A general method for rotational averages

Abstract

The theory of nonlinear spectroscopy on randomly oriented molecules leads to the problem of averaging molecular quantities over 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.

Three-Dimensional Rotational Averaging Using Irreducible Sets of Linearly Independent Fundamental Isotropic Cartesian Tensors: A Computational Approach

“…The relative orientations of these two Cartesian frames are defined through the Euler angles ϕ, θ, and ψ, here represented by using the z−y−z convention. 43 In particular, 0 ≤ ϕ ≤ 2π, 0 ≤ θ ≤ π, and 0 ≤ ψ ≤ 2π. molecular frame can assume any possible orientation, with respect to the laboratory frame.…”

“…Trigonometric Approach. The trigonometric approach represents the first mathematical approach for computing the rotational averaging of Cartesian tensors of rank n. In this approach, the rotational average of direction cosines, ⟨l id 1 λd 1 ...l id n λd n ⟩, is computed by means of the following triple trigonometric integral: 3,28,43,46,48,50,51 (4)…”

“…Laboratory, ( X , Y , Z ), in black, and the molecular, ( x , y , z ), in green, Cartesian frames. The relative orientations of these two Cartesian frames are defined through the Euler angles ϕ, θ, and ψ, here represented by using the z – y – z convention . In particular, 0 ≤ ϕ ≤ 2π, 0 ≤ θ ≤ π, and 0 ≤ ψ ≤ 2π.…”

“…The trigonometric approach represents the first mathematical approach for computing the rotational averaging of Cartesian tensors of rank n . In this approach, the rotational average of direction cosines, ⟨ l i 1 λ 1 ... l i n λ n ⟩, is computed by means of the following triple trigonometric integral: ,,,,,, ⟨ l i 1 λ 1 ... l i n λ n ⟩ newline= 1 8 π 2 ∫ 0 2 π ∫ 0 π ∫ 0 2 π l i 1 λ 1 ... l i n λ n nobreak0em0.25em⁡ sin ( θ ) nobreak0em0.25em⁡ normald ϕ nobreak0em0.25em⁡ normald θ nobreak0em0.25em⁡ normald ψ In principle, the rotational average obtained from this approach requires the computation of the 3 2 n integrals. This is a formidable computational task even for tensors with rather small rank n (e.g., for n = 3, one needs to consider 3 6 = 729 integrals, while, for n = 6, one needs to compute a total of 3 12 = 531441 integrals).…”

“…This is a formidable computational task even for tensors with rather small rank n (e.g., for n = 3, one needs to consider 3 6 = 729 integrals, while, for n = 6, one needs to compute a total of 3 12 = 531441 integrals). The trigonometric approach was solved in 2021 by Nessler and Begzjav . In particular, the authors showed that this problem can be rewritten in such a way that scales polynomially with n , i.e., ( n + 8 8 ) , instead of exponentially .…”