 2 IKYILZ sine pi Jkl t2 cosine omega k t2 and then we have IKY is cosine pi Jkl t2 minus 2 IKXILZ sine pi Jkl t2 sine omega k t2. This comes as a result of evolution of the IKX operator first under the influence of the chemical shift Hamiltonian so that gives you the cosine omega k and sine omega k here and then under the influence of the coupling Hamiltonian that gives the terms which are in the interior brackets here. So IKX term will give you these terms IKY term gives you these terms as a result of coupling evolution during the period t2 and this is the one which is present here the t1 evolution contribution FDt1 is cosine pi Jkl t1 sine omega k t1 that is this here cosine pi Jkl t1 and sine omega k t1. So now if you look at this in a little bit more detail if you assume you can choose one of these to say we are going to get Y magnet decode Y magnetization or the X magnetization so you can choose one of those if for demonstrating the principles. Now if you see either case if I take this so I have here cosine cosine and here I have take this and this one gives me cosine sine in the t2 period cosine cosine and this is cosine sine in the KYLZ. Now out of these terms these terms are not observable because this is anti-phase magnetization the anti-phase magnetization is not observable therefore what we will restricted to only this or this. So assuming that we record the Y magnetization then we will only have this particular term Iky cosine pi Jkl t2 sine omega k t2 into FDt1 and FDt1 is cosine pi Jkl t1 sine omega k t1. So now if you expand this further as separate frequencies what this gives you omega k and pi Jkl cosine pi Jkl t2 sine omega k t2 this will produce you two sine terms with the frequency that omega k plus pi Jkl and another sine term with omega k minus pi Jkl in t2 and similarly in the t1 period also if you take the Fourier transformation of this one this will produce you two frequencies omega k plus pi Jkl along t1 and omega k minus pi Jkl along t1 which means after Fourier transformation you will get those frequencies in your spectrum along the F1 axis these ones you will get them along the F2 axis. So this is represented here so you will get a total of 4 peaks after a real Fourier transformation along the t1 in the t2 dimension leads to 4 peaks with dispersive line shapes at the following frequencies why did we say dispersive line shapes because they were sine terms the sine terms as we have seen before will give you dispersive line shapes and the frequencies that are present are nu k plus Jkl by 2 now it is written in terms of hertz not in terms of the radians so the 2 pi part has been taken out so it is omega k plus pi Jkl is the same as 2 pi into nu k plus Jkl by 2 so the nu k is 2 pi is taken out here. So you have here nu k plus Jkl by 2 along the t1 and nu k plus Jkl by 2 along the t2 so this is a positive peak and it will have the dispersive line shape and this one is nu k plus Jkl by 2 along the t1 and nu k minus Jkl by 2 along the t2 and this will be again positive and dispersive line shape and then you will have correspondingly for the nu k minus Jkl by 2 along the t1 and nu k plus Jkl by 2 along the t2 positive dispersive and finally nu k minus Jkl by 2 and nu k minus Jkl by 2 which is also positive and has a dispersive line shape so you get 4 peaks along the diagonal and we said this will produce peaks like this and these are the 4 peaks which are present originating from the k spin as we started calculation from the k spin similarly we will also get 4 peaks from the l spin if we started calculation from the l spin magnetization to begin with so all of these are dispersive line shapes and all of them have the same sign. So now we continue the discussion for the other term in the density operator which was the second term in the rho 4 density operator and that was 2ikz ily sin pi Jkl t1 sin omega kt1 this was the second term in your rho 4 density operator. Now what we do here is now we have to evolve this term in the t2 time period so in this case we shall evolve the J first under the influence of the J coupling we will evolve this operator so we call this as rho 5c and I get here 2ikz ily cosine pi Jkl t2 minus ilx sin pi Jkl t2 fct1 fct1 is this term here sin pi Jkl t1 sin omega kt1 I represent it as fct1 and then after this J evolution we consider the shift evolution. So each one of these terms will evolve under the chemical shift. So this gives me rho 5c dash and that I write it as 2ikz ily evolves now with the frequency of omega l so ily cosine omega lt2 minus ilx sin omega lt2 and this cosine pi Jkl t2 comes from here and then this term is minus ilx cosine omega lt2 plus ily sin omega lt2 sin pi Jkl t2 and to the whole thing we have this fct1. So this is the one inside the bracket is due to the evolution under the chemical shift. Now assuming that we measure the y magnetization the observable signal is given by trace of rho 5c dash ily. So and we get here because we are looking at the L spin magnetization in the we are looking at the cross peak. So this gives me sin omega lt2 sin pi Jkl t2 sin omega kt1 sin pi Jkl t1. So now you break this up into its components this one will give me sin omega kt1 sin pi Jkl t1 will give me cosine omega k plus pi Jkl t1 minus cosine omega k minus pi Jkl t1 and this one will give me cosine omega l plus pi Jkl t2 minus cosine omega l pi Jkl t2. So if you see there are four terms here this into this, this into this, this into this and this into this. So because of this minus signs here we do get combinations of positive negative peaks here. So this into this gives me cosine omega k plus pi Jkl t1 into cosine omega l pi Jkl t2 and this into this will give me minus cosine omega k plus pi Jkl t1 into cosine omega l minus pi Jkl t2 and now this into this once again with a minus sign gives me cosine omega k minus pi Jkl t1 into cosine omega l pi Jkl t2. Then finally this into this gives me plus cosine omega k minus pi Jkl t1 and cosine omega l minus pi Jkl t2. So after two dimensional Fourier transformation each one of these will produce a peak. So therefore where will these peaks appear along the F1 dimension this will be at omega k plus pi Jkl and along the F2 dimension it will be omega l plus pi Jkl and this will have a plus sign and this one will produce me a peak at omega k plus pi Jkl and omega l minus pi Jkl and this will have a negative sign and this term will again be negative in sign and this will be at omega k minus pi Jkl and omega l plus pi Jkl along the F1 and the F2 dimensions. Finally this will be again positive and this will appear at omega k minus pi Jkl along F1 and omega l minus pi Jkl along F2. So therefore this is what is indicated here this leads to four absorptive peaks at the following coordinates. Why did we say absorptive? Because you notice here all of them are cosine terms here since all of them are cosine terms I get absorptive peaks and then the signs are as indicated here. This is positive, this is negative, this is negative and this is positive and all of them are absorptive peaks. So that is indicated here as four peaks here you can see these are the four peaks originating from k spin and this is the so-called cross peak. So the cross peak has absorptive line shapes in the all the four components and this is indicated as minus plus plus minus. This is an important feature of the COSY and you will see therefore this is called as the differential transfer because if you took at the total integral of this, the total sum of this it is 0. So therefore there is no net transfer of magnetization it is called as the differential transfer of magnetization. The coherent transfer is differential in nature and that produces me positive negative contributions in the cross peak. So this comes from k spin and similarly if you start it from the L spin again you will get four peaks in the cross peak here fine structure. Now here is an example, experimental example you can see this is the molecule what we have. So this is taken from this book Harald Gunther Anomar spectroscopy Vellice H and it is directly taken from there because this one has two protons. This is an example, this one has two protons and therefore it is an ideal AX spin system and if you look at the one-dimensional spectrum that is indicated here this has two doublets along the F2 dimension also two doublets, F1 dimension also two doublets. This is one of the spins and this is the second spin. Now you notice here this is the diagonal peak, this is the cross peak, this is the diagonal peak and this is the cross peak and you see here all of these have same sign and they are virtual line shape and these ones are absorptual line shapes and different signs. However this is indicated more explicitly by taking cross sections here. So you take the cross section at this point that is shown here. So this comes from here so you have a negative positive peak these are absorptual line shapes, absorptual line shapes, anti-phase in nature we call this as anti-phase in nature. So you have two peaks the two components which are anti-phase in nature you have negative and positive and the two peaks which are here you see these ones are dispersive in nature they are in phase they have the same sign. So you see here this goes negative positive this is one dispersive component the other one is again negative positive again dispersive component these both have the same sign therefore they go in this manner. This is the dispersive line shape. Now you take the next one here so what was negative here now becomes positive so negative positive this is positive negative therefore this becomes positive and this one is negative and this goes in this positive positive fashion once more and then here you have the positive negative and positive negative both have the same sign. So now you come here this is originating from the L spin so you have now the dispersive components here so the dispersive components are here in this manner you have this line shape this is going there is two dispersive components in phase in nature and the two peaks which are present here they are anti-phase in nature so you have the negative positive going up here and this one is the next component which is present here and that gives you dispersive component in this manner. And these are the absolute what was negative here now becomes positive and negative here. This resolution is quite high if you see here the cross peaks in all of these components the cross peaks are very good the intensity distribution in the two components whereas in the diagonal peaks that intensity distribution is not uniform simply because these dispersive peaks have big tails and this tails interfere with the other peak and therefore there is contribution of one peak to the other peak in the eventual appearance of the fine structure of the peak and that is the contour diagram is indicated here while these are the cross sections now you take the contour diagrams of the four peaks this is one of the cross peaks which is here so and therefore you have here this cross peak fine structure this is the fine structure of the diagonal peak. So that was so much for the two spins we can extend of course all your molecules are not only two spins there will be more than two spins and there can be three spins. So what happens if there is a three spin so there are different situations of course the different complexities will be there depending upon the nature of the coupling network of the spin systems. So here if I take three spins I can call them as AMX there are different ways the spins can be coupled if the geometry of this coupling is like this that suppose this is a linear three spin system then you have the A here M here and X here. So the A spin is coupled to the M spin M spins coupled to the X spin but there is no A to X coupling. So how will the one-dimensional spectrum of this look like we have just shown it by a stick diagram so see and with the different coupling constant they can all have different coupling constants. So A spin will be doublet because of coupling to the M spin and that is indicated as a doublet here and the M spin will be a doublet of a doublet because it has two couplings so it has one coupling due to the A spin and the other coupling due to the X spin that is indicated here. So this coupling is the MX coupling and then from here to here is the AM coupling this coupling has to be the same as this so this is the AM coupling this is the MX coupling. Likewise the X spin has the coupling due to the M and this coupling appears here as JMX and therefore we have here a doublet of the X spin. So therefore this what sort of a Cauchy spectrum you expect here in a schematic so we will have the AMX so A will produce a peak to the M spin so this is the AM cross peak and the M will produce a peak to the X spin cross peak and this is the MX cross peak and this symmetrically appearing here as well as there and all of these will have final structures in the case of the two spins. Suppose we have a three spin which is the coupling network is like this is the triangle so A coupled to X, A is coupled to M and M is coupled to X as well therefore each one of them has two couplings therefore each one of them will be doublet of a doublet. So there are four lines for each one of those and these four structures of individual spins is indicated here the A spin now has a doublet of a doublet the one spin is the AX coupling here the AX coupling is assumed to be larger than AM coupling therefore from here to here it is AX from here to here it is the AM coupling. And the M spin has the MX coupling and the AM coupling here the AM coupling is assumed to be larger than the MX coupling therefore this one is like this and this one is like this of course these structures can change depending upon the relative magnitudes of the coupling constants. So now the X spin likewise has a MX coupling here notice this coupling has to be the same what is present here and the AX coupling of course appears here as well and appears here therefore this coupling has to be the same as this AX coupling here so then you put that together so all of these things look different in their fine structures. Now in the two dimensional spectrum of course in the schematic you have here the correlations showing up here as A to M and then you have A to X as well and then you have from the M you have M to X and of course the M to A so therefore you have this complete plane full you have all the couplings are displayed in the form of cross peaks so this enable you to identify what sort of a network of coupled spins we might have okay. So let me just repeat this feature once more that for the case of three spins we will have the fine structure in the individual peaks this we will discuss later but at the moment it is sufficient to note that you have a network of coupled spins and this network of coupled spin will be displayed in the form of cross peaks appropriately depending upon the nature of the couplings we might have and the fine structure will be present in each of these diagonal peaks as well as in the cross peaks and the features of this we can see in the next class. So I think we can stop here just to recap what we did we looked at the fine structures of the diagonal peak and we explained how the diagonal peak has dispersive character and in phase character and the cross peaks have absorptive character and anti-phase character in the case of the two spins the similar thing will happen for the three spin as well that we will see in the next class and these fine structure calculation we did explicitly using the product operator formalism which was extremely useful in defining what sort of a frequency will appear what will be the line shapes and what will be the advantages and disadvantages of this we will see in the next class. So I think we will stop here we will continue in the next class.