Comparing, Modeling, and Assigning Chemical-Shift Tensors in the Cartesian, Irreducible Spherical, and Icosahedral Representations

Abstract

A measure of the difference between two chemical-shift tensors is developed by defining the scalar distance between them. Chemical-shift tensors are treated as functions whose domain is the surface of a sphere and the mathematical definition of the quadratic distance between two functions is invoked. Expressions for the distance between two chemical-shift tensors are developed in the Cartesian and irreducible spherical representations and in a new icosahedral representation. A representation wherein the chemical-shift tensor is specified by the shifts when the magnetic field is along six directions defined by the vertices of an isosahedron is developed and its properties are discussed. The expression for the distance between two tensors is found to be particularly attractive and useful in this icosahedral representation. The distance between tensors computed in the icosahedral representation is useful in fitting linear models to tensor data. It is shown how such fitting can contribute to the assignment of tensors obtained from single-crystal studies. A quantitative figure of merit useful for comparing multiple assignment possibilities is developed. The results derived are applicable to any physical phenomenon described by real zero-rank and second-rank tensors.