 In the last class we discussed about a concept called scaling in two dimensional NMR. The idea was to see that the parameters like the chemical shifts and the coupling constants appear modified in the spectra, this is two particular advantages. We can make the J's values appear larger in which case some advantages will be there with respect to the cancellation of the intensities of the positive and negative signals which will not happen. And we also talked about the downscaling of chemical shifts, it will decrease effective spectral width and therefore it will increase the dwell time between the two points and it will improve the resolution in the spectra. The resolution in the sense it will improve the resolution in the fine structures of the cross peaks and therefore the cross peaks appear better resolved. However, of course the separation between two cross peaks in the spectrum along the F1 dimension will get reduced because you are downscaling the chemical shifts and that of course if one can afford it then one can use downscaling of the chemical shifts and get better resolved multiplied structures in your cross peaks. So we will continue the discussion today but now we will discuss the upscaling of chemical shifts, how to increase the separation between the cross peaks in the two dimensional spectra. This will happen along the F1 dimension only as before because that is where we are going to do all kinds of manipulations of the delays and we will get a better results because there is no constraint of the acquisition of data and we can manipulate the delays of the pulses and things like that in the indirect dimension. So T1 evolution is played with and that will give you the desired results. So today we are going to discuss the upscaling of chemical shifts. The pulse sequence for this experiment is shown here and this is called as cos, this is correlation with shift scaling. So it is very similar to the previous pulse sequences except that we play around with the delays, the evolution periods of the different Hamiltonians. So here we start with the 90 expulse again and here we have the T1 evolution period. Now we have an extended T1 evolution period here we call it as XT1 and then from this point onwards there is the so-called spin echo sequence tau 180 tau sequence from here to here. Now we also keep this period from here to here constant and that period is delta. So therefore you can see as you increase this period here this tau values will decrease and therefore this 180 degree pulse keeps moving along with the T1 increments. So the net result should be that I should have the time period from here to here constant delta. Now what all things happen in this? So this we can see immediately that during this period during the tau 180 tau period the chemical shifts will get refocused because this is the spin echo sequence whatever evolution happens during this time period will get refocused during this time period. Therefore chemical shift will not evolve during this 2 tau period but it will evolve during this T1 and XT1. So from here to here chemical shifts will evolve. Now the coupling constants are not affected by the 180 degree pulse and they will evolve during the entire period T1 plus delta. So that is what is indicated here and then we will actually do explicit calculation using the product operator formalism. We will illustrate this we will not go through in as much detail as we did for the COSY and other experiments because the same way the calculations can be done and we will illustrate this with one someone particular evolution and we will give the end results. So product operator calculation for the two spin system K and L both are spin half systems. So at the beginning the row 1 is the Z magnetization. So it is ikz plus ilz and after the first 90 degree pulse I get iky plus ily with a minus sign here because the pulse is applied along the X axis. Now during the next time period T1 plus delta the following evolutions will occur. This is what I explained to you just now. The chemical shift evolution will occur for the period 1 plus X times T1 because these chemical shifts are refocused by the spin echo sequence during the period 2 tau. Coupling evolution will however occur for the entire period T1 plus delta and delta is a constant. So therefore the pulse is smooth 180 degree pulse especially it moves through the experiment and in order to maintain this delta constant the tau period has to be decreased and as T1 is incremented. Now we will not go through as full calculations as we did for the COSY because we have already indicated how this calculation has to be done and one can go through the same procedure and calculate the density operator, remove the non-observable parts and keep only the observable parts of the density operator. Now going through the all of this calculation you arrive at row 4. This is the density operator at the beginning of the detection period. Now here the calculation is done taking both K spin evolution as well as L spin evolution and only those parts of the density operator retain which are observable. These ones will evolve in the T2 period to produce observable magnetization and so that is what is retained here. So at the end of the T1 evolution or we will say at the beginning of the detection period because the last pulse has already been applied soon after this one they will evolve in T2. Now what does this row 4 contain? The row 4 contains these terms iky sin omega k alpha T1 plus ily sin omega l alpha T1 and multiplied by cosine pi jkl T1 plus delta. Two terms here and additional two terms here plus 2ikz ilx sin omega k alpha T1 plus 2ilzikx sin omega l alpha T1 sin pi jkl T1 plus delta. Now let us look at this carefully. Now this first two terms here they have the J modulation for the period T1 plus delta and delta is a constant. Therefore T1 dependence will happen with the coupling constants during the period T1. Now with regard to the chemical shifts here you see this come from the chemical shift evolution omega k alpha T1 and omega l alpha T1. These come from the chemical shift evolutions of the k spins and the l spins. Now you see this is omega k alpha T1 and what is alpha? Alpha is 1 plus x. So because the chemical shift evolution has occurred for the period 1 plus x times T1 call it as alpha T1. So therefore you can see in this full sequence this from here to here is 1 plus x times T1. The chemical shift evolution has happened for this entire period and that is why it is appearing there as alpha T1 omega k alpha T1. And what comes from ily it is giving you ily sin omega l alpha T1. And both these are in phase magnetizations. This is the in phase magnetization of the k spin and this is the in phase magnetization of the l spin. So this comes from the k spin evolution during the T1 period. Now I have omega k here and I have iky here and this will evolve in T2 period with the frequency of the k spin with omega k. Therefore this will be responsible for one diagonal peak because in the T2 period it will evolve with the k spin frequencies. Now this term ily sin omega l alpha T1 this came from the evolution of the l spin during the T1 period during the period T1 plus delta. And therefore this has the frequency here omega l and now this also is ily here ily evolves in T2 period with the frequency of the l spin. Therefore this also produces a diagonal peak. So these two terms produce diagonal peaks in the final 2D spectrum. But the chemical shifts are multiplied by the by the factor alpha. So and that alpha is 1 plus x, x is the number which you can choose. Whatever value you choose if x is equal to 1 then it will be multiplied by a factor 2. Chemical shifts will be scaled by a factor 2. If x is 0.5 then it will be scaled by the factor 1.5 and so on. This you can choose depending upon how much you want to increase the separation between the peaks. But notice if you want to do that your spectral width will be increased. So if your normal spectral width was 5000 hertz let us say total spectral width was 5000 hertz. Now if you put x is equal to 1 alpha becomes equal to 2. Therefore your total spectral frequency width will be multiplied by a factor 2. Therefore 5000 hertz will go to 10000 hertz. So which means when you do incrementation the T1 incrementation this will have to account for 10000 hertz. So therefore your increment will be half of what it was earlier in this case. So this would mean that your T1 max will be half what it would be for the same number of T1 increments as compared to the previous case without the scaling effect. But the peaks will appear great more separated because of the increased chemical shift difference. Now what about the second term here? The second term consists of anti-phase magnetizations here. This is an anti-phase term and this is also an anti-phase term. So this will now is omega sin omega k alpha T1. This came from the evolution of the k spin during the T1 plus delta period. But now it has this operator term Lx which is L spin magnetization anti-phase to the k spin. And therefore this will evolve during the T2 period with the frequency of the L spin. And therefore this will produce me a cross peak along the f1 dimension it will have the frequency of k spin f2 dimension it will have the frequency of the L spin. But now because of this anti-phase operator here I will get plus minus terms in the fine structure in the cross peak along the f2 period. And this one is k spin magnetization and this came from the evolution of the L spin in the T1 plus delta period. So therefore this will also be modulated by this alpha T1. This will also be modulated by alpha T1 and this term will evolve during the f1 axis it will have the L spin chemical shifts and the f2 axis it will have the k spin chemical shifts and therefore this will also produce a cross peak. Therefore I separate this rho4 into two terms as rho4 is equal to rho4C plus rho4D. This is responsible for the diagonal peaks in the 2D spectrum. This is responsible for the cross peaks in the 2D spectrum. So the diagonal peak terms we explicitly write it here iky sin omega k alpha T1 plus ily sin omega l alpha T1 cosine pi jkl T1 plus delta. And likewise the cross peak terms will be these two 2ikz ilx sin omega k alpha T1 plus 2ilzikx sin omega l alpha T1 and multiplied by sin pi jkl T1 plus delta. Now let us look at the diagonal peaks first. So here we have iky sin omega k alpha T1 plus ily sin omega l alpha T1 cosine pi jkl T1 plus delta. This expanded it further this term especially the other thing is the same this term is expanded. So we have here these two terms though the sin term will be multiplied by this cosine term. So which will produce some kind of the two peaks and there will also be modulated by the sin multiplied by the sin term. Now each one of these if you multiply this so you will have one product here it is the sin cosine and this will be sin sin. Now they will both produce two peaks so therefore there will be superposition of two terms from here and two terms from here. So the sin sin product will produce you cosine cosine terms which are absorptive line shapes with plus minus and this will produce me sin cosine this will produce dispersive line shape with plus plus in phase. Therefore there will be superposition of this kind of splitting plus plus splitting with the dispersive line shape and plus minus splitting with absorptive line shape. Therefore this introduces as indicated here in the F1 dimension some interface character will be introduced along the diagonal this is in the diagonal piece. Of course this is the kind of a modulation of the amplitude of these ones depending upon the value of delta you have this contributions of the two components different. So depending upon what value of cosine pi jkl delta you have and sin pi jkl delta you have the contribution of the dispersive line shapes and the absorptive line shapes from these two terms will be different and that although introduces the mixed phases it will have some advantages as you can see here. Similar arguments appear for ily as well ily also will have two terms here one product of this and other product of this. So this will produce you dispersive two components this will produce you absorptive two components this will be in phase this will be anti-phase and therefore there is a superposition of the peaks of dispersive and absorptive line shapes in the diagonal peak of the L spin as well. And the contributions of the two will depend upon this value delta what value of delta you can choose and how much one can choose that will depend upon of course your T2 how much you can afford because delta remember it is a fixed time period it will be the same for every T1 increment and therefore the signal will decay during that time period and that causes loss of signal. So depending upon your system how much loss you can afford how much decay of the signal afford you can choose a value of delta and in relation to jkl this jkl delta this product whatever the value that is that will determine the contributions of these two terms to the diagonal peak. And you can see here for a typical experiment in one of the samples which one has recorded and this you see as a significant improvement in the diagonal peak the diagonal many of this diagonal peaks have vanished and all the peaks which are close to the diagonal are now becoming much more clearer. These peaks are much more clearer compared to what they are present here. So this is the particular advantage of recording the shift scale spectrum. Another example is shown here because of the upscaling then you have cross peak separation is getting increased and you see here this is the particular region of the spectrum of the DNA molecule and therefore this is a so called 1 prime 2 prime to double prime region of the sugar rings in the DNA and you see here a significant improvement in the separation of the cross peaks in this area these cannot be resolved very well but these ones are got resolved very well. Similarly you can see here there is a better separation of the cross peaks overall the peaks appear more compressed here compared to this and that is because of the increased overall spectral width in this but the important point is that the cross peaks are looking more resolved in this spectrum. So therefore that is the advantage of doing upscaling of chemical shifts. Now so now we will take up another experiment which is called as total correlation spectroscopy it is called TOXI. So this is an experiment which is very commonly used in assigning spin systems and this is a very popular experiment which has very important sensitivity advantages and we will see as we discuss further. So the pulse sequence is very simple here so you have the 190 degree pulse that is the excitation pulse, magnetization is excited and it evolves for the period T1 and after that the magnetization is locked. Now you remember this magnetization is in the transverse plane and you apply 90x pulse you get magnetization in the transverse plane in the y axis because it will evolve during this period but now you apply a spin lock, spin lock is applied along the x or the y or whatever axis and then the entire magnetization is locked along that axis. So here this is a very complex spin sequence here spin lock it will consist of several pulses there are many many different schemes for spin locking and there are many different pulses inside here. So we will not go into the discussion of the details of the spin lock how it is achieved but suffice it to say that during this period the chemical shift evolution is completely removed the Zeeman Hamiltonian is completely removed and the only Hamiltonian that is present is the coupling Hamiltonian. So therefore the effective Hamiltonian during this period is the coupling Hamiltonian. Now since the chemical shifts are not there we will have to consider the full Hamiltonian not just a ikz ilz as was done in the previous cases where you have a weakly coupled spin systems. So when you have a weakly coupled spin systems coupling Hamiltonian will be ikz ilz here 2 pi j ikz ilz but when the chemical shifts are removed the j coupling becomes the only part of the Hamiltonian and then you will have the full Hamiltonian that has to be considered and that is the effective Hamiltonian represented here the effective Hamiltonian is 2 pi j ik dot il for the two-spin system you see and you expand it here so you get 2 pi j ikz ilz plus ikx ilx plus iky ily. So we have seen the effects of this early in course when we talked about the analysis of spectra this was discussed by Professor Ashutosh Kumar and you see when there is a strong coupling you have to include all these terms in the coupling Hamiltonian if there is a weak coupling then you will have only this term in the coupling Hamiltonian and the product operator calculations were done for the weakly coupled situation here. You can also do a detailed calculation of the evolution of the density operator for a strongly coupling Hamiltonian but certain tricks are to be used and this is more complex and we will not go into that detailed discussion here and we will simply accept the results that have been obtained. So once that is done evolution of the magnetization under the influence of this effective Hamiltonian is simply given by this expression here. So if you start with the ikx, ikx evolves during the spin log period how does it evolve for the period t and this is the period of the spin log. So during this it will produce me ikx into 1 plus cosine 2 pi j t divided by 2 plus ilx to 1 minus cosine 2 pi j d divided by 2 plus 2 interface terms here 2 iky ilz minus 2 ily ikz and this is multiplied by sine 2 pi j t by 2 and notice so during the we are starting here from the kx because this is what came from the evolution during that t1 period is started from the kx and gave me kx and it also gave me a lx that is the interesting part. So compared to the Kozy you had this kx here and the lx part that why what this was 2 ily ikz this was producing an anti-phase term in the case of the Kozy and here we have the in phase term for the ilx as well. So we have kx which is in phase k magnetization ilx which is in phase l magnetization and now these are multiplied by different coefficients and then you have an anti-phase term which is multiplied by another coefficient. Notice here if t is equal to 1 by 2j what happens if t is equal to 1 by 2j this becomes cosine pi this is minus 1 therefore this becomes 1 1 plus 1 by 2 this is 1 whereas this 1 goes to 0 this 1 1 minus 1 divided by 2 this goes to 0 which means there will be complete transfer of magnetization from ikx to ilx and this becomes 1 so and this anti-phase terms will be there but there is a complete transfer of in phase magnetization from kx to lx. So now another important point to be noticed here is that you have both are the same phase this is the diagonal peak which will has the x along the x axis and this is also ilx both are x therefore these have the same phase and therefore they will produce the same phase in the final spectrum therefore both the diagonal and cross peaks have the same phases. Now if you started with ilx suppose you start with ilx and do the same evolution here so which means after the even evolution you are having the ilx term and you are considering the evolution of that under the influence of the spin locked so you get ilx i plus cosine 2 pi jt by 2 plus ikx 1 minus cosine 2 pi jt by 2 and plus you have the same anti-phase terms here and sine 2 pi jt by 2. Now once again ilx in phase transfer to k that is the important part in the case of cosy and double quantum filtered cosy is we have anti-phase terms for both in the cosy you have the anti-phase terms from the cross peaks in phase for the diagonal peaks and the double quantum filtered cosy you have anti-phase terms in both the diagonal and cross peaks and here you have in phase components for both the diagonal and cross peaks and they have the same phases because you have both lx and kx for the two cases. Now if you added these two equations this is equation 2 and the earlier one was equation 1 which originated from k spin if we add this you notice here this term will completely vanish and you have ikx plus ilx going to ikx plus ilx that means the x magnetization is completely conserved so it is just distributed between the two spins kx plus lx and kx plus lx here that after the spin lock it is just the same. So therefore nothing has happened kx has gone to lx and lx has gone to kx. Suppose you started with ky same thing will happen ky will go to ly and ly we go to ky if you started with kz same thing will happen it will go from kz to lz and lz to kz. So therefore this mixing by the spin lock is called as isotropic mixing because it retains the phase and that is called isotropic mixing and the Hamiltonian is termed as isotropic Hamiltonian. After the mixing the magnetization components are detected in the T2 time period and two-dimensional Fourier transformation of the collected signal results in the 2D spectrum. So what is the consequence of this in phase isotropic mixing? There will be relay of magnetization. Suppose during the mixing period you had multiple spins which are connected so you had let us consider a 3-spin network here kl and m, k is coupled to l, l is coupled to m. There is no km coupling but now so during the mixing period we saw that if you start from kx it went to lx. When you had only two spins lx starting with lx it went to kx but suppose there is a third spin here the lx is coupled to the m therefore what stops it from going to m. Therefore during the spin lock period what originated from k came to l and then from l it can go to m. There is no chemical shift evolution here anyway it is only coupling evolution all the spins have essentially lost their identity therefore it is called as the only the coupling part the individual chemical shifts are not there therefore what came from here went to l and l can go to m so it can be transferred from l to m as well. So transfer can occur from k to l and l to m and if there are more it can go further m to p or p to q and things like that. So it can relay through the entire magnetization the coupling network and that is the strength of the Toxy experiment that is why it is called as total correlation spectroscopy. Look at the example here, here is an experimental example this is the Cauchy spectrum of a particular molecule whose one dimensional spectrum looks like this because it is put on both sides in the same manner and this is the corresponding Toxy spectrum of the same molecule. So you notice here that in the Cauchy spectrum you got a correlation from this diagonal peak to this produces a cross peak here that is to this diagonal peak and it produces a cross peak here to this diagonal peak here. So correlations are established between these three spins in the Cauchy spectrum. So this proton is therefore coupled to this proton and also to this proton. Now you see here in the Toxy spectrum you have those two same peaks present here these two are the two Cauchy peaks which were present here and here but you get additional peaks here this peak here and this peak here these have two additional peaks which are coming. So how did they come? They came because of the relay of the magnetization through the network of the coupled spins. So because these ones are further coupled to other spins and that is how you generated those two. See for example here this diagonal peak is here. So this diagonal peak is coupled to this you can see this cross peak here but this diagonal peak here is also showing a cross correlation here and therefore that is the same which is the relay here. So you have this relay coming here through the relay. So therefore you have this kind of correlations appearing in the Toxy spectrum through the relay mechanism. So therefore it is called two dimensional total correlation spectroscopy or the Toxy. Similarly here you start with this here you had only one cross peak showing one correlation one coupling from here to here. But this diagonal peak here this proton may be coupled to several others. This one is coupled to several others and that is shown by these three peaks which are additionally appearing here. This one is appearing additional and these ones are appearing additional which means this respective diagonal peaks are connected to more spins and the magnetization is relate to those spins in the spin lock period and they will evolve during the T2 time period with their respective frequencies and therefore they will appear here in the 2D spectrum as cross peaks and the respective chemical shifts of this spin which had evolved during the T1 period. So therefore this generally contains more number of peaks but it is extremely useful and it has more information. So complete spin networks can be identified from the Toxy spectrum. So with that we will stop here. So we will complete it the two dimensional correlation experiments homonuclear correlation experiments with this.