 So, let us continue the discussion of two dimensional NMR spectroscopy and this slide here is a summary of the previous few lectures which we have heard. Essentially it is giving you a concept the so called selection of coherence transfer for pathways. It considers we discussed three experiments here the COSY, double quantum filtered COSY and also the NOZY which is not listed here but we discussed it in the last class. In the COSY we had magnetization transfer in the following way we started with a single quantum coherence of a particular spin let us say K spin and the K spin magnetization is transferred directly to single quantum L spin magnetization and similarly for the L spin magnetization is also transferred to the K spin. So, in the two spin system the K L spin system same thing was applicable to more spins a three spin system or multiple spin systems the same principle will apply. I will discuss the various features of this spectrum we said the diagonal peak has dispersed your line shapes and all the components have in phase character the cross peaks have anti-phase character and they are absorbed to a line shapes. The dispersive line shapes was a disadvantage for the COSY spin because it reduces the resolution in the spectra peaks which are close to the diagonal become masked and will not be observed. To circumvent this problem we developed this double quantum filtered COSY and here the pulse sequence is slightly modified instead of the two pulse experiment here this becomes a three pulse experiment we start with initial single quantum K spin magnetization as before and this is now converted into double quantum coherence it is not that the double quantum coherence is not created here it is created after the first two pulses but we actually ignored that one saying that is not observable and therefore that is we do not need to consider that and here we actually want to select that we do not want to select the single quantum coherence as was selected here here we want to select the double quantum coherence which is created between the two spins by the application of the first two pulses and we use a strategy called as phase cycling where the phase of the pulses is cycle as X Y minus X minus Y for the same T 1 increment and the receiver phase is appropriately adjusted so that the other components which are created here the single quantum is eliminated the Z magnetization which is also created is eliminated and only the double quantum coherence is preserved but the double quantum signal is not observable as we have seen before discussed it and therefore it has to be converted into observable magnetization and that is done by the third pulse which we apply in the double quantum coherence. And then we observe the magnetization the result of this is that the diagonal peak also gets in phase character and it has absorptive line shapes the cross peaks remain in phase and as anti phase character and also remain absorptive line shape therefore the diagonal and the cross peaks have both anti phase magnetization and they have absorptive line shapes this substantially improves the resolution in the spectrum and therefore peaks which were lying very close to the diagonal could be observed in the double quantum spectroscopy. In the third experiment which we considered that was the nosy experiment where the Z magnetization is selected we throw away the single quantum coherence that is created after the first two pulses and the double quantum coherence is also thrown away and zero quantum coherence which remains actually decays because of the long mixing time which is used in the nosy pulse sequence and by suitable adjustment of the mixing time one could also eliminate the zero quantum coherence which was created in the three pulse experiment this the nosy was also three pulse experiment and that only the Z magnetization is retained at the end of the mixing time and this is then converted into single quantum coherence for observation. So here again the phase cycling is used but the phase cycling that is used is different from the phase cycling that is used here and that is how we achieve the selection of different pathways of magnetization transfer through the pulse sequence that was an important concept which we developed and this is basically what I said so far is a summary of all of that in words. Now we go forward to explore other two dimensional experiments and in that the one thing which comes most important which is used in many many different kinds of multi-dimensional experiments and that is called as the constant time cozy and the pulse sequence for that is indicated here. So you start with a 90 degree X pulse it could be 90 Y it does not matter so we just chosen here 90 X pulse and this you have 180 X pulse here and a 90 X pulse here. So we have here the first a 90 degree pulse then you have a time period which is tau and this is also tau and now you notice that this is a spin echo. So 90 tau 180 tau is a spin echo and this period is then followed by T1 and the total from here to here is constant. Total time from here to here is constant that is delta and therefore tau will become equal to delta minus T1 divided by 2. This is an interesting experiment. You see the time from here to here is kept constant total time from here to here is kept constant and here is a component which is T1 which is incremented from experiment to experiment and which also means that the tau also changes with T1. This also is changing which means this 180 degree pulse keeps moving. At time T1 is equal to 0 there is a very first experiment you will see that this time is 0 and this 180 pulse will be in the middle of this delta period because at time T1 is equal to 0 tau is equal to delta by 2 and this will be in the middle of this whole period delta period. Now as T1 starts increasing say this tau starts decreasing which means this 180 degree pulse keeps moving towards the left from the middle which was it was it starts moving towards the left. So from experiment to experiment this 180 degree pulse keeps moving. Now how many increments then you can have here? See that is limited then by the delta period. So once you reach a certain value of T1 which is almost equal to delta then this 180 pulse will be almost adjacent to this first initial 90 degree pulse. You cannot increase T1 beyond that. Therefore the number of T1 increments you can do here is limited by what delta you choose. And how do you choose this delta? We choose this delta depending upon how much signal you will have. So depending upon the relaxation times of this spin system what you might have the k spin or the l spin or whatever of the system what you might have the shortest T2 and it will be dictated by that shortest T2. So the signal will decay the transverse magnetization will decay from here to here. We actually select the single quantum coherence here. In this experiment we select the single quantum coherence. This is very similar to the Cauchy experiment except that we have 180 degree pulse here and that creates what is called the spin echo part from here to here. So it is the single quantum magnetization as in the case of the Cauchy. But the period from here to here is constant that is delta and that limits how many increments you can have. The maximum value of T1 is limited by the total delta period what you might have. When T1 max is such that this tau becomes equal to 0 delta minus T1 max divided by 2 that is what the tau value if that becomes equal to 0 you cannot have T1's incrementing after that. So that certainly will have an implication for the resolution along the F1 axis in your experiment because the resolution in the X1 axis will depend upon what is the acquisition time in the T1 dimension. But what are the advantages why do we do it? And this we will see as you start looking at the explicit product operator calculation. This is the spin echo period here. The spin echo means is actually refocuses the chemical shifts. Therefore the chemical shift evolution happens only for this period here T1. Whereas the coupling evolution for the homonuclear system when you are considering the homonuclear system the coupling evolution happens all the way from here to here because that is not dependent on the spin echo and for homonuclear system this 180 pulse is applied to both the spins or all the spins and therefore the coupling evolution continues to happen here and it will happen during the T1 period as well. So you will have the coupling evolution happening for the entire delta period which means the coupling evolution time is not dependent on the T1 because it is constant for every T1 incremented is the same whereas the chemical shift evolution happens only for the particular T1 period. So let us look at the product operator calculations for a two spin system as we have done before for the other pulse sequences. For illustration we consider a two spin system the initial density operator at rho1 is z magnetization of the two spins ikz plus ilz and once you apply the first 90 x pulse you create the y magnetizations of the two spins minus iky plus ily. Since these two are independent spins they evolve independently during the next evolution periods and to demonstrate the evolution we will consider the k spin evolution and the same thing can be applied to the l spin as well. Let us look at this iky, iky is the y magnetization of the k spin. Now during the next delta period which is constant the following evolution takes place. Evolution under j coupling occurs for the entire period delta as I mentioned to you the spin echo does not have any influence on that because it continues to evolve except it will have some implications for the line weights. The 180 degree pulse in the middle of the delta minus T1 period the chemical shift gets refocused thus evolution under chemical shift happens only for the time period T1. Thus the evolutions need to be calculated as per these time evolutions. Let us look at that explicitly consider the iky and we will first evolve j coupling evolution. It does as I said we have seen before that it does not matter which one you consider first and which you consider later let us consider here the j evolution first. The Hamiltonian for j is hj and which is 2 pi ikz ilz and the k magnetization evolves like this. So it is iky cosine pi jkl delta minus 2 ikx ilz sin pi jkl delta. So clearly the j evolution does not depend on T1 and it is the same for all the increments in the 2D experiment. So now let us consider the shift evolution of these 2 terms. The first term here which is iky cosine pi jkl delta if I consider the evolution under chemical shift this is the Ziemann Hamiltonian hz cosine pi jkl delta I keep it out here and inside the bracket I keep the chemical shift evolution of the k spin. So I get here iky cosine omega kT1 minus ikx sin omega kT1. The second term which is here this is minus 2 ikx ilz sin pi jkl delta under the shift evolution. So I keep this 2lz sin pi jkl delta outside and inside the ikx term is evolving under chemical shift as ikx cosine omega kT1 plus iky sin omega kT1. Now to simplify the symbols or the equations we use let us use some abbreviations here. So let ckl delta when I write it like this it means cosine pi jkl delta and similarly I write skl delta for sin pi jkl delta and I write ckT1 to represent cosine omega kT1 and skT1 is sin omega kT1. So with this notation so what happens now we consider the evolution at rho 3 density operator at time 0.3 in the pulse sequence. What is the time 0.3 in the pulse sequence that we have seen here this is the time 0.3 in the pulse sequence. We consider it up till now the evolution under j until here and we also consider the chemical shift evolution during this period here and therefore we consider the density operator at rho 3 and the rho 3 now is using that simplified notation we have ckl delta iky ckT1 minus ikx skT1 and this part the second term is minus 2 ilz skl delta and inside the bracket we have the shift evolution terms ikx ckT1 plus iky skT1. Now this is the z magnetization. Now at this point we have this z magnetization here and this one is actually double quantum and zero quantum coherence because they have this term 2 ily ikx term. So these ones are not directly observable. So what happens here I mean this is the rho 4 this is the rho 4 density operator. The rho 4 density operator is obtained after applying the 90x pulse to this rho 3 density operator and when we do 90x pulse iky goes to ikz and ikx remains as ikx and here ilz goes to ily and this ikx remains ikx and this iky goes to ikz. Now here you see this is z magnetization which is not observable because we are not doing any more pulses after that we are directly detecting the signal this term is not observable this term is single quantum coherence of the k spin this is observable and this is double quantum and zero quantum mixture which we do not observe which is not observable therefore we do not consider this for further treatment and this is the ily ikz this is l magnetization anti phase 2 k spin and this also is as such it is not observable when it evolves in the T2 period it becomes observable and therefore we retain this. So keeping these observable parts we write this observable part of rho 4 is ikx skt1 ckl delta that is this here and this one is 2 ily ikz then I have skl delta skt1. So this will evolve during the T2 period with the frequency of k spin and this will evolve during the T2 period with the frequency of l and therefore this will produce me in the end a cross peak whereas this will produce me a diagonal peak fine. So let us look at these evolutions individually let us look at the first term which is the diagonal peak so minus ikx skt1 ckl delta so this will evolve under the coupling now we first consider once again the coupling evolution here skt1 ckl delta and the operator part is the ikx which is evolving in T2 gives me ikx ckl T2 plus 2 iky ilz sklt2. So this has now generated therefore a term which is 2 iky ilz this is after j evolution the shift evolution is not going to make any difference in this with regard to whether it is anti phase it will not convert a anti phase term in phase term therefore we can safely ignore this saying okay this is not observable in the T2 period but this term is observable. So this diagonal peak term keeps this ikx here this is observable and this term is not observable so therefore we can ignore this. Now let us consider the shift evolution of this after that so I have the same here skt1 ckl delta and ckl T2 comes from here and the ikx operator part evolves under the chemical shift as ikx ckl T2 plus iky skt2 using this abbreviation which we indicated there. So this is cosine omega k T2 and this is sine omega k T2. Now if you assume without losing any generality we assume y detection in which case the diagonal peak signal is given by this term we only keep this and we ignore this and therefore I will have here minus skt1 ckl delta ckl T2 and skt2 okay. So now we notice that during the T1 period I only have the chemical shift part right the sk and there is no coupling because of the coupling part is evolving for a constant time and therefore there is no modulation of the signal due to coupling evolution during the delta period. Therefore this will only appear as amplitude modulation of the signal detected whereas T1 dependent term results in a frequency. When a Fourier transform it the T1 dependence produces a frequency and this constant time evolution will not produce a frequency it appears only as the amplitude modulation of this detected signal during the T1 period. Along the T2 period of course you have both the chemical shift evolution here and the coupling evolution and both are modulating the detected signal as a function of T2. Therefore when a Fourier transform along the T2 dimension then I will have both the chemical shift and the coupling information and both ones these have the sine terms okay let us expand this. So I have here the same expression diagonal signal is I will remove this iky because when you take the trace with iky suppose that iky square goes to 1 therefore the signal is only dependent on this coefficients. So the skt1 ckl delta ckl T2 skt2 now if I expand this I have here ckl delta keep out this constant part which is the kind of an amplitude modulation as I said and this will be sine omega k T1 explicitly writing now sine omega k T1 and sine omega k T2 cosine pi jkl T2 that is this one here. From this we see that j splitting will occur only along the F2 dimension and F1 dimension will only have the chemical shift. After expanding this as before we have considered that I am writing here the final frequency distribution in the 2 dimensional plane. So how many peaks I will get here along the F1 dimension I will get 2 peaks because this one is a sine omega k T1 and I will get at the frequency of omega k so single frequency at the frequency of omega k. Along the F2 dimension I will have 2 peaks because when I expand this there will be 2 terms therefore I will get 2 peaks and this will be therefore F1 F2 will have the frequencies like this the nu k here and this is the F1 dimension and this is nu k plus jkl by 2 and the second peak will be nu k along the F1 dimension and F2 dimension will be nu k minus jkl by 2 therefore there is a splitting here and the separation between them is the coupling constant. And notice these both will be positive peaks and they will have the sine peak shape. Since they are sine modulation they will have dispersive line shapes here. So both of them will have dispersive line shapes along the F2 and the F1 dimension. Just now let us keep it that way and we will see that this actually can be changed by phase correction. We do a phase correction of course this can go to the absorbed line shape but let us do it afterwards after we see the cross peak as well. Now what happens to the cross peak? The cross peak is we have written the same function here as it evolves in the T2. This is the initial term at the end of the T1 period and after applying the 90 degree pulse and this is at the beginning of the detection period and we do the j coupling evolution here first along the T2. So this will give me I take the 2 ily ikz evolution here. So skl delta and skt1 as here I keep it here and this operator part evolves under the coupling as 2 ily ikz ckl T2 minus ilx skl T2. Once again we notice after the j evolution this is the anti phase term which is not observable. It does not have a non-zero trace and therefore this will be non-observable and only this term will be observable. Fine, let us take that one here. So skl delta skt1 skl T2 of course we have the minus into minus of course this gives you the plus sign here I keep this plus sign and now consider the chemical shift evolution. This gives me skl delta skt1 skl T2 operator part ilx. Now this will evolve with the frequency of L. So therefore right here skl T2 and ily skl T2. The ilx term is evolving as ilx cl T2 plus ily skl T2. Once again assuming y detection because we have to be consistent always the same as to y detection in one case it cannot have x detection in the other case. So assume y detection there use to y detection here as well. So this cross signal will now be skl delta skt1 skl T2 sl T2. So therefore this will produce amplitude modulation because of the coupling evolution during time delta and along the f1 dimension when you Fourier transform you will only have the frequency skt1 and this will produce me peaks along the f2 dimension. So now this one f1 f2 if I expand it I will get here like this nu k along the f1 dimension and nu l plus jkl by 2 along the f2 at the frequency of this and nu k and nu l minus jkl by 2 along the f2 dimension. So we will have two peaks here but this because of the sign there will these two will have opposite phases. So if this is positive this will be negative but both are cosine terms and therefore this will produce me an absorptive peak along the f2 dimension. And notice this is absorptive along f2 but this will be dispersive along f1 because it was a sin term sin omega k T1 it was and because along the f1 dimension I will have this omega k frequency. This is the cross peak you will have this omega k frequency is dispersive line shape because that was the sin term and it is absorptive line shape along f2 absorptive line shape along f2 for both of these peaks. And what was in the previous case for the diagonal peak I had here the dispersive line shape along both f2 and f1 dimension. They were both positive here and the dispersive line shape along both and f1 dimension also it was dispersive. So therefore since there is a dispersive line shape along f1 dimension for both the diagonal and the cross peak I can apply the same phase correction for the f1 dimension. So apply a 90 degrees phase shift along f1 this will convert this dispersive line shape into absorptive line shape same true for the diagonal peak but we focus on the cross peak here we are interested in looking at the cross peak. So if I do that then what I will have I will have absorptive line shape along f1 and absorptive line shape along f2. I do not do any phase correction along the f2 dimension therefore I will have absorptive line shape here and absorptive line shape along the f1 dimension here. Now look what is the consequence of that you look at a comparison this is an experimental spectrum a particular cross peak region the cross peak region this is the cosy spectrum and this is the constant time cosy spectrum and you see here these peaks are completely overlapping you cannot separate these and here of course there is a splitting here along the f1 dimension as well and that has gone because we have removed the coupling along the f1 dimension so this multiplicity along the f2 dimension remains so the same 4 peaks are present but they peak up here at the centre here of course that chemical shift is remaining and the coupling is removed along the f2 and here there are 2 peaks which are overlapping and those got separated in this constant time cosy. Therefore this one is called as omega 1 decoupled cosy you may call it as constant time cosy but sometimes you can also call it as omega 1 decoupled cosy and this improves the resolution along the f1 dimension in the cross peak area because we want to look at the cross peak fine structure and wherever we can measure this we can actually measure the coupling constants very well that is the advantage of the constant time cosy and therefore this component has been used in many many multidimensional experiments this is used as an entity which is inserted at various places to obtain higher resolution along the indirect dimension and this becomes important even though I have limited t1 max because I have removed the coupling information here this becomes quite beneficial and improve the resolution in the spectra and measure the chemical shift positions precisely and the coupling constants you can measure along the f2 dimension. So with that we will stop here and we continue with the other experimental sequences in the next class.