Convolution, correlation and Patterson function

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 61 | 1 | 2 |

Section 1.3.4.2.1.6. Convolution, correlation and Patterson function

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.4.2.1.6. Convolution, correlation and Patterson function

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Let $[\rho\llap{$-\!$} = r * \rho\llap{$-\!$}^{0}]$ and $[\sigma\llap{$-$} = r * \sigma\llap{$-$}^{0}]$ be two electron densities referred to crystallographic coordinates, with structure factors $[\{F_{{\bf h}}\}_{{\bf h} \in {\bb Z}^{3}}]$ and $[\{G_{{\bf h}}\}_{{\bf h} \in {\bb Z}^{3}}]$ , so that $[\eqalign{\rho\llap{$-\!$}_{\bf x} &= {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} F({\bf h}) \exp (-2 \pi i {\bf h} \cdot {\bf x}), \cr \sigma\llap{$-$}_{\bf x} &= {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} G({\bf h}) \exp (-2 \pi i {\bf h} \cdot {\bf x}).}]$

The distribution $[\omega = r * (\rho\llap{$-\!$}^{0} * \sigma\llap{$-$}^{0})]$ is well defined, since the generalized support condition (Section 1.3.2.3.9.7) is satisfied. The forward version of the convolution theorem implies that if $[\omega_{\bf x} = {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} W({\bf h}) \exp (-2 \pi i {\bf h} \cdot {\bf x}),]$ then $[W({\bf h}) = F({\bf h}) G({\bf h}).]$

If either $[\rho\llap{$-\!$}^{0}]$ or $[\sigma\llap{$-$}^{0}]$ is infinitely differentiable, then the distribution $[\psi = \rho\llap{$-\!$} \times \sigma\llap{$-$}]$ exists, and if we analyse it as $[\psi_{\bf x} = {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} Y({\bf h}) \exp (-2 \pi i {\bf h} \cdot {\bf x}),]$ then the backward version of the convolution theorem reads: $[Y({\bf h}) = {\textstyle\sum\limits_{{\bf k} \in {\bb Z}^{3}}} F({\bf h}) G({\bf h} - {\bf k}).]$

The cross correlation $[\kappa [\rho\llap{$-\!$}, \sigma\llap{$-$}]]$ between $[\rho\llap{$-\!$}]$ and $[\sigma\llap{$-$}]$ is the $[{\bb Z}^{3}]$ -periodic distribution defined by: $[\kappa = \breve{\rho\llap{$-\!$}}^{0} * \sigma\llap{$-$}.]$ If $[\rho\llap{$-\!$}^{0}]$ and $[\sigma\llap{$-$}^{0}]$ are locally integrable, $[\eqalign{\kappa [\rho\llap{$-\!$}, \sigma\llap{$-$}] ({\bf t)} &= {\textstyle\int\limits_{{\bb R}^{3}}} \rho\llap{$-\!$}^{0} ({\bf x})\sigma\llap{$-$}({\bf x} + {\bf t}) \;\hbox{d}^{3} {\bf x} \cr &= {\textstyle\int\limits_{{\bb R}^{3} / {\bb Z}^{3}}} \rho\llap{$-\!$}({\bf x})\sigma\llap{$-$}({\bf x} + {\bf t}) \;\hbox{d}^{3} {\bf x}.}]$ Let $[\kappa ({\bf t}) = {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} K({\bf h}) \exp (-2\pi i {\bf h} \cdot {\bf t}).]$ The combined use of the shift property and of the forward convolution theorem then gives immediately: $[K({\bf h}) = \overline{F({\bf h})} G({\bf h})\hbox{;}]$ hence the Fourier series representation of $[\kappa [\rho\llap{$-\!$}, \sigma\llap{$-$}]]$ : $[\kappa [\rho\llap{$-\!$}, \sigma\llap{$-$}]({\bf t}) = {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}} \overline{F({\bf h})}} G({\bf h}) \exp (-2\pi i {\bf h} \cdot {\bf t}).]$ Clearly, $[\kappa [\rho\llap{$-\!$}, \sigma\llap{$-$}] = (\kappa [\sigma\llap{$-$}, \rho\llap{$-\!$}]){\breve{}}]$ , as shown by the fact that permuting F and G changes $[K({\bf h})]$ into its complex conjugate.

The auto-correlation of $[\rho\llap{$-\!$}]$ is defined as $[\kappa [\rho\llap{$-\!$},\rho\llap{$-\!$}]]$ and is called the Patterson function of $[\rho\llap{$-\!$}]$ . If $[\rho\llap{$-\!$}]$ consists of point atoms, i.e. $[\rho\llap{$-\!$}^{0} = {\textstyle\sum\limits_{j = 1}^{N}} \;Z_{j}\delta_{({\bf x}_{j})},]$ then $[\kappa [\rho\llap{$-\!$}, \rho\llap{$-\!$}] = r * \left[{\textstyle\sum\limits_{j = 1}^{N}} \;{\textstyle\sum\limits_{k = 1}^{N}}\; Z_{j}Z_{k}\delta_{({\bf x}_{j} - {\bf x}_{k})}\right]]$ contains information about interatomic vectors . It has the Fourier series representation $[\kappa [\rho\llap{$-\!$}, \rho\llap{$-\!$}]({\bf t}) = {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} |F({\bf h})|^{2} \exp (-2\pi i {\bf h} \cdot {\bf t}),]$ and is therefore calculable from the diffraction intensities alone. It was first proposed by Patterson (1934, 1935a,b) as an extension to crystals of the radially averaged correlation function used by Warren & Gingrich (1934) in the study of powders.

References

Patterson, A. L. (1934). A Fourier series method for the determination of the components of interatomic distances in crystals. Phys. Rev. 46, 372–376.Google Scholar

Patterson, A. L. (1935a). A direct method for the determination of the components of interatomic distances in crystals. Z. Kristallogr. 90, 517–542.Google Scholar

Patterson, A. L. (1935b). Tabulated data for the seventeen plane groups. Z. Kristallogr. 90, 543–554.Google Scholar

Warren, B. E. & Gingrich, N. S. (1934). Fourier integral analysis of X-ray powder patterns. Phys. Rev. 46, 368–372.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 61