Computer Assistance in the Analysis of High-Resolution NMR Spectra

Nuclear magnetic resonance spectroscopy, which has evolved only within the last 20 years, has become one of the very important tools in chemistry and physics. The literature on its theory and application has grown immensely and a comprehensive and adequate treatment ofall branches by one author, or even by several, becomes increasingly difficult. by experts workinginvarious This seriesis planned to present articles written fields of nuclear magnetic resonance spectroscopy, and will contain review articles as well as progress reports and original work. Its main aim, however, is to fill a gap, existing in literature, by publishing articles written by specialists, which take the reader from the introductory stage to the latest development in the field. The editors are grateful to the authors for the time and effort spent in writing the articles, and for their invaluable cooperation. The Editors Computer Assistance in the Analysis of High-Resolution NMR Spectra P. DIEHL and H. KELLERHALS Departmentof Physics, University ofBasle, Switzerland E. LUSTIG Food and Drug Administration, Washington, D.C., U.S.A.

Foreword.- Notation and Symbols.- A. The Calculation of an NMR Spectrum.- A.1. Some Postulates of Quantum Mechanics.- A.2. Quantum Mechanics of a Spin System.- A.2.1. The High-Resolution NMR Hamiltonian Operator.- A.2.2. Spin Functions.- A.3. Representation of Operators.- A.3.1. Representation of an Operator Ô by a Matrix.- A.3.2. Effect of Spin Operators Î on Spin Functions.- A.3.3. Representation of the Hamiltonian Operator ?.- A.3.4. Some Properties of the Matrix ?.- A.3.5. Representation of the Lowering Operator Î_ 14.- A.3.6. Expansion of Wave Functions.- A.3.7. Basis Transformations.- A.4. Solution of the Schrödinger Equation.- A.4.1. Representation of the Schrödinger Equation by Vectors and Matrices.- A.4.2. Example: The Two-Spin System.- A.5. The Schrödinger Equation as a Basis Transformation.- A.6. Calculation of a Normal Spectrum.- A.6.1. Transition Frequencies.- A.6.2. Transition Intensities.- A.6.3. The Spectrum.- A.7. Breakdown of Hamiltonian Matrices.- A.7.1. Breakdown with Simple Product Spin Functions: Weak Coupling.- A.7.2. Breakdown with Linear Combinations of Product Spin Functions.- B. Analysis of Spectra, Unaided by a Computer.- B.1. “Hand” Methods of Analysis.- B.2. Determination of Starting Parameters for Computer-Aided Analysis.- C. Basic Computer Methods.- C.1. Simulation Programs.- C.2. Basic Iterative Programs.- C.2.1. The Swalen=Reilly Method.- C.2.2. The Hoffman Method.- C.2.3. The Granger Method.- C.2.4. The Castellano=Bothner-By/Braillon Method (Programs LAOCN3 and SUPERMAN).- C.2.5. The Arata=Shimizu=Fujiwara Method.- C. 3. Auxiliary Methods.- C.3.1. Spectral Decomposition (Program DECOMP).- C.3.2. Spectral Assignment (Program ASSIGN).- D. Developments of the Basic Computer Programs.- D.1. Developments of the Swalen=Reilly Programs.- D.1.1. Magnetic Equivalence Factoring.- D.1.2. “Missing” Energy Levels (Program NMREN2). A Unified Swalen=Reilly Method (Program NMRENIT).- D.1.3. Complete Analysis by the Lusebrink Method (Program DENOUE).- D. 2. Developments of the Castellano―Bothner-By/Braillon Method.- D.2.1. Magnetic Equivalence Factoring.- D.2.2. Weak Coupling, Symmetry Factoring. Weighting of Observables and Printout of Derivatives.- D.2.3. Analysis of Spectra of Partially Oriented Molecules.- (Programs LAOCOONOR. LAOCOONORW and LEQUOR).- E. Parameter Errors.- E.1. Description of Errors.- E.2. Significance of Covariances.- E.3. Incomplete Description of Errors.- E.4. Calculation of Errors in the Swalen=Reilly Method as Modified by FERGUSON and MARQUARDT.- E. 5. Calculation of Errors in the Castellano=Bothner-By Method and its Modifications.- F. Uniqueness of Solutions.- F.1. The Number of ELD’s Derived from an Experimental Spectrum.- F.2. The Number of Parameter Sets Derived from an ELD.- F.3. The Number of Assigned Transitions Necessary for a Solution.- G. Comparison of the Iterative Methods.- G.1. Mathematical Structure and Convergence.- G.2. Computational Effort per Iteration Cycle.- G.3. Prior Knowledge of Parameters.- G.4. Conclusions.- H. Computer Assistance in NMR Studies of Rate Processes.- H.1. The Calculation of Spectra of Spin Systems Undergoing Rate Processes.- H.1.1. The Modified Bloch Equations.- H.1.2. The Density Matrix Treatment of Exchange.- H.1.3. The Unified Theory of Exchange.- H. 2. Computer Programs for Spectra of Spin Systems Undergoing Rate Processes.- H.2.1. Early Programs.- H.2.2. The Programs DNMR.- H.2.3. Computer Programs for the Study of Quadrupolar Relaxation.- I. Double Resonance.- I. 1. The Effects of Double Irradiation on NMR Spectra.- I.2. Simulation of Double Resonance Spectra (Program NMDRS).- V. Vectors and Matrices.- V.1. Vectors.- V.2. Matrices.- V.3. Operations with Matrices.- V.4. Some Rules of Matrix Algebra.- W. Diagonalization of Symmetrical Matrices.- W.1. Orthogonal Transformations.- W.2. Diagonalization of Symmetric Matrices.- W.3. The Jacobi Method.- W.4. Diagonalization via the Tridiagonal Form.- X. Least-Squares Fitting.- X.1. The Fitting Problem.- X.2. Linearization of the Fitting Problem.- X.3. The Normal Equations.- X.4. Iteration.- Y. Parameter Errors from Least-Squares Fits.- Y.1. Some Concepts from Statistics.- Y.2. The Variance-Covariance Matrix and Experimental Errors.- Y.3. Propagation of Errors.- Y.3.1. Errors of Parameters Obtained by Least-Squares Methods, Determined with Approximate Derivatives.- Y.3.2. Rigorous Calculation of the Derivatives for the Determination of Parameter Errors.- Y.4. Calculation of Linear Combinations of Parameters, Leading to Extreme Errors.- Z. The Density Matrix.- Z.1. Basic Concepts.- Z.2. Calculation of Expectation Values of an Operator.- Z.3. The Equation of Motion for ?.- Acknowledgments.- References.