 We are now going to discuss a new concept in two-dimensional NMR. So far we discuss different types of 2D experiments, 2D separation of interactions namely the chemical shifts in the coupling constants along the two dimensions of the 2D NMR spectrum. We discuss correlation experiments where spins which are coupled in one way or the other, they can be understood by looking at the correlations in the two-dimensional spectra. The cross peaks in the 2D correlation spectra reflect the correlations or the interactions between the spins. And we also looked at how to improve the line shapes and the resolution in the spectra by choosing appropriate coherence transfer pathways. So these are new concepts. So we are going to introduce now another important concept namely scaling in two-dimensional NMR. What do we scale? We scale the two parameters which appear in the spectra namely the coupling constants or the chemical shifts. We can do that at will. It is not that the chemical shifts of the coupling are actually changed for the sample. It is just that they will appear as modified in your spectra. It will have its advantages with regard to the separation of the peaks, with regard to the resolution in the peaks, with regard to the intensities of the peaks and so on so forth. And therefore that is a concept which are going to discuss now. So as we see here the objective is to scale selectively either the coupling constants or the chemical shifts. This can be done in the indirect dimension of the 2D experiment where there is explicitly no detection of the signal. During the T2 period when the signal is actually detected the receiver is on you will be limited with regard to the manipulations you can perform. Therefore it is rather hard to do such manipulations of the coupling constants or the chemical shifts during the detection period. Because obviously if you want them to appear differently in your spectrum you will have to do some manipulations in the evolution and that is little harder. Whereas in the T1 period which is the before the detection period where there is no actual detection of the signal we can play around with the pulse sequence and get some information of the type you want. We did the same thing in constant time cosy where we had decoupled along the F1 axis. Here we are now going to discuss how we can make the J coupling constants changed. In this particular case we will see that the J's will appear upscaled. And the pulse sequence is as follows you have the 90 degree pulse and this will be followed by the T1 evolution period and that will be followed by an extension of the evolution period which consists of tau 180 tau and then you have the detection pulse the 90 degree x pulse as in the cosy. So this part of the pulse sequence which is tau 180 tau this is like a spin echo sequence. So and therefore this is a kind of an extension of the evolution period. Notice also here that tau is actually proportional to T1 it is alpha times T1 alpha is a positive number and is obviously greater than 0 and therefore you have an extended evolution period here and as you perform the experiment as T1 is getting incremented this period is also getting incremented. Therefore the separation between these two pulses will go on increasing as you increase the T1 period. Of course that is the same as in cosy as well as we increase the T1 period this separation go on increasing and also this 180 degree pulse keeps moving further and further as we increase the T1 value because it is equal to alpha T1 and alpha is a constant. Let us see how it works let us try and understand this. We will do a product operator calculation for two spin system as usual consider two spins k and l and these are the density operator time points where we will actually look at the density operator. At time point 1 so it is basically z magnetization of the two spins you have ikz plus ilz and when you apply the 90 degree pulse long x axis you get minus iky plus ily. These are independent spins so they will evolve independently during the next evolution time period. For our purpose we will only illustrate evolution of the k spin and the same thing will apply for the l spin as well. During the next period T1 plus 2 tau what happens what kind of evolutions happen? Notice the chemical shift evolution will occur for the period T1 only because they are refocused by the spin echo sequence during the period 2 tau. As you remember the spin echo refocus is the chemical shifts but it does not affect the coupling constants it does not affect the coupling evolution. Therefore coupling evolution will continue to happen during the 2 tau period therefore the total coupling evolution period will be T1 plus 2 tau the chemical shift evolution will be restricted to T1 whereas the coupling evolution will be for the period T1 plus 2 tau. So let us write these evolutions here consider iky and here we write the chemical shift evolution therefore we have the Ziemann Hamiltonian here and that gives you iky cosine omega k T1 minus ikx sine omega k T1. Now let us use these abbreviations here cosine omega k T1 is written as Ck T1 sine omega k T1 is written as sk T1. If you write that then of course you will write this expression once more here and evolve further under the influence of the coupling constant. Take the first term here the iky cosine omega k T1 under the influence of the coupling constant which will give you Ck T1 that is this one and inside the bracket you will have the coupling evolution iky cosine pi jkl T1 plus 2 tau minus 2 ikx iLz sine pi jkl T1 plus 2 tau notice here this is the important point the coupling evolution is happening for the entire period T1 plus 2 tau therefore the time dependent term here is T1 plus 2 tau and this one you explicitly put the value of tau then you get here Ck T1 iky cosine pi jkl 1 plus 2 alpha times T1 minus 2 ikx iLz sine pi jkl 1 plus 2 alpha T1. Similarly for the second term here ikx sine omega k T1 under the coupling evolution will give you sk T1 this is sk T1 is this here and ikx cosine pi jkl 1 plus 2 alpha T1 plus 2 iky iLz sine pi jkl into 1 plus 2 alpha T1 therefore your total density operator rho 3 at the end of the period 2 tau is the sum of these 2 evolutions and that is Ck T1 iky cosine pi jkl 1 plus 2 alpha T1 minus 2 ikx iLz sine pi jkl 1 plus 2 alpha T1 and then you have here minus sk T1 that is ikx cosine pi jkl 1 plus 2 alpha T1 plus 2 iky iLz sine pi jkl 1 plus 2 alpha T1. So now therefore you see the coupling term is multiplied by the factor 1 plus 2 alpha so therefore when you Fourier transform this total thing along the f1 dimension the coupling constant will therefore appear as 1 plus 2 alpha times j this is the coupling constant appears scaled by a factor 1 plus 2 alpha along the f1 axis in the 2D spectrum. Now we will not go through the rest of the calculation along the T2 evolution and things like that because that remains the same as in the cosy and we do not want to repeat that here. The idea here was to show that during the T1 period the coupling constant appears scaled by the factor 1 plus 2 alpha and this is because of that introduction of the spin echo sequence tau 180 tau. Now in this calculation what we have done we actually have not considered the relaxation that happens during the evolution period. We did not consider that in the previous cases as well but here it is more important to consider that because there is a spin echo sequence in the evolution period. So what happens during this period? So if you look at this sequence here during this spin echo if you recall the previous discussions the field in homogeneity effects will also be refocused. The chemical refocusing means the field in homogeneity will be refocused as well therefore during this period the transverse relaxation happens with the time constant which is T2 star which includes T2 plus the contribution from the field in homogeneity whereas here during this period the relaxation will be dictated by T2 alone there will be no field in homogeneity contributions. Therefore the signal will decay in the following manner up till here it will decay with the time constant of T2 star and from here to here it will decay with the time constant of T2. Therefore we have to calculate that explicitly and that is what we will show here. So this T2 relaxation would cause scaling of the line width as well. So that is what we would like to show what is the effect of T2 relaxation during this whole period. Now if I want to write the density operator as rho 3 star for this and I have this density operator rho 3 which is without the relaxation considered and then I will have to multiply this by the relaxation factors. So if during the T1 period the time constant of relaxation is T2 star therefore I have this e to the minus T1 by T2 star. So this is the multiplication and this is also the multiplication here. So therefore you get rho 3 into e to the minus T1 by T2 star and multiplied by e to the minus 2 tau by T2 because this total is the relaxation factor. This is the relaxation factor. This is the first relaxation during the time T1 and this is the relaxation with the time constant T2. So when you put this together then of course you will get this in the superscript e to the minus inside bracket 1 plus 2 alpha T2 star by T2 into T1 by T2. This is the same as this and this one of course is coming from this term here. So therefore if I were to call this as a factor so rho 3 star is equal to rho 3 into e to the minus beta into T1 by T2 star then with beta is equal to 1 plus 2 alpha T2 star by T2. So therefore the relaxation causes a modulation of the line widths as well. What is the line width? The line width is 1 by T2 star. If you had simply e to the minus T1 by T2 star then 1 by T2 star corresponds to the line width. 1 by T2 star represents the normal line width. Now that is now multiplied by a factor beta. Therefore the line width will appear modified and along the f1 dimension therefore the line width will be scaled by a factor beta that is this one here. Now in the event T2 star is equal to T2 the line width scaling factor will be identical to the J scaling factor 1 plus 2 alpha that is if T2 star is equal to T2 this will cancel and then you have 1 plus 2 alpha here. So the J scaling then does not necessarily increase the resolution in the fine structure of the peaks. The individual components in the cross peak of the diagonal peaks will have a line width and we are trying to increase the separation between those components by scaling the J values and therefore they appear more resolved. However if the line widths also increase by the same factor then you do not achieve a significant improvement in the resolution. The J scaling does not necessarily increase the resolution defined. However in most of the cases this condition is not satisfied and T2 star is never equal to T2. So therefore the relaxation time constants in the two cases are different. Therefore this will be less than T2. T2 star will be less than T2 therefore this factor is less than 1. Therefore you will have the line width will be scaled by a smaller factor than the J's and then you will get the benefit of upscaling of the J values and then it will improve the resolution in the spectrum. The benefit of the J scaling along the F1 dimension is to increase the separation between positive and negative signals in the cosy spectrum and thus lead to reduction in the cancellation of the intensities. Remember in the cosy the cross peaks will have plus minus character in the cross peaks and they would tend to cancel their intensities in the event of insufficient resolution. Therefore if you are able to increase the separation then cancellation will be reduced and then increase the sensitivity of the cross peaks. And here is an experimental example. This is a small molecule uracil. It has two protons here and these two protons are J coupled and the coupling constant between them is about 7.3 hertz. And this coupling constant appears in the F2 dimension because we have done nothing in the F2 dimension. During the T2 period the evolution happens as with the normal cosy and it will appear in the fine structure as well. So there is one cross peak here in the diagonal peak here and this cross peak for example or this cross peak or this cross peak is actually blown up here in this one. We can see along the omega 2 dimension or the same as F2 dimension this coupling constant remains the same. This is 7.3 hertz and the same one now appears scaled along the F1 dimension. Here the scaling factor is 4, 1 plus 2 alpha is equal to 4. Therefore this appears multiplied by a factor 4. So this is a clear indication of the enhancement of the resolution and you can see the positive negative components here very clearly although there is same color is used here but these positive negative components all the 4 components can be seen very clearly in this spectrum. So therefore this helps you to improve the resolution in the spectra and enable measurement of the coupling constants wherever they are not sufficiently well resolved in your spectra. Now we will see the next scaling experiment and here is a simple modification of the previous one. So this is called shift scale cosy. In the earlier case we scaled the coupling constant. Now we will scale the chemical shifts. What do we do? Simple modification here. Now the T1 period is all the way from here to here. Earlier T1 period was from here to here and this was additional and we put this as alpha T1. So the whole period was 1 plus 2 alpha T1. So now what we are doing is well we are keeping this whole period from here to here as T1 and in between we introduce this. So for tau 180 tau this remains the same and 2 tau is equal to alpha T1 here just a small change in their representation. So this whole period is called alpha T1. So therefore the chemical shifts will now evolve only for this period from here to here and that is T1 minus alpha T1 that is 1 minus alpha into T1 and the coupling constants will evolve for the whole period T1 whereas the chemical shifts will evolve for the period 1 minus alpha T1. So the pulse sequence is the same. We have just played around with the timing how you adjust the timings of the various pulses. Now this produces different scaling effects observed by delay manipulations. So what is the consequence here? We just do the same calculation once more. So during the time period T1 the following evolutions will happen. Again we show only for the case spin the chemical shift evolution will occur for the period 1 minus alpha times T1 only as they are refocused by the spin echo sequence during the period 2 tau while j coupling evolution will happen for the entire period T1. So we do this calculation iky under chemical shift gives you iky cosine omega k1 minus alpha times T1 minus ikx sin omega k1 minus alpha T1. Now you hold this under the coupling iky cosine omega k1 minus alpha T1 gives you this cosine omega k1 minus alpha T1 will keep it out and inside this bracket this is the j evolution iky cosine pi jkl T1. Now you notice here for T1 I have only pi jkl because the coupling constant evolves for the period T1 and there is no multiplication of here. Therefore it happens for the cosine pi jkl T1 a minus 2 ikx iLz sin pi jkl T1. And the second term gives you ikx sin omega k1 minus alpha T1 evolves under the coupling as sin omega k1 minus alpha T1 and inside bracket you have ikx cosine pi jkl T1 plus 2 iky iLz sin pi jkl T1. Thus your density operator row 3 will be cosine omega k1 minus alpha T1 inside bracket iky cosine pi jkl T1 minus 2 ikx iLz sin pi jkl T1 and the second term gives you minus sin omega k1 minus alpha T1 and inside bracket you have ikx cosine pi jkl T1 plus 2 iky iLz sin pi jkl T1. So therefore the chemical shift appears at downscale by factor 1 minus alpha along the f1 dimension. Now this enables improving the resolution in the multiplied structure along the f1 dimension by increasing the T1 max for the given number of T1 increments. How does that happen? Because now since you have scaled down the chemical shifts, suppose you have the chemical shift range of 5000 hertz then you scale it by a factor of half then you will have if alpha is equal to half then it will be 1 minus half therefore it is 0.5. So if you have that one then your chemical spectral range will be 2500 hertz in which case your increment from experiment to experiment in the T1 will be twice that in the normal cosy where you are 5000 hertz. Therefore for the same number of T1 increments your T1 max will go up and therefore your inherent resolution in the spectrum will increase because you remember that will depend upon what is the acquisition time, resolution will depend upon the acquisition time. So this will be the acquisition time and this will get increased because you have changed the chemical shift spectral range. So therefore again you will get an improvement in the resolution but however the separation between the peaks overall that will get reduced. So so long as you can afford that you can do this and so long as you can do it in such a way that the peaks do not overlap on each other you can do this and you can achieve the enhanced resolution in the fine structure of the cross peaks or the diagonal peaks. Now once again here we have not included the relaxation effects we can include the relaxation effects here. So relaxation during the period is not explicitly included. Now if we include this it will cause the scaling of the line widths as in the case of j scaling. Now if we include the relaxation effects we will get the rho 3 star explicit calculation will go in the same manner as was done for the j scaling effect. So you get here rho 3 into e to the minus T1 by T2 star 1 minus alpha this is from the chemical shift and this is from the coupling alpha T2 star by T2. So therefore clearly the line widths are scaled by the factor. So this whole thing if I take away T1 by T2 star this whole factor is the so called beta here and that is 1 minus alpha plus alpha T2 star by T2. In the event T2 star is equal to T2 the line widths will not get scaled because this will become alpha and this 1 minus alpha plus alpha will cancel and this will remain as T1 by T2 star. In that case it will not appear scale but if T2 star is less than T2 as it happens in most practical cases the line widths will be scaled down and this will result in better resolution in the multiplied fine structure. And this condition will always be satisfied that T2 star will be less than T2. You get improvement when you have this condition satisfied the fine structure will appear much better when you have downscaled the chemical shifts. So here is an experimental example practical example although you cannot see the fine structures here but this was used to improve the resolution in the fine structure of the peak. This is a spectrum taken from Wittrick's book NMR of Proteins and Nuclear Gases and this is a spectrum of a DNA molecule. You have so many cross peaks here what is shown here is only the cross peak region of a DNA segment. Along this axis you have particular portion of the protons and this axis you have certain segments. These are basically 2 double prime protons in the DNA and this is the 1 prime protons in the DNA in the sugar ring and these are the 3 prime protons in the sugar ring and you can see in every case the fine structure here is improved along the omega 1 axis. And one can actually use this to measure this individual coupling constants and then when you do that you can obtain information about the geometry of the sugar ring in the DNA segments and that is the application why this was done. So with this application you could actually determine the coupling constants in the sugar ring and that will allow you to determine the geometry of the sugar ring in the DNA segment. So therefore in summary I have shown you here today a new concept that is how to manipulate the parameters in your spectrum. The parameters are this coupling constants and the chemical shields. We have considered the upscaling of the coupling constants which will allow you to improve the sensitivity in the spectra and improve the separation between the multiplets in the fine structure and thereby you can measure the coupling constants from their separations in the cross peaks. And I have shown you the downscaling of the chemical shifts. You can afford to do this if the peaks are not too close by in the 2D spectrum. So long as they do not overlap with each other by downscaling you can do it and that will improve the acquisition time along the F1 dimension, the T1 dimension and that will improve the resolution along the F1 dimension or the omega 1 dimension as is indicated here. In the early days one used to use call as omega 1 and omega 2 but often people use F1 and F2 as well. So this is the application and we will see later how to increase the chemical shifts. Appearance of in the spectrum that can we make the chemical shifts enhance or scaled up. So we will see that later and so with that we will stop here.