Wigner D-matrix

The Wigner D-matrix (also known as the Wigner rotation matrix) is a square matrix, of dimension Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle 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 Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \alpha ,\;\beta ,} and Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \gamma \;} are Euler angles, and where Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle d_{m'm}^{j}(\beta )} , known as Wigner's reduced d-matrix, is given by (Ref. 2 Eq. 4.11 and 4.13)

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\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}}}

The sum over $\chi$ is restricted to those values that do not lead to negative factorials. This function 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 (initial 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) 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