Spherical harmonics

Navaza, J.

doi:10.1107/97809553602060000682

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 13.2, p. 274

Section A13.2.1.3. Spherical harmonics

J. Navaza^a ^*

^aLaboratoire de Génétique des Virus, CNRS-GIF, 1. Avenue de la Terrasse, 91198 Gif-sur-Yvette, France
Correspondence e-mail: jnavaza@pasteur.fr

A13.2.1.3. Spherical harmonics

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The $[Y_{\ell, \, m}]$ 's, with $[-\ell \leq m \leq \ell]$ and $[0 \leq \ell \lt \infty]$ , constitute a complete set of functions of the unit vector u, having the following properties (Brink & Satchler, 1968):

(1) Transformation under rotations: $[\eqalignno{Y_{\ell, \, m}({\bf R}^{-1}{\bf u}) &= {\textstyle\sum\limits_{m'=-\ell}^{\ell}} D^{\ell}_{m, \, m'}({\bf R}^{-1}) Y_{\ell, \, m'}({\bf u})\cr &= {\textstyle\sum\limits_{m'=-\ell}^{\ell}} Y_{\ell, \, m'}({\bf u}) \overline{D^{\ell}_{m', \, m}({\bf R})}. &\hbox{(A13.2.1.8)}}]$
(2) Orthogonality condition: $[{\textstyle\int} \overline{Y_{\ell, \, m}({\bf u})} Y_{\ell',m'}({\bf u})\ \hbox{d}^{2}{\bf u} = \delta_{\ell, \, \ell'} \delta_{m, \, m'.} \eqno\hbox{(A13.2.1.9)}]$
(3) Inversion: $[Y_{\ell, \, m}({\bf -u}) = (-1)^{\ell} \ Y_{\ell, \, m}({\bf u}). \eqno\hbox{(A13.2.1.10)}]$
(4) Relation with rotation-matrix elements: $[Y_{\ell, \, m}(\theta,\varphi) = i^{\ell} [(2\ell+1)/4\pi]^{1/2} D^{\ell}_{m, \, 0}(\varphi,\theta,0), \eqno\hbox{(A13.2.1.11)}]$ where (θ, φ) are the polar coordinates of u.

References

Brink, D. M. & Satchler, G. R. (1968). Angular momentum, 2nd ed. Oxford University Press.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 13.2, p. 274