Relations between angular and Cartesian orientational expansions

Michael Te Vrugt,Raphael Wittkowski

doi:10.1063/1.5141367

Abstract

Orientational expansions, which are widely used in natural sciences, exist in angular and Cartesian forms. Although these expansions are orderwise equivalent, it is difficult to relate them in practice. In this article, both types of expansions and their relations are explained in detail. We give explicit formulas for the conversion between angular and Cartesian expansion coefficients for functions depending on one, two, and three angles in two and three spatial dimensions. These formulas are useful, e.g., for comparing theoretical and experimental results in liquid crystal physics. The application of the expansions in the definition of orientational order parameters is also discussed.

Highlights

INTRODUCTIONThe conversion rules can be used in any field of the natural sciences where orientational distributions are relevant
Orientational expansions, i.e., expansions of the angular dependence of a function f (Ω), with Ω denoting an orientational variable, are used in many fields of natural sciences, such as liquid crystal physics,1–6 active matter physics,7–13 polymer physics,14 electrostatics,15,16 optics,17–19 geophysics,20 astrophysics and cosmology,21,22 general relativity,23,24 quantum mechanics,25 chemistry,26,27 engineering,28,29 machine learning,30 and medicine.31 Important examples for orientational expansions are the Fourier expansion for Ω = φ ∈ [0, 2π), the expansion in spherical harmonics for Ω = (θ, φ) ∈ [0, π] × [0, 2π), and the expansion in outer products of a normalized orientation vector û for Ω = û
The expansions can be classified in two main categories, which differ in the way the expansion coefficients transform under rotations: angular expansions and Cartesian expansions

Summary

INTRODUCTION

The conversion rules can be used in any field of the natural sciences where orientational distributions are relevant They allow us to convert Cartesian data from a computer simulation or an NMR experiment on a liquid crystal into a form that can be compared with theoretical calculations that use angular functions. These equations allow, e.g., to calculate the dipole vector corresponding to data that are given in the form of an expansion in spherical harmonics. A list of elements of the Wigner D-matrices can be found in Appendix B

UNIAXIAL EXPANSIONS

Angular multipole expansion

Cartesian multipole expansion

Order parameters

BIAXIAL EXPANSIONS

CONCLUSIONS