 So, there have been detailed theoretical calculations for NOE, we will not go into those detailed theories, but I just want to show you the results, NOE as I said this will depend upon the relaxation times and this will depend upon the molecular motion. How fast the molecule is tumbling inside the solution and that is characterized by a particular time constant known as the correlation time. For small molecules correlation times is of the order of 10 to the minus 10 to 10 to the minus 12, this is for nanoseconds, seconds are nanoseconds therefore 10 to the minus extremely small time, seconds so that means is the order of nanoseconds and for this is for small molecules and correlation time can be of the order of 10 to the minus 8 to 10 to the minus 9 for large molecules. So, this basically reflects how fast the molecule is tumbling inside your solution and the relaxation times will depend upon this. Now, according to that so the theoretical calculations on NOE have been done and it turns out that the NOE is equal to if we want to represent this as NXA and A is the one which is perturbed in the earlier case I wrote it as NiX, A is the one which is perturbed and NX is the X is the one which we are going to monitor and this is typically equal to half gamma A by gamma X where gamma is the gyromagnetic ratio and it all depends upon the individual spins also and this will be different for different situations, this is for this condition, this is for this condition. So, therefore what will be the kind of NOE, this is the steady-state NOE, steady-state NOE means I do an experiment in the as I indicated to you before, the experiment is done in this manner that I have a FID collected here and I do a pre saturation perturbation, I do a perturbation here and one particular line on a particular line let us I do it for the spin A and I monitor the spin X here. So, when the perturb here I monitor the intensity and without the perturbation I monitor the intensity and then I take the ratio get the NOE and this will now by calculations one can show that this is proportional to half gamma A by gamma X, what is the advantage here? So, the advantage is suppose A is proton, suppose A is proton and X is let us say carbon 13, what is the ratio gamma A by gamma X? It is 4. So, therefore and this NOE will be a factor of 2, if gamma A is equal to gamma X then of course it is a factor half. So, this is 0.5, this is an enhancement, how much is the enhancement in the intensity of the lines then when you perturb one spin and observe the other spin and if you are perturbing proton and observing carbon 13 then means you are getting a factor of 2 enhancement in the intensity of the line. So, and that is the significant advantage, that is the significant advantage because the factor of 2 enhancement meaning it is like a factor of 4 in terms of the time with respect to the signal averaging. So, therefore the steady state NOE, this is the steady state NOE which we call steady state NOE and in the this limit is called as the extreme narrowing limit. This is the small molecules very rapid motions and this is also called as extreme narrowing limit. Here the tumbling is very rapid, the motions are very fast. So, the averaging happens much much better and the relaxation times are also very different. So, therefore without going into the theoretical detail this is to just to show you what kind of a thing is it will look like. Now what will be the situation when we are in this larger molecule range? The large molecule range it turns out that this is equal to minus gamma A by gamma X for large molecules NXA is equal to minus gamma A by gamma X. So, therefore you can see for proton if A is equal to X then this NXA is equal to minus 1. And if I want to plot here the NOE as a function of the correlation time here it will be it will look like this and this is my 0.5 and this is my minus 1. So, this is the case for homonuclear systems. So, you have minus 1 so the negative NOE you will get this is called as negative NOE. But nonetheless it is it is a kind of a significant change in the intensity and when you take the difference you will get a substantial enhancement and for large molecules you will get a minus 1 as a this one. But if it is not homonuclear then of course you are gamma A by gamma X that is proton and nitrogen or something then you get a factor of half also and then you will get minus 5. So, you get heteronuclear once of course you will have a different number for it will be minus 4 something like that minus 5 for proton or nitrogen because this is well nitrogen has another complications in the sense of the Garam ironic ratio for nitrogen is negative. So, therefore, there even for the small molecules there that is the gamma A by gamma X in the case of nitrogen it will be negative. So, this is what I showed here in the previous slide when I said half gamma A by gamma X you see here if you are talking about nitrogen proton nitrogen NOE and then let me also draw that here if let us say A is equal to proton then X is equal to a nitrogen 15 then this nitrogen 15 has gamma of nitrogen is negative. So, therefore, you will get a negative enhancement there and that will be a factor of 5 because gamma A by gamma X there is 10 and therefore, divided by multiplied by half then you will get a minus 5. So, even in the extreme narrowing limit when you are talking about heteronuclear NOE proton nitrogen it is minus 5 and proton carbon it is plus 2. So, these are the factors one has to remember one studying these experiments in the steady state NOE. Now there is another experiment so far I talked about what is called as the saturation pre saturation this is called as one saturation was done on the spin A and we are monitoring the spin X. There is another way one can do and that is called as transient NOE. Transient NOE what you do here is your perturbation will be like this you do the experiment like this let me draw this by a different color this one I will put like this. So, this is a selective perturbation selective inversion and this is a hard 90 degree pulse this is a 90 hard pulse what does that mean? So, then let us say I have a spectrum something like this. So, what I will do is I will selectively invert this line I will selectively invert this line after 180 degree pulse this is 180 degree. So, I will selectively invert this so then I will get here this one like this and this remains like that and whatever happens here and I give a time here called tau m and tau m is the is the called the so called mixing time what I have achieved here let me explain what we have achieved here. So, this is the experiment what happens? So, let us assume that I have the initial magnetization of all the spins all of these have different magnetization components right let me draw the X, Y, Z. So, draw the magnetization of this particular spin here and the magnetization of the other spins also will be here let us say. So, during the period when I apply selective 180 degree pulse what I am achieving is what I am achieving is I am inverting this magnetization I am selective applying a selective 180 degree pulse on the black line which means the magnetization which was here is now like this when you measure it of course I will see it as a negative signal now this is the perturbation this is the perturbation this perturbation has to recover back to equilibrium the system will have to recover back to equilibrium what it will do it will relay this perturbation to some other spins which are close by in space. So, then this will relay so as it recovers as it recovers like this it will relay this magnetization to some other spins. So, therefore what will happen is these components which are present here they will also get little bit of that component the red one which was there red one we will also get certain contribution from the let us say like this a contribution and the black of course is recovering down there. So, the black is slowly recovering so it is it has come let us say then of course certain contribution of the magnetization has gone do not assume that the certain portion of the black it is transferred that to the red. So, therefore this will show up when I measure it it will show up in this some intensity of the red one if this were a red one that would have gone down or it would have gone up either way this can happen for both positive and negative NOEs you saw already that for small molecules it is positive NOE which means that is an enhancement for large molecules it is the negative NOE means that will decrease intensity will decrease. So, therefore there can be changes in this other portions of the spectrum as a result of the of the inversion here and I give certain amount of time here tau m because I am allowing it the magnetization to transfer this is this is dependent on the relaxation time depending upon the relaxation time t1 the efficiency of transfer will vary. Therefore, this can go to different ones as a result of the dipolar coupling between the two spins. So, therefore when I measure this I will get different intensities of these lines here not all of them but some some I will get some I do not get no something will be perturbed some will not be perturbed. So, now if I take the difference here if I take the difference between these two this fellow has not perturbed this has perturbed this has perturbed and maybe this is not perturbed let me let me take it up there this is not perturbed. So, then I take the difference difference is equal to what I will get I will get a thing here not much larger let us say this minus this then I will have a 0 there and this I will see minus there because this is increased this is small and this also I get a minus here and then I get 0 here again no perturbation. These are the perturbed ones these are the NOEs and this will depend upon the mixing time this will depend upon the mixing time how much time do I give why does it depend on the mixing time because it depends on the relaxation time of the perturbed spin and the relaxation times as other ones also. Now once I have done this will it always remain there will it remain at that point only question is let us say I have a molecule like this and I have a proton here a proton there a proton here a proton here a proton there so on so forth I perturbed one of these fellows I perturbed this fellow then it will relay from here to here because it is close directly during the T1 T1 but this is also now perturbed as a result of this this is also perturbed therefore it will also have to relax what it will do it will also transfer to something which it is close by so let us say I have I take that so from here it will go here one more or it can go from here to here or it can go from here to here and so on so forth. So what is this phenomenon called this phenomenon is called as spin diffusion. Spin diffusion means the magnetization which is perturbed at one point it will keep diffusing through your network of coupled spins and that will obviously depend upon the time tau m so this will depend upon proportional to tau m. Larger the tau m you give longer will be the spin diffusion and therefore if you restrict the tau m to very small value then it will go only to the directly neighboring spin so if I want to plot so if I want to plot the NOE as a function of tau m then it will go like this it will come down. So this is the linear this is called the linear regime during which time it is directly proportional to the neighboring spin only going to the neighboring spin only and it is in this region one can calculate inter proton distances from the NOE intensity okay the here the NOE intensity is proportional to what is called as the cross relaxation rate which is represented as sigma and this is proportional to 1 over r ij to the power 6. So if I say sigma ij so this is proportional to 1 over r ij to the power 6 in the in this approximation in this region in this region it is proportional if we go beyond that because the intensity NOE intensity is no longer just restricted to that it will start diffusing to other ones so the relaxation individual coupling between all the spins will come into the picture and you may not be able to do it. Therefore transient NOE is an experiment which is done with a controlled manner you can control the extent of diffusion that can happen from one spin to another spin and that is the extremely useful parameter for calculating the structures of molecules and this is the basis of structure calculations this is for short tau m okay and there it is restricted to the near neighbor interactions only for short tau m the transfer is is restricted near neighbor interactions therefore so polarization transfer has two objectives right so two objectives polarization transfer by NOE has two objectives so therefore NOE in summary of the time NOE has two gains one is enhance signal intensity less sensitive nuclei this is like carbon 13 nitrogen 15 and the second is derive structural parameters macromolecules so these are two important applications of NOE and we will not go into the theoretical details of this much more than this I think that is that is required for our purpose of structural biology which is a focus of this because otherwise it can get very complex okay so now let me take one more topic okay so we will take another topic here which is called as selective population inversion consider a two spin system AX A and X this can be heteronuclear mnemonuclear also and what I will have I will have here a simple 2x 2x spin system will have A will have two lines and X also will have two lines okay let me draw the energy level diagram for this I will have four states let's say the population of this is delta plus delta population of this is minus delta plus small delta and this one is delta minus delta and this one is minus delta minus delta okay so now I call this transition as A1 this transition as A2 call this as X1 and this as X2 okay now let me draw the intensities of the A transitions intensity of the A transition is population this population minus this so therefore that will be 2 delta and similarly the other one also A2 so if I call this as A1 this is A2 and both these have intensities of 2 delta and what about X1 and X2 this is this minus this so I will have smaller intensities and what is here delta is greater than delta okay and this is 2 delta and this is also 2 delta and this is my X1 and this is my X2 so now what I do is I invert this transition invert selective population inversion this is what I say selective population inversion so now what will be my intensities of the four levels now I will have to draw this levels again here this population becomes this so this will be minus delta plus delta this will go there delta plus delta okay I am inverting only this I am not changing anything else so this will be delta minus delta and this will be delta minus delta I am not doing anything here I am not perturbing these populations at all okay now I see what happens to the transitions what happens to the transitions the four transitions these were my A1 A2 transitions and the X1 X2 transitions are these X1 X2 what will be the intensities here let me draw the A1's first and this is now minus 2 delta okay the A1 will become minus 2 delta change the color here okay this became minus 2 delta and this one is this minus this this will be plus 2 delta okay so this is my A1 and this is my A2 what about the red ones red ones are these so delta plus delta minus of delta minus delta so minus delta plus delta minus delta minus delta so that is minus 2 delta plus 2 delta so this is minus 2 delta plus 2 delta and the X2 is 2 delta plus 2 delta 2 capital delta plus 2 small delta right so that is this plus this minus this so 2 capital 2 delta plus 2 delta see look at this what was small 2 delta here 2 delta here has become this large quite a substantial enhancement in the although the gain is not the same in both cases that is but the nonetheless each one of them has gained substantially in terms of the intensity of the lines you have at the cost of the magnetization of the A1 transition you have got the enhanced intensity for the X transitions enhanced intensity for the X transitions notice here these A1 and A2 transitions have become completely opposite to each other anti-phase this is called as the anti-phase they have become anti-phase the A transitions have become anti-phase because they inverted only one of them and therefore this kind intensity is changed so one is minus 2 delta other one is plus 2 delta in the earlier case both were 2 delta 2 delta and I have inverted one of them therefore that became one of them became minus 2 delta other one became remain the same as 2 delta and with the X1 and the X2 transitions which were initially both equal as 2 delta and 2 delta now one of them has become minus 2 capital delta plus 2 delta and the other one have become plus 2 capital delta plus 2 small delta and this is substantial gain in the intensity right. So therefore this obviously will depend upon how much is the individual difference between capital delta and small delta and that is dependent on the gamma of the gamma A gamma of A and gamma of X okay so that is proportional to the ratio of the gyromagnetic ratio. So the gain is proportional to this is this depends upon upon gamma A by gamma X okay so the gain will be proportional to the ratio of this gyromagnetic ratios of the two so substantial gain can be obtained if it is heteronuclear for single homonuclear systems you may not be that much of an advantage but heteronuclear it will be great advantage this principle is used in other techniques which will take up next time and that is particularly useful for all multidimensional multiples experiments so the important thing of course to notice here is that you may call this as a disadvantage while there is an advantage which you already discussed the disadvantage could be that you have positive negative signals you can say that disadvantages would be you have one positive and negative signals second you cannot decouple A and X because if you decouple the two will collapse when the collapse you will get back the same intensity okay only when the coupling is present the two lines are separately observable then you gain the advantage so therefore and you want to get rid of both of these kind of situations and that leads us brings us to other techniques which is which is called as insensitive net and of course in the third disadvantage also could be selective inversion requires long pulses requires long pulses that means it will not be 10 microsecond pulse it may be millisecond pulse okay if depends on the selectivity what you will want to observe so the better the selectivity you want the longer the pulse length will be therefore and this is a difficult thing to achieve and then the four selectivities selectivity may be difficult complex spectra so while the gains are there with regard to the intensities you also have certain disadvantages here to get over this we go into some other techniques which are called as inept and that will be taken up next and in fact that will be the most crucial thing for all multidimensional NMR experiments so I think we will stop here.