Classical and Quantum Molecular Dynamics in NMR Spectra (eBook)

Slawomir Szymanski is Professor at the Institute of Organic Chemistry, Warsaw, Poland.  His research centres on NMR spectroscopy, namely molecular structure and dynamics in condensed phases studied by NMR spectroscopy methods, and quantum mechanical effects in the stochastic dynamics of hindered molecular rotators.  He is a recipient of the Award of the Mathematical, Physical and Chemical Sciences Division of the Polish Academy of Sciences.   Piotr Bernatowicz is Head of the NMR Laboratory in the Institute of Physical Chemistry, Polish Academy of Sciences. 

Preface 5
Contents 7
1 Introduction 12
References 16
2 Principles of NMR Spectroscopy 17
2.1 Nuclear Magnetic Dipole Moment in an External Magnetic Field 17
2.2 The Statistical Operator of One-Spin System 22
2.3 A Single-Pulse Experiment of PFT NMR Spectroscopy in the Vector Model 23
2.3.1 The Radiofrequency Pulse in the Rotating Frame 24
2.3.2 The FID Signal 29
2.3.3 The Quadrature Detection of the FID Signal 30
2.3.4 The Spectrum 32
2.3.5 Summary 35
2.4 Coupled Spin Systems: NMR Spectra Beyond the Vector Model 36
2.4.1 Multi-spin Systems 36
2.4.2 Spin Hamiltonian of Coupled Multi-spin Systems 38
2.4.3 The Spectrum of Coupled Multi-spin System. Part One 41
2.4.4 The Notion of Quantum Coherence 44
2.4.5 The Spectrum of Coupled Multi-spin System. Part Two 46
2.4.6 Weakly Coupled Systems 50
2.4.7 Molecular Symmetry in Spectra 52
2.4.8 Magnetic Equivalence 59
2.5 Introduction to Liouville Space Formalism 62
2.5.1 One-Spin Systems 62
2.5.2 Coupled Multi-spin Systems 66
2.5.3 Operator Product Bases 68
2.6 Remarks on the Solid State Systems 69
2.6.1 Secular and Nonsecular Spin Interactions in Solids. CSA Tensor 70
2.6.2 Secular Part of CSA Tensor. Angular Dependence 71
2.6.3 Nuclei with Electric Quadrupole Moments 75
2.6.4 Dipole Interactions 77
2.6.5 Spin Systems with Different Anisotropic Interactions 79
2.6.6 Single-Crystal Spectra 80
2.6.7 Example of Bandshape Modeling in Wide-Line Spectra of Solids 81
2.6.8 Wide-Line Spectra of Powders 82
2.6.9 Magic Angle Spinning Spectra of Powders 84
2.7 Spin Echo 87
2.8 Two Dimensional Spectra 91
References 92
3 NMR Spectroscopy and Molecular Dynamics - An Outlook 94
3.1 Nuclear Spin Relaxation and Molecular Motion. Introductory Remarks 94
3.1.1 Semiclassical Approach 95
3.1.2 Quantum Mechanical Approach 103
3.1.3 Justification of the Bloch Equations 109
3.1.4 Explicit Evaluation of Relaxation Rates for CSA Interactions 111
3.1.5 Nuclear Spin Interactions Leading to Relaxation. Temperature Effects 114
3.1.6 More on Dipolar Relaxation. Nuclear Overhauser Effect 118
3.2 Dynamic Line Shape Effects in the Vector Model 119
3.2.1 Stochastic Picture 122
3.2.2 Heuristic Approach 126
3.2.3 The FID Signal and the Line Shape Equation 128
3.2.4 The Pulse Offset Effects 136
3.2.5 DNMR Spectra of Solids and the Vector Model 138
3.2.6 Selective Population Inversion 140
3.2.7 EXSY - A 2D Experiment 144
References 151
4 Nuclear Spin Relaxation Effects in NMR Spectra 153
4.1 Theory 153
4.1.1 Irreducible Spherical Tensor Description of Anisotropic Interactions 154
4.1.2 Derivation of BWR Relaxation Matrix 160
4.1.3 Heteronuclear Systems 166
4.1.4 General Properties of the BWR Relaxation Matrix 168
4.2 Molecular Tumbling in Isotropic Fluids 171
4.2.1 Angular Correlation Functions in Rotational Diffusion Model 172
4.2.2 BWR Relaxation Matrix in Isotropic Systems 177
4.2.3 Local Dynamics. Other Models of Molecular Motion 178
4.3 Nuclear Permutation and Magnetic Equivalence Symmetries 181
4.3.1 Permutation Symmetry in Liouville Space. Macroscopic Symmetry 182
4.3.2 Microscopic Symmetry 187
4.3.3 Violation of the Magnetic Equivalence Symmetry 191
4.4 Relaxation Effects in Spectral Line Shapes 195
4.4.1 A Survey of Line Shape Effects 195
4.4.2 Numerical Calculations of Spectra With Relaxation Effects 198
4.5 Nuclear Spin Relaxation in Solids 200
References 200
5 Discrete Molecular Dynamics and NMR Line Shape Effects. Intramolecular Exchange 202
5.1 Basic Notions 202
5.1.1 Monte Carlo Approach 204
5.1.2 DNMR Equation in Liouville Space 205
5.1.3 Degenerate Rearrangements 209
5.2 DNMR Theory for Intramolecular Rearrangements of Symmetric Molecules 211
5.2.1 Molecular Symmetries as Feasible Symmetries. Topomers as Cosets of Feasible Groups 213
5.2.2 Exchange Networks in Group Theory Language 217
5.2.3 Macroscopic Conservation of Symmetry in Intramolecular Dynamic Equilibria 219
5.2.4 DNMR Line Shape Equation for Symmetric Systems 223
5.2.5 DNMR Line Shape Equation in Symmetry Adapted Liouville Bases 234
5.2.6 Microscopic Conservation of Symmetry 236
5.2.7 Magnetic Equivalence and Exchange 243
5.3 Quantitative Interpretation of DNMR Spectra. Methodological Aspects 244
5.3.1 Proton Exchange in a Corrole Molecule. Temperature-Dependent Chemical Shifts 245
5.3.2 Conformational Equilibrium in [3.3]-Paracyclophane 248
5.3.3 Inversions of Aliphatic Bridges in [4.3]paracyclophane 250
5.3.4 General Remarks 253
References 254
6 Discrete Molecular Dynamics and NMR Line Shape Effects. General Exchange 256
6.1 Problem Outline 256
6.2 Intermolecular Rearrangements in the Vector Model 257
6.3 Density Matrix Description of Intermolecular Equilibria 261
6.3.1 Retrospective Picture of Intermolecular Equilibria 262
6.3.2 Reference Molecules and Exchange Superoperators 265
6.3.3 Bilinear Equations of Motion for Exchanging Systems 271
6.3.4 Macroscopic Symmetry in Intermolecular Processes 273
6.3.5 Linear Approximation 280
6.3.6 General Case of Exchange in Linear Approximation 283
6.4 Exchange of Fragments 284
6.4.1 Additional Conventions in Notation 285
6.4.2 Exchange Superoperators in Bilinear Equations of Motion 288
6.4.3 Exchange Superoperators in Linear Equations of Motion 294
6.5 Examples 295
6.5.1 Proton Exchange in Methanol 295
6.5.2 Proton Exchange in an Ammonium Salt. Symmetry-Equivalent Reactions 302
6.5.3 Self-Exchange with No Unique Fragmentation Pattern 308
6.5.4 Degenerate Exchange with No Unique Fragmentation Pattern 309
References 310
7 Rotational Tunneling in Stick NMR Spectra of Solids 311
7.1 Introductory Remarks 311
7.2 The Effective Spin Hamiltonian 312
7.2.1 Hindered Rotators in Solids 313
7.2.2 The Librational Hamiltonian in the Pocket Basis 316
7.2.3 Inclusion of Spin-Dependent Interactions 323
7.3 Tunneling Splittings of the Torsional Bands 327
7.4 A Glimpse into Temperature Effects 329
7.5 Rotational Tunneling in Experimental NMR Spectra of Solids 330
References 336
8 Quantum Molecular Dynamics in Liquid-Phase Stick NMR Spectra 338
8.1 The Symmetrization Postulate in Liquid-Phase NMR. Introductory Remarks 338
8.2 Transition Metal Polyhydrides 339
8.2.1 Experimental Evidences 339
8.2.2 The Effective Spin Hamiltonian for the Di- and Trihydrides 341
8.2.3 Temperature Effects on Exchange Couplings 348
8.3 Strongly Hindered Methyl Groups 349
References 352
9 Quantum Mechanical Rate Processes in NMR Spectra 354
9.1 Three-Fold Rotators in Solids 354
9.1.1 An Outline of the DQR Theory 355
9.1.2 Temperature Effects on the DQR Quantities 366
9.1.3 DQR Effects in Experimental Solid State DNMR Spectra 371
9.2 DQR Theory for Planar n-Fold Rotators 378
9.3 DQR Effects in Liquid Phase Spectra 382
9.3.1 Discrimination Between Similar Line-Shape Models 383
9.3.2 DQR Effects in Methyltriptycene Derivatives 385
9.4 Temperature Effects in the Spectra of the Metal Polyhydride Complexes 389
9.5 Proton-Transfer Reactions 393
References 393
10 Correction to: Classical and Quantum Molecular Dynamics in NMR Spectra 395
Correction to: S. Szyma?ski and P. Bernatowicz, Classical and Quantum Molecular Dynamics in NMR Spectra, https://doi.org/10.1007/978-3-319-90781-9 395
A Selected Properties of Matrices 396
A.1 Similarity Transformations of Matrices. Diagonalization 396
A.2 Matrix Functions of Matrices 397
A.3 Kronecker Multiplication of Matrices 398
A.4 Block Inversion of Matrices 400
B Derivation of a General DNMR Lineshape Equation 401
C Nuclear Permutation Symmetry in NMR Spectra 403
C.1 Symmetry Selection Rules for Matrix Elements of Operators 403
C.2 Decomposition of the Totally Symmetric Group Superprojector into Symmetry-Parentage Superprojectors 404
C.3 Double Cosets 405
C.4 Double Cosets and Projection Superoperators 405