CONTENTS
Preface
Preface to the first edition
Dedication
1: What this book is about and who should read it
1.1 How this book is organized
1.2 Scope and limitations
1.3 Context and further reading
1.4 On-line resources
1.5 Abbreviations and acronyms
2: Setting the scene
2.1 NMR frequencies and chemical shifts
2.2 Linewidths, lineshapes and integrals
2.3 Scalar coupling
2.4 The basic NMR experiment
2.5 Frequency, oscillations and rotations
2.6 Photons
2.7 Moving on
2.8 Further reading
2.9 Exercises
3: Energy levels and NMR spectra
3.1 The problem with the energy level approach
3.2 Introducing quantum mechanics
3.3 The spectrum from one spin
3.4 Writing the Hamiltonian in frequency units
3.5 The energy levels for two coupled spins
3.6 The spectrum from two coupled spins
3.7 Three spins
3.8 Summary
3.9 Further reading
3.10 Exercises
4: The vector model
4.1 The bulk magnetization
4.2 Larmor precession
4.3 Detection
4.4 Pulses
4.5 On-resonance pulses
4.6 Detection in the rotating frame
4.7 The basic pulse-acquire experiment
4.8 Pulse calibration
4.9 The spin echo
4.10 Pulses of different phases
4.11 Off-resonance effects and soft pulses
4.12 Moving on
4.13 Further reading
4.14 Exercises
5: Fourier transformation and data processing
5.1 How the Fourier transform works
5.2 Representing the FID
5.3 Lineshapes and phase
5.4 Manipulating the FID and the spectrum
5.5 Zero filling
5.6 Truncation
5.7 Further reading
5.8 Exercises
6: The quantum mechanics of one spin
6.1 Introduction
6.2 Superposition states
6.3 Some quantum mechanical tools
6.4 Computing the bulk magnetization
6.5 Summary
6.6 Time evolution
6.7 RF pulses
6.8 Making faster progress: the density operator
6.9 Coherence
6.10 Further reading
6.11 Exercises
7: Product operators
7.1 Operators for one spin
7.2 Analysis of pulse sequences for a one-spin system
7.3 Speeding things up
7.4 Operators for two spins
7.5 In-phase and anti-phase terms
7.6 Hamiltonians for two spins
7.7 Notation for heteronuclear spin systems
7.8 Spin echoes and J-modulation
7.9 Coherence transfer
7.10 The INEPT experiment
7.11 Selective COSY
7.12 Coherence order and multiple-quantum coherences
7.13 Summary
7.14 Further reading
7.15 Exercises
8: Two-dimensional NMR
8.1 The general scheme for two-dimensional NMR
8.2 Modulation and lineshapes
8.3 COSY
8.4 DQF COSY
8.5 Double-quantum spectroscopy
8.6 Heteronuclear correlation spectra
8.7 HSQC
8.8 HMQC
8.9 Long-range correlation: HMBC
8.10 HETCOR
8.11 TOCSY
8.12 Frequency discrimination and lineshapes
8.13 Further reading
8.14 Exercises
9: Relaxation and the NOE
9.1 The origin of relaxation
9.2 Relaxation mechanisms
9.3 Describing random motion – the correlation time
9.4 Populations
9.5 Longitudinal relaxation behaviour of isolated spins
9.6 Longitudinal dipolar relaxation of two spins
9.7 The NOE
9.8 Transverse relaxation
9.9 Homogeneous and inhomogeneous broadening
9.10 Relaxation due to chemical shift anisotropy
9.11 Cross correlation
9.12 Summary
9.13 Further reading
9.14 Exercises
10: Advanced topics in two-dimensional NMR
10.1 Product operators for three spins
10.2 COSY for three spins
10.3 Reduced multiplets in COSY spectra
10.4 Polarization operators
10.5 ZCOSY
10.6 HMBC
10.7 Sensitivity-enhanced experiments
10.8 Constant time experiments
10.9 TROSY
10.10 Double-quantum spectroscopy of a three-spin system
10.11 Further reading
10.12 Exercises
11: Coherence selection: phase cycling and field gradient pulses
11.1 Coherence order
11.2 Coherence transfer pathways
11.3 Frequency discrimination and lineshapes
11.4 The receiver phase
11.5 Introducing phase cycling
11.6 Some phase cycling ‘tricks’
11.7 Axial peak suppression
11.8 CYCLOPS
11.9 Examples of practical phase cycles
11.10 Concluding remarks about phase cycling
11.11 Introducing field gradient pulses
11.12 Features of selection using gradients
11.13 Examples of using gradient pulses
11.14 Advantages and disadvantages of coherence selection with gradients
11.15 Suppression of zero-quantum coherence
11.16 Selective excitation with the aid of gradients
11.17 Further reading
11.18 Exercises
12: Equivalent spins and spin system analysis
12.1 Strong coupling in a two-spin system
12.2 Chemical and magnetic equivalence
12.3 Product operators for AXn (InS) spin systems
12.4 Spin echoes in InS spin systems
12.5 INEPT in InS spin systems
12.6 DEPT
12.7 Spin system analysis
12.8 Further reading
12.9 Exercises
13: How the spectrometer works
13.1 The magnet
13.2 The probe
13.3 The transmitter
13.4 The receiver
13.5 Digitizing the signal
13.6 Quadrature detection
13.7 The pulse programmer
13.8 Further reading
13.9 Exercises
A: Some mathematical topics
A.1 The exponential function and logarithms
A.2 Complex numbers
A.3 Trigonometric identities
A.4 Further reading
Index
Preface
I am very pleased to have the opportunity to produce a second edition of Understanding NMR Spectroscopy, not least as I have been encouraged by the many kind comments that I have received by users of the first edition. For all its undoubted flaws, the book has clearly been found to be useful in helping people to get to grips with the theory of NMR.
I have resisted the temptation to add a great deal of additional material or to make the discussion more technical. However, I have included a new chapter which covers two topics which, in retrospect, seemed to be serious omissions from the first edition. The first topic is how product operators can be extended to describe experiments in AX2 and AX3 spin systems, thus making it possible to discuss the important APT, INEPT and DEPT experiments often used in 13C spectroscopy.
The second topic is spin system analysis i.e. how shifts and couplings can be extracted from strongly coupled (second-order) spectra. In the early days of NMR this kind of analysis was all but essential since the low field strengths then available meant that spectra were often strongly coupled. The current use of much higher fields means that strong-coupling effects are less common, but they have not gone away completely. It therefore remains important to be aware of such effects and their consequences for the appearance of spectra. In a related topic, I also discuss how the presence of chemically equivalent spins leads to spectral features which are somewhat unusual and possibly misleading. In contrast to strong-coupling effects, these features are independent of the field strength and so are not mitigated by the move to higher fields.
The chapter on relaxation has been reorganised, and a discussion of chemical exchange effects has been introduced in order to help with the explanation of transverse relaxation. Finally, I have added a short section on double-quantum spectroscopy to Chapter 10.
The use of two-colour printing will, I hope, both improve the clarity of many of the diagrams and improve the appearance of the printed pages.
I am very much indebted to Dr Andrew Pell (now at the École Normale Supérieure de Lyon) who found time between completing his PhD and starting a postdoctoral position to help me in the preparation of this edition. Andy worked on adding colour to the figures, produced some additional simulations for Chapter 9, recorded all of the experimental spectra and commented on the new sections. I am also grateful to Dr Daniel Nietlispach (Department of Biochemistry, University of Cambridge) for once again providing very useful and perceptive comments on the new material.
Cambridge, January 2010
Preface to the first edition
I owe a huge debt of gratitude to Dr Daniel Nietlispach and Dr Katherine Stott (both from the Department of Biochemistry, University of Cambridge) who have read, corrected and commented on drafts of the entire book. Their careful and painstaking work has contributed a great deal to the final form of the text and has, in my view, improved it enormously. I am also grateful to them for their constant enthusiasm, which sustained and encouraged me throughout the project. I could not have wanted for two more constructive and helpful readers.
Special thanks are also due to Professor Nikolaus Loening (Lewis and Clark College, Portland, Oregon) who, at short notice and with great skill, provided all of the experimental spectra in the book. His good humoured response to my pernickety requirements is much appreciated. Andrew Pell (Selwyn College, Cambridge) also deserves special mention and thanks for his skilled assistance in producing the solutions manual for the exercises.
I would like to acknowledge the support and advice from my collaborator and colleague Dr Peter Wothers (Department of Chemistry, University of Cambridge): he remains both my sternest critic and greatest source of encouragement. I am also grateful to Professor Jeremy Sanders (Department of Chemistry, University of Cambridge) for his much valued support and advice.
My appreciation and understanding of NMR, such that it is, has been very much influenced by those I have been fortunate enough to work alongside, both as research students and collaborators; I thank them for their insights. I would also like to thank Professor Malcolm Levitt (Department of Chemistry, University of Southampton), Professor Art Palmer (Columbia University, New York) and Dr DavidNeuhaus (MRC LMB, Cambridge) for tirelessly answering my many questions.
This book grew out of a series of lecture notes which, over a number of years, I prepared for various summer schools and graduate courses. On the initiative of Dr Rainer Haessner (Technische Universität, Munich), the notes were made available on the web, and since then I have received a great deal of positive feedback about how useful people have found them. It was this, above all, which encouraged me to expand the notes into a book.
The book has been typeset by the author using LATEX, in the particular implementation distributed as MiKTEX (http://www.miktex.org). I wish to express my thanks to the many people who develop and maintain the LATEX system. All of the diagrams have been prepared using Adobe Illustrator (Adobe Systems Inc., San Jose, California), sometimes in combination with Mathematica (Wolfram Research Inc., Champaign, Illinois).
Finally, I am delighted to be able to dedicate this book to Professor Ray Freeman. I was lucky enough to have started my NMR career in Ray's group, and what I learnt there, both from Ray and my fellow students, has stood me in good stead ever since. My appreciation for Ray has continued as strong as ever since those early days, and I am pleased to have this opportunity to acknowledge the debt I owe him.
Cambridge, July 2005
Dedicated to
Professor Ray Freeman FRS
1
What this book is about and who should read it
This book is aimed at people who are familiar with the use of routine NMR for structure determination and who wish to deepen their understanding of just exactly how NMR experiments ‘work’. It is one of the great virtues of NMR spectroscopy that one can use it, and indeed use it to quite a high level, without having the least idea of how the technique works. For example, we can be taught how to interpret two–dimensional spectra, such as COSY, in a few minutes, and similarly it does not take long to get to grips with the interpretation of NOE (nuclear Overhauser effect) difference spectra. In addition, modern spectrometers can now run quite sophisticated NMR experiments with the minimum of intervention, further obviating the need for any particular understanding on the part of the operator.
You should reach for this book when you feel that the time has come to understand just exactly what is going on. It may be that this is simply out of curiosity, or it may be that for your work you need to employ a less common technique, modify an existing experiment to a new situation or need to understand more fully the limitations of a particular technique. A study of this book should give you the confidence to deal with such problems and also extend your range as an NMR spectroscopist.
One of the difficulties with NMR is that the language and theoretical techniques needed to describe it are rather different from those used for just about all other kinds of spectroscopy. This creates a barrier to understanding, but it is the aim of this book to show you that the barrier is not too difficult to overcome. Indeed, in contrast to other kinds of spectroscopy, we shall see that in NMR it is possible, quite literally on the back of an envelope, to make exact predictions of the outcome of quite sophisticated experiments. Further, once you have got to grips with the theory, you should find it possible not only to analyse existing experiments but also dream up new possibilities.
There is no getting away from the fact that we need quantum mechanics in order to understand NMR spectroscopy. Developing the necessary quantum mechanical ideas from scratch would make this book rather a hard read. Luckily, it is not really necessary to introduce such a high level of formality provided we are prepared to accept, on trust, certain quantum mechanical ideas and are prepared to use these techniques more or less as a recipe. A good analogy for this approach is to remember that it is perfectly possible to learn to add up and multiply without appreciating the finer points of number theory.
One of the nice features we will discover is that, despite being rigorous, the quantum mechanical approach still retains many features of the simpler vector model often used to describe simple NMR experiments. Once you get used to using the quantum mechanical approach, you will find that it does work in quite an intuitive way and gives you a way of ‘thinking‘ about experiments without always having to make detailed calculations.
Quantum mechanics is, of course, expressed in mathematical language, but the mathematics we will need is not very sophisticated. The only topic which we will need which is perhaps not so familiar is that of complex numbers and the complex exponential. These will be introduced as we go along, and the ideas are also summarized in an appendix.
1.1 How this book is organized
The ideas we need to describe NMR experiments are built up chapter by chapter, and so the text will make most sense if it read from the beginning. Certain sections are not crucial to the development of the argument and so can be safely omitted at a first reading; these sections are clearly marked as such in the margin.
Chapter 6, which explains how quantum mechanics is formulated in a way useful for NMR, is also entirely optional. It provides the background to the product operator formalism, which is described in Chapter 7, but this latter chapter is written in such a way that it does not rely on anything from Chapter 6. At some point, I hope that you will want to find out about what is written in Chapter 6, but if you decide not to tackle it, rest assured that you will still be able to follow what goes on in the rest of the book.
The main sequence of the book really ends with Chapter 8, which is devoted to two-dimensional NMR. You should dip into Chapters 9–13 as and when you feel the need to further your understanding of the topics they cover. This applies particularly to Chapter 10 which discusses a selection of more advanced ideas in two–dimensional NMR, and Chapter 11 which is concerned with the rather ‘technical’ topic of how to write phase cycles and how field gradient pulses are used.
Quite deliberately, this book starts off at a gentle pace, working through some more–or–less familiar ideas to start with, and then elaborating these as we follow our theme. This means that you might find parts of the discussion rather pedestrian at times, but the aim is always to be clear about what is going on, and not to jump over steps in calculations or arguments. The same philosophy is followed when it comes to the more difficult and/or less familiar topics which are introduced in the later chapters. If you are already familiar with the vector model of pulsed NMR, and are happy with thinking about multiplets in terms of energy levels, then you might wish to jump in at Chapter 6 or Chapter 7.
Each chapter ends with some exercises which are designed to help your understanding of the ideas presented in that chapter. Tackling the exercises will undoubtedly help you to come to grips with the underlying ideas.
1.2 Scope and limitations
In this book we are going to discuss the high-resolution NMRof liquid samples and we will concentrate, almost exclusively, on spin-half nuclei (mainly 1H and 13C). The NMR of solids is an important and fast-developing field, but one which lies outside the scope of this book.
The experiments we will choose to describe are likely to be encountered in the routine NMR of small to medium-sized molecules. Many of the experiments are also applicable to the study of large biomolecules, such as proteins and nucleic acids. The special multi-dimensional experiments which have been devised for the study of proteins will not be described here, but we note that such experiments are built up using the repertoire of pulsed techniques which we are going to look at in detail.
The existence of the chemical shift and scalar coupling is, of course, crucial to the utility of NMR spectroscopy. However, we will simply treat the values of shifts and coupling constants as experimentally derived parameters; we will have nothing to say about their calculation or interpretation – topics which are very well covered elsewhere.
1.3 Context and further reading
This is not a ‘how to’ book: you will find no advice here on how to select and run a particular experiment, nor on how to interpret the result in terms of a chemical structure. What this book is concerned with is how the experiments work. However, it is not a book of NMR theory for its own sake: rather, the ideas presented, and the theories introduced, have been chosen carefully as those most useful for understanding the kinds of NMR experiments which are actually used.
There are many books which describe how modern NMR spectroscopy is applied in structural studies, and you may wish to consult these alongside this text in order to see how a particular experiment is used in practice. Two useful texts are: J. K. M. Sanders and B. K. Hunter, Modern NMR Spectroscopy (2nd edition, OUP, 1993), and T. D.W. Claridge, High-Resolution NMR Techniques in Organic Chemistry (Elsevier Science, 1999).
There are also a number of books which are at roughly the same level as this text and which you may wish to consult for further information or an alternative view. Amongst these, R. Freeman, Spin Choreography (Spektrum, 1997) and F. J. M. van de Ven, Multidimensional NMR in Liquids (VCH, 1995) are particularly useful. If you wish to go further and deeper into the theory of NMR, M. H. Levitt, Spin Dynamics (2nd edition, John Wiley & Sons, Ltd, 2008) is an excellent place to start.
The application of NMR to structural studies of biomolecules is a vast area which we will only touch on from time to time. A detailed account of this important area, covering both theoretical and practical matters, can be found in J. Cavanagh, W. J. Fairbrother, A. G. Palmer III, M. Rance and N. J. Skelton, Protein NMR Spectroscopy (2nd edition, Academic Press, 2007).
At the end of each chapter you will also find suggestions for further reading. Many of these are directions to particular chapters of the books we have already mentioned.
1.4 On–line resources
A solutions manual for the exercises at the end of each chapter is available on–line via the spectroscopyNOW website:
http://www.spectroscopynow.com/nmr
follow the ‘Education‘ link from this page
A list of corrections and amendments will also be available on this site, as well as other additional material. It will also be possible to download all of the figures (in ‘jpeg’ format) from this book.
1.5 Abbreviations and acronyms
| ADC | analogue to digital converter |
| APT | attached proton test |
| COSY | correlation spectroscopy |
| CTP | coherence transfer pathway |
| DEPT | distortionless enhancement by polarization transfer |
| DQF COSY | double-quantum filtered COSY |
| FID | free induction decay |
| HETCOR | heteronuclear correlation |
| HMBC | heteronuclear multiple-bond correlation |
| HMQC | heteronuclear multiple-quantum correlation |
| HSQC | heteronuclear single-quantum correlation |
| INEPT | insensitive nuclei enhanced by polarization transfer |
| NMR | nuclear magnetic resonance |
| NOE | nuclear Overhauser effect |
| NOESY | nuclear Overhauser effect spectroscopy |
| RF | radiofrequency |
| rx | receiver |
| ROESY | rotating frame Overhauser effect spectroscopy |
| SHR | States–Haberkorn–Ruben |
| SNR | signal-to-noise ratio |
| TOCSY | total correlation spectroscopy |
| TPPI | time proportional phase incrementation |
| TROSY | transverse relaxation optimized spectroscopy |
| tx | transmitter |