Numerical evaluation of orientation averages and its application to molecular physics

TL;DR

This paper reviews and compares numerical quadrature methods for averaging over molecular orientations in physics, providing guidelines and a Python tool to optimize computational efficiency in simulations.

Contribution

It offers a comprehensive comparison of spherical quadrature methods, derives practical guidelines for their selection, and introduces a versatile Python package for orientation averaging.

Findings

Gauss quadratures enable exact integration in many cases.

Different quadrature methods have specific advantages depending on the application.

The guidelines can be extended to higher-dimensional and other geometrical domains.

Abstract

In molecular physics, it is often necessary to average over the orientation of molecules when calculating observables, in particular when modelling experiments in the liquid or gas phase. Evaluated in terms of Euler angles, this is closely related to integration over two- or three-dimensional unit spheres, a common problem discussed in numerical analysis. The computational cost of the integration depends significantly on the quadrature method, making the selection of an appropriate method crucial for the feasibility of simulations. After reviewing several classes of spherical quadrature methods in terms of their efficiency and error distribution, we derive guidelines for choosing the best quadrature method for orientation averages and illustrate these with three examples from chiral molecule physics. While Gauss quadratures allow for achieving numerically exact integration for a wide range of applications, other methods offer advantages in specific circumstances. Our guidelines can also by applied to higher-dimensional spherical domains and other geometries. We also present a Python package providing a flexible interface to a variety of quadrature methods.