Wigner D-matrix

The Wigner D-matrix (also known as the Wigner rotation matrix) is a square matrix, of dimension $2j+1$, given by (Ref. 2 Eq. 4.12)

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

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 (Ref. 2 Eq. 4.11 and 4.13)

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

This represents a rotation of $\beta$ 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)

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

References

1. Eugene Paul Wigner "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931).
2. M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967)
3. Miguel A. Blanco, M. Flórez and M. Bermejo "Evaluation of the rotation matrices in the basis of real spherical harmonics", Journal of Molecular Structure: THEOCHEM 419 pp. 19-27 (1997)
4. Holger Dachsel "Fast and accurate determination of the Wigner rotation matrices in the fast multipole method", Journal of Chemical Physics 124 144115 (2006)

External links

• Wigner D-matrix page on Wikipedia