Cartesian and Spherical Tensors in NMR Hamiltonians

Abstract

NMR Hamiltonians are anisotropic due to their orientation dependence with respect to the strong, static magnetic field. They are best represented by product of two rank‐2 tensors: one is the space‐part tensor T and the other is the spin‐part tensor A. We reformulated the dot product of Cartesian tensors and the dyadic product of spherical tensors in NMR Hamiltonian as the double contraction of these two tensors. As the double contract has two definitions (double inner product and double outer product of two rank‐2 tensors), there are two sets of spherical tensor components in terms of Cartesian tensor components for any rank‐2 tensor, two Cartesian Hamiltonians, and two spherical Hamiltonians. We succeeded in determining the spherical components of tensors A and T that verify the Cartesian Hamiltonian defined by the double inner product of two rank‐2 tensors and the spherical Hamiltonian defined by the double outer product of two rank‐2 tensors. In particular, we established the spherical components of space‐part tensor T in terms of Cartesian tensor components provided by Cook and De Lucia. Throughout the article, Wigner active rotation matrix is used to illustrate the active rotation of spherical vector, spherical harmonics, and spherical tensor as well as the passive rotation via their rotational invariants. © 2014 Wiley Periodicals, Inc. Concepts Magn Reson Part A 42A: 197–244, 2013.