Two-Dimensional (2D) NMR Methods E-Book

• Herausgeber: John Wiley & Sons
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1200

Veröffentlichungsjahr: 2023

Beschreibung

TWO-DIMENSIONAL (2D) NMR METHODS Practical guide explaining the fundamentals of 2D-NMR for experienced scientists as well as relevant for advanced students Two-Dimensional (2D) NMR Methods is a focused work presenting an overview of 2D-NMR concepts and techniques, including basic principles, practical applications, and how NMR pulse sequences work. Contributed to by global experts with extensive experience in the field, Two-Dimensional (2D) NMR Methods provides in-depth coverage of sample topics such as: * Basics of 2D-NMR, data processing methods (Fourier and beyond), product operator formalism, basics of spin relaxation, and coherence transfer pathways * Multidimensional methods (single- and multiple-quantum spectroscopy), NOESY (principles and applications), and DOSY methods * Multiple acquisition strategies, anisotropic NMR in molecular analysis, ultrafast 2D methods, and multidimensional methods in bio-NMR * TROSY (principles and applications), field-cycling and 2D NMR, multidimensional methods and paramagnetic NMR, and relaxation dispersion experiments This text is a highly useful resource for NMR specialists and advanced students studying NMR, along with users in research, academic and commercial laboratories that study or conduct experiments in NMR.

Leseprobe

Two-Dimensional (2D) NMR Methods

Edited by

K. Ivanov‡ International Tomography Center, Novosibirsk, Russia

P.K. Madhu Tata Institute of Fundamental Research, Hyderabad, India

G. Rajalakshmi Tata Institute of Fundamental Research, Hyderabad, India

This edition first published 2023.

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Cover Images: Centre and bottom right-hand image courtesy of Nathaniel J. Traaseth. Bottom right-hand image adapted from Figure 2 from dx.doi.org/10.1021/jp303269m | J. Phys. Chem. B 2012, 116, 7138–7144; Bottom left-hand images courtesy of Pramodh Vallurupalli. Bottom left-hand image adapted from Figure 1 from DOI: 10.1039/c5sc03886c, Chem. Sci., 2016, 7, 3602

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Dedication

Dedication and tribute to Kostya

K. Ivanov at the meeting of the Alexander von Humboldt foundation scholarship holders in Novosibirsk, 2012.

Our esteemed colleague and good friend Konstantin L’vovich (“Kostya”) Ivanov became one of the first victims of the Covid-19 pandemic in the NMR community. He passed away at a hospital in Novosibirsk on March 5, 2021. He was not only a great scientist, but also a good human being, always sincere, honest, joyous, and considerate. In addition, he was a great citizen of the scientific community. Aside from his demanding job as the Director of the International Tomography Center (ITC), Novosibirsk, he kept his research at a very high level and organized a multitude of meetings, seminars, and webinars.

Alexandra Yurkovskaya met Kostya for the first time in 1998 at the ITC when he was a master student in the theoretical chemistry group headed by Nikita Lukzen. Having defended his Ph.D. thesis in 2002 at the ITC, Kostya teamed up with the experimental group of Alexandra Yurkovskaya in solving theoretical problems related to chemically induced dynamic nuclear polarization. Since 2005 both of them worked in Hans-Martin Vieth group at the Free University of Berlin as part of the large EU project "Bio-DNP" under the leadership of Thomas Prisner. In 2007, Kostya became a fellow of the Alexander von Humboldt Foundation at the Free University of Berlin and Alexandra had a Marie Curie International fellowship with Hans-Martin Vieth as the host professor. The scientific collaboration had since continued for almost 20 years and led to over 100 joint publications on nuclear hyperpolarization and polarization transfer processes, combining modern theoretical and sophisticated experimental techniques. In particular, over the last decade, Alexey Kiryutin from the Yurkovskyaya group at the ITC pushed the limits by developing unique high-resolution fast field-cycling NMR apparatus over nine orders of magnitude of magnetic field. Kostya proposed multiple smart applications of the technique to hyperpolarization and relaxation, establishing Kostya as one of the leaders of a new scientific direction. In 2007, Kostya defended his second doctoral thesis, which is the second scientific degree in Russia, awarded for a significant and sustainable contribution to scientific knowledge, which is analogous to the habilitation degree in Germany. The thesis title was: “Kinetics of multi-stage liquid-phase processes involving particles with spin degrees of freedom”. He completed his degree at the Institute of Chemical Physics in Moscow, which is the leading institute of the Russian Academy in the field of chemical physics. His scientific work was honored with several important prizes and awards, including the Medal of the European Academy of Science in 2010, Voevodsky prize in 2012, a Fellowship of the Japanese Society for Promotion of Science in 2016, and the Laukien Prize in 2020 for his contribution to the SABRE research field.

The cooperation between French (ENS) and Russian (ITC) groups doing fast field-cycling NMR started in 2017 with a visit of Kostya to École Normale Supérieure (ENS, Paris) organized by Geoffrey Bodenhausen. This led to several collaborations with Bodenhausen, Daniel Abergel, and Fabien Ferrage on topics as diverse as long-lived states, mechanisms of dynamic nuclear polarization, and coherence effects in field-cycling experiments. A comparative study of ZULF spectroscopy with NMR detection using an atomic magnetometer and inductive detection at high magnetic field was inspired by Kostya’s tutorial talk at the ZULF seminar organized by the Marie Curie ZULF-NMR Innovative Training Network in Mainz in the fall of 2019. During those several unforgettable days, followed by fantastic musical evenings including Kostya and Dima Budker singing together with a band formed by Dima’s research group members, the atmosphere of warm mutual cooperation was created and the fruitful although dramatically short joint work started.

Dima Budker’s acquaintance and collaboration with Kostya were both relatively short, some two-three years, depending on where one marks the beginning. Nevertheless, this collaboration, consisting of several joint projects that have resulted in at least five joint papers and book chapters, has had a profound effect on Dima’s research interests. In most cases, the collaboration was initiated by Kostya and was centered around an idea formulated by him with such clarity and enthusiasm that it was impossible not to embrace the project wholeheartedly.

This was also the case with the chapter for this 2D-NMR book. On the February 3, 2021, Dima received an e-mail from Kostya that said: “Madhu and myself have got an offer from Wiley to edit a book on 2D-NMR. As a part of this initiative, I want to cover some aspects of field-cycling NMR and zero-field NMR. Although 2D methods are not that widespread in these branches of NMR, they are used as well and can be of interest to NMR people. Would you like to write such a chapter together with me and Fabien Ferrage, with whom I cooperate on field-cycling NMR?” This was followed by a detailed outline of the chapter proposed by Kostya. Kostya and Dima then met on Skype to discuss this suggestion, including the scope and whom to invite as co-authors. Their last communication was on the February 11, 2021, and they agreed to talk again very soon… That conversation was not destined to occur.

Fabien Ferrage’s first acquaintance with Kostya was by reading his work, particularly the series of articles he published with Hans-Martin Vieth, Rob Kaptein, and Alexandra Yurkovskaya on level anti-crossings and coherent effects in field-cycling experiments. As Fabien was making his first steps in relaxometry, these were enlightening contributions that defined very clearly what he should not do if he wanted to measure pure relaxation rates at low fields. Geoffrey Bodenhausen and Fabien discussed several of Kostya’s articles before meeting him, often with praise, in particular the introduction of SABRE at high field. Fabien met Kostya in person in May 2015, when Kostya came to Paris for a COST meeting on hyperpolarization. Never wasting any time, Kostya came to ENS to visit the lab and give a seminar during the lunch break of the meeting. Over the years, several meetings and an invited professorship at ENS (an honour rarely given to scientists of such a young age), Kostya and Fabien discussed many topics on field cycling, relaxometry, and hyperpolarization. If Kostya was always asking for chemical insight from Fabien side, Fabien was always impressed by Kostya’s command of theory, in quantum mechanics in general as well as, of course, in both nuclear and electron magnetic resonance, the latter subject being invariably challenging for many NMR specialists. The collaboration between Fabien and Kostya really took off when Ivan Zhukov, a talented graduate student supervised by Alexandra Yurkovskaya, one of Kostya’s mentors, came to work on field cycling with the Ferrage group in 2018. As often happens, Ivan’s work with at ENS ended up being guided by serendipity and led us to an investigation of interesting effects of scalar couplings at low fields discussed in this book. This work inspired Kostya to propose the ZULF-TOCSY experiment, such a beautiful idea, that they have just started exploring.

Fabien Ferrage recalls: Kostya was a true force, initiating and leading many projects with intelligence, ambition and intensity while, at the same time, being the nicest, kindest and most generous colleague. Such a rare combination. Kostya and I were born the same year. His daughter and my son are about the same age. I considered him as a compass, setting a path to inspire others. In February 2021, we exchanged about the scope of this book chapter. One of the last emails I received from Kostya, on February 15, 2021, was about a conversation he had had with Dmitry Budker to write this book chapter together, which I am glad we accomplished. On March 5, my colleague Daniel Abergel called me to inform me that Kostya had died. The compass was broken, but the inspiration lives on.

Madhu recollects his association with Kostya which started in 2016 at the Windischleuba School on NMR, a series being organised by Joerg Matysik: We had been regular teachers in this workshop series in 2017, 2018, and 2020. Somewhere in 2018, we started discussing the prospects of writing a book on solid-state NMR. We met up in Leipzig in June 2019 for a discussion regarding this, having done some amount of writing already. During this meeting Kostya had this wonderful spark and told me that why do not we write a review on Floquet theory with new ideas embedded and with more examples than existing reviews. This review kept us busy till he passed away when we were refining it after comments from the editors. In fact, we in between took up the assignment with Wiley to bring out this book and Kostya was, as usual, very enthusiastic and full of ideas about what to be included and how they should be. Writing the review with him, in particular, was very rewarding. I remember the large number of extensive discussions we had from which I learned quite a bit. We actually had the Skype window always active in our computers, and either of us would call the other informally in case of any questions. We were also discussing some of the solid-state NMR experiments that my group was carrying out which were the topics of discussion just before he got hospitalized. Needless to say, the other major thing Kostya did for the magnetic resonance community was starting the Friday Intercontinental Seminar Series on April 8, 2020 (which was a Wednesday). A week before that as we were talking about the review, he as usual wanted to do something for the community. We got in touch with Daniel Abergel and Gerd Buntkowsky and the seminar series has been running since then smoothly with great talks. As Kostya wanted, it combines talks by senior and younger researchers. He also started the ICONS conference in 2020 and just before he passed away, we had the second of that meeting. ICONS conferences have been continuing since then on a regular basis with the third held in September 2021, the fourth in February 2022, the fifth in August-September 2022, and the sixth in January 2023. All these clearly reveal the vitality in Kostya with a sharp eye for details. Besides science, we had fun discussing politics, literature, music, and many other things.

Muslim Dvoyashkin recalls the first meeting with Kostya on the way to Windischleuba NMR school in 2020, picking up Kostya at the train station in Leipzig. Kostya made the first impression of a very intelligent and very modest person, which later turned out to be quite true. As his countryman, it was very interesting for Muslim to find out how he and his colleagues manage to remain at the forefront of fundamental science given its very limited financial support in Russia. Therefore, Kostya will remain in Muslim’s memory as a real hero, and thanks to whom, many students were able to find their way to the academia later on.

Malcolm Levitt recalls that his close association with Kostya started around 2017 when Malcolm expressed his appreciation of Kostya’s conference talk and his insightful theoretical approach. This led to extensive discussions and the identification of various possible themes for collaboration. Vigorous discussions led to several collaborative papers on the theme of singlet NMR. In 2019 Malcolm visited Novosibirsk and had the pleasure of getting to know Kostya and his colleagues, in their “natural habitat”. It was a very pleasant and instructive visit. Malcolm’s most striking impression of Kostya was how he managed this large and prestigious institute, full of competing egos, with the minimum of perceptible friction, and always with remarkable good humour. He was basically an extremely rare person who combined refined political skills with a sharp and creative scientific mind as well as possessing remarkable patience and energy, a great sense of humour, and legendary powers of concentration. His loss is extraordinarily tragic.

We shall deeply miss Kostya as an exceptional human being. He was a creative and rigorous scientist, a generous and attentive friend, and a considerate and eminently civilized colleague. We dedicate our contributions to him and the whole book itself, which was initiated by Kostya, that will be a brilliant scientific testament to his memory.

Contents

Cover

Title page

Copyright

Dedication

List of Contributors

Preface

1 Basics of Two-dimensional NMR

1.1 Introduction

1.1.1 Time-domain NMR

1.1.2 Hans Primas and the “Correlation Function of the Spectrum”

1.2 Spin Dynamics

1.2.1 Density Operator

1.2.2 Spin Hamiltonian

1.2.3 Liouville Space

1.2.4 Liouvillian

1.2.5 Propagation Superoperator

1.3 One-dimensional Fourier NMR

1.3.1 The One-dimensional NMR Experiment

1.3.2 One-dimensional NMR Spectrum

1.4 Two-dimensional NMR

1.4.1 The Two-dimensional NMR Experiment

1.4.2 Two-dimensional NMR Signal

1.4.3 Two-dimensional NMR Spectrum

1.4.4 Two-dimensional Experiments

1.5 Summary

Acknowledgments

References

2 Data Processing Methods: Fourier and Beyond

2.1 Introduction

2.2 Time-domain NMR Signal

2.3 NMR Spectrum

2.4 The Most Important Features of FT

2.5 Distortion: Phase

2.6 Kramers-Kronig Relations and Hilbert Transform

2.7 Distortion: Truncation

2.8 Distortion: Noise and Multiple Scans

2.9 Distortion: Sampling and DFT

2.10 Quadrature Detection

2.11 Processing:Weighting

2.12 Processing: Zero Filling

2.13 Fourier Transform in Multiple Dimensions

2.14 Quadrature Detection in Multiple Dimensions

2.15 Projection Theorem

2.16 ND Sampling Aspects and Sparse Sampling

2.17 Reconstructing Sparsely Sampled Data Sets

2.18 Deconvolution

References

3 Product Operator Formalism

3.1 Introduction

3.2 Product Operators and Time Evolution

3.2.1 Advantages of Product Operators

3.3 Time Evolution of the Product Operators

3.3.1 Effect of Pulses

3.3.2 Effect of Evolution Under the Hamiltonian

3.4 Applications

3.4.1 Spin-echo Experiments

3.4.2 Multiple-quantum Coherence

3.4.3 Composite Pulses

3.5 Two-dimensional Experiments

3.5.1 Two-dimensional J-Resolved

3.5.2 COSY

3.5.3 Two-dimensional NOE

3.5.4 Double-quantum Filtered COSY

3.5.5 Two-dimensional Double-quantum Spectroscopy

3.5.6 Relayed-COSY

3.5.7 TOCSY or Homonuclear Hartmann-Hahn Transfer

3.5.8 INEPT and HSQC

3.5.9 HMQC and HMBC

References

4 Relaxation in NMR Spectroscopy

4.1 Introduction

4.2 Theory

4.2.1 Bloch Equations

4.2.2 Transition-rate Theory

4.2.3 Semi-classical Relaxation Theory

4.2.4 Quantum-mechanical Relaxation Theory – Lindblad Formulation

4.3 Relaxation in Spin-1/2 Systems: Dipolar and CSA Relaxation

4.3.1 Longitudinal Relaxation in a Two-spin System

4.3.2 Transverse Relaxation in a Two-spin System

4.3.3 Double-quantum Relaxation

4.3.4 Relaxation in Larger Spin Systems

4.4 Other Relaxation Mechanisms

4.4.1 Quadrupolar Relaxation

4.4.2 Scalar Relaxation

4.5 Concluding Remarks

References

5 Coherence Transfer Pathways

5.1 Coherence Transfer Pathways: What and Why?

5.2 Principles of Coherence Selection

5.2.1 Precession of a coherence about the z-component of a magnetic field

5.2.2 Effect of changing the phase of a radiofrequency pulse that converts one coherence order term into another

5.3 Coherence Transfer Pathway Selection by Phase Cycling

5.3.1 CYCLOPS

5.3.2 EXORCYCLE

5.4 Cogwheel Phase Cycling

5.5 Coherence Transfer Pathway Selection by Pulsed-field Gradients

5.6 Comparison Between Phase Cycling and Pulsed-field Gradients

5.7 CTP Selection in Heteronuclear Spin Systems

5.8 Additional Approaches to Coherence Selection

References

6 Nuclear Overhauser Effect Spectroscopy

6.1 Introduction

6.2 Nuclear Overhauser Effect

6.2.1 Qualitative Picture

6.2.2 NOE: Quantitative Picture

6.2.3 NOE and Distance Dependence: Many-spin System

6.2.4 NOE Comparison and Distance Elucidation

6.2.5 Indirect NOE Effects

6.3 Measurement of NOE

6.4 Heteronuclear NOE

6.5 NOE Kinetics

6.5.1 Initial-Rate Approximation

6.6 Nuclear Overhauser Effect Spectroscopy, NOESY

6.6.1 NOESY Pulse Scheme

6.6.2 NOESY Theory

6.7 Rotating-frame NOE, ROE

6.8 Relative Signs of Cross Peaks

6.9 Generalised Solomon’s Equation

6.10 NOESY and ROESY: Practical Considerations and Experimental Spectra

6.11 Conclusions

Acknowledgements

References

7 DOSY Methods for Studying Non-equilibrium Molecular and Ionic Systems

7.1 Introduction

7.2 Spatial Spin “Encoding” Using Magnetic Field Gradient

7.3 Formation of NMR Signal and Spin Echo in the Presence of Field Gradient

7.4 NMR of Liquids in An Electric Field: Electrophoretic NMR

7.4.1 Measurement of Drift Velocities

7.4.2 Technical Development

7.4.3 Application Areas: From Dilute to Concentrated Electrolytes

7.4.4 Methods of Transformation and Processing: MOSY

7.4.5 Is eNMR a non-equilibrium experiment or a steady-state experiment?

7.5 Ultrafast Diffusion Measurements

7.6 Ultrafast Diffusion Exchange Spectroscopy

References

8 Multiple Acquisition Strategies

8.1 Introduction

8.2 Types of Multiple Acquisition Experiments

8.3 Utilization of Forgotten Spin Operators

8.4 Application of Multiple Acquisition Techniques

8.4.1 Solution NMR Spectroscopy

8.4.2 Solid-State NMR Spectroscopy

8.5 Modularity of Multiple Detection Schemes and Other Novel Approaches

8.6 Future of Multiple Acquisition Detection

Acknowledgments

References

9 Anisotropic One-dimensional/Two-dimensional NMR in Molecular Analysis

9.1 Introduction

9.2 Advantages of Oriented Solvents

9.2.1 Description of Orientational Order Parameters

9.2.2 The GDO Concept

9.3 Description of Useful Anisotropic NMR Parameters

9.3.1 Residual Dipolar Coupling (RDC)

9.3.2 Residual Chemical-shift Anisotropy (RCSA)

9.3.3 Residual Quadrupolar Coupling (RQC)

9.3.4 Spectral Consequences of Enantiodiscrimination

9.4 Adapted 2D NMR Tools

9.4.1 Spin-1/2 Based 2D Experiments

9.4.2 Spin-1 Based 2D Experiments

9.5 Examples of Polymeric Liquid Crystals

9.5.1 Polypeptide or Polyacetylene-based Systems

9.5.2 Compressed and Stretched Gels

9.5.3 Polynucleotide-based Chiral Oriented Media

9.5.4 Some Practical Aspects of Polymer-based LLCs Preparation

9.6 Contribution to the Analysis of Chiral and Prochiral Molecules

9.6.1 Analysis and Enantiopurity Determination of Chiral Mixtures

9.6.2 Discrimination of Enantiotopic Elements in Prochiral Structures

9.6.3 Dynamic Analysis by 2H NMR

9.7 Structural Value of Anisotropic NMR Parameters

9.7.1 From the Molecular Constitution to Configuration of Complex Molecules

9.7.2 Contribution of Spin-1/2 NMR

9.7.3 Configuration Determination Using Spin-1 NMR Analysis

9.7.4 Determining the Absolute Configuration of Monostereogenic Chiral Molecules

9.8 Conformational Analysis in Oriented Solvents

9.9 Anisotropic 2H 2D NMR Applied to Molecular Isotope Analysis

9.9.1 The Natural (2H/1H) Isotope Fractionation: Principle

9.9.2 Case of Prochiral Molecules: The Fatty Acid Family

9.9.3 New Tools for Fighting Against Counterfeiting

9.10 Anisotropic NMR in Molecular Analysis: What You Should Keep in Mind

References

10 Ultrafast 2D methods

10.1 Introduction

10.2 UF 2D NMR Principles: Entangling the Space and the Time

10.2.1 Spatial Encoding

10.2.2 Reading Out the Spatially Encoded Signal

10.2.3 Processing Workflow in UF Experiments

10.3 Specific Features of UF 2D NMR

10.3.1 Line-shape of the Signal

10.3.2 Resolution and Spectral Width

10.3.3 Sensitivity Considerations

10.4 Advanced UF Methods

10.4.1 Improving the Sensitivity

10.4.2 Improving Spectral Width and Resolution

10.5 UF 2D NMR: A Versatile Approach

10.5.1 Accelerating 2D NMR Spectroscopy Experiments

10.5.2 Accelerating Dynamic Experiments (UF pseudo-2D)

10.6 Overview of UF 2D NMR Applications

10.6.1 Reaction Monitoring

10.6.2 Single-scan 2D Experiments on Hyperpolarized Substrates

10.6.3 Quantitative UF 2D NMR

10.6.4 UF 2D NMR in Oriented Media

10.6.5 UF 2D NMR in Spatial Inhomogeneous Fields

10.7 Conclusion

References

11 Multi-dimensional Methods in Biological NMR

11.1 Introduction

11.2 Experimental Approaches

11.2.1 NMR Spectroscopic Information on Structural Features

11.2.2 Spectroscopic Information on Dynamical Features

11.2.3 NMR Spectroscopic Information Obtained from Interaction Studies

11.2.4 Quench Flow Methodology in Combination with NMR – Hydrogen-to-deuterium Exchange

11.2.5 Expanding Multi-dimensional NMR Spectroscopy from in vitro to in vivo Applications

11.2.6 Multi-Dimensional NMR Spectroscopy as an Integrated Approach in Structural Biology

11.3 Case Studies

11.3.1 Determining Thermodynamic Stability of Biomolecules at Atomic Resolution

11.3.2 Exotic Heteronuclear NMR Spectroscopy Correlating 31P with 13C

11.3.3 Following Biomolecular Dynamics by Homonuclear and Heteronuclear ZZ Exchange

11.3.4 Probing Structural Features by Solvent PREs

11.3.5 Discerning Protein Dynamics by Probing Fast Amide Proton Exchange

11.3.6 Integrated Approaches Utilizing Structural Information from NMR Spectroscopy

11.3.7 Multi-dimensional NMR Spectroscopy on ex vivo Samples

References

12 TROSY: Principles and Applications

12.1 Introduction

12.2 The Principles of TROSY

12.2.1 The Physical Picture of TROSY

12.2.2 Theory of TROSY

12.3 Practical Aspects of TROSY

12.3.1 Field Strength Dependence of TROSY for 1H–15N Groups

12.3.2 Peak Pattern of 1H-15N TROSY Spectrum

12.4 Applications of TROSY

12.4.1 Two-Dimensional [1H,15N]-TROSY

12.4.2 [1H,15N]-TROSY for Backbone Resonance Assignments in Large Proteins

12.4.3 [1H,15N]-TROSY for Assignment of Protein Side-chain Resonances

12.4.4 Application of [1H,15N]-TROSY for RDC Measurements

12.4.5 [1H,15N]-TROSY-based NOESY Experiments

12.4.6 Studies of Dynamic Processes Using the [1H,15N]-TROSY Concept

12.5 Transverse Relaxation-optimization in the Polarization Transfers

12.6

15

N Direct Detected TROSY

12.7 [1H,13C]-TROSY Correlation Experiments

12.7.1 Methyl-TROSY NMR

12.8 Applications to Nucleic Acids

12.9 Intermolecular Interactions and Drug Design

12.10 Conclusion

12.A Appendix

Acknowledgement

References

13 Two-Dimensional Methods and Zero- to Ultralow-Field (ZULF) NMR

13.1 Introduction and Motivation

13.2 Early Work

13.3 Two-dimensional NMR Measured at Zero Magnetic Field

13.4 Nuclear Magnetic Resonance at Millitesla Fields Using a Zero-Field Spectrometer

13.5 Field Cycling NMR and Correlation Spectroscopy

13.6 ZERO-Field - High-Field Comparison

13.7 Conclusion and Outlook

Acknowledgments

References

14 Multidimensional Methods and Paramagnetic NMR

14.1 Introduction

14.2 NMR Methods for Paramagnetic Systems in Solution

14.2.1 Homonuclear Correlations

14.2.2 Heteronuclear Correlations

14.2.3 Long-Range Paramagnetic Effects

14.2.4 Heteronuclear Detection Strategies

14.3 NMR Methods for Paramagnetic Systems in Solids

14.3.1 Adiabatic Pulses

14.3.2 Homonuclear Correlations

14.3.3 Heteronuclear Correlations

14.3.4 Long-Range Paramagnetic Effects

14.3.5 Separation of Shift and Shift-anisotropy Interactions

14.3.6 Separation of Shift-anisotropy and Quadrupolar Interactions

Acknowledgments

References

15 Chemical Exchange

15.1 Introduction

15.2 Bloch-McConnell Equations

15.2.1 Slow Exchange

15.2.2 Fast Exchange

15.2.3 Dependence of the Linewidth On Magnetic Field Strength

15.2.4 Exchange in the Absence of Chemical-Shift Differences

15.2.5 Multi-State Exchange

15.3 Studying Exchange Between Visible States

15.3.1 Lineshape Analysis

15.3.2 ZZ-Exchange Experiment

15.4 Studying Exchange Between a Visible State and Invisible State(s)

15.4.1 CPMG Experiments

15.4.2 CEST and DEST Experiments

15.4.3 R

1ρ

Relaxation Dispersion Experiment

15.5 Summary

Acknowledgments

References

Appendix A Proton-Detected Heteronuclear and Multidimensional NMR

Index

End User License Agreement

List of Tables

CHAPTER 03

Table 3.1 Effect of RF...

Table 3.A.1 Description of...

CHAPTER 04

Table 4.1 Definition of eigenoperators...

Table 4.2 Definition of eigenoperators...

Table 4.3 Definition of eigenoperators...

Table 4.4 Allowed relaxation pathways...

Table 4.5 Allowed relaxation pathways...

Table 4.6 Definition of eigenoperators...

Table 4.7 Definition of eigenoperators...

CHAPTER 05

Table 5.1 Some spherical product...

Table 5.4 Four-step CYCLOPS...

Table 5.5 Four-step EXORCYCLE...

CHAPTER 09

Table 9.1 Examples of kij...

CHAPTER 10

Table 10.1 Typical analytical performances...

CHAPTER 12

Table 12. A.1 Transverse...

Table 12. A.2 Selected TROSY...

APPENDIX 01

Table A.1a Relative...

Table A.1b Relative...

Table A.2 Ranges of...

List of Illustrations

CHAPTER 01

Figure 1.1 (a) A general 1D...

Figure 1.2 A general 2D...

CHAPTER 02

Figure 2.1 Following the radio...

Figure 2.2 A perfect peak...

Figure 2.3 The NMR peak...

Figure 2.4 The phase error...

Figure 2.5 The visualization of...

Figure 2.6 The effect of...

Figure 2.7 The average FID...

Figure 2.8 The sampling of...

Figure 2.9 The sampling of...

Figure 2.10 The concept of...

Figure 2.11 Four examples of...

Figure 2.12 How weighting allows...

Figure 2.13 How weighting...

Figure 2.14 The effect of...

Figure 2.15 The digital resolution...

Figure 2.16 The quadrature...

Figure 2.17 The projection...

Figure 2.18 The sampling schemes...

Figure 2.19 The spectral..

CHAPTER 03

Figure 3.1 Effect of RF...

CHAPTER 04

Figure 4.1 Coherent numerical simulation...

Figure 4.2 Simulated time evolution...

Figure 4.3 Energy levels of...

Figure 4.4 Structure of the...

Figure 4.5 (a) The dipolar...

Figure 4.6 (a) The CSA...

Figure 4.7 (a) The dipolar...

Figure 4.8 (a) The steady...

Figure 4.9 (a) The steady...

Figure 4.10 (a) The dipolar...

Figure 4.11 (a) The dipolar...

Figure 4.12 (a) The dipolar...

Figure 4.13 The...

Figure 4.14 (a) The dipolar...

Figure 4.15 Structure of the...

Figure 4.16 (a) The quadrupolar...

Figure 4.17 (a) The quadrupolar...

Figure 4.18 The quadrupolar...

CHAPTER 05

Figure 5.1

1

H NMR spectrum of...

Figure 5.2 Subsets of experimental...

Figure 5.3 Pulse sequence diagram...

Figure 5.4 Graphic depicting how...

Figure 5.5 CTPs that are...

Figure 5.6 Graphic illustrating the...

Figure 5.7 Graphic depicting the...

Figure 5.8 Attenuation of the...

Figure 5.9 Two examples of...

CHAPTER 06

Figure 6.1 (a) A homonuclear...

Figure 6.2 Plot of the...

Figure 6.3 Plot of the...

Figure 6.4 (a) Plot of...

Figure 6.5 (a) Linear arrangement...

Figure 6.6 (a) Pulse scheme...

Figure 6.7 Plot of the...

Figure 6.8 Pulse schemes for...

Figure 6.9 The trNOE build...

Figure 6.10 The NOESY pulse...

Figure 6.11 The NOESY pulse...

Figure 6.12 The (a) ROE...

Figure 6.13 The plot of...

Figure 6.14 Schematic...

Figure 6.15 (a) NOESY...

CHAPTER 07

Figure 7.1 (a) Illustration of...

Figure 7.2 Time evolution of...

Figure 7.3 Representation of the...

Figure 7.4 Illustration of basic...

Figure 7.5 Hahn echo with...

Figure 7.6 Selected...

Figure 7.7 Double stimulated echo...

Figure 7.8 (a)

1

H MOSY...

Figure 7.9 Ion drift and...

Figure 7.10 (a) The pulse...

Figure 7.11 Ultrafast DOSY magnetization...

Figure 7.12 Pulse sequence for...

CHAPTER 08

Figure 8.1 Schematic of multiple...

Figure 8.2 Combined two-dimensional...

Figure 8.3 PANACEA (left) and...

Figure 8.4 Method for detecting...

Figure 8.5 Afterglow solid-state...

Figure 8.6 PHRONESIS pulse sequence...

Figure 8.7 A schematic of...

CHAPTER 09

Figure 9.1 (a) Schematic description...

Figure 9.2 (a) Variation of...

Figure 9.3 The origin of...

Figure 9.4 The origin of...

Figure 9.5 Isosurface plot of...

Figure 9.6 The origin...

Figure 9.7 Examples of R...

Figure 9.8 Schematic description of...

Figure 9.9 Principle of the...

Figure 9.10 (a) Pulse diagram...

Figure 9.11 (a) Principle of...

Figure 9.12 Schematic description of...

Figure 9.13 (a) Examples of...

Figure 9.14 (a) Structure of...

Figure 9.15 (a) Pictures of...

Figure 9.16 Arbitrary scaling of...

Figure 9.17 First compression...

Figure 9.18 (a) More recent...

Figure 9.19 Example of modern...

Figure 9.20 (a) Structure of...

Figure 9.21 (a) Schematic pulse...

Figure 9.22 (a) 100.3...

Figure 9.23 Selected examples of...

Figure 9.24 Selected examples of...

Figure 9.25 (a) Example of...

Figure 9.26 (a and b...

Figure 9.27 (a) Experimental...

Figure 9.28 Two examples of...

Figure 9.29 Three examples of...

Figure 9.30 (a) (top) C2...

Figure 9.31 (a) Enantiomeric...

Figure 9.32 (a) Examples of...

Figure 9.33 (a) Pulse scheme...

Figure 9.34 3D structures of...

Figure 9.35 (a) Schematic description...

Figure 9.36 Simplified principle of...

Figure 9.37 (a) Two possible...

Figure 9.38 (a) Structure of...

Figure 9.39 Structure of the...

Figure 9.40 Experimental conditions leading...

Figure 9.41 Potential structures of...

Figure 9.42 Conformational exchange of...

Figure 9.43 (a) Chemical structure...

Figure 9.44 Chemical transformation of...

Figure 9.45 (a) Structure of...

Figure 9.46 (a) Scaling the...

Figure 9.47 (a) CLIP-HSQC...

Figure 9.48 Correlation plot of...

Figure 9.49 The DFT 3D...

Figure 9.50 Stretching (left) and...

Figure 9.51 Variation of 13C...

Figure 9.52 (a) The DFT...

Figure 9.53 Variation of Q...

Figure 9.54 (a) Structure of...

Figure 9.55 Example of variations...

Figure 9.56 Stereochemical...

Figure 9.57 Structure of dictyospiromide...

Figure 9.58 Clear discrimination of...

Figure 9.59 (a) Bar charts...

Figure 9.60 Gels stretching device...

Figure 9.61 Working flow for...

Figure 9.62 (a) 92.1...

Figure 9.63 Experimental strategy for...

Figure 9.64 (a) Structure of...

Figure 9.65 Structure of briarane...

Figure 9.66 (a) Simplified flowchart...

Figure 9.67 (a) Molecular structure...

Figure 9.68 (a) Isotropic 92...

Figure 9.69 (a) Atomic numbering...

Figure 9.70 (a) Structure of...

CHAPTER 10

Figure 10.1 (a) Conventional and...

Figure 10.2 Comparison between...

Figure 10.3 (a) Slice selection...

Figure 10.4 (a) The spatial...

Figure 10.5 Pulse sequences for...

Figure 10.6 Cartoon showing the...

Figure 10.7 Two types of...

Figure 10.8 (a) Scheme representing...

Figure 10.9 (a) Example of...

Figure 10.10 Spatial apodization of...

Figure 10.11 Pulse sequence of...

Figure 10.12 (a) Example of...

Figure 10.13 (a) Standard...

Figure 10.14 Examples of UF...

Figure 10.15 (a) UF MQ...

Figure 10.16 Single-scan measurement...

Figure 10.17 Single-scan 2D...

Figure 10.18 UF 2D TOCSY...

Figure 10.19 Real-time monitoring...

Figure 10.20 Hyperpolarized UF 2D...

Figure 10.21 Example of UF...

Figure 10.22 Chemically selective...

Figure 10.23 Example of UF...

Figure 10.24 Two single-scan...

Figure 10.25 UF 2D NMR...

Figure 10.26 2D

J

-resolved...

CHAPTER 11

Figure 11.1 Pattern of one...

Figure 11.2 Superimposition of 2D...

Figure 11.3 Progression of chemical...

Figure 11.4 Two-dimensional (H...

Figure 11.5 Homonuclear 2D 1H...

Figure 11.6 Two-dimensional 1H...

Figure 11.7 Heteronuclear 1H-15N...

Figure 11.8 Two-dimensional 1H...

Figure 11.9 Solvent PRE studies...

Figure 11.10 Characterization...

Figure 11.11 Modified MEXICO...

Figure 11.12 Modified MEXICO NMR...

Figure 11.13 Schematic workflow...

Figure 11.14 Metabolomics NMR study...

Figure 11.15 NMR spectroscopy on...

Figure 11.16 Comparison of ex...

CHAPTER 12

Figure 12.1 Sketch on the...

Figure 12.2 Interactions of the...

Figure 12.3 Energy level diagram...

Figure 12.4 Experimental scheme for...

Figure 12.5 Contour plots of...

Figure 12.6 TROSY-type and...

Figure 12.7 [1H,15N]-TROSY...

Figure 12.8 Comparison of...

Figure 12.9 Two-dimensional [1H...

Figure 12.10 Exact NOEs measured...

CHAPTER 13

Figure 13.1 Principle of a...

Figure 13.2 (a) Schematic diagram...

Figure 13.3 (a) Energy level...

Figure 13.4 ZULF J-spectrum...

Figure 13.5 Schematic for acquisition...

Figure 13.6 2D ZULF spectrum...

Figure 13.7 (a) 2D ZULF...

Figure 13.8 (a) Energy level...

Figure 13.9 (a) Experimental protocol...

Figure 13.10 Demonstration of selective...

Figure 13.11 2D NMR with...

Figure 13.12 (a) Protocol of...

Figure 13.13 ZULF-TOCSY spectrum...

Figure 13.14 (a) Protocol of...

Figure 13.15 Protocol of a...

Figure 13.16 2D HF-ZULF...

Figure 13.17 Comparison of J...

CHAPTER 14

Figure 14.1 Expected hyperfine shifts...

Figure 14.2 Pulse sequences for...

Figure 14.3 (a) Experimental 2D...

Figure 14.4 (a, b) Pulse...

Figure 14.5 Overlay of a...

Figure 14.6 ...

Figure 14.7 Illustration of the...

Figure 14.8 (a) Pulse sequence...

Figure 14.9 (a, c) Pulse...

Figure 14.10 1H-detected measurement...

Figure 14.11 (a) aMAT pulse...

Figure 14.12 Pulse sequences and...

Figure 14.13 Pulse sequences for...

Figure 14.14 (a) The PASS...

Figure 14.15 (a) Structure of...

Figure 14.16 (a) MAT spectrum...

CHAPTER 15

Figure 15.1 The N-methyl...

Figure 15.2 Spectra of a...

Figure 15.3 Spectra of a...

Figure 15.4 Deciphering mechanism from...

Figure 15.5 (a) Pulse sequence...

Figure 15.6 Principle behind the...

Figure 15.7 Studying conformational...

Figure 15.8 The sign of...

Figure 15.9 a) Schematic illustration...

Figure 15.10 (a) Large R

2B

values can have

Figure 15.11 Simulated on-resonance...

APPENDIX 01

Figure A.1 Representation of the evolution of...

Figure A.2 Vector diagram representation...

Figure A.3 Evolution of the coupling...

Figure A.4 HXXH spin system:...

Figure A.5 Building blocks...

Figure A.6 180° pulses...

Figure A.7 (a) Pulse sequence...

Figure A.8 (a) Pulse sequence...

Figure A.9 (a) Schematic multiplet...

Figure A.10 HMBC spectrum...

Figure A.11 HMBC experiment...

Figure A.12 Comparison of...

Figure A.13 Comparison of...

Figure A.14 Comparison of...

Figure A.15 HSQC with...

Figure A.16 Constant-time HSQC....

Figure A.17 Comparison of...

Figure A.18 Editing delay...

Figure A.19 Expansion of the ID HSQC...

Figure A.20 Expansion of the...

Figure A.21 Folding characteristics...

Figure A.22 BIRD-HSQC pulse...

Figure A.23 Four spectra...

Figure A.24 HSQC of 15N-labeled...

Figure A.25 Operator transformations...

Figure A.26 Application...

Figure A.27 BIRD/2 pulses...

Figure A.28 (a) Amplitude-modulating...

Figure A.29 (a) Conventional...

Figure A.30 ω2 traces...

Figure A.31 Pulse sequence...

Figure A.32 1D slices from...

Figure A.33 Application of...

Figure A.34 (a) Conventional...

Figure A.35 Application of B1...

Figure A.36 Heteronuclear filters...

Figure A.37 Pulse sequence...

Figure A.38 Selective decoupling...

Figure A.39 Pulse sequences...

Figure A.40 (a) The spectrum...

Figure A.41 Bloch–Siegert...

Figure A.42 Schematic 3D...

Figure A.43 Schematic 3D...

Figure A.44 Homonuclear...

Figure A.45 (a) Part of...

Figure A.46 Assignment...

Figure A.47 Pulse sequence...

Figure A.48 Coupling topology...

Figure A.49 Various implementations...

Figure A.50 HNCO experiment derived...

Figure A.51 (a) 4D HNCAHA...

Figure A.52 (a) 3D H(N)COCA...

Figure A.53 Sequential assignment...

Figure A.54 (a) HC(C)H-COSY...

Figure A.55 Principle of the...

Figure A.56 Pulse sequence of...

Figure A.57 (a)Example for the...

Figure A.58 (a) 13C, 1H HSQC...

Figure A.59 ω1 traces through...

Figure A.60 Fitting of the spectrum...

Figure A.61 Illustration of the...

Figure A.62 Various implementations...

Figure A.63 (a) Pulse sequence...

Figure A.64 Contour plot of the...

Figure A.65 (a) Pulse sequence...

Figure A.66 (a) Schematic multiplet...

Figure A.67 (a) Cross peak of...

Figure A.68 Application of mirror...

Figure A.69 Cross peak of the...

Figure A.70 HNCA-E.COSY experiment...

Figure A.71 Comparison of the...

Figure A.72 Example of the...

Guide

Cover

Title Page

Copyright

Dedication

Table of Contents

List of Contributors

Preface

Begin Reading

Appendix A Proton-Detected Heteronuclear and Multidimensional NMR

Index

End User License Agreement

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List of Contributors

John W. BlanchardQuantum Technology Center, University of Maryland, Maryland, USA

Rolf BoelensDepartment of Chemistry, Utrecht University, The Netherlands

Dmitry BudkerHelmholtz Institut Mainz, Johannes Gutenberg-Universität Mainz, Germany

Department of Physics, University of California, Berkeley, California, USA

Muslim DvoyashkinInstitute of Chemical Technology, Leipzig University, Leipzig, Germany

Matthias ErnstPhysical Chemistry, ETH Zürich, Switzerland

Fabien FerrageLaboratoire des Biomolécules, LBM,Département de chimie, École normale supérieure,PSL University, Sorbonne Université, Paris, France

Roberto R. GilDepartment of Chemistry, Carnegie Mellon University, Pittsburgh, USA

Boris GouilleuxUniversité Paris-Saclay, laboratoire ICMMO, Orsay, France

K. IvanovInternational Tomography Center SB RAS, Novosibirsk, Russia

Alexej JerschowDepartment of Chemistry, New York University, New York, USA

Harindranath KadavathLaboratory of Physical Chemistry, ETH Zurich, Switzerland

Robert KapteinDepartment of Chemistry, Utrecht University, The Netherlands

Pawel KasprzakFaculty of Physics, University of Warsaw, Warsaw, Pasteura, Poland

Centre of New Technologies, University of Warsaw, Warsaw, Poland

Krzysztof KazimierczukFaculty of Physics, University of Warsaw, Warsaw, Poland

Alexey KiryutinInternational Tomography Center SB RAS, Novosibirsk, Russia

David KorenchanAthinoula A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital Boston, Massachusetts, USA

Michael KovermannDepartment of Chemistry, Universität Konstanz, Konstanz, Germany

Graduate School Chemical Biology KoRS-CB, Universität Konstanz, Konstanz, Germany

Philippe LesotUniversité Paris-Saclay, RMN en Milieu Orienté, France

Malcolm H. LevittDepartment of Chemistry, University of Southampton, Southampton, UK

P.K. MadhuDepartment of Chemical Sciences,Tata Institute of Fundamental Research Hyderabad, India

Vladislav OrekhovDepartment of Chemistry and Molecular Biology, University of Gothenburg, Gothenburg, Sweden

Andrew J. PellCRMN, Centre de RMN à Très Hauts Champs de Lyon Université de Lyon, Villeurbanne, France

Guido PintacudaCRMN, Centre de RMN à Très Hauts Champs de Lyon Université de Lyon, Villeurbanne, France

Roland RiekLaboratory of Physical Chemistry, ETH Zürich, Switzerland

Thomas RobinsonCRMN, Centre de RMN à Très Hauts Champs de Lyon Université de Lyon, Villeurbanne, France

Kevin J. SandersCRMN, Centre de RMN à Très Hauts Champs de Lyon Université de Lyon, Villeurbanne, France

Tobias SchneiderDepartment of Chemistry, Universität Konstanz, Konstanz, Germany

Graduate School Chemical Biology KoRS-CB, Universität Konstanz, Konstanz, Germany

Monika SchönhoffInstitute of Physical Chemistry, University of Münster, Münster, Germany

Ashok SekharMolecular Biophysics Unit, Indian Institute of Science, Bengaluru, Karnataka, India

Tobias SjolanderDepartment of Physics, University of Basel, Klingelbergstrasse 82, Basel, Switzerland

Ville-Veikko TelkkiNMR Research Unit, University of Oulu, Oulu, Finland

Nathaniel J. TraasethDepartment of Chemistry, New York University,New York, USA

Pramodh VallurupalliTata Institute of Fundamental Research Hyderabad, Hyderabad, India

Alexandra YurkovskayaInternational Tomography Center SB RAS, Novosibirsk, Russia

Yvan ZhukovInternational Tomography Center SB RAS, Novosibirsk, Russia

Preface

NMR spectroscopy has grown tremendously since its inception in the 1940’s when NMR signals were first recorded by Felix Bloch and Edward Purcell. The developments in the field have been fuelled by advancements in various fields, such as, superconducting magnets, spectrometer hardware and software, isotope labelling, and the theory behind understanding the various NMR experiments. The applications have been vast from small molecules, materials, structure and dynamics of biomolecular complexes to medical applications of NMR and MRI. NMR has become a standard characterisation tool in many Chemistry laboratories and widely used in various industrial applications, for example, in the pharmaceutical industry.

One of the most important milestones in the time line of NMR is the development of 2D NMR methods, which provide an elegant way to get around the problem of “spectral crowding” and provide versatile and highly detailed information about molecular structure and dynamics through correlation spectroscopy. This book attempts to give a rigorous account of the basic concepts in 2D NMR methods, both in terms of theory and applications. Whilst the list of content covered here is not exhaustive, we believe it certainly gives the reader a good flair and comprehension of the field and necessary background to grasp more advanced topics, including higher-dimensional NMR methods.

The book has Chapters on the basics of NMR, analysis of pulse sequences, zero-to ultralow-field NMR, NMR methods in biomolecules in solution state, methods on data processing, pushing the frontiers of resolution, and speeding up of 2D data acquisition, diffusion spectroscopy, NMR of anisotropic and paramagnetic systems, and chemical exchange. We have also included a Chapter as Appendix that we hope will complement many of the other Chapters, and in particular Chapters 11 and 12. The Chapters are all written by leading researchers in the field who have also contributed significantly to the topic of their respective Chapter. We hope that this book will be useful to established as well as beginning researchers who want to explore the benefits of NMR spectroscopy and new areas of its multidimensional facets.

We thank all the authors who have contributed to this book and also in a timely fashion. The significant amount of care that has gone into each Chapter merits attention and deep acknowledgement. We also thank the people at Wiley for putting up this book very nicely, some of them being, Jenny Cossham, Sarah Higginbotham, Elke Morice-Atkinson, and Richa John.

Finally, this book also has a tribute section written for Kostya (K. Ivanov) by some of the authors. The concept of this book was Kostya’s idea, however, unfortunately he passed away on March 5, 2021, at the young age of 44. A great scientist, a fantastic human being, we dedicate this book to his memories.

1 Basics of Two-dimensional NMR

Malcolm H. Levitt

Department of Chemistry, University of Southampton, SO17 1BJ, Southampton, UK

1.1 Introduction

1.1.1 Time-domain NMR

The introduction of pulse-Fourier transform (pulse-FT) NMR in 1966 by Ernst and Anderson [1] represented a paradigm shift, not only in nuclear magnetic resonance but also in many other forms of spectroscopy. Prior to this seminal experiment, there were two forms of NMR, which were generally viewed as being quite distinct and practiced mainly by chemists on the one hand, and physicists on the other.

Chemical applications of NMR spectroscopy used a “continuous-wave” (cw) method, in which chemical shifts and spin-spin couplings were probed, either by (i) varying the frequency of applied radiofrequency irradiation and detecting a change in the nuclear magnetic response when the applied frequency matches a nuclear energy level spacing, or (ii) by applying radiofrequency irradiation of fixed frequency and varying the applied magnetic field while monitoring the response. For technical reasons, the latter method (fixed frequency, variable field) was easier to perform and more common. A residue of this historical method still persists in the common nomenclature of modern NMR, where the terms “high field” and “low field” are still often used (despite a jarring logical inconsistency) to characterize nuclei in electron environments, which are relatively weakly shielded from the external magnetic field (low field!), and for those that are relatively strongly shielded from the external magnetic field (high field!).

In 1966, many important time-domain NMR experiments and phenonema also existed, but they were largely developed and used by physicists. These include the spin echo [2], the modulation of spin echoes by spin-spin couplings [3], the use of spin echoes to probe molecular diffusion in a field gradient [3], the development of multiple spin echoes [3, 4], and Hartmann-Hahn cross-polarization between different nuclear species in solids [5].

The introduction of pulse-FT NMR established a permanent link between the time-domain “physics” phenomena and the continuous-wave “chemistry” procedures.

The perturbation of nuclear spins by a strong, short, resonant radiofrequency pulse elicits an extended time-domain electrical response termed a free-induction decay, following Bloch [6]. Ernst realized that Fourier transformation of the time-domain NMR signal s(t) yields a frequency-domain function , which under certain assumptions, is identical to the NMR spectrum generated far more slowly by a continuous-wave frequency-domain NMR experiment:

(1.1)

In one stroke, Ernst unified time-domain NMR, with its central object the free-induction decay s(t), and frequency-domain NMR, with its central object the NMR spectrum at fixed field . Furthermore, this groundbreaking theoretical unification was accompanied by a large increase in signal-to-noise ratio, of great practical importance. The timing of this advance was exceptionally favorable. Practical application of the Fourier transform requires numerical computation. Sufficiently powerful computational hardware was becoming widely available, and furthermore, a fast algorithm for numerical Fourier transformation had just been developed [7]. As they say, the rest is history. NMR was revolutionized, with a huge impact on many other sciences, as recognized by Ernst’s Nobel prize in 1991. It was probably one of the least contentious Nobel prizes in history.

1.1.2 Hans Primas and the “Correlation Function of the Spectrum”

The seminal contribution of Ernst and Anderson is well known to most NMR spectroscopists. Less well known is that an Equation very similar to Equation 1.1 had been published by a close colleague of Ernst at ETH-Zürich a few years earlier. This colleague was Hans Primas, who enjoyed a particularly remarkable career. Primas was a self-taught genius who rose to become a Professor of Physical and Theoretical Chemistry at the ETH-Zürich and an authority in quantum theory and the philosophy of science, despite the absence of secondary school education or a university degree of any kind [8]. The 1963 paper by Banwell and Primas [9] introduces a quantity called the “correlation function of the spectrum” and given by (omitting an unimportant normalization factor). Since the Fourier transform is reversible, Equations 1.1 and 1.2 are entirely equivalent (overlooking a technical difference in integration limits). Banwell and Primas introduced the term K(t) as an object of theoretical interest and presumably missed its significance as being identical to the free-induction decay s(t) generated by a nuclear spin system subjected to pulse excitation – and was certainly not aware of its practical significance.

(1.2)

I mention Primas’ work not to disparage Ernst’s achievement – on the contrary, I feel that the work by Primas and colleagues helps us understand better the historical provenance of Ernst’s insight. Ernst’s skill was in marrying the contemporary thinking at the ETH-Zürich with the highly practical motivation of Anderson at the Varian Corporation, with whom Ernst developed the concept and application of FT-NMR [1].

Remarkably, Equation 1.2 is far from being the only important insight in reference [9]. This paper also introduces the concepts of superoperators and Liouville space and also anticipates Cartesian product operator techniques, brought to fruition by Sørensen et al. [10] and other groups nearly 20 years later (see Chapter 3).

1.2 Spin Dynamics

1.2.1 Density Operator

Time-domain Fourier NMR is most conveniently analyzed by the modern operator description of NMR, as expounded in the textbook by Ernst, Bodenhausen, and Wokaun [11].

The quantum state of the nuclear spins in the sample is described by a spin density operator ρ, defined:

(1.3)

where is the spin state of an individual spin system and the overbar indicates an ensemble average.

1.2.2 Spin Hamiltonian

In many cases, the spin Hamiltonian may be written as the sum of a coherent term, which is identical for all members of the spin ensemble, and a fluctuating term, which is strongly time-independent, has a different value for different ensemble members, and a zero average over the ensemble:

(1.4)

The coherent part of the spin Hamiltonian contains terms, which are responsible for major features of the NMR spectrum such as chemical shifts, spin-spin couplings, and residual dipolar couplings in anisotropic phase [12].

In general, the coherent spin Hamiltonian possesses a set of eigenstates and eigenvalues , satisfying the following eigenequation:

(1.5)

When multiplied by , the eigenvalues correspond to the quantum energy levels of the spin system.

The fluctuating part of the spin Hamiltonian is responsible for dissipative phenomena, including relaxation processes, which bring the spin system back to equilibrium with the molecular environment as well as important dissipative effects such as the nuclear Overhauser effect, which underpins many applications of NMR to structural biology [13].

1.2.3 Liouville Space

The principles of time-domain NMR are expressed most compactly in Liouville space, meaning the space of all orthogonal operators for the spin system [11, 14, 15]. The dimension of this space is .

A convenient basis for Liouville space is given by all possible products of the kets and bras of the coherent Hamiltonian eigenstates, each operator having the form

(1.6)

The operators with r = s are known as population operators:

(1.7)

The operators Crs with r ≠ s are known as coherence operators.

In this representation, the density operator may be written as a -dimensional vector, termed a Liouville ket, as follows:

(1.8)

The diagonal elements of the density operator, called populations, are defined by

(1.9)

where the Liouville bracket is defined [11, 14]:

(1.10)

The off-diagonal elements of the density operator, called coherences, are defined by

(1.11)

where, by symmetry, .

In a high magnetic field and in the absence of additional applied fields, the secular approximation may be made. This implies that all components of the spin Hamiltonian are neglected if they do not commute with the total angular momentum operator along the static magnetic field. In these circumstances the coherence operators obey the following eigenequation:

(1.12)

where is the commutation superoperator of the total angular momentum operator in the field direction (by convention the z-axis), defined by the following property:

(1.13)

The eigenvalues prs in Equation 1.12 are real integers called coherence orders. Coherence orders play a prominent role in the principles of 2D spectroscopy (see Chapter 5).

1.2.4 Liouvillian

The equation of motion of the density operator is a first-order differential equation

(1.14)

where the superoperator is called the Liouvillian. Under suitable conditions the Liouvillian may be written as the sum of two terms:

(1.15)

where is the commutation superoperator of the coherent Hamiltonian, defined as follows:

(1.16)

and is the relaxation superoperator, representing the dissipative behavior of the spin system, generated by the fluctuating Hamiltonian . If necessary, the effects of chemical exchange, diffusion, molecular transport, and mechanical motion may also be included in the Liouvillian by using an extended Liouville space (see Chapters 7 and 15) [11, 16].

The construction of the relaxation superoperator from the correlation function of the fluctuating Hamiltonian occupies a large body of theory by itself, which is beyond the scope of this chapter (see Chapter 4). Lindbladian techniques may be used to construct in such a way that the approach to thermal equilibrium is treated correctly, even for spin systems, which are far from equilibrium [15, 17].

In those cases that do not require an extended Liouville space, the Liouvillian superoperator may be represented by a matrix.

In general, the Liouvillian has right eigenkets or right eigenoperators