High-precision evaluation of Wigner's d-matrix by exact diagonalization

TL;DR

The paper introduces a highly precise method for calculating Wigner's d-matrix by expanding it into a Fourier series and using exact diagonalization, enabling accurate computations for large spins.

Contribution

It presents a novel approach that avoids numerical instability by diagonalizing the angular-momentum operator to compute the d-matrix and its derivatives with high precision.

Findings

Achieves about 10^{-14} precision for spins up to 100

Enables calculations for spins up to a few thousand

Provides a simple and effective computational method

Abstract

The precise calculations of the Wigner's d-matrix are important in various research fields. Due to the presence of large numbers, direct calculations of the matrix using the Wigner's formula suffer from loss of precision. We present a simple method to avoid this problem by expanding the d-matrix into a complex Fourier series and calculate the Fourier coefficients by exactly diagonalizing the angular-momentum operator $J_{y}$ in the eigenbasis of $J_{z}$. This method allows us to compute the d-matrix and its various derivatives for spins up to a few thousand. The precision of the d-matrix from our method is about $10^{-14}$ for spins up to $100$.