Concepts in Magnetic Resonance - up to May 10, 2010

Concepts in Magnetic Resonance Part A
Volume 36A Issue 2, Pages 84 - 126
Published Online: 21 Apr 2010

Analysis of electric field gradient tensors at quadrupolar nuclei in common structural motifs
Jochen Autschbach, Shaohui Zheng, Robert W. Schurko

Keywords
electric field gradients • quadrupolar coupling • quantum chemistry • localized orbitals

Abstract
This article is concerned with the analysis of electric field gradients (EFGs) using first-principles theory along with model calculations. Simple atomic orbital (AO )models for the EFG are developed in the spirit of the Townes-Dailey (TD) analysis and applied to various sets of spn hybrid orbitals and to atomic d orbital shells. These AO models are then combined with modern analysis methods rooted in first principles theory which provide accurate localized molecular orbital contributions to the EFG. It is shown by density functional computations how such analyses of the EFG for a variety of typical structural motifs can provide an intuitive way of understanding the chemical origin of the magnitude and the sign of EFG tensors at atomic nuclei, as well as of their orientation with respect to the molecular coordinate frame. The utility of graphical visualizations of EFG tensors is also emphasized. The systems that are investigated span the range from very small molecules (carbon and sulfur EFGs in CO, CS, OCS) to small- and medium-sized molecules (nitrogen and aluminum EFGs in ammonia, methyl-cyanide and -isocyanide, aluminum AlX3 model systems and various alumino-organic systems), to the metal atom field gradient in transition metal complexes with Ru and Nb and a variety of ligands. © 2010 Wiley Periodicals, Inc. Concepts Magn Reson Part A 36A: 84-126, 2010.
Received: 9 December 2009; Revised: 15 February 2010; Accepted: 2 March 2010

Digital Object Identifier (DOI)
10.1002/cmr.a.20155

--

Concepts in Magnetic Resonance Part A
Volume 36A Issue 2, Pages 49 - 83
Published Online: 21 Apr 2010

The product operator formalism: A physical and graphical interpretation
David P. Goldenberg *

Keywords
product operators • scalar coupling • quantum correlations • vector diagrams

Abstract
The product-operator formalism is the most commonly used tool for describing and designing multidimensional NMR experiments. Despite its relative simplicity and sound theoretical underpinnings, however, students and practitioners often find it difficult to relate the mathematical manipulations to a physical picture. In an effort to address this pedagogical challenge, the present article begins with a quantum-mechanical treatment of pure populations of scalar-coupled spin pairs, rather than the equilibrium population of spin pairs in different quantum states, which is the usual starting point for treatments based on the density matrix and product operators. In the context of pure populations, the product operators are shown to represent quantum correlations between the nuclei in individual molecules, and a new variation on the classical vector diagram is introduced to represent these correlations. The treatment is extended to mixed populations that begin at thermal equilibrium, and the density matrix is introduced as an efficient means of carrying out quantum calculations for a mixed population. Finally, it is shown that the operators for observable magnetization and correlations can be used as a basis set for the density matrix, providing the formal justification for the widely used rules of the product-operator treatment. Throughout the discussion, the vector diagrams are used to help maintain a connection between the mathematics and the sometimes subtle physical principles. An electronic supplement created with the Mathematica computer program is used to provide additional mathematical details and the means to carry out further calculations. © 2010 Wiley Periodicals, Inc. Concepts Magn Reson Part A 36A: 49-83, 2010.
Received: 19 June 2009; Revised: 5 March 2010; Accepted: 5 March 2010

Digital Object Identifier (DOI)
10.1002/cmr.a.20156