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Simulation of one-dimensional NMR spectra

1. V I I I I I I I 190 000 185 000 180 000 175 000 170 000 165 000 160 000 Simulation and spectrum analysis 3 Figure 4 O bserved and simulated spectrum of complex 2a and parameters used in the simulation Example 4 Dynamic behaviour of 1 6 8 13 anti bis methano 14 annulene A few trial simulations show that the spectrum can be explained by an AA BB X system with X Rh and accurate coupling constants can be obtained by iteration see Figure 4 Attempts to reproduce the spectrum using A2B2X or AA BB XX systems were unsuccessful This in combination with the numerical values of the coupling constants shows that the product is a cis bis diphosphine complex 2a De ieee eaten coupling constants Hz 48 P P PP 449 P Py 34 P P 40 RhP 185 RhP 166 Chapter 1 PR Compound 3 has a temperature dependent NMR spectrum Figure 5 1 It seems reasonable to explain this behavior by freezing out of the double bond shift in 3 at low temperature Is temperatures ee ee eer ere ee nd e ces 190 000 185 000 180 000 175 000 170 000 165 000 160 000 M MAN H H H H APATE f _ 4 HH H H 3 this explanation correct and if so can we extract the rates at different Simulation and spectrum analysis Chapter 1 Figure5 Temperature dependent spectrum of annulene 3
2. and rate constants Y ou can even fit away imperfections in the spectrum like baseline and phasing errors A draw back of the method is that it is very sensitive to the quality of the observed spectrum Small amounts of impurities the presence of humps in the baseline intensity distortions or incorrect phasing may trip up thefitting process and prevent you from finding an acceptable solution Therefore you should always try to obtain the best possible spectrum if you plan to do a full lineshape iteration and pay careful attention to phasing Even so it may be necessary to do some editing of the experimental spectrum before you start the full lineshape iteration 8 3 Strategy The analysis of an NMR spectrum is usually undertaken to extract a set of parameters shifts and coupling constants There are three distinct phases in this process e Arriving at a set of parameters e Refining the parameters e Checking for correctness and or uniqueness Full lineshape least squares analysis uses a least squares procedure to obtain the best fit between the observed and calculated spectrum One of the most attractive features of this method is that the precise numerical values of the final parameters do not depend on the way you arrive at them within limits there may be several distinct acceptable solutions Thus it is allowable to use any number of tricks to arrive at a reasonable set of parameters as long as you use a complete
3. 1981 G Binsch Band Shape Analysis in Dynamic Nuclear Magnetic Resonance Spectroscopy L M Jackman and F A Cotton eds Academic Press London 1975 p 45 ff A Steigel Mechanistic studies of Rearrangements and Exchange Reactions by Dynamic NMR Spectroscopy in NMR Basic Principles and Progress P Diehl E Fluck and R Kosfeld eds vol 15 Springer Verlag Berlin 1978 p 1 G M Whitesides and H L Mitchell J Am Chem Soc 91 1969 5348 M Eisenhut H L Mitchell D D Traficante R J Kaufman J M Deutch and G M Whitesides J Am Chem Soc 96 1974 5385 J P Fackler Jr J A Fetchin J Mayhew W C Seidel T J Swift and M Weeks J Am Chem Soc 91 1969 1941 10 M L H Green L L Wong and A Sella Organometallics 11 1992 2660 11 S Alexander J Chem Phys 32 1960 1700 References 73 12 13 14 15 16 17 18 19 74 References J D Swalen and C A Reilly J Chem Phys 37 1962 21 S Castellano and A A Bothner By J Chem Phys 41 1964 3863 B Braillon and J Barbet C R A cad Sci 261 1965 1967 B Braillon J M ol Spectrosc 27 1968 313 R Lozag h and B Braillon J Chim Phys 67 1970 340 Y Arata H Shimizu and S Fujiwara J Chem Phys 36 1962 1951 S Stephenson and G Binsch J Magn Res 32 1978 145 and references cited therein G H gele M Engelhardt and W Boenigk Simulation und automatisierte Analyse von Kernresonanzspektren VCH Verlag Weinhei
4. Contents Table of Contents Table of Contents eese n iii 1 Theroleof simulation in spectrum analysis 1 Ll Introduction teer titre tr Penn 1 12 2 OVEtVION 3 rera Nee 6 2 Thespin system iiiecaickiica ii co naa dnV IVa ga dein Ska anna 7 21 Introduction nennen nnne nnne 7 22 Magnetic equivalence sssessseeeeeennenenees 8 2 3 Chemical equivalence sse 10 24 Temperature dependent equivalence 11 25 Anisotropic spectra and full equivalence 12 26 Shifts and coupling constants sss 14 2 7 Thesigns of coupling constants sssesssss 15 28 Isotopic substitution sesseeeeeeenneees 16 3 Simple simulation nere enne nennen nnn nnns 19 31 Linewidths and lineshapes sees 19 3 2 First order spectra 21 3 3 Second order effects essent 22 4 Prediction of parameters from molecular structure 25 5 Simulating large systems seen 27 51 Onthescaling of NMR calculations sssss 27 52 Simplification by the simulation program 28 5 3 Simplification by the user 28 5 4 Approximate calculations sees 30 6 Chemical exchange 3
5. symmetry carbon atoms 2 and 6 have become equivalent and only a single resonance is observed for these atoms The spin system 11 Chapter 2 A simpler example is the methyl group of an ethyl compound In any static structure it can have at most C symmetry which would give riseto two separate resonances in the ratio 2 1 However the barrier to methyl rotation is usually extremely low 4 kcal mol so the rapid rotation occurring under most terrestrial conditions results in effective magnetic equivalence of the three methyl protons Similarly the three methyl groups of a t butyl or trimethylsilyl group are usually equivalent 2 5 Anisotropic spectra and full equivalence So far we have assumed that coupling constants are simply numbers In fact they aretensors and have an orientation dependent term In non viscous solutions however the molecules tumble rapidly and have no preferred orientation so we only see the average over all orientations the trace of the coupling tensor which is the number we call the indirect coupling constant J Itis also possibleto record NMR spectra of compounds dissolved in liquid crystals anisotropic media hence the term anisotropic spectra In such a medium the molecules will not tumble completely randomly but will have a preferred orientation with respect to the medium and to the external field Because of this the averaging of the coupling tensor is incomplete and we also see a
6. theoretical methods 25 R References 73 S Second order systems examples 61 Second order spectra 22 when to expect 22 Shielding and chemical shift 14 Shift definition 13 Simulation approximate calculations 30 large systems 27 simplification 27 why simulate 1 Singular variance analysis 55 76 Index Singular value analysis 54 Spin system AA BB 67 AA X 63 AnBm 61 definition 7 Standard deviation 53 Symmetry effective 11 T Transformation 58 Triangular 19 V Variance 53 Variance covariance matrix 54 W Weighting 58 Index
7. 00 QVE Chor a Ck 9 C UL UTO 000057 980880 038811 000050 006159 047396 138888 028805 4118968 000051 030019 980739 000048 005717 047676 141295 028754 011204 000014 003530 003562 000502 241583 159830 163774 9829310 0 X2 ER uM a GE X o 031482 000284 003734 000574 000602 969685 038923 025715 237759 053600 000026 138671 143841 000095 014302 610098 729864 4228216 349033 000174 006726 O 000368 0 s 0 024732 088840 019402 733241 575855 919820 000501 132936 089717 000790 026405 241990 262590 000584 Occasionally a certain linear combination of parameters is not determined at all by the data This may happen for example when a certain shift or coupling constant does not affect the appearance of the spectrum You will then see a zero singular value in the SV matrix Take care The corresponding direction has had to be excluded from the calculation of the normal variance covariance matrix because including it would mean dividing by zero So you may see a small or even zero estimated standard deviation for such an undetermined parameter The moral please read the singular value analysis Error analysis 55 Chapter 10 10 1 D NMR data processing 10 1 Introduction The main focus of this booklet is on using simulation to analyze NMR spectra Before doing this however you need to havean NM
8. 120 Hz An attempt to prepare 1 1 1 4 4 4 Rd hexafluoro 2 butene gave a a m B product with the IH NMR on a spectrum shown in Figure 2 Did 2 e Xx the synthesis succeed A nd if so is the produc the cis isomer the cis trans trans isomer or a mixture 2 Simulation and spectrum analysis Figure2 M ixture of cis and trans hexafluorobutenes Example 3 An unknown rhodium complex Figure 3 Rh complex of phosphine 2 Chapter 1 cl th a O a a u u a u a u A M5 11r 6 500 6 400 6 300 6 200 6 100 6 000 5 900 5 800 5 700 Both isomers are AA X 3X 3 systems which always give rise to symmetrical spectra Since the spectrum contains two symmetrical multiplets it seems likely that it is a mixture of thetwo isomers But which is which Even though the multiplets look complicated their appearance is governed by only four coupling constants 2 uu JJur ur and 5Jer A bit of trial and error simulation followed by iterative optimization will yield values for all four parameters The most important one is probably Ju which turns out to be 11 Hz for the low field multiplet and 15 5 Hz for the high field multiplet This isa strong indication that the major component is the cis isomer Reaction of diphosphine ligand 2 with a rhodium complex resulted in a compound with the 31P 4H PoP NMR spectrum shown in Figure 3 Is it possible to deduce anything about the stoichiometry and structure of the complex PR Ito
9. Particular orientations of bonds or z systems relative to a nucleus can cause longer range effects on chemical shifts and particular shapes of the bond path connecting two nuclei sometimes result in abnormally large long range couplings The prediction of NMR parameters from molecular structures is discussed briefly chapter 4 2 7 The signs of coupling constants NMR resonances are dueto transitions between different spin states of nuclei Coupling constants are a measure of the influence that the spin state of one nudeus has on the energy levels of another nucleus A positive coupling constant implies that the nuclei prefer to have their spins antiparallel a or Ba and a negative coupling constant implies that they prefer to have their spins parallel o or Bp 5 In general it is difficult to determine the absolute sign of a coupling constant but rdative signs i e relative to the signs of other coupling constants can often be determined by several types of 1 D or 2 D experiments It is possibleto give rules for the signs of sometypes of coupling constants For example the geminal coupling of an aliphatic methylene group is usually negative vicinal HCCH couplings are nearly always positive For other types of couplings however the signs can vary from compound to compound If coupling constants can have either sign the question arises whether these signs affect the appearance of the NMR spectrum In general spectra that are complet
10. TEE BrE Br a 6 3 Interpretation of exchange rates It will be dear that band shape analysis can be a powerful mechanistic probe There are however a number of potential pitfalls 40 Small line broadenings as observed near the slow and fast exchange limits can be caused by a large number of factors and exchange is only one of them Therefore rate constants determined near these limits are necessarily rather inaccurate Chemical shifts often show a marked temperature dependence If the signals that are coalescing in the exchange process are dose together to begin with this may result in large errors in the fitted rate constants In principle it is possibleto fit chemical shifts and rate constants simultaneously but near coalescence there will always be a high correlation between the two which makes such an optimization risky Coupling constants are much less temperature dependent they should be determi ned from the slow exchange spectrum and fixed for subsequent fits The predicted differences in coalescence behavior for different mechanisms are seldom as obvious as those illustrated above One should not be overly optimistic in distinguishing between mechanisms Small amounts of impurities may have a large effect on reaction rates Also impurities may cause new exchange mechanisms competing with the one you aretrying to observe This may lead to completely erroneous interpretations of the results Occasionally y
11. group second order effects will appear when coupling constants to nuclei outside the group become comparable to chemical shift differences between these nuclei Thus the second order effects in an A2B3 ethyl group depend on the ratio JApg A Ag both Jaa and Jgg are irrelevant If there are groups of chemically equivalent nudei in the molecule you can expect problems The shift difference between the nuclei in the group is zero by symmetry so there is no J A rule to use Instead you can expect second order effects when for any nudeus X outside the group and two nucle Y and Z inside the group the ratio rx Jyz lJxv Jxz is in the order of 1 If rx is very small you will see separate XY and XZ coupling constants if rx is very large you will only see an average virtual coupling and if rx 1 you will see second order complications You can also expect second order effects if rx is very small for some X and very largefor others even if there is no X for which rx 1 To illustrate this Figure 10 shows the 1H spectrum of ODCB at different magnetic field strengths At low field theinner lines are much more intense than the outer lines this second order effect is caused by the small chemical shift difference between the two types of protons For fields higher than ca 300 MHz this effect has largely disappeared the two multiplets are each approximately symmetrical However they are not simple doublets of doublets of doublets and
12. handle the interaction between these groups Perturbation theory does not result in large savings but likethe other techniques mentioned above it can make the difference between a feasible simulation and an impossible one 5 3 Simplification by the user Unlike a simulation program you as user know what is really interesting about a particular spectrum Therefore you can take more drastic measures to reducethe size of a simulation e Delete parts of the molecule remote to the fragment of interest e f you areinterested in a molecule having several equivalent fragments use only one such fragment and if necessary terminate it with an innocent end group e Setvery small couplings between nuclei in different fragments to zero so that the simulation program can divide the molecule into uncoupled fragments These measures will all change the simulated spectrum unlike the ones mentioned in the previous section Therefore it would be 28 Large systems Chapter 5 unwiseto let the program apply them automatically And if you apply them yourself you should alwaystry to check whether the simplifications were justified For the correct simulation of second order systems you often need to indude more than just the nuclei that couple directly to the fragment of interest As an example let us try to reproducethe Ph cp cu P methylene group signals of a N pet bis benzylphosphine rhodium complex Ph P ds pes t Figure 11A Thetw
13. isotope Y are related to the original coupling constants Jz via Jxy Jxz w yz These relationships between isotopomers are not exact because the presence of an isotope changes the vibrational levels of a molecule and the populations of different conformers Obviously substitution of a single isotopic nucleus for one member of a magnetic equivalence group destroys the equivalence Couplings to the isotope can now be observed and the above relationship can be used to estimate the coupling constants within the original group of equivalent nuclei For example substitution of one proton of a methyl group by deuterium allows observation of up and therefore estimation of Jyp of the original methyl group as Jyp 65xJup The presence of an isotope can also destroy the symmetry of a molecule in a more subtle way For example ethylene has four equivalent 1H atoms and the 1H NMR spectrum shows just a singlet no H H coupling constants can be extracted H owever the presence of a single BC atom in this molecule lowers the symmetry and produces an AA BB X type spectrum from which all H H and C H coupling constants can be determined 16 The spin system Chapter 2 Symmetry reduction is particularly C3 important in natural abundance BC spectroscopy when one usually looks at molecules having a single BC atom Even if the original all 12C molecule is symmetrical many of its BC isotopomers will not be symmetrical because the 13C atom does no
14. least squares fit to the actual observed spectrum to obtain your final refined values We will discuss the three phases of the process finding a solution refining it and checking for alternatives separately in the following sections 48 Full lineshape iteration Chapter 8 8 4 Finding a solution The chances of obtaining a reasonable solution from a full lineshape iteration depend critically on the quality of the experimental spectrum A wavy baseline impurity peaks or incorrect relative intensities will send the procedure way off in its initial phase and it will probably never get back on track So start with a good well phased spectrum Use availabletricks of your spectrum processing software to get a good baseline Displaying the integral can be very helpful here since errors in the baseline show up as non constant integrals in regions not containing any peaks If your processing program allows it you may also wantto remove impurity peaks and noisy areas not containing any peaks from the spectrum After you are satisfied with this manipulated spectrum save it and set up the iteration If you are fitting only a single multiplet you can do the iteration in a single window If there are large empty areas between the parts of the spectrum you are interested in it is usually better to define several windows one for each occupied part of the spectrum In this way you will ge a higher accuracy and avoid useless fitting to baseli
15. looks good the chances of this kind of error are rather small However there are examples of AA XX and AA A A XX systems giving very similar spectra for the A nudeus In general you should be careful if you are fitting the spectrum for a single nucleus eg 31P in a system containing several NM R active nuclei e g 31P and 103Rh 50 Full lineshape iteration Chapter 8 e Some parameters or combinations of parameters may not affect the spectrum at all and can therefore not be determined by iteration Fitting will still give you a valuefor these butthe value will be meaningless Careful inspection of the error analysis see next section can alert you to such situations e Thespectrum may not contain enough detail for a complete determination of all parameters For example if the linewidth of the observed spectrum is 2 Hz coupling constants cannot be determined to a much greater accuracy than this This can be especially important in rate processes chapter 6 There may be several solutions giving very similar spectra Often these alternatives differ only in the signs of one or more coupling constants It is important that you try to find out whether such alternatives really exist If there are only a few independent coupling constants in the system you can easily try out all combinations by hand If there are more your simulation program may be able to test them in a systematic fashion The alternative solutions may give ri
16. not affect the spectrum and can be ignored This means less typing for you since you do not have to enter them It can also bea disadvantage since these constants cannot be determined from the experimental spectrum unless you reduce the symmetry of the molecule e g by isotopic substitution For example the SF4 spectrum is completely determined by two shifts 81 and 63 and one coupling constant J13 J 12 and J34 do not affect the spectrum and cannot be determined In contrast there are six relevant parameters in the ODCB system 61 52 J12 J 13 J14 and J23 and they can all be determined from the observed spectrum The greater complexity of the AA BB system is clearly illustrated in Figure 6 The spin system 9 Chapter 2 Figure 6 Spectra of SF ODCB SFy left and o dichlorobenzene right EM S JV rt APA 2 3 Chemical equivalence Two or more nuclei are called chemically equivalent when they have the same chemical shift for reasons of symmetry The values of coupling constants are not relevant to this definition but the symmetry will in general imply some relationship between coupling constants involving chemically equivalent nuclei Magnetic equivalence implies chemical equivalence but not vice versa Asan example consider the four protons in the ODCB molecule discussed in the previous section The molecule has Cz symmetry which causes Hj and H4 to hav
17. the file stays on the NMR machine but if you try to export it to other processing software that software may not be able to handle the filtered FID If you have to use custom filtering we NMR data processing 57 Chapter 10 recommend you remove the filter sometimes called converting the FID to analog form which is a misnomer before exporting the data 10 3 Standard processing Normally an FID is multiplied with one or more weighting functions Fourier transformed and phased The optimal choice of weighting function depends on the intended use of the spectrum For full lineshape iteration you want to have peaks without broad feet and a good signal to noise ratio This is best achieved by a Gaussian multiplication function For assignment iteration sharp peaks are important but some noise is tolerable as long as you can distinguish between real peaks and noise or spikes by eye An unweighted FT or modest resolution enhancement may be best here Zero filling by a factor of 2 may be useful but anything beyond that is merely cosmetic and will not produce better iteration results Correct phasing is extremely important for full lineshape iteration The reason for this is that the imaginary or dispersion component is much broader and has a much larger area than the real or absorption component If the automatic phasing function of your NMR softwareis any good we recommend that you use it for all spectra intended for full lin
18. virtually exact For molecules containing several nuclei of the same type small deviations are usually observed mostly intensity changes Spectra that are nearly first order are best Cl interpreted by hand Chemical shifts are assigned C from the centers of multiplets and J couplings from N the splittings Comparison of splittings in different multiplets can be used to assign couplings to a specific pair of nuclei small thatch effects may also be helpful here As an example Figure 9 shows the first order analysis of the 1H spectrum of 2 i propyl 3 chloro pyridine In principle this process could be automated However analysis programs get confused easily by partially overlapping lines in multiplets and they also have a tendency to miss the weak outer lines of e g septets which makes such automatic analysis unreliable Simulation is generally not needed to analyze simple first order spectra In fact the time required to set up the simulation may well exceed that needed to interpret the spectrum by hand Simple simulation 21 Figure9 First order analysis of IH NMR spectrum of 2 i propyl 3 chloro pyridine Chapter 3 1 32 ppm 7 Hz e 3 24 ppm 7Hz AL JA 8 62 ppm 7 64 ppm 6 82 ppm Js 4 7 Hz Jas 8 Hz Jug 8 HzH J4 72 8Hz de Jeg 4 7 Hz 46 N Ah Al J 4500 3500 2500 700 0500 3 3 Second order effects Second order effects are all deviations from the sim
19. will not become so at any field the small outer lines of each multiplet really belong to the spectrum and will not disappear The criterion for second order effects here r1 J23 J12J13 7 47 8 14 1 49 1 is fulfilled regardless of the external field Therefore interpretation of the splittings as coupling constants is not allowed and will in fact produce completely incorrect values Simple simulation 23 Chapter 3 Figure 10 100 MHz 300 MHz Calculated spectra of ODCB at different field strengths T M ans v 1000 MH JAM There is nothing mysterious about second order effects Their origin is completely understood and any decent simulation program will producethe correct spectrum given the right parameters H owever interpretation of second order spectra without a simulation program is difficult since the human mind and eye are simply not well suited to the recognition of patterns of matrix eigenvalues Therefore simulation is an indispensable tool for the interpretation of second order spectra 24 Simple simulation 4 Chapter 4 Prediction of parameters from molecular structure It would be niceif it were possible to predict chemical shifts and coupling constants from a given molecular structure U nfortunately thisis not generally possible at present although some significant advances have been made in recent years There are tw
20. 0 17 00 TheAA part of an AA X spectrum always consists of two AB quart Second order systems with the same coupling constant J A4 but different 65 Appendix A apparent shifts none of the other four relevant parameters 84 Sa Jax Jax can be extracted directly from peak positions in the spectrum If the chemical shift difference A A 4 is large it may be difficult to determine which left half of one AB belongs to which right half the peak positions will be the same only the intensities are different Even if this choice has been made correctly there are always two possible solutions giving riseto identical spectra this corresponds to switching left and right halves of one of theAB quartets Determining which solution is correct requires either measurement of the X part of the spectrum or re recording the AA part at a different spectrometer frequency The examples below illustrate the two independent solutions for one spectrum AA X 11 AA X 12 solutions for two alternative choices of the A B halves AA X 13 AA X 14 an example where the AB halves are all interspersed AA X 15 and onein which one of the AB quartets has an effective chemical shift difference dose to zero AA X 16 As for the X part only the relative signs of J ax and Ja x are important and the sign of Jaq does not affect the spectrum AA X 11 6 J Hz Nucleus ppm 1 2 1 1H 0 000 2 1H 0 100 3 00 3 31P 0 000 15 00 20 00 AA X 12
21. 3 61 The effects of chemical exchange sss 33 6 2 Intra and inter molecular exchange 35 6 3 Interpretation of exchange rates sss 40 7 Iteration with assignments eene 43 JA Descriptors ec mea ee 43 Contents iii Contents 7 2 Pros and cons of assignment iteration sss 43 7 3 Why the computer cannot do the assignments 45 8 Full lineshape iteration erre 47 Sly JDescriplon ee ae 47 8 2 Pros and cons of full lineshape iteration 47 8 3 19trategy centeno i er t ederet Re te dni rain 48 84 Finding a solution esee 49 85 Thefinal refinement 50 86 Checking your solution seen 50 9 Error analysis unsanssnonsonunnunnunnunsnnnunnununnunnnnunnunnnnnnnunnunnnnnnnnannannnnnnnn 53 10 1 D NMR data processing rennen 57 10 1 Introduction e e retener 57 10 2 Recording the spectrum seen 57 10 3 Standard processing n nerennnennnnennnnnnnnennnnnnnnnnnnnnnnnnnnnnnn nn 58 10 4 Custom processing sss 58 10 5 Linear prediction and other processing techniques 59 A Examples of typical second order systems 61 A 1 The AnBm Systems eese enne nnn
22. C symmetry with a mirror H Ho Me plane bisecting the OCO angle Reflection in this 9 plane interchanges H 4 and H y so these two Me hydrogens must be chemically equivalent However there is no symmetry operation that interconverts H and H2 These protons are diastereotopic They not only have different chemical shifts but will also differ in other chemical properties for example the rates of abstraction by a strong base will be different 2 4 Temperature dependent equivalence The above discussion suggests that the classification of nudei as chemically or magnetically equivalent is absolute i e only dependent on the overall molecular structure H owever there are many examples of molecules which have a static low temperature structure but acquire a higher effective symmetry at elevated temperature usually through rapid inversion or rotation processes or chemical exchange rate processes are discussed in more detail in chapter 6 Consider a molecule of G 6 I dicyclohexylphosphine This has only Cs m ya 2 symmetry the carbon atoms 2 and 6 of each cydohexyl ring are diastereotopic H inequivalent and the BC spectrum of a carefully purified sample at low temperature shows two disti nct resonances for these two carbons Addition of a trace of acid or raising the temperature results in rapid inversion at phosphorus via a protonation deprotonati on pathway In the fast exchange limit the molecule has acquired effective C2
23. J Hz Nucleus ppm 1 2 1H 0 037 1H 0 063 3 00 3 3P 0 000 746 27 54 AA X 13 J Hz Nucleus ppm 1 2 1 1H 0 135 2 1H 0 035 3 00 66 Second order systems Appendix A 3 31P 0 000 13 15 8 35 AA X 14 J Hz Nucleus ppm 1 2 1 1H 0 036 2 1H 0 136 3 00 3 31P 0 000 8 05 12 91 AA X 15 o J Hz Nudeus ppm 1 2 1 1H 0 047 2 1H 0 063 3 00 3 31P 0 000 1946 1954 AA X 16 J Hz Nucleus ppm 1 2 1 1H 0 027 2 1H 0 070 3 00 3 31P 0 000 20 44 14 06 A 3 The AA BB system Six independent parameters 84 p Jaa Jag Jag and Ja g determine the appearance of this type of spectrum Therefore there are many possible patterns H ere we just illustrate a few of the most common ones ethylene groups with hindered or free rotation coordinated ethylene and o and p substituted benzene When analyzing these spectra it is important to realize that one cannot distinguish between Jaa and Jgg or between Jag and Ja g on the basis of the spectra alone The relative signs of Jag and Ja g are important but changing the signs of Jaq and Jgg usually has only a small effect on the spectrum Second order systems 67 Appendix A anti staggered constrained X CH2 CH2 Y AA BB 1 X B B J H2 he Nucleus ppm 1 2 3 2 Y A 1 1H 1 000 2 JH 1 000 14 00 3 1H 3 000 3 00 1250 4 1H 3000 1250 300 16 00 J EL syn edipsed constr
24. R spectrum Moreover it has to be of sufficient quality to let you do the desired analysis Itisimpossible to do justice to the topic of recording and processing NMR spectra in the space of a few pages M any books have been written on the subject for a recent one that gives an excellent overview of established and new techniques see ref 19 Nevertheless it might be useful to go through some of the most important steps of the process here 10 2 Recording the spectrum If you are planning to do a full lineshape iteration you need a good field homogeneity Ill adjusted high order shims usually cause peaks to have broad feet The spectrum will still look good enough to the eye but the intensity hidden in the baseline is likely to throw the iteration off the track especially if the feet are asymmetrical Such feet are less of a problem for assignment iteration where the primary concern is high resolution near the tops of peaks Make sure you use enough data points when recording a spectrum In these days of cheap storage media there is no good reason to record 8K or 16K 1 D NMR spectra Resolution lost at this stage can never befully recovered Several brands of NMR machines now use digital filtering techniques by default There is nothing against this and the resulting spectra may be of significantly higher quality However some machines store and process FID s still containing filter functions This is nota problem as long as
25. Simulation of one dimensional N M R spectra a companion to the gN M R U ser M anual E nii Lit A V V UUN D SEPS Py A i t 72 E A e W Peter H M Budzelaar Cherwell Scientific Limited The Magdalen Centre Oxford Science Park Oxford OX4 4GA United Kingdom Copyright Disclaimer Trademarks Author Publisher Copyright 1995 1999 IvorySoft All rights reserved No part of this manual and the associated software may be reproduced transmitted transcribed stored in any retrieval system or translated into any language or computer language in any form or by any means electronic mechanical magnetic optical chemical biological manual or otherwise without written permission from Cherwell Scientific ISBN 0 9518236 47 Simulation of one dimensional NMR spectra a companion to the gNMR User M anual Cherwell Scientific make no representations or warranties with respect to the contents hereof and specifically disclaims any implied warranties of merchantability or fitness for any particular purpose All trademarks and registered trademarks are the property of their respective companies Peter H M Budzelaar This booklet is a companion to the manual of the gN MR package for NMR simulation It provides general background about the use of simulation for spectrum analysis gNMR is published by Cherwell Scientific Limited The Magdalen Centre Oxford Science Park Oxford OX4 4GA gNMR
26. a number of different terms which correspond as follows low value high value low frequency high frequency high field low field high shielding low shielding shielded deshielded diamagnetic shift paramagnetic shift The coupling constant between two nudei A and B is the energy difference between the situations where the two nuclei have parallel and antiparallel spins M ore precisely the energy contribution to the Hamiltonian is gt Eag h Jag m A m B From this equation it is apparent that J 2 O0 implies the situation with parallel spins is higher in energy than the one with antiparallel spins The energy difference is independent of the external field so couplings are expressed in Hz It is important to realize that in contrast to e g infrared force constants there is no general connection between coupling constants and bond strengths Shifts and couplings can usually be regarded as molecular properties They are somewhat sensitive to temperature and solvent but variations caused by the environment are usually small compared to the differences between different molecules The most notable exceptions are observed for the chemical shifts of protons involved in hydrogen bridges Both chemical shifts and couplings can also usually be related to the direct environment 1 3 bonds of the nucleus or pair of nuclei in question In that sense they are local probes of chemical structure 14 The spin system Chapter 2
27. ained X CH2 CH2 Y AA BB 2 Yx 5 J Hz Be Nucleus ppm 1 2 3 B 1 1H 1 000 2 1H 1000 14 00 3 1H 3000 1100 450 4 1H 3000 450 11 00 16 00 C z 68 Second order systems Appendix A gauche staggered constrained X CH 2 CH gt Y AA BB 3 sje s s 8 J Hz X y A A 3 X Nucleus ppm 1 2 3 1 1H 1 000 2 1H 1 000 14 00 3 1H 3 000 3 00 8 00 4 1H 3000 800 300 16 00 pe ER gauche eclipsed constrained X CH 2 CH 2 Y YA Ya AA BB 4 E n kG ee J Hz CES B X Nudeus ppm 1 2 3 1 IH 1 000 2 IH 1 000 14 00 3 IH 3 000 450 12 00 4 IH 3 000 12 00 4 50 16 00 PES zu Second order systems 69 Appendix A rotation averaged X A A unconstrained X CH gt B B B B B B CH gt Y AA BB 5 OE a x J Hz Nucleus ppm 1 2 3 1 1H 1 000 2 1H 1 000 14 00 3 1H 3 000 6 00 7 00 4 1H 3 000 7 00 6 00 16 00 Je E Freely rotating coordinated ethylene B AA BB 6 ae AR 5 J Hz M j meer Os Nudeus ppm 1 2 3 B n p 1 1H 1 000 2 1H 1 000 13 00 3 1H 3 000 2 00 8 00 4 1H 3 000 8 00 2 00 1100 J L 70 Second order systems Appendix A Static coordinated ethylene with mirror plane through C C bond AA BB 7 J Hz Nudeus ppm 1 2 3 1 IH 1 000 2 IH 1 000 1 50 3 1H 3 000 800 12 00 4 1H 3 000 12 00 8 00 2 50 AR N Static coord
28. ameters Let us walk through a few examples where simulation might play a rolein the analysis These examples illustrate different questions one can have about a spectrum and therefore different applications of simulation Sometimes you just want to know whether a spectrum can belong to a certain compound 1 3 Sometimes you are interested in the numerical values of parameters because they can tell you something about the structure of a compound 2 And sometimes simulation may even be used to extract some mechanistic information from a spectrum 4 Simulation and spectrum analysis 1 Example 1 Synthesis of a new triphosphine Figure 1 31P 1H NM R spectrum 80 96 M Hz IH 200 M Hz of phosphine 1 Example 2 cis and or trans isomers Chapter 1 An attempt to prepare compound 1 produced PPh a white solid with the P H NMR spectrum prp PPh shown in Figure 1 Could this really be the desired product If so what are the shifts and coupling constant needed for publication Im 13 500 14 500 15 500 16 500 17 500 18 500 19 500 20 500 Simulation quickly shows that this spectrum can indeed be explained completely by a strongly coupled A2B system with 84 17 5 ppm g 16 ppm and JAg 120 Hz Without simulation you might have thought that you had a mixture of several compounds Note that there are no peaks in this spectrum with a separation of
29. contribution of a second coupling called the direct or dipolar coupling D Dipolar couplings are usually much larger than indirect couplings Because they provide information on the spatial positions of atoms analysis of anisotropic spectra can yield direct structural information This is a rather specialized topic see Reference 4 for a more detailed discussion To simulate anisotropic spectra you will haveto supply direct D as well as indirect J coupling constants if possible you should extract the indirect couplings from isotropic spectra and fix them in anisotropic calculations In our discussion of magnetic equivalence earlier in this Chapter we stated that couplings within a magnetic equivalence group do not affect the spectrum This is no longer truefor anisotropic spectra The indirect couplings J within the group are irrelevant but the direct couplings D do contribute to the spectrum and must be included in the simulation So for anisotropic spectra the rules for equivalence are stricter 12 The spin system Figure7 Anisotropic spectrum of benzene obtained with J couplings of 8 2 0 5Hz and D couplings of 333 64 42 Hz Chapter 2 e All of the Nz N have the same chemical shift e For every individual nucleus M not belonging to the N N 4 the coupling constants Jw m Jnnm and the Dy m Dnnm are equal e All D couplings within the group N N are equal Groups of nuclei satisfying these criteria are call
30. e simulated behavior Note that at equilibrium the forward and backward reaction rates are equal This implies that the rate constant of disappearance of the cis isomer Kis 5trans Rate cis is much larger than the rate constant of disappearance of the trans isomer Ktrans gt cis Rate trans Therefore line broadening for the cis isomer starts at a lower temperature than for the trans isomer the process does not look very symmetric The high temperature effective chemical shift is an average weighted by the concentrations of the separate low temperature chemical shifts if there were any coupling constants these would become weighted averages as well Finally we will consider an example of what is commonly called intermolecular exchange using a hypothetical metal bis phosphine complex as an example Chemical exchange 37 Chapter 6 This example which shows a curious rate dependence of the NMR signal was first described by Swift Reference 9 Figure 14 shows the theoretical BC resonance of a carbon atom of the phosphine ligand as afunction of the exchange rate of the phosphines At low exchange rates the spectrum is a virtual triplet because J pp is large At high exchange rates the 13C atom only sees the phosphorus atom in its own ligand molecule so the spectrum is a nice doublet At intermediate exchange rates something curious happens it looks as if there is only a broad singlet N ot all intermolecular e
31. e 61 A 2 The AA X system sse nennen nnn tnn 63 A 3 The AA BB system eseensesnnensnnennennnnnnenennnnennnnnnnnnenennn na 67 References ssssssssssssssasunnnnnnanannnnnnananannnnanananannnnananannnnanananannananananannanananana nna 73 Ip 75 iv Contents Chapter 1 1 Therole of simulation in spectrum analysis 1 1 Introduction NMR spectra are usually recorded in order to analyze a sample The desired analysis can be quite simple if you have a mixture of two compounds each having a single NMR resonance integration of the area of the two peaks can be used to determine the relative concentrations Usually NMR spectra are more complicated than this and the analysis can become correspondingly more difficult In such cases simulation can often be very helpful Simulation in the strict senseis the calculation of an NMR spectrum from a set of parameters shifts coupling constants Theterm simulation is also used frequently to denote the calculation of a spectrum from a molecular structure which involves prediction of the parameters from the structure as an intermediate step In some cases first order spectra a few simple rules suffice to predict the appearance of an NMR spectrum and simulation is not necessary There are many cases however where these rules do not hold second order spectra and then computer simulation is the only practical way to predict the appearance of a spectrum from its basic par
32. e M eN PF4 example discussed above also showed such an exchange in its 31P spectrum Chemical exchange 35 Chapter 6 Figure 13 Effect of hindered C N rotation on A A B HCON CH3 2 and B H CON R 1 R2 36 Chemical exchange Chapter 6 Now consider the BC H C Hs H CH N N spectrum of the same C_N C N compound At low N ol Se temperature we actually have CHg C Hs two different molecules one with a BC atom trans to oxygen and one with the BC atom cis to oxygen We are ignoring molecules containing two or more BC atoms because their abundance will be negligible This type of exchange is called intramolecular non mutual exchange For this particular case the resulting spectrum will still be rather similar to the IH example described above but the distinction between mutual exchange within a single species and non mutual exchange exchange of species is important We can carry this point further by H Ry H R2 RUE N looking at the isomerization of an CN QN amide with different organic N groups at the nitrogen Let us R2 9 Ry consider only the BC resonance of the carbonyl carbon Since the two organic groups in our hypothetical amide are different in size there will be an energy difference between the cis and trans isomers the equilibrium will contain say 10 cis and 90 trans Figure 13B shows th
33. e the same chemical shift the same holds for H2 and H3 Thus ODCB contains two groups of chemically but not magnetically equivalent protons The molecular symmetry also implies that J13 J24 and J14 J23 The use of chemical equivalence or symmetry in general in NMR simulation can significantly reduce the computation involved However the full exploitation of symmetry is less trivial than that of magnetic equivalence so not all simulation programs use full symmetry factorization If nuclei are magnetically equivalent they can be specified in groups since they all have the same coupling constants to nuclei outside the group Thus you only have to specify a single entry for each magnetic equivalence group instead of for each individual nucleus Such a simplification is not possible for chemical equivalence since different nuclei in a chemical equivalence group may have different coupling constants to a single nucleus outside the group Therefore you will have to supply a separate entry for each nucleus in a 10 The spin system Chapter 2 chemical equivalence group Y ou can however enforce symmetry by linking parameters shifts coupling constants to ensure that when you change one parameter all symmetry related parameters will also be changed Itis not always trivial to decide whether two Me nudei are chemically equivalent Consider the HL 0 methylene groups of acetaldehyde diethylacetal Hy Yel This molecule has
34. ed fully equivalent If you wantto simulate anisotropic spectra usethe full equivalence criterion to divide your spin system into equivalence groups For example the six protons of benzene are not fully equivalent since D12 z D13 2 D a you have to enter oriented benzene as a system of six separate chemically equivalent protons However ethane could be specified as two full equivalence groups of three protons each As an example Figure 7 shows the simulated spectrum for benzene in an anisotropic medium calculated with parameters given in Reference 4 JUU UJ WVU W UWL 2 6 Shifts and coupling constants The chemical shift 6 of a nucleus is its resonance frequency relative to that of a particular reference compound The shift is proportional to the external magnetic field which is why shifts are usually expressed in ppm of the field for different fields they are constant when expressed in ppm not when expressed in Hz By convention The spin system 13 Chapter 2 the sign of 8 is chosen in such a way that higher 5 values correspond to higher resonance frequencies Also by convention NMR spectra are written with 6 values increasing from right to left In principle the chemical shift is a tensor but in liquid NMR one usually just observes its trace which is a scalar or number The magnitudes of chemical shifts are often discussed using
35. ed for an accurate description In practice however having a single relaxation time per nucleus is usually satisfactory exceptions occur in cases with chemical exchange see chapter 6 or with quadrupolar relaxation There is no clean way of assigning a different relaxation timeto each nucleus short of the relaxation matrix treatment which we want to avoid because it is too computationally expensive Therefore gN MR uses a more pragmatic solution and assigns to each peak a linewidth based on the composition of the corresponding transition using a kind of population analysis This appears to give satisfactory results even for strongly coupled nuclei with very different natural linewidths 20 Simple simulation Chapter 3 3 2 First order spectra In simple cases the appearance of an NMR spectrum can be predicted easily using the following rules e Every nucleus has a peak at its resonance frequency given by the chemical shift 5 The area of the peak is proportional to the number of nudei e For every pair of spin nuclei between a coupling exists both peaks are split into two components with the same splitting J If one of the nuclei has a spin I different from Y it splits up the other peak into 21 1 components Repeated application of these rules produces the familiar doublets triples quartets etc of high resolution liquid NMR spectroscopy If the nuclei are all of different types eg 1H and 1P these rules are
36. ely first order are not affected by the signs of coupling constants However sign changes affect the peak labeling which may be important in iteration In spectra showing second order effects signs may be important It is often true that there are groups of coupling constants which can change signs simultaneously without affecting the spectrum whereas individual sign changes may produce a different spectrum Before reporting the results of an iteration it is important to check how many alternative sign combinations would also produce an acceptable possibly identical solution The spin system 15 Chapter 2 2 8 Isotopic substitution Molecules of the same chemical composition but having a different isotopic composition are usually called isotopomers The presence of different isotopes of a single element can give rise to a number of interesting effects in NMR spectroscopy To avery crude first approximation the presence of an isotope does not disturb the shifts and coupling constants of the other nuclei in the molecule This is really a rather crude approximation Especially for nearest neighbors the effect is often significant Typical one bond isotope shifts A8 are 0 5 ppm in BC for CH gt CD and 0 03 ppm in 31P for P1 2C gt P15c Also the chemical shift of the isotope expressed in ppm will be approximately the same as that of the original nucleus in the original molecule and coupling constants J xy of any nucleus X to the
37. em near the limit of what most simulation programs can handle but the pattern finally looks correct Of course the addition of a single P H coupling to represent the effect of one phenyl and one pyridyl ring looks a bit like fudging Clearly any coupling constant fitted for it will be meaningless and some other parameters may not be very significant either H owever the exercise illustrates that you really can Large systems 29 Figure 11 M ethylene resonance of a bis benzylphosphine rhodium complex A simulated with increasingly complicated spin systems B D Chapter 5 get the curious resonance shape of Figure 11A from the structure shown above 2A d E As mentioned earlier for really large molecules exact simulation is impossible so oneis forced to resort to some sort of approximate calculation The most drastic approach is simple first order calculation see section 3 2 possibly with some cosmetic corrections to reproduce thatch effects This is certainly extremely fast but is only good for near first order spectra for which one usually doesn t need simulation anyway 5 4 Approximate calculations Here we propose an intermediate scheme based on a divide and conquer It has been implemented in gNMR and appears to work satisfactorily in most cases We call this method chunking The design of the algorithm is based on the way one normally does the analysis of a spectrum W
38. erns in intermediate situations even though the fast exchange limits are the same If these differences are large enough as they arein Figure 12 it will be possibleto distinguish between such mechanisms in the particular case discussed here the reaction was clearly shown to follow a two pair exchange pathway 6 2 Intra and inter molecular exchange Actually theterms intra and inter molecular exchange are slightly misleading because their normal chemical meaning is not entirely appropriate to NMR The distinctions needed to understand dynamic behavior are more subtle Four typical examples are illustrated below We will start with the simplest case which is often H CH3 called intramolecular mutual exchange and will use Yen dimethylformamide as an example The dynamic N behavior shown by this molecule Figure 13A is CHs hindered rotation around the amide bond At low temperature you will seetwo different methyl resonances in the 1H spectrum On raising the temperature they broaden and then coalesce to a single peak In effect all six protons of the methyl groups have become magnetically equivalent on the NMR time scale The position of the single peak corresponds approximately to the average of the chemical shifts of the individual methyl groups If there had been any observable couplings from the methyl groups to other parts of the molecule the high temperature limit would also show averages of these coupling constants Th
39. eshape iteration it may be a good idea to do a rough phasing by hand first The quality of the phasing is easily judged from the spectrum integral it should not dip immediately before or after peaks Phasing is somewhat less of an issue for assignment iteration although phasing errors above 20 may introduce significant errors in peak positions 10 4 Custom processing Most processing software allows you to do a lot of special processing At the very least there will be options for baseline correction This is important for full lineshape iteration as mentioned earlier The quality of the baselineis easily judged from the integral which should be strictly horizontal in regions not containing any peaks 58 NMR data processing Chapter 10 Be very careful with baseline corrections in chemical exchange spectra These spectra usually have broad lines and it is easy to correct away the feet of such lines resulting in poor matches between experimental and simulated spectra In such cases it may be useful to add an innocent compound having peaks outside of the region of interest and to use these sharp peaks for phasing and baseline correction Custom processing may also indude various smoothing techniques options to remove parts of the spectrum containing impurities etc While such tricks can be useful at times one should not normally use them to generate spectra for presentations that is simply too dose to cheating However u
40. etic equivalence and then use a few examples to illustrate the concept A group of two or more nuclei N N are called magnetically equivalent if and only if e All of the nuclei have the same chemical shift e For every individual nucleus M not belonging to the set N 1 N 5 the coupling constants Jw 1M JN nw are equal However different couplings within the se are allowed e g JN1N2 JN 1N3 In principle such a situation could occur by chance but the term magnetic equivalence is usually reserved for those cases where there is a symmetry reason for the above conditions to hold Let us consider two examples sulfur tetrafluoride and o dichlorobenzene SF has atrigonal bipyramidal structure with one Fi equatorial position occupied by a lone pair orbital As a ei consequence it has two types of fluorine atoms apical 1 Fs and 2 and equatorial 3 and 4 Thetwo apical fluorines F2 have the same chemical shift 61 62 as do the equatorial ones 83 84 but 81 will be different from 83 Also by symmetry all coupling constants between an apical and an equatorial fluorine are identical Therefore there are two groups of magnetically equivalent nuclei the group of apical fluorines and the group of equatorial fluorines This spin system is called an A 2Bz system Generally a group of magnetically equivalent nuclei in a spin system e g the group of two apical fluorines is denoted by a capital letter A and a subscript 2 i
41. gN MR uses unscaled values The variance covariance matrix conventionally shows parameter variances squares of the estimated standard deviation es d or o on its diagonal and covariances as off diagonal elements Large covariances imply strong dependencies between parameters and indicate that it may be dangerous to cite the singl e parameter o s as independent error limits Covariances can be either positive or negative but variances are always positive because they are squares The example below shows a large standard deviation for parameter 2 and a sizable correlation between parameters 1 and 2 A part from that everything looks normal Error analysis 53 Variance Covariance Oo OA DO F amp F WYN EF matrix Chapter 9 O S56 S E9L 0 000 0 001 019 8 9 001 0 003 018 0 056 000 0 000 000 0 000 000 0 000 000 0 000 000 0 002 020 0 002 002 0 107 The most important part of the error analysis and also the part that is least looked at and not even printed by some programs is the singular value SV analysis This can show you how well the parameters you tried to optimize are determined by the experimental data The SV analysis is printed as a square matrix see below The columns are headed by the singular values and the rows are labeled by the scaled parameters There are usually some minor or major dependencies between the parameters you try to optimize the SV analysis transforms your set of parame
42. hereas a simulation program calculates the whole spectrum at once a chemist will look at each individual multiplet in turn Direct couplings to the nudeus in question are considered first the first shell If there are other nuclei that don t couple directly with the target nucleus but do couple strongly to other nuclei in the first shell second order effects will occur e g virtual triplets and these nudei are also required to understand the spectrum the second shell One could go further but in 30 Large systems Chapter 5 practice two shells are usually sufficient to explain the shape of a multiplet This suggests that the simulation should also calculate the multiplets one by one using only as much of the environment as is needed to reproduce the patterns The problem is that simulation of a part of a molecule will not only produce the target multiplet which is presumably accurate but also resonances due to the shells which are probably very inaccurate The key point of the approximate approach is that the simulation is indeed done in chunks but from each chunk spectrum everything is thrown away that is not due to its target nucleus then the chunk spectra are added to give the final spectrum The technical details can become a bit complex but are not important here The key advantage of this scheme is that it scales linearly in system size which means that future increases in CPU speed will immediately result
43. ies of the bond path zigzag or W paths or with very short through space contacts prediction of the magnitude of such couplings is difficult 26 Parameters and structure Chapter 5 5 Simulating large systems 5 1 Onthe scaling of NMR calculations In principle NMR simulation is simple Set up the H amiltonian diagonalizeto get the energy levels multiply eigenvectors with transition moments to get intensities evaluate a Lorentzian for every calculated peak Unfortunately the scaling of the calculation is rather unpleasant The size of the calculation dimension of the H amiltonian scales as 272 the storage requirements as the square of this and the computing time as its cube For every nucleus added to the system thetime required increases with a factor of 8 This makes calculations for large molecules gt 12 15 atoms rather difficult For example on a 100 MHz Pentium a particular 6 spin problem took 0 1 seconds to simulate an analogous 8 spin problem 0 73 seconds the 10 spin problem 22 seconds and the 12 spin problem 27 minutes With the current rates of increase of CPU speed a factor of 2 every 1 2 years it will be several decades before we can do 25 spin systems This is the reason many NMR simulation programs won t let you simulate systems larger than 8 spins Or if they do the spectrum is often evaluated by first order methods which are rarely good enough There are several methods for reducing the co
44. in the ability to simulate significantly larger systems The minimum chunk size needed to obtain correct multiplet patterns is usually 8 9 nuclei so the break even point of this method appears to bein the range of 11 12 nuclei For smaller systems exact simulation is still the method of choice Large systems 31 Chapter 6 6 Chemical exchange 6 1 The effects of chemical exchange In contrast to many other spectroscopic methods where kinetics can only be used to study irreversible reactions NMR can also be used for kinetic studies of systems in equilibrium This is becausethe NMR time scale of the order of milliseconds or microseconds is conveniently close to our own time scale Reversible processes with activation energies of the order of 5 20 kcal mol can be studied by band shape analysis explained below For reactions with slightly higher barriers techniques like polarization transfer may be more appropriate Asan illustration of an exchange process let us consider Me2N PF 4 which has been studied by Whitesides We have modified a few parameters from the data given by Whitesides to make the example more illustrative This has a trigonal bipyramidal structure with the amino group in the equatorial plane There are two groups of magnetically equivalent fluorine atoms as in the SF4 example discussed earlier Since the phosphorus atom is also magnetically active we can characterize this molecule as an A2B2X system ignor
45. inated ethylene with mirror a B plane bisecting C C bond AA BB 8 hid M a J Hz Le Nucleus ppm 1 2 3 N 1 1H 1 000 2 1H 1 000 7 00 3 1H 3 000 2 00 12 00 4 1H 3 000 12 00 2 00 9 00 J n Second order systems 71 Appendix A o Disubstituted benzene AA BB 9 A 5 J Hz B Nudeus ppm 1 2 3 1 1H 6 500 2 2 1H 6500 0 25 3 1H 8500 800 120 n 4 1H 8500 120 800 700 E t J ls p Disubstituted benzene AA BB 10 i A A 8 J Hz Nudeus ppm 1 2 3 1 1H 6 500 B B 2 1H 6500 150 Y 3 1H 8500 750 025 4 1H 8500 025 750 080 U U 72 Second order systems References References 1 9 E Vogel U Haberland and H G nther Angew Chem 82 1970 510 R G Jones The Use of Symmetry in Nudear Magnetic Resonance in NMR Basic Principles and Progress P Diehl E Fluck and R Kosfeld eds vol 1 Springer Verlag Berlin 1969 p 100 F A Cotton Chemical A pplications of Group Theory Wiley Interscience 2nd ed New York 1971 P Diehl and C L Khetrapal NMR Studies of Molecules Oriented in the N ematic Phase of Liquid Crystals in NMR Basic Principles and Progress P Diehl E Fluck and R Kosfeld eds vol 1 Springer Verlag Berlin 1969 p 1 M H Levitt J Magn Res 126 1997 164 E Pretsch J Seibl W Simon and T Clerc Tabellen zur Strukturaufkl rung organischer Verbindungen mit spektroskopischen Methoden 2Nd ed Springer Verlag Berlin
46. ing the dimethylamino group The low temperature 31P spectrum a triplet of triplets Figure 12A can indeed be interpreted in this way However at higher temperatures the fluorine atoms start to exchange In the high temperature limiting spectrum also called the fast exchange limit the spectrum shows just the quintet of an A 4X system Figure 12D the fluorine atoms have become equivalent on the NMR timescale What happens is that the exchange is so much faster than the actual NMR experiment that we observe the time averaged situation Chemical exchange 33 Figure 12 One pair left and two pair right exchange 1P spectra for MeNPF Chapter 6 F F Y LY MeN m 3 Me N PT F E aliu alu Neither the low temperature or slow exchange limit nor the high temperature limit is particularly interesting the interesting things happen in between As thetemperature is raised the initially sharp lines Figure 12A broaden and coalesce Figure 12B C until in the fast exchange limit a sharp spectrum is obtained again Figure 12D For the intermediate situations it is possibleto determine a rate constant from the line broadening by fitting The temperature dependence of the rate constant can then be used to extract activation 34 Chemical exchange Chapter 6 energies and entropies M oreover different exchange mechanisms may give riseto different line broadening patt
47. is problem has been developed by Binsch1 and also used by H gele 7 These authors use a generalization of the least squares formalism to flatten the y2 function and so remove the local minima This strategy helps convergence to a reasonable solution even from poor starting values However the flattening prevents accurate determination of parameters Therefore once a solution has been found the flattening is decreased in stages allowing progressively more accurate determination of the parameters while staying near the global minimum The final stage with no flattening at all isatrueleast squares fit To usethis kind of iteration the user must supply experimental spectra the spin system and some reasonable starting values for the shifts and coupling constants but does not haveto do any peak assignments A related approach has been implemented by Laatikainen H e uses an integral transformation to introduce an artificial broadening of the spectrum in the initial stages of the fitting procedure this broadening is then reduced in several stages In the final refinement stages this may then be followed by a standard assignment iteration 19 8 2 Pros and cons of full lineshape iteration One of the main advantages of full lineshape analysis is that you can useit to optimize any parameter that affects the appearance of the spectrum not just shifts and coupling constants but also linewidths Full lineshape iteration 47 Chapter 8
48. l as for normal organic molecules At least as important as the computational problems of the theoretical approach are the chemical ones NMR parameters are thetime averaged values over all accessible conformations of a molecule and often also include significant contributions due to Parameters and structure 25 Chapter 4 interaction with the solvent Therefore accurate prediction from theory alone is at least an order of magnitude more complicated than just a single IGLO or GIAO calculation As an alternative to the rather expensive ab initio method prediction using semi empirical methods has also been attempted Extensive parametrization is required to make this work induding dassification of atom types Therefore this method is again unsuitable for unusual bonding situations or non standard nuclei However it may be a useful addition to the database approach mentioned above If one doesn t set the sights too high simple additivity rules for chemical shifts can produce quite reasonable results We have found the rules given by Pretsch6 quite useful but other good collections exist Coupling constants are strongly conformation dependent but for most common cases this dependence is well documented if not completely understood so if the 3D structure of a moleculeis known short range couplings can be estimated with reasonable accuracy Abnormally large long range couplings are nearly always associated with particular geometr
49. m 1987 R Laatikainen J M agn Res 92 1991 1 R Laatikainen M Niemitz U Weber J Sundelin T Hassinen and J Veps l inen J Magn Res A 120 1996 1 J C Hoch and A S Stern NMR Data Processing Wiley Liss New York 1996 Second order systems Index A A2B2 8 AA BB 9 Anisotropic spectra 12 Approximate calculations 30 chunking 30 Assignments 43 Band shape analysis 33 Baseline 48 Baseline correction 58 C Chemical equivalence 10 Chemical shift 13 prediction 25 Coupling constant 14 and bond strength 14 dipolar 12 direct 12 indirect 12 sign of 15 D Data processing 57 linear prediction 59 Databases for prediction 25 Diastereotopic 11 Equivalence and anisotropic spectra 12 and isotopic substitution 16 chemical 10 full 13 References References magnetic 8 temperature dependent 11 Error analysis 53 singular variance 55 variance covariance matrix 54 Exchange 33 intermolecular 37 intramolecular 35 mechanisms 35 Exchange rates interpretation of 40 F First order spectra 21 Fourier transformation 58 Full equivalence 13 Full lineshape iteration 47 strategy 48 Isotopomers 16 Iteration assignments 43 full lineshape 47 L Linear prediction 59 Lineshape 19 Gaussian 19 Lorentzian 19 Linewidth 20 M Magnetic equivalence 8 Maximum entropy 59 P Phasing 58 75 Prediction 25 empirical methods 25
50. meter the ratio J Ad Figures 15 and 16 show these spectra for values of 0 1 0 3 1 0 and 3 0 of this ratio Second order systems 61 Appendix A Figure 15 AB and AB A B spectra J AS 0 1 ul J AS 1 0 i m J AS 3 0 62 Second order systems Appendix A Figure 16 A B and AB AB A 5B spectra J A8 0 1 l ll DUM i E Cd J AS 3 0 A 2 The AA X system This consists of two nuclei with nearly identical chemical shifts A and A coupling to a third with a very different shift X Jax and Jax are different if they were not this would be an A2X system Systems of this type are often encountered as a consequence of the presence of isotopes For P Pa eR example the C4 resonance of 1 3 1 E 3 diphosphinopropane is the X part of an AA X Second order systems 63 Appendix A system The presence of the BC nudeus induces a small isotope shift A6 for Py and J pic Jpac The X part of an AA X system is always symmetric It consists of two lines of intensity 0 25 and an set of four lines with total intensity 0 5 There arefive independent parameters that influence the spectrum appearance Jax Jax Jaa 9x and Aaa and only four independent peak positions so you will need to use intensity data to determine all parameters Sometimes you may already know the value of one of the parameters e g J ax 0 i
51. mposition to the peak With that information the program can do the iteration Now it will also be dear why you need good starting values for assignment iteration without a good start you will not be ableto recognize patterns and make the right assignments The convergence of the iteration after assignments are done is much less of a problem If you changethe signs of one or more coupling constants in the system the overall spectrum often remains nearly the same but the compositions of individual transitions change Therefore you will have to redo assignments after such a change if you want to try different sign combinations Simply changing a sign and restarting the iteration usually doesn t produce a new solution but only restores the original sign 46 Assignment iteration Chapter 8 8 Full lineshape iteration 8 1 Description An obvious alternative to the method of assignment iteration described above would beto do a direct least squares iteration on the full experimental spectrum However unless you have an extremely good initial guess of parameters a direct least squares fit of an observed to a calculated NMR spectrum is unlikely to converge to the correct parameter values The reason for this is that usually the y error function has many local minima surrounding the global minimum and the direct optimization is likely to get trapped in such a local minimum before it ever reaches the global minimum One solution to th
52. mputation requirements of a simulation Some of these can be carried out automatically by the simulation program and some can be done by the user as detailed in the next two sections But none of these will help with the simulation of really large systems say larger than 15 nuclei To handle such systems one must resort to approxi mate calculations and that is the topic of the final section of this chapter 5 2 Simplification by the simulation program Thefollowing techniques can be applied automatically to reduce the size of an NMR simulation without any loss of accuracy e Splitting of the system into uncoupled fragments if possible Large systems 27 Chapter 5 e Detection of magnetic equivalence and treating of groups of magnetically equivalent nuclei as composite particles e Detection and use of full molecular symmetry chemical equivalence e Division of the system into X groups for nuclei of different types Splitting a system can result in huge savings of computation time The other techniques will only result in a modest reduction of the size of the calculation Nevertheless it is worthwhile to exploit them whenever possible If the result need not be exact but still rather good it is possible to usethe technique of X group division between nudei of the same type This will introduce errors but as long asthe groups are only weakly coupled most errors can be eliminated by the use of perturbation theory to
53. n which case the other four parameters can be determined from peak positions alone The low intensity pair of combinations lines which are essential for the determination of Jar frequently have a larger linewidth than the other lines which can make them hard to see The sign of Ja does not affect the spectrum The relative signs of J ax and JA x are important but changing both will leave the spectrum unchanged In the limit of large Ja a the spectrum looks like an A2X system AA X 2 virtual triplet the A atoms become effectively equivalent In the limit of large A6 it becomes an AM X spectrum AA X 7 doublet of doublets AA X 1 J Hz Nudes ppm 1 2 1 1x 0 000 2 g3P 0010 2000 Jk al 31P 0 000 200 15 00 x D AA X 2 J Hz Nudes ppp 1 2 1 13x 0 000 2 31P 0000 1100 J 31P 0 000 2 00 21 00 LAGGY 64 Second order systems AA X 3 Nucleus ppm 13C 0 000 31P 0 010 3 31P 0 000 AA X 4 Nucleus ppm 13C 0 000 31P 0 020 3 31P 0 000 AA X 5 Nucleus ppm 13C 0 000 31P 0 100 3 31P 0 000 AA X 6 Nucleus ppm 13C 0 000 31P 0 100 3 31P 0 000 AA X 7 Nucleus ppm 1 13C 0 000 2 31P 0 200 31P 0 000 J Hz Appendix A 1 2 15 00 800 20 J Hz T 2 15 00 ML 100 400 l l J Hz 1 2 15 00 J l U 8 00 6 00 EAR pepe J Hz 1 2 6 00 6 00 pom J Hz 1 2 15 00 A A 20
54. ndicating the size of the group 2 8 The spin system Chapter 2 o Dichlorobenzene ODCB also has two groups of H4 nuclei with identical chemical shifts two orthotoa ci H chlorine 1 and 4 and two para to a chlorine 2 and 3 However nucleus 1 cannot be magnetically equivalent with 4 since J12 an ortho coupling Cl differs from J 24 a meta coupling It is not relevant H4 here that J12 J34 and J13 J24 as long asthereisa single nucleus i for which Jj Ja nudei Land 4 cannot be magnetically equivalent They are however called chemically equivalent as explained below The ODCB type spin system is usually called an AA BB or AB system Inequivalent nuclei that are related by a symmetry operation are usually indicated by a notation using primes e g AA for hydrogens 1 and 4 Note that the overall molecular symmetry of SF and ODCB is the same C3y 3 so overall symmetry is not enough to determine magnetic equivalence Hs We will not discuss symmetry notations in detail here for an excellent discussion see Reference 3 Czy indicates the presence of two mirror planes and a twofold axis C means just a single mirror plane and C1 means no symmetry at all Magnetic equivalence is important because it allows considerable simplification in the calculation of NMR spectra One of the reasons for this is a theorem which states that for any group of magnetically equivalent nuclei in a system couplings within the group do
55. ne noise If despite the above precautions the procedure does not converge to a meaningful solution you can restart it with a different set of parameters Y ou can also let the program itself generate more or less random start values for coupling constants in that way you can do a large series of trials overnight and inspect the resulting solutions one by onein the morning If your initial guess looked reasonable but the full lineshape iteration seems to make it worse you could try starting with less flattening which tends to keep you closer to the start values of course it may also prevent you from finding the correct solution Do not break off an iteration too soon it will almost always drift away in thefirst few cydes but often come back later on The above guidelines especially the part about removing impurities and baseline noise may look like cheating Remember however that we are only trying to find a solution at this stage Once this has been done it istimeto do a definitive refinement without any cheating Full lineshape iteration 49 Chapter 8 8 5 The final refinement Objectively the only correct way of refining parameters is a direct least squares fit of observed to calculated spectrum without any fudging except phasing and baseline correction Y ou should always finish your lineshape analysis by doing such a refinement becausethis is the only way to obtain a meaningful set of error limits To d
56. ning There are already some fairly sophisticated computer algorithms for automatic peak assignment 18 However they are far from foolproof and doing it yourself is still the best way Thefirst problem is that there is seldom a 1 1 correspondence between calculated and experimental peaks A single observed peak may be due to a number of contributing elementary transitions The simulation program yields them as distinct peaks but there is no general way to tell from an experimental spectrum whether a peak is single or composite Also some peaks may have such a low intensity that you simply do not see them in the experimental spectrum The second problem is that even if the computer recognizes the correct number of peaks in the experimental spectrum assigning them in order may not be correct Every peak in the spectrum consists of a well defined transition for example aaa a e The ordering of the peaks depends on the parameter values the a xo o o transition might be at higher field than aao poa for a certain J1 value and at lower field for another value of this constant Thereis no direct way to tell from the experimental spectrum which is which but the iteration algorithm needs this information compare this with the phase problem in X ray crystallography This is where human input is needed by telling the program which experimental Assignment iteration 45 Chapter 7 peak corresponds to a calculated peak you assign a co
57. o different ways to approach the problem empirical methods based on measured data and theoretical methods based on quantum chemical calculations Empirical methods Using a database containing many known compounds with their NMR data it is possible to estimate data for related but unknown compounds using various statistical methods and structure property relationships The accuracy of this method depends strongly on the quality and size of the set of reference data Clearly itis impossibleto predict data for compounds with very abnormal structures or interactions in this way Accurate prediction is now possible for 1H and BC shifts and couplings of normal organic compounds and database based prediction programs for 19F and 31P have recently started to appear Predictions of metal NMR shifts is not yet available partly because of the lack of sufficient reference data and partly because there is not enough commercial interest Theoretical methods Abinitio calculation of coupling constants is possible but requires large basis sets and advanced electron correlation treatments Chemical shifts can be calculated with reasonable accuracy a few ppm for heavy atoms or 0 5 ppm for hydrogen but this requires the use of optimized structures and polarized basis sets These calculations are therefore mostly limited to simple molecules with up to ca 15 non hydrogen atoms However the method allows predictions for exotic structures as wel
58. o protons of each Py E methylene group are diastereotopic so we will need at least these two protons a phosphorus atom and the rhodium atom the Rh H couplings are not zero This gives a 4 spin H3PRh system However even the best simulation Figure 11B comes nowhere near the experimental result The 2 pp coupling is fairly large 43 Hz so we may have to include the second phosphorus atom In that case we will also haveto include the second CH group If we did not do so the phosphorus atoms would become very different and the results might not be meaningful The system is now a 7 spin H 4P2Rh system already rather large but the simulation Figure 11C is still unableto reproducethe curious pattern of the experi mental spectrum although it starts to look reasonable What can be missing here There are no significant couplings from the methylene group to the benzylic phenyl group so the problem must be somewhere else It turns out that extra couplings to the phosphorus atoms are needed to get the pattern of Figure 11A These couplings are really there the phenyl and pyridyl protons all have significant phosphorus couplings What is surprising is that you would need them to reproduce the benzylic methylene signal Luckily you do not need all the phenyl and pyridyl protons in the simulation Figure 11D shows the simulation of Figure 11C with just one hydrogen atom added per phosphorus atom Jp 20 Hz This is a 9 spin H gP4Rh syst
59. o this takethe solution obtained in the last section but set up a new iteration this time using the raw observed spectrum Set the flattening parameter to zero to do a normal least squares fit and start the iteration Even in the presence of impurities and baseline errors this fit will seldom run wild it will remain trapped in its current local minimum If the iteration has converged savethe data and print the error analysis Use these results for any illustration you plan to create not the ones showing edited experimental spectra 8 6 Checking your solution Once you have obtained reasonable looking fit results either by assignment or full lineshape iteration you might sit back and think you have solved the problem However your reasonable looking solution may still not bethe right one There may be other combinations of parameters that give rise to exactly the same calculated spectrum and are therefore also candidates there may even be solutions that give a better fit to the observed spectrum So it is important to ask whether you have a solution or the solution Unfortunately there is no general way to answer this question There are a number of possible sources of error in any solution you obtain e You may have chosen the wrong spin system In that case any parameters you have obtained via a fitting procedure will probably be meaningless since they have nothing to do with the parameters of the actual system If your fit
60. or is seldom observed The actual linewidth is often dominated by field inhomogeneities in which casethe lineshape tends to resemble a Gaussian S f fof fa Nee Even under idealized conditions both lineshape functions are strictly applicable only to either CW scans or to FT spectra without weighting In practice cleverly chosen weighting schemes are widely used to improve the appearance of NMR spectra and such weighting may occasionally produce bizarre results including lines complete with fake wiggles Imperfect phasing may result in mix in of dispersion components of the lineshape functions Typical absorption and dispersion lineshapes Lorentzian Gaussian and triangular are illustrated in Figure 8 In particular note the extremely slow fall off of the dispersion component of a Lorentzian away from its centre Simple simulation 19 Figure 8 Examples of Lorentzian Gaussian and Triangular lineshapes Chapter 3 in phase out of phase E Lorentzian Gaussian Triangular For systems consisting of many nuclei most NMR simulation programs use just a single linewidth for the whole spectrum which is often unsatisfactory In practice different nuclei can have very different relaxation times Strictly speaking it is not correct to assign a single relaxation time to each nucleus relaxation processes of nuclei are often connected and a relaxation matrix treatment is need
61. ost peaks correctly and the optimization is unlikely to produce useful results e For large systems assigning peaks can become very tedious For example a 6 spin system without symmetry will have about 200 peaks not counting the combination lines and assigning even the majority of these will be rather awkward and time consuming Assignment iteration 43 Chapter 7 however helpful the software tries to be in the process Moreover the chances are that many of these lines will partly overlap so the assignment is not likely to be very accurate This introduces an arbitrariness in the results and the final optimized parameters will contain systematic errors which are not reflected in an error analysis You can iterate only on shifts and coupling constants not on linewidths or rate constants Thus on completion of the iteration your result may not look as good as when you had carried out a full lineshape analysis next chapter even though the agreement in peak positions is perfect Intensity data are not used in the calculation There are cases where peak positions alone do not determine all relevant parameters the X part of the simple AA X system is an example The last objection is not insurmountable Arata al previously proposed including intensity data in the iteration scheme although they did not actually implement such a scheme gN MR is probably the first simulation program to incorporate this possibility Moreimpor
62. ou may encounter dynamic behavior in a situation where an equilibrium strongly favors one side You may never directly observe the minority species because its concentration is too low at all temperatures and still see some kind of coalescence behavior in the majority species Such spectra can be very difficult to interpret correctly Band shape analysis produces pseudo first order rate constants How these actually relate to the real rate constants for the process you are interested in depends on the model you Chemical exchange Chapter 6 use for the reaction The relation can already be nontrivial for intra molecular mutual exchange processes for inter molecular processes it be even more complicated e There may be more than one dynamic process occurring in the system It is often easy to distinguish between an inter and an intra molecular process but if you suspect the occurrence of several intra molecular processes only the difference in computed rate constants may be able to prove your case Since the errors in rate constants are always rather large regardless of what an optimistic fit program may tell you you should be very careful not to assume several processes where only oneis really needed Occam s razor Note that a difference in coalescence temperatures does not imply a difference in rate constants Chemical exchange 41 Chapter 7 7 Iteration with assignments 7 1 Description Iterative optimization of
63. ple rules for spectrum appearance mentioned above The use of higher field strengths is often cited as the remedy for all second order effects in NMR Chemical shift differences become large compared to coupling constants so second order effects will surely disappear While this is an attractive argument for buying higher frequency spectrometers and for avoiding delving into NMR simulation it is incorrect As a general rule you will see second order effects when the chemical shift difference between two nuclei is of the same order of magnitude as the coupling constant between them say to within a factor of 10 either way If the coupling constant is very small the nuclei are weakly coupled and will give riseto a simplefirst order spectrum If the coupling constant is very large the nuclei become effectively equivalent again giving riseto a first order spectrum Second order effects are expected in the intermediate range of strong coupling The first signs of second order effects are usually 22 Simple simulation Chapter 3 small intensity distortions inner lines become more intense at the expense of outer lines If the coupling becomes stronger the distortions become larger and extra splittings may appear Also second order effects may appear on the multiplets of other nudei in the molecule even though these are not strongly coupled to any spin in the molecule If two nuclei are magnetically equivalent you can treat them as a
64. pter 5 gives hints on how to simulate spectra of large molecules Chapter 6 explains what happens when the system being studied is undergoing chemical reactions on the NMR time scale Chapters 7 and 8 discusses the two iterative methods for obtaining accurate parameters from experimental spectra and chapter 9 describes the error analysis applicable to both Simulation is generally only useful when you already have an experimental spectrum Nowadays NMR data are always recorded as FID signals This means that they haveto be processed in some way to convert them to a spectrum meaningful to humans At the very least this requires a Fourier transformation apodization resolution enhancement and corrections for various filters may also be needed Data processing is described briefly in Chapter 10 Finally we have collected in Appendix A a number of frequently encountered second order spectrum types that may help you in the interpretation of your own spectra 6 Simulation and spectrum analysis Chapter 2 2 The spin system 2 1 Introduction The information that is needed for an NMR simulation consists of a qualitative part and a quantitative part Together they form the spin system The qualitative part is the composition of the system the number and types of NMR active nuclei and their symmetry relations If the structure of the molecule being studied is known this part can usually be written out easily When the molecular structu
65. r systems The appearance of second order spectra can be complicated and often bears no obvious relation to the original spectral parameters This makes setting up the initial simulation difficult since you don t know where to start Once you recognize the pattern of a multiplet and can reproducethis in a simulation obtaining more accurate parameter values by eg iteration is easy The examples in this chapter are intended to help you recognize such patterns In each section you will see spectra calculated for a particular type of system A 2B3 AA BB AA X and several sets of parameter values shifts and couplings The parameters have been chosen to illustrate typical spectrum patterns and do not necessarily represent realistic values All spectra have been calculated for a spectrometer 1H frequency of 100 MHz The filenames mentioned with the examples refer to sample files distributed with gQNMR In general it is impossible to deduce the absolute signs of coupling constants from NMR spectra In many cases however relative signs may affect the spectrum appearance Therefore you will see examples of both positive and negative coupling constants in the examples below Changing the signs of all coupling constants simultaneously will never change the spectrum appearance but changing the sign of only one coupling may have a large effect A 1 The A Bn systems The appearance of these spectra is completely determined by a single para
66. re is not known classification of the system is more difficult In simple cases thetype of spin system can be recognized directly from the N MR spectrum e g the distinctive pattern of an ethyl group or the typical 6 line pattern of the X part of an AA X system But most types of spin systems have too many independent parameters to have a distinctive easily recognizable pattern If you want to simulate a complicated spectrum of a completely unknown compound you will often have to go through some trial and error as far as the type of spin system is concerned The quantitative part is the set of shifts and coupling constants and possibly other relevant parameters like exchange rates Guessing accurate values for shifts and coupling constants is not easy see also chapter 4 But once you are close enough too see correspondences between calculated and experimental spectra further optimization can usually be done by the computer Itisimportant to note here that the appearance of the spectrum depends only on the spectral parameters shifts and couplings not directly on the structure If two completely different chemical structures would accidentally give riseto the same set of spectral parameters they would also produce the same NMR spectrum The spin system 7 Chapter 2 2 2 Magnetic equivalence The concepts of magnetic and chemical equivalence are very important in NMR Therefore we will start with a formal definition of magn
67. sc 3 k gt TFG k 106 Hz ss k 17 Hz 138 C k gt 0 2 600 2 400 2 200 2 000 1 800 1 500 1 400 1 200 1 000 0 800 Simulation can be used to predict the appearance of the spectrum at different exchange rates given the parameters for the non exchanging system The results show that the proposed process is indeed consistent with the observed spectra Fitting produces the rates at different temperatures from which the activation parameters can be deduced These examples demonstrate the usefulness of simulation in the analysis of NMR spectra Simulation is by no means necessary for every analysis But if you are uncertain whether a spectrum you have measured may really correspond to a particular structure simulation can be an easy way of obtaining confirmation Simulation and spectrum analysis 5 Chapter 1 1 2 Overview The remainder of this manual provides some background on the simulation of NMR spectra It is not a textbook on NMR if you do not understand the principles of NMR you should consult a textbook before trying to read further However most of the aspects of NMR spectroscopy that are relevant to simulation will be touched upon Chapter 2 discusses the spin system the basic unit that determines the type of NMR spectrum Chapter 3 then describes how a spectrum can be calculated from this basic information Chapter 4 touches briefly on the prediction of spectral parameters from molecular structures Cha
68. se to slightly different spectra in which case you may be ableto judge from the quality of thefit which solution is the most likely one Often however there are different sign combinations that produce exactly the same spectrum In that case you can only try to rule out some possibilities on the basis of general knowledge see section 2 7 If you are not completely sure it is often better to report several possibilities Full lineshape iteration 51 Chapter 9 9 Error analysis Once you havefinished your iterative simulation you will probably want to report the results There are standard ways to report results of least squares fits we will discuss a gN MR error analysis as an example but other programs will produce very similar output Inaleast squares analysis it is important that variables are scaled so that similar variations in different parameters have similar effects The scaling does not have to be perfect but differences in parameter sensitivity in the order of 106 will wreak havoc in most least squares fits If the program does scaling the scaling factors for the different parameters will be printed somewhere The part of the output you are probably most interested in is called the variance covariance matrix This is a square symmetric matrix the rows and columns are numbered for the parameters The matrix may be expressed in either scaled or unscaled parameters be sureto check on this before using the results
69. shifts and coupling constants by computer was first implemented by Alexander and Swalen and Reilly using a scheme based on the determination of energy levels Several modifications to the scheme were subsequently implemented but the most important improvement was introduced by Bothner By and Castellano and Braillon 14 they decided to use the observed frequencies as the basis for a least squares optimization Various refinements of the method have been described since including the use of magnetic equivalence molecular symmetry and anisotropy but the principle of the method has hardly changed The user must start with an initial guess of shifts and coupling constants calculate a spectrum and then decide which lines in the calculated spectrum correspond to which lines in the experimental spectrum this phase is called the assignment phase After that the computer performs least squares minimization and the user checks whether the results seem reasonable either by comparing the calculated and experimental spectra or by inspecting the list of calculated and observed frequencies 7 2 Pros and cons of assignment iteration The assignment iteration method has been in use for many years and is still useful especially for small molecules However it has a number of disadvantages e trequires a good guess of starting values for the shifts and coupling constants If the initial guess is not good enough you will not be able to assign m
70. sing cooked spectra to help along the initial stages of an iteration is perfectly legitimate as discussed in section 8 4 10 5 Linear prediction and other processing techniques Apart from the standard Fourier transformation there are a few other techniques for generating a spectrum from an FID The most important of these are linear prediction and maximum entropy Linear prediction effectively does a direct fit of a set of decaying sinusoids to the FID There are several variations of this method Some are merely designed to improve the quality of the transformed spectrum by throwing away noise components while others generate alist of peaks directly without even going through a transformed spectrum Linear prediction is more computationally intensive than FFT but the difference is not prohibitive and we believe the technique will become more important in the future Maximum entropy is a statistical method of generating a transformed spectrum from an FID which achieves a better S N than standard FFT This is also computationally expensive and moreover there are too many parameters that can be varied and not enough experience to let this play an important role in routine spectrum processing at the moment An important disadvantage in the current context is that it is nonlinear which makes it less suitable for use in combination with full lineshape iteration NMR data processing 59 Appendix A A Examples of typical second orde
71. t lie on all symmetry elements This has noticeable consequences particularly if there are other magnetically active nudei in the molecule For example consider the diphosphine 1 3 bis diphenylphosphino propane and its 1 3C and 2 BC isotopomers In the all 12C species the phosphorus atoms are equivalent They are also equivalent in the 2 BC isotopomer and the BC resonance of C5 will bea nice triplet In the 1 3C isotopomer however the phosphorus atoms are inequivalent since the BC atom destroys the symmetry The Jpc coupling constant will be different from 3Jpc and there will also be a small shift difference between the two phosphorus atoms Therefore the BC peak for C1 will have a more complex splitting pattern Very complex patterns can also be observed in 4H coupled BC spectra of symmetrical molecules C PhP cf PPh The spin system 17 Chapter 3 3 Simple simulation 3 1 Linewidths and lineshapes In the case of a single nucleus resonating at a So far we have been discussing NMR spectra as if they were stick spectra that could be fully characterized by a set of peak positions and intensities Actually peaks also have a particular lineshape frequency fo with a relaxation behavior characterized by a single transverse relaxation time T gt in the absence of saturation the absorption lineshape is a pure Lorentzian with a width at half height of Wy RT W RJ suf In practice however ideal relaxation behavi
72. tantly for some spectra there are several distinct well determined solutions giving exactly the same set of peak positions but with different distributions of intensities Clearly there is no way that iteration on peak positions is going to distinguish between such solutions The assignment iteration scheme also has some advantages 44 It is fast It gives the user a fair degree of control over where the iteration is going You do not have to import an experimental spectrum to start the analysis a peak list is enough For large systems typing in a peak list may be tedious but for small systems retyping a few numbers may be more efficient than transferring the whole spectrum Also the spectrum is sometimes not available in electronic form Assignment iteration Chapter 7 e You can iterate on very noisy spectra or spectra showing impurities and baseline errors where full lineshape analysis would not work at all So for small systems with not too many lines assignment iteration can bethe method of choice For larger systems with many independent parameters where a good initial guess is difficult to obtain anyway full lineshape analysis is recommended 7 3 Why the computer cannot do the assignments The assignment phase of assignment iteration seems rather trivial you could just let the computer assign the peaks in order of their occurrence in the spectrum So why doesn t this work and why do you have to do the assig
73. ters into an orthogonal set of linear combinations that represent independent search directions Each column shows one such linear combination the numbers above each column are a measure of the precision with which the movement in that particular direction is determined the larger the better If all singular values are comparable in magnitude say to within a factor of 1000 your parameters are apparently all well determined by the data In the example shown below however there is one combination that is clearly not very well determined The lowest singular value 1 5x10 corresponds to a search direction that consists mainly of parameter 2 so this is the culprit In the present case this parameter also stands out in the conventional error analysis above Sometimes however you may find that the sum of two or more parameters is ill determined whiletheir difference is well determined by the data In such cases you may not see any unusually large single parameter errors but you still have a problem with your data 54 Error analysis Singular Value Analysis Note wo OAT DN UO FF MN FB wo OAD UO i WN FB is columns refer to 54e 05 is 94e 02 SCALED variables 4 43e 02 7 78e 02 08e 00 Chapter 9 326 00 000546 2999998 000006 000012 001630 000262 000001 000018 000055 66e 00 000895 001631 000038 000069 999999 000674 000036 000140 000318 77e
74. xchange processes show such strange behavior but it is important to renember that predicting the appearance of dynamic spectra can be difficult The loss of coupling constant information is often taken as proof of an intermolecular process For example if you observe the disappearance of the 183W satellites on the 31P signal of a tungsten phosphine complex you may well be looking at a phosphine exchange process This is not an absolute proof since intramolecular averaging may also lead to near zero values but it is a reasonably strong indication 38 Chemical exchange Figure 14 A part of exchanging AX X system with Jax 10 Jax 3 and Ixx 50Hz Chapter 6 Asfar as NMR is concerned the meaning of intermolecular only relates to the collection of nuclei you are observing in a specific reaction The reaction would be called intramolecular if this collection stayed together regardless of whether the reaction is caused by intermolecular exchange involving other parts of the molecule For example the allylic bromine exchange shown below is intramolecular as far as NMR is concerned since bromine is not NM R active However the dependence of exchange rate on bromine concentration could reveal the bimolecular nature of the reaction This once again illustrates that one should be very careful in discussing the nature of rate processes using NMR data Chemical exchange 39 Chapter 6 Br Br

Simulation of one-dimensional NMR spectra

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