Wigner D-matrix

The Wigner D-matrix (also known as the Wigner rotation matrix) is a square matrix, of dimension $2j+1$, given by

$D_{m'm}^{j}(\alpha ,\beta ,\gamma ):=\langle jm'|{\mathcal {R}}(\alpha ,\beta ,\gamma )|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma }$

where $\alpha ,\;\beta ,$ and $\gamma \;$ are Euler angles, and where $d_{m'm}^{j}(\beta )$, known as Wigner's reduced d-matrix, is given by

${\begin{array}{lcl}d_{m'm}^{j}(\beta )&=&D_{m'm}^{j}(0,\beta ,0)\\&=&\langle jm'|e^{-i\beta j_{y}}|jm\rangle \\&=&[(j+m')!(j-m')!(j+m)!(j-m)!]^{1/2}\sum _{s}{\frac {(-1)^{m'-m+s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\\&&\times \left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}\end{array}}$

This represents a rotation of $\theta$ about the (inital frame) $Y$ axis.

Relation with spherical harmonic functions

The D-matrix elements with second index equal to zero, are proportional to spherical harmonics (normalized to unity)

$D_{m0}^{\ell }(\alpha ,\beta ,\gamma )^{*}={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell }^{m}(\beta ,\alpha )$

External links

• Wigner D-matrix page on Wikipedia

References

1. E. P. Wigner, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg Verlag, Braunschweig (1931).
2. Holger Dachsel "Fast and accurate determination of the Wigner rotation matrices in the fast multipole method", Journal of Chemical Physics 124 144115 (2006)