Editing Wigner D-matrix
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The '''Wigner D-matrix''' (also known as the Wigner rotation matrix) | The '''Wigner D-matrix''' (also known as the Wigner rotation matrix) is a square matrix, of dimension <math>2j+1</math>, given by (Ref. 2 Eq. 4.12) | ||
:<math> D^j_{m'm}(\alpha,\beta,\gamma) := \langle jm' | \mathcal{R}(\alpha,\beta,\gamma)| jm \rangle = | :<math> D^j_{m'm}(\alpha,\beta,\gamma) := \langle jm' | \mathcal{R}(\alpha,\beta,\gamma)| jm \rangle = | ||
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where <math>\alpha, \; \beta, </math> and <math>\gamma\;</math> are [[Euler angles]], and | where <math>\alpha, \; \beta, </math> and <math>\gamma\;</math> are [[Euler angles]], and | ||
where <math>d^j_{m'm}(\beta)</math>, known as Wigner's reduced d-matrix, is given by ( | where <math>d^j_{m'm}(\beta)</math>, known as Wigner's reduced d-matrix, is given by (Ref. 2 Eq. 4.11 and 4.13) | ||
:<math>\begin{array}{lcl} | :<math>\begin{array}{lcl} | ||
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:<math>D^{\ell}_{m 0}(\alpha,\beta,\gamma)^* = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^m (\beta, \alpha )</math> | :<math>D^{\ell}_{m 0}(\alpha,\beta,\gamma)^* = \sqrt{\frac{4\pi}{2\ell+1}} Y_{\ell}^m (\beta, \alpha )</math> | ||
==References== | ==References== | ||
#Eugene Paul Wigner "Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren", Vieweg Verlag, Braunschweig (1931). | |||
#M. E. Rose "Elementary theory of angular momentum", John Wiley & Sons (1967) ISBN 0486684806 | |||
#[http://dx.doi.org/10.1016/S0166-1280(97)00185-1 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)] | |||
#[http://dx.doi.org/10.1063/1.2194548 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== | ==External links== | ||