# A general method for rotational averages

## TL;DR

This paper introduces a general, closed-form method for calculating rotational averages of tensor quantities in nonlinear spectroscopy, significantly simplifying the computational process for randomly oriented molecules.

## Contribution

It derives a universal formula for rotationally invariant tensors of averaged direction cosine products, applicable to tensors of any rank.

## Key findings

- Provides a closed-form expression for rotational averages
- Simplifies calculations in nonlinear optics involving random molecular orientations
- Offers new insights into the properties of averaged tensors

## Abstract

The theory of nonlinear spectroscopy on randomly oriented molecules leads to the problem of averaging molecular quantities over the random rotation. We solve this problem for arbitrary tensor rank by deriving a closed-form expression for the rotationally invariant tensor of averaged direction cosine products. From it we obtain some useful new facts about this tensor. Our results serve to speed the inherently lengthy calculations of nonlinear optics.