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

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 \(100\).