Rotation functions

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, pp. 273-274 | 1 | 2 |
https://doi.org/10.1107/97809553602060000682

Appendix A13.2.1. Formulae for the derivation and computation of the fast rotation function

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

This appendix aims to present a complete set of formulae which allow the derivation and computation of the fast rotation function. They involve a particular convention for the definition of the irreducible representations of the rotation group suitable for crystallographic computations.

A13.2.1.1. Euler parameterization

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By applying the group property of rotations [equation (13.2.2.2)], the Euler parameterization may be expressed as rotations around fixed axis (see Fig. 13.2.2.1b): $[\eqalignno{{\bf R}(\alpha,\beta,\gamma) &= {\bf R}(\gamma,{\bf p}) {\bf R}(\beta,{\bf n}) {\bf R}(\alpha,{\bf z})\cr &= \left[{\bf R}(\beta,{\bf n}) {\bf R}(\gamma,{\bf z}) {\bf R}(\beta,{\bf n})^{-1}\right] {\bf R}(\beta,{\bf n}) {\bf R}(\alpha,{\bf z})\cr &= {\bf R}(\beta,{\bf n}) {\bf R}(\gamma,{\bf z}) {\bf R}(\alpha,{\bf z})\cr &= \left[{\bf R}(\alpha,{\bf z}) {\bf R}(\beta,{\bf y}) {\bf R}(\alpha,{\bf z})^{-1}\right] {\bf R}(\gamma,{\bf z}) {\bf R}(\alpha,{\bf z})\cr &= {\bf R}(\alpha,{\bf z}) {\bf R}(\beta,{\bf y}) {\bf R}(\gamma,{\bf z}). &(\hbox{A}13.2.1.1)}]$

A13.2.1.2. The $[D^{\ell}_{m, \, m'}]$ matrices

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A linear representation of dimension n of the rotation group is a correspondence between rotations and matrices of order n. The matrices $[D^{\ell}_{m, \, m'}({\bf R})]$ , with $[-\ell \leq m,m' \leq \ell]$ , are associated with the irreducible representation of dimension $[2\ell + 1]$ ( $[0 \leq \ell \lt \infty]$ ). They have the following properties (Brink & Satchler, 1968):

(1) Group multiplication: $[D^{\ell}_{m, \, m'}({\bf RR'}) = {\textstyle\sum\limits_{n=-\ell}^{\ell}} D^{\ell}_{m, \, n}({\bf R}) D^{\ell}_{n, \, m'}({\bf R'}). \eqno\hbox{(A13.2.1.2)}]$
(2) Complex conjugation: $[D^{\ell}_{m, \, m'}({\bf R}^{-1}) = \overline{D^{\ell}_{m', \, m}({\bf R})}. \eqno\hbox{(A13.2.1.3)}]$
(3) Euler parameterization: $[D^{\ell}_{m, \, m'}(\alpha,\beta,\gamma) = d^{\ell}_{m, \, m'}(\beta) \exp [i(m\alpha+m'\gamma)]. \eqno\hbox{(A13.2.1.4)}]$
(4) Recurrence relation for the reduced matrices: $[\eqalignno{d^{\ell}_{m-1, \, m'}(\beta) &= {[m'-m \cos(\beta)]^{2} \over [(\ell-m+1)(\ell+m)]^{1/2} \sin(\beta)} d^{\ell}_{m, \, m'}(\beta)\cr &\quad - \left[{(\ell-m)(\ell+m+1) \over (\ell-m+1)(\ell+m)}\right]^{1/2} d^{\ell}_{m+1, \, m'}(\beta). &\hbox{(A13.2.1.5)}}]$
(5) Initial values (bottom row of $[d^{\ell}]$ ): $[d^{\ell}_{\ell, \, m}(\beta) = (-1)^{\ell-m} \left[{(2\ell)! \over (\ell-m)!(\ell+m)!}\right]^{1/2} \sin(\beta/2)^{\ell-m} \cos(\beta/2)^{\ell+m}. \eqno\hbox{(A13.2.1.6)}]$
(6) Symmetry relations: $[d^{\ell}_{-m, \, -m'}(\beta) = d^{\ell}_{m', \, m}(\beta) = (-1)^{m-m'} d^{\ell}_{m, \, m'}(\beta). \eqno\hbox{(A13.2.1.7)}]$

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.

A13.2.1.4. Spherical Bessel functions

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The $[j_{\ell}]$ 's, with $[0 \leq \ell \lt \infty]$ , constitute a complete set of functions having the following properties (Watson, 1958):

(1) Recurrence relation: $[j_{\ell-1}(x) - (2\ell+1) [\ j_{\ell}(x)/x] + j_{\ell+1}(x) = 0. \eqno\hbox{(A13.2.1.12)}]$
(2) Initial values: $[\eqalignno{j_{0}(x) &= \sin(x)/x\cr j_{1}(x) &= [\sin(x) - x \ \cos(x)]/x^{2}.&(\hbox{A}13.2.1.13)}]$
(3) Integral of a product of spherical Bessel functions: $[\eqalignno{U^{\ell}(\;p, q) &= {\textstyle\int\limits_{0}^{1}} j_{\ell}(\;px) j_{\ell}(qx) x^{2}\ \hbox{d}x\cr &= \cases{[\ j_{\ell}(p)j_{\ell - 1}(q)q - j_{\ell}(q)j_{\ell - 1}(\;p)p] / (\;p^{2} - q^{2})\;\; \hbox{if } {p} \neq {q} \cr {1 \over 2} [\ j_{\ell}(p)^{2} - j_{\ell - 1}(\;p)j_{\ell + 1}(\;p)] {\hbox to 5.3pc{}} \hbox{if } {\it p} = {\it q} \cr}\cr &= (2\ell + 3) [\ j_{\ell + 1}(\;p) j_{\ell + 1}(q)]/pq + U^{\ell + 2}(\;p, q)\cr &= {\textstyle\sum\limits_{n = 1}^{\infty}} [2(\ell + 2n)-1] [\ j_{\ell + 2n-1}(\;p)j_{\ell + 2n-1}(q)]/pq.\cr& &(\hbox{A}13.2.1.14)}]$

A13.2.1.5. Expansion of $[\exp (2\pi i{\bf sr})]$

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This is also called the plane-wave expansion or Laplace's expansion (Landau & Lifschitz, 1972): $[\exp (2\pi i{\bf sr}) = 4\pi {\textstyle\sum\limits_{\ell = 0}^{\infty}} i^{\ell} {\textstyle\sum\limits_{m = -\ell}^{\ell}} j_{\ell}(2\pi sr) \overline{Y_{\ell, \, m}({\hat {\bf s}})} Y_{\ell, \, m}({\hat {\bf r}}). \eqno(\hbox{A}13.2.1.15)]$

A13.2.1.6. Expansion of the interference function

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$[\eqalignno{{\cal \chi}_{b}({\bf h - hR}^{-1}) &= (3/4\pi b^{3}) {\textstyle\int\limits_{0}^{b}} {\textstyle\int\limits_{0}^{\pi}} {\textstyle\int\limits_{0}^{2 \pi}} \exp[2\pi i({\bf h - kR}^{-1}){\bf r}] r^{2} \sin (\theta)\ \hbox{d}r\ \hbox{d}\theta\ \hbox{d}\varphi\cr &= {\textstyle\sum\limits_{\ell, \, \ell' = 0}^{\infty}}\ {\textstyle\sum\limits_{m = -\ell}^{\ell}}\ {\textstyle\sum\limits_{m'= -\ell'}^{\ell'} i^{\ell - \ell'}} \overline{Y_{\ell, \, m}({\hat {\bf h}})} Y_{\ell', \, m'}({\hat {\bf k}})\cr &\quad \times (12\pi/b^{3}) {\textstyle\int\limits_{0}^{b}} j_{\ell} (2\pi hr)\ j_{\ell'} (2\pi kr)r^{2}\ \hbox{d}r\cr &\quad \times {\textstyle\int\limits_{0}^{\pi}} {\textstyle\int\limits_{0}^{2\pi}} Y_{\ell, \, m} (\hat{{\bf r}}) \overline{Y_{\ell', \, m'} ({\bf R}^{-1} \hat{{\bf r}})} \sin (\theta)\ \hbox{d}\theta\ \hbox{d}\varphi\cr &= {\textstyle\sum\limits_{\ell = 0}^{\infty}}\ {\textstyle\sum\limits_{m=-\ell}^{\ell}}\ {\textstyle\sum\limits_{\ell'=0}^{\infty}}\ {\textstyle\sum\limits_{m'=-\ell'}^{\ell'} i^{\ell - \ell'}} \overline{Y_{\ell, \, m} (\hat{{\bf h}})} Y_{\ell', \, m'} (\hat{{\bf k}})\cr &\quad \times (12\pi/b^{3}) {\textstyle\int\limits_{0}^{b}} j_{\ell} (2\pi hr)\ j_{\ell'} (2\pi kr) r^{2}\ \hbox{d}r\cr &\quad \times {\textstyle\sum\limits_{m''=-\ell'}^{\ell'}} {\textstyle\int\limits_{0}^{\pi}} {\textstyle\int\limits_{0}^{2\pi}} Y_{\ell, \, m} (\hat{{\bf r}}) \overline{Y_{\ell', \, m''} (\hat{{\bf r}})} \sin (\theta)\ \hbox{d}\theta\ \hbox{d}\varphi \ D_{m'', \, m'}^{\ell'} ({\bf R})\cr &= {\textstyle\sum\limits_{\ell = 0}^{\infty}}\ {\textstyle\sum\limits_{m'=-\ell}^{\ell}} \overline{Y_{\ell, \, m} (\hat{{\bf h}})} Y_{\ell, \, m'} (\hat{{\bf k}}) 12\pi U^{\ell} (2\pi hb, 2\pi kb) D_{m, \, m'}^{\ell} ({\bf R}).\cr &&(\rm A13.2.1.16)}]$

References

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

Landau, L. D. & Lifschitz, E. M. (1972). Théorie quantique relativiste, pp. 109–196. Moscow: Editions MIR.Google Scholar

Watson, G. N. (1958). A treatise on the theory of Bessel functions, 2nd ed. Cambridge University Press.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 13.2, pp. 273-274
https://doi.org/10.1107/97809553602060000682

Appendix A13.2.1. Formulae for the derivation and computation of the fast rotation function

A13.2.1.1. Euler parameterization

A13.2.1.2. The matrices

A13.2.1.3. Spherical harmonics

A13.2.1.4. Spherical Bessel functions

A13.2.1.5. Expansion of

A13.2.1.6. Expansion of the interference function

References

A13.2.1.2. The $[D^{\ell}_{m, \, m'}]$ matrices

A13.2.1.5. Expansion of $[\exp (2\pi i{\bf sr})]$