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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \gamma\;} are Euler angles, and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d^j_{m'm}(\beta)} , known as Wigner's reduced d-matrix, is given by (Ref. 2 Eq. 4.11 and 4.13)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{array}{lcl} d^j_{m'm}(\beta) &=& D^j_{m'm}(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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} about the (inital frame) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D^{\ell}_{m 0}(\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