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)

$D_{m^{\prime }m}^j(\alpha ,\beta ,\gamma ):=⟨j{m}^{\prime }|\mathcal{R}(\alpha ,\beta ,\gamma )|jm⟩=e^{-i{m}^{\prime }\alpha }{d}_{m^{\prime }m}^j(\beta )e^{-im\gamma }$

where $\alpha ,\beta ,$ and $\gamma$ are Euler angles, and where $d_{m^{\prime }m}^j(\beta )$, known as Wigner's reduced d-matrix, is given by (Ref. 2 Eq. 4.11 and 4.13)

$\begin{array}{ccc}{d}_{m^{\prime }m}^j(\beta )& =& D_{m^{\prime }m}^j(0,\beta ,0)\\ & =& ⟨jm\text{'}\left|e^{-i\beta j_y}\right|jm⟩\\ & =& [(j+m)!(j-m)!(j+m\text{'})!(j-m\text{'})!{]}^{1/2}{\sum }_{\chi }\frac{(-1{)}^{\chi }}{(j-m^{\prime }-\chi )!(j+m-\chi )!(\chi +m^{\prime }-m)!\chi !}\\ & & \times {(\cos \frac{\beta }{2})}^{2j+m-m^{\prime }-2\chi }{(-\sin \frac{\beta }{2})}^{m^{\prime }-m+2\chi }\end{array}$

The sum over $\chi$ is restricted to those values that do not lead to negative factorials. This function represents a rotation of $\beta$ about the (initial 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) ISBN 0486684806
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