Sphecerix

Figure: Demonstration of rotating a spherical harmonic by means of the Wigner D-matrix. In the above image, a 3dz2 and a 4fz3 atomic orbital are rotated by an angle $\pi$ over an axis with coordinates $(1,1,1)$. Note that the image has not been constructed by rotating the isosurfaces but by calculating the scalar field after the rotation.

Purpose

Rotation of spherical harmonics using Wigner-D matrices

Background

The canonical spherical harmonics $Y_{lm}$ subject to a rotation in $\mathbb{R}^{3}$ will always mix among each other. As such, we can represent the act of rotating the spherical harmonics by the following matrix-vector equation

\vec{Y}_{l}\prime = \mathbf{D}\vec{Y}_{l}

wherein $\mathbf{D}$ is the Wigner D-matrix and $\vec{Y}_{l}$ a vector composed of the canonical spherical harmonics of order $l$. The vector $\vec{Y}_{l}\prime$ is the linear combination that represents the result of the rotation upon the (linear combination) of spherical harmonics prior to the rotation.

Usage

In the script below, a dz2 spherical harmonic is rotated over an axis with coordinates $(1,1,1)$ by an angle $\pi$. The result of this rotation is a linear combination of spherical harmonics all with $l=2$.

from sphecerix import tesseral_wigner_D
from scipy.spatial.transform import Rotation as R
import numpy as np

def main():
    # build rotation axis and set angle
    axis = np.ones(3) / np.sqrt(3)
    angle = np.pi
    Robj = R.from_rotvec(axis * angle)
    
    # construct tesseral Wigner D matrix
    D = tesseral_wigner_D(2, Robj)
    Y = np.zeros(5)
    Y[2] = 1
    
    # calculate linear combination of the spherical harmonics after rotation
    Yp = D @ Y
    print(Yp)
    
if __name__ == '__main__':
    main()