Concepts in Magnetic Resonance Part A

Vol: 28A, Issue: 6, November 2006

pp. 369 - 383

Title: Operator formalisms: An overview

Author: Bain, Alex D.a

Affiliations: a. Department of Chemistry, McMaster University, 1280 Main Street West, Hamilton, Ontario, Canada L8S 4M1

Keywords: operators; density matrix theory; spherical tensors; Cartesian tensors; single transition operators; quadrupolar echo; pulse sequences

Abstract (English):

Operator formalisms are mathematical recipes for simplifying the manipulations of the density matrix. Because the density matrix is a vital tool in describing and analyzing magnetic resonance experiments, many approaches have been developed. This article gives an overview of a number of different methods: spherical tensors, fictitious spin-1/2, single-transition operators, product operators, superspin methods, and others. In principle, they all must give the same answer, because an exact description of magnetic resonance phenomena is usually within reach. The choice of the formalism for the user, therefore, depends on various personal decisions. Among these decisions is the choice between spherical and Cartesian tensors, between Hilbert space and Liouville space, between commutators and matrix elements, and so on. We do not go into the details of any of the formalisms but rather try to compare their approaches at a fairly general level. The quadrupolar echo pulse sequence is used as an example of the application of the formalisms. The aim of this overview is to give readers enough of a picture so that they can make an intelligent choice for themselves. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part A 28A: 369–383, 2006

Concepts in Magnetic Resonance Part A

Vol: 28A, Issue: 6, November 2006

pp. 384 - 409

Title: On the physical interpretation of density operators at the atomic scale: A thorough analysis of some simple cases

Author: Paniagua, Juan C.a

Affiliations: a. Departament de Química Física, Universitat de Barcelona, C. Martí i Franquès 1, E‐08028 Barcelona, Spain

Keywords: density operator; density matrix; coherences; phase coherence; physical interpretation

Abstract (English):

The physical meaning of the concepts underlying the density operator formalism are analyzed from different points of view. In particular, the diversity of ways of expressing the density operator of the simplest NMR sample is exploited to show that there are many molecular-scale physical pictures compatible with each macroscopic state, some of them being more useful than others for specific aims. Those corresponding to diagonal representations of the density matrix conform closely to classical-like pictures, which allow us to ignore the subtle effects of quantum interferences that are implicit in the concept of coherence. A widespread biconical picture that does not rely on a sound physical basis is shown to be quantitatively valid provided that ad hoc populations are chosen for the involved quantum states. The interpretation of the coefficients of the density operator expansion in terms of observable-related basis elements is discussed to show that the identification of these with the corresponding physical properties can be misleading in certain cases. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part A 28A: 384–409, 2006

Concepts in Magnetic Resonance Part A

Vol: 28A, Issue: 5, September 2006

pp. 307 - 320

Title: A simple analytical model to describe dynamic magic-angle spinning experiments

Author: Hirschinger, Jérômea

Affiliations: a. Institut de Chimie, UMR 7177 CNRS, Université Louis Pasteur, BP 296, 67008 Strasbourg Cedex, France

Keywords: memory-function approach; Anderson-Weiss theory; magic-angle spinning; molecular motion; rotational-echo double-resonance (REDOR); solid-state NMR

Abstract (English):

A simple analytical method based on the memory-function approach and the Anderson-Weiss theory is presented for studying in a uniform way the spin dynamics in several NMR experiments performed under the conditions of magic-angle spinning (MAS). A series of 1H MAS spectra of adamantane serve as an example for a sample with homogeneous line broadening. A good agreement between experimental and calculated integral spinning sideband intensities for a spinning frequency ranging from 5 to 33 kHz is obtained using a sixth-order approximation of the theory. Moreover, this model, which is able to describe the NMR spectra in the presence of a rotationally diffusive isotropic and anisotropic Gaussian-Markoff process, allows the treatment of the combined effects of sample spinning and molecular motion. Finally, the application of MAS recoupling methods in the intermediate motional regime is examined by using the concept of refocusing pulses. It is shown that recoupling experiments such as rotational-echo double-resonance (REDOR) can still be applied in the presence of anisotropic molecular motion. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part A, 28A: 307–320, 2006

Concepts in Magnetic Resonance Part A

Vol: 28A, Issue: 5, September 2006

pp. 347 - 368

Title: The embedded ion method: A new approach to the electrostatic description of crystal lattice effects in chemical shielding calculations

Author: Stueber, Dirka

Affiliations: a. Department of Chemistry, Washington University, St. Louis, Missouri 60607‐7061

Keywords: NMR; nuclear magnetic shielding tensor; quantum mechanical shielding calculations; embedded ion method (EIM); intermolecular interactions; Ewald method; point charge arrays; electrostatic; crystal lattice effects

Abstract (English):

The nuclear magnetic shielding anisotropy of NMR active nuclei is highly sensitive to the nuclear electronic environment. Hence, measurements of the nuclear magnetic shielding anisotropy represent a powerful tool in the elucidation of molecular structure for a wide variety of materials. Quantum mechanical ab initio nuclear magnetic shielding calculations effectively complement the experimental NMR data by revealing additional structural information. The accuracy and capacity of these calculations has been improved considerably in recent years. However, the inherent problem of the limitation in the size of the systems that may be studied due to the relatively demanding computational requirements largely remains. Accordingly, ab initio shielding calculations have been performed predominantly on isolated molecules, neglecting the molecular environment. This approach is sufficient for neutral nonpolar systems, but leads to serious errors in the shielding calculations on polar and ionic systems. Conducting ab initio shielding calculations on clusters of molecules (i.e., including the nearest neighbor interactions) has improved the accuracy of the calculations in many cases. Other methods of simulating crystal lattice effects in shielding calculations that have been developed include the electrostatic representation of the crystal lattice using point charge arrays, full ab initio methods, ab initio methods under periodic boundary conditions, and hybrid ab initio/molecular dynamics methods. The embedded ion method (EIM) discussed here follows the electrostatic approach. The method mimics the intermolecular and interionic interactions experienced by a subject molecule or cluster in a given crystal in quantum mechanical shielding calculations with a large finite, periodic, and self-consistent array of point charges. The point charge arrays in the EIM are generated using the Ewald summation method and embed the molecule or ion of interest for which the ab initio shielding calculations are performed. The accuracy with which the EIM reproduces experimental nuclear magnetic shift tensor principal values, the sensitivity of the EIM to the parameters defining the point charge arrays, as well as the strengths and limitations of the EIM in comparison with other methods that include crystal lattice effects in chemical shielding calculations, are presented. © 2006 Wiley Periodicals, Inc. Concepts Magn Reson Part A 28A: 347–368, 2006

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