 So, today we are going to start a new topic namely multidimensional NMR spectroscopy. This is basically an extension from the Fourier transform NMR which came up in 1966 and that caused a revolution in NMR applications. Things which could not be handled earlier could be handled because of the sensitivity enhancement it allowed. Fourier transform NMR had another significant advantage which I mentioned earlier when we were discussing Fourier transform NMR that is it separated the excitation of the spin system from detection of the response to the excitation. So, the time in between was available for various kinds of manipulations and two dimensional NMR and various other kinds of polarization transfer experiments which we discussed earlier were a consequence of this important concept. So, we will go through this development and this I consider as another major revolution in the application of NMR. Multidimensional NMR spectroscopy it started with the two dimensional NMR then it went on to more dimensions, three dimensions, four dimensions etc. And the basic foundation was already late when the schematic for the two dimensional NMR was developed. The ideas basically originated also quite early. So, it originated somewhere in 1971 and this was put forward by a Belgian physicist called Jane Jenner in a summer school where he proposed the idea of doing two dimensional NMR spectroscopy. Since then it has grown unparalleled and the kind of applications which it has generated is quite mind blowing. It allowed entry into new disciplines like biology, structural biology became possible because of two dimensional NMR in a big way and therefore we are going to go through this exciting period, exciting development in a systematic manner. All those would like to pursue chemistry in big detail or structural biology as a profession this becomes an important topic to learn. So, we are going to discuss two dimensional NMR to begin with and over then we go forward to the higher dimensions. The basic concept of two dimensional NMR as I mentioned to you is in this segmentation of the time axis. This is called the segmentation of the time axis. Here you have the time axis and now you actually divide this time into four different periods. I will explain these to you what these one periods are and just to note to take note of this. This is the so called one preparation period then this is followed by an evolution period then a mixing period and a detection period. Preparation period prepares the system in a suitable initial state. This may include the excitation. So, when you do that you create transverse magnetization from the spin system and then you allow this magnetization to evolve during the period t1 which is now a variable and therefore we call it as an evolution time and this is a variable time and the mixing is the period which allows mixing of spin systems. So, it can it depends on various kinds of interactions between the spins and it will cause transfer of information from this state of the spin system to this state of the spin system. Eventually we collect the data only during the t2 period but however it will have a modulation from the evolution of the spin system during the period t1. Therefore, here this time period is often called as indirect detection period and this is the direct detection period. The signal which is detected here will be modulated by what happens during this period and the mixing actually causes transfer of information from this period to this period and then you generate a two-dimensional spectrum which displays the information present here and present here and we will see this. Now, each of these periods, preparation period, the mixing period or even the evolution period can have a variety of possibilities. They may be simple delays or they can have pulses in built which will help create a suitable initial state and here also it can help create a suitable information in your spin system evolution and mixing depends upon what sort of interactions you want to reflect in your final spectrum and what sort of information transfer you want to achieve from here to here and that determines the so-called mixing and data is collected explicitly during the t2 period only. Now, so how is the experiment actually performed? Experimental protocol is given here. So, you do a series of experiments. In other words, you collect series of FID as typically the data you collect is in the form of FID and that happens in the t2 period. So, here you have the preparation period and the evolution time here it is 0. So, we call it a 0 time period here and is immediately followed by the mixing and you collect the data as a function of time t2. Now, you introduce a time period in between, you call it as delta t1. This is the evolution, this is the evolution time for the mechanization that has been created at this point in time and this one is then mixed by suitable interactions. The spin system transfer of information happens from one spins to another spins and so on and then the data is collected here as a function of time. Notice the initial condition of this FID is not the same as the initial condition of this FID and this is because of the mixing and what has happened during the delta t1 period. The magnetization has evolved if it is a transfer of magnetization here it evolved for a time delta t1 and then you do a mixing and you collect the data here as a function of time t2. Now, you increase this time to 2 delta t1 here it is an increment let us say this may be about 100 microseconds. We will see what determines this time increment and this you systematically increment this time increment. So, here you have delta t1 and you do this experiment once more with 2 delta t1 this part remains the same this remains the same these do not change the only this changes you increment this time. So, it is 2 delta t1 and you collect an FID once more here. So, once again notice the initial time point here is not the same as the initial time point here and this is actually the crucial part of this experiment and also although the FIDs are looking the same here and here the similarly it may not be the case it may have more complicated FIDs here here and here and therefore that also we will see as we go along. Now, you do the third experiment 3 times delta t1 do the same collect the data here as t2 4 times delta t1 do the same collect the data as a function of t2. Therefore, you repeat this several times how many times you do it and this will depend upon it the resolution you require in your spectrum. Now, notice that I mean of course you remember that this FID is collected as a function of time and in a digitized manner you collect various data points although it is shown as an analog signal here but you actually collect it in a digital manner right in the Fourier transform NMR this data is collected in a digital manner. So, here you are actually also generating time systematically incrementing the time. So, as the time is incremented here you are collecting data as a function of t2. So, here you are actually generating the second time variable where you have first point at 0 the second point at delta t1 the third point is a 2 delta t1 3 delta t1 4 delta t1 is the systematically incremented and you can collect as many FIDs here as many increments you want here typically how many you collect and this will depend upon the spectral resolution you want in your spectrum. Here in the t2 period often we know that in 1D spectrum you collect something like about 2048 points or 4096 or 8192 and which is easy. So, this will go on for a short period of time which is like about few hundred milliseconds. So, in few hundred milliseconds all this data are collected. Now, here if you increase them as you want to increase the number of points each data point is a separate experiment. So, you will have to perform that many experiments 2048 or 4096 experiments you will have to perform if you want to have that many data points here as a function of in the time evolution here. So, typically the number of data points here will be less than the number of data points here this is limited by the practical considerations and if we need to have more of course you can collect more and in each of these period of course you follow the same rules as we use in the case of Fourier transform and MR if you wanted to signal averaging for each of that time increment here you must do the signal averaging 8 scan 16 scan etc whatever you have to do for each of these experiments and you collect a two dimensional data body. Now, let us see what happens afterwards. So, here is a actual demonstration here you have the first data point the time t is equal to 0 you have the first FID and then delta T1 you have the second FID the third to delta T1 this is the third FID but there is no signal here as I said there will be modulation and depending upon what the initial point is and how the system evolves what the signal is present you will have this different kinds of FIDs. So, each of these FIDs is different so 3 delta T1 now it is actually starting from the negative value here if you see here this was positive here reduced to positive value 0 here now it is negative 0 and then it is further negative and then this starts decreasing again therefore the signal amplitude at the time t is equal to 0 is getting modulated as you are going down with the increments. So, you can keep collecting such data of course it will be repetitive when it is repetitive and you get time dependent variation. So, let us say you collect something about 256 data points or something like that so that many that much of time you will have to give but let us look what happens afterwards. Now, you collected the FID here and you Fourier transform this along this time axis this is called as the T2 time axis this is the T2 you get a spectrum which is like this. Now, the second one you notice the first data point intensity is reduced therefore what happens this if it were a single line spectrum here it will be a single line spectrum and it would produce a line with a reduced intensity because the intensity is reduced here if you remember correctly the first data points is the integral of the spectrum F of 0 is the integral of the spectrum we discussed this in the Fourier transform in MR F of 0 is equal to integral FWDW right. So, therefore that whatever is the initial point here and that is what you will get here in this one. Now, for the third FID for the fourth FID this starting negative and the signal is now inverted. So, it is going negative here and this is fourth point is also negative but with the enhanced intensity because this data point is initial point is higher than this and phi delta 1 again it reduces 6 1 is back to 0 7 1 it is again going positive with the positive signal and so on. So, therefore this keeps getting modulated the signal here is getting modulated by the evolution during the period T1. Now, let us look at this in more detail this is the spectrum here now what you do this is all also digital right. So, when you do a Fourier transformation you get a digital spectrum this is the spectrum is drawn as continuous analog signal but actually appears as a digital spectrum this is each points here there are points 1 2 3 4 5 things like that various points are there. Now, these will pick up a few of those points here. So, this is a let us say this point is called A this is this is the time axis now this is the frequency axis here this frequency axis along the F2 after due to Fourier transformation here you get the F2 frequency axis right this is the transformation along T2. So, you got the F2 frequency axis now this is the frequency A and this is frequency B then you have a frequency C frequency D then you have a frequency E all these C D E they are all here then you have F and you have a point G you could have chosen any other points also it does not matter what is this done here is you have chosen two points where there is no signal in the frequency spectrum and you have chosen a few points which are in the signal in this spectrum. Now, what you do is you take each this point in each of these FIDs each of this spectra and plot them here. So, let us look at what is point A. So, point A this here this here this here this here and you put all these points here okay there is no variation here right because it is all the same now we come to point B the point B it is this one is here and here it is 0 and then it is here then you take this here here here and here and you plot that here. So, initially therefore this is going like down see along this axis what is it now this axis is T1 because what is varying as you are going along is the T1 increment therefore I want to call this as the T1 variable this is the T1 time axis here. So, this point B has got a small dip and there is an oscillation here it is going at this point. Now, you take the third point which is well in your signal in your line. So, now you talk the third point C then you will see there is a greater dip here. So, there is a greater intensity variation here in this point okay. So, and you take the fourth point it is even greater. So, there is more variation here then you go to the E point then it is decreasing then you go to the F point further down and you go to G and again it is very down. So, what we have got here therefore this is also an FID and like here what you had the FID was going point by point you just showing variation these ones are also FIDs. These FIDs are different depending upon which portion of this frequency spectrum here you are looking at. Therefore, I can do a Fourier transformation along this time axis now right this is the time variable T1. See if I do a Fourier transformation along this time against this time T1 there is no signal here therefore there is a 0. If I do here there is a small signal there is a small frequency component here therefore you have a small signal with a particular intensity which is very small. Now, if you come down further it is the same line peaks at this point now the intensity is more. So, intensity is higher. So, this signal FID is increasing the signal is higher here Fourier transformation again decreases goes to 0. Therefore, you see here I have created a time variable which contains frequency information. So, this one now is called as the F1 axis. So, I got an F1 frequency axis and I have an F2 frequency axis these two things are different. So, I have generated a 2 frequency axis by doing this sort of a manipulation. So, this is very important this point must be very clear before we go forward. So, we may be I will try to repeat it here. So, that you things become absolutely clear. So, we do various experiments with increments along the T1 axis 0 delta T1 2 delta T1 3 delta T1 and at each of these delta T1 values you collect an FID and the FID has different behaviors because of the evolution during the delta T1 and something may happen during the mixing also which can also affect this FID. So, therefore you generate a whole series of FID is here and if you Fourier transform this along the T2 dimension you generate a frequency domain spectrum and you can see the frequency domain spectrum has modulations here because the FID was modulated by the evolution period. So, this you take all of these frequency points and join the individual frequency points and plot them here and the generates the time axis this variation is a representation of the variation due to the time T1. So, this also has a frequency component therefore if a Fourier transform each one of these FIDs here these are now indirect FIDs right these are not direct FID you do not actually collect the signal during this. So, these are all indirect points after you have collected the signal and Fourier transformed it by looking at these frequency points you generated a frequency dependent variation. So, therefore this is a time variable which has a frequency component here frequency it displays the frequency and your Fourier transform here therefore you get a frequency component whatever is present here. So, in this particular case it is the single frequency it is the same frequency which is present and it is intensity is getting modulated by the evolution during the time T1. So, therefore these frequencies may be different may be the same. So, here you have just shown you have not indicated what this frequency is it may be the same frequency it can be different frequency it can be combination of both. So, now let us put it little bit more formally here you have S, T1, T2 you have generated a two dimensional data body as a function of time variables T1 and T2. Therefore, you can do a two dimensional Fourier transformation and it will generate a two dimensional frequency spectrum F1, F2 whatever is the information present in the T1 period will appear along F1 and whatever is the frequency information present during the T2 period will appear along the F2 axis. So, this is your two dimensional frequency domain spectrum and this is your two dimensional time domain signal. Now, consider a spin K whose X, Y magnetization has been created by the preparation period. So, in the preparation period what you did is to create a transverse magnetization of the spin K like for example it may could have been I K X or I K Y whatever. So and assume that you created the transverse magnetization here and this magnetization evolves with the frequency omega K during the evolution period because of the frequency evolution right the chemical shift evolution during the period T1 it evolves with the frequency omega K. And of course there can be more than one frequencies as well here but let us assume one of those there can be a frequency omega K it can be some other modulated frequency chemical shift plus minus coupling if it is there and things like that but let us consider one particular evolution which is a omega K frequency. Now, at the end of the period T1 the magnetization has components MK0 cosine omega K T1 and MK0 sine omega K T1. So, let me explicitly state this to you you have created the transverse magnetization after the preparation period. So, here this is the preparation you have created a transverse magnetization here let us say it is I K Y and here during this is the evolution period T1 and this is our mixing period and here you have the T2 period T2. So, during this period this magnetization you holds with its characteristic frequency I K Y cosine omega K T1 minus I K X sine omega K T1 and of course these are the operators here and I will put here a magnetization component this is the to indicate the magnitude of the magnetization which is represented as MK0. So, but this is the operator term here we will ignore this. So, the magnetization actually will be MK0 cosine omega K T1 and MK0 sine omega K T1 that is the detection during the you have this right. So, you have this is the Y and this is X and this is Z the magnetization which is here as MK0 and this as it is evolving. So, you get magnetization moves here. So, this is cosine omega K T1 and this is the sine omega K T1 right. So, these are the frequency components you will have. So, we will have these components MK0 cosine omega K T1 and MK0 sine omega K T1 along let us say the Y and the X axis this because these will be the two components MK0 is the magnetization at the beginning of the evolution period. Now, let us now assume that the mixing period transfers part of the magnetization to spin L. I said what happens during the mixing is dependent on the interactions between the various spins. Consider the spins K and L and if there is an interaction between them of some sort it may be either a J coupling interaction or it could be a dipolar coupling interaction whatever it is. But if this mixing allows transfer of magnetization from K to L then a part of this magnetization may get transferred to the spin L in which case what happens during the period T2. So, there will be some magnetization which remains on the spin K and some magnetization will be on spin L. So, and the component which is present there are represented by this coefficients here A and B. A is the component which remains on spin K itself and the B is the component which gets transferred to the spin L. Now, during the period T2 what happens this magnetization continues to evolves with its own characteristic frequency omega K right. So, this is omega K this is the because with the K magnetization. So, during T2 also it is goes with the same frequency omega K. I mean I am giving a very simplistic picture here the actual calculation will may have to involve the Hamiltonian evolutions consideration under the influence of the different Hamiltonians and you are considering a very simplistic picture to bring out the concepts. So, if it is going with the same frequency omega K then we have the signal which you are going to detect which you are going to detect here will be a MK0 cosine omega K T1 cosine omega K T2. So, this and the one which has gone on to the L that will evolve with the frequency omega L during the period T2 right. So, therefore, in the T1 period this was cosine omega K T1 that was the component which had come as a result of evolution during the T1 period and at the end of the T1 period this was partly transferred to B and some of it remained on A itself. So, therefore, this portion is the end of at the end of the T1 period and this is the result of evolution during the T2 period. Now, if I do a two dimensional Fourier transformation of this signal of course it will be combined we are going to detect it together. There will be both the components present in the signal what you will detect in the FID what you will get as a function of T2 it will be superposition of these two both these will be present. Therefore, when you do a Fourier transformation you have to consider Fourier transformation of both of this both of these terms and when you do this consider this term what will it generate it will generate a frequency omega K along the F1 dimension and the frequency omega K along the F2 dimension. Therefore, we call this as the diagonal called as the diagonal in the frequency domains in the two dimensional frequency domain spectrum this is the F1 is equal to F2 is equal to omega K. Now, in this particular case if I do a Fourier transformation then along the F1 dimension I will have this frequency omega K but along the F2 dimension I will have this frequency omega L. Therefore, this is F1 is equal to omega K F2 is equal to omega L. Therefore, this is called as the cross peak. Here it is assumed that only the Y component of the magnetization is detected during T2 period but this is a very simplistic view as I mentioned to you A and B are some coefficients representing the relative contributions these equations represent of course oversimplification as I mentioned to you already to bring out the concepts. Now, schematically if you want you to represent in a diagrammatic form this is your two dimensional frequency domain spectrum and these are the two 1D signals here. So, one frequency omega K other frequency omega L we started the calculation with the omega K frequency we said omega K frequency part of it remains at omega K and this produces the diagonal and part of it is transferred to the spin L and this becomes your cross peak. So, I generated from omega K this frequency and this frequency but there is no stopping if you are exciting both the spins at the same time then the same thing can happen for the spin L as well. So, if you started with the spin L then during the T1 period it will evolve with the frequency omega L and during the mixing some of it will get transferred to omega K. Therefore, this will produce a cross peak here and a diagonal peak here. So, therefore, each one of these will produce a diagonal peak and a cross peak. So, this will depend upon the interactions between these two spins whatever the interactions are and that will be reflected in the two dimensional plane and this was the phenomenal effect because where there are lots of signals which are present underneath at various one particular place you could filter out those signals which are interacting. So, you can figure out from the cross peaks which spins in this spectrum are interacting if you notice this is essentially a one dimensional spectrum. So, the diagonal essentially represents a one dimensional spectrum you take a projection here or here it is the same frequency right here or here it is the same frequency but when you have lots of signals here present then you actually filter out those which are interacting. Therefore, this was a major revolution because you could display the interactions between the spins in a two dimensional plane and that allowed resolution enhancement in the spectrum and extracting information which one could not do earlier. So, here the this one is F1 is equal to F2 this is the diagonal and this is also diagonal and this this color references F1 is not equal to F2 and it could this will where this peak will appear depend upon what are the interactions between the spins and how the mixing transfers the magnetization between the spins. So, we will stop here and we will continue with the theory of this in the next class.