Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, p. 67   | 1 | 2 |

Section Differential syntheses

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France Differential syntheses

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Another particular instance of the convolution theorem is the duality between differentiation and multiplication by a monomial (Sections[link],[link]).

In the present context, this result may be written[\eqalign{&\bar{\scr F}\left[{\partial^{m_{1} + m_{2} + m_{3}} \rho \over \partial X_{1}^{m_{1}} \partial X_{2}^{m_{2}} \partial X_{3}^{m_{3}}}\right] ({\bf H}) \cr &\quad = (-2 \pi i)^{m_{1} + m_{2} + m_{3}} H_{1}^{m_{1}} H_{2}^{m_{2}} H_{3}^{m_{3}} F ({\bf A}^{T} {\bf H})}]in Cartesian coordinates, and[\bar{\scr F}\left[{\partial^{m_{1} + m_{2} + m_{3}} \rho\llap{$-\!$} \over \partial x_{1}^{m_{1}} \partial x_{2}^{m_{2}} \partial x_{3}^{m_{3}}}\right] ({\bf h}) = (-2 \pi i)^{m_{1} + m_{2} + m_{3}} h_{1}^{m_{1}} h_{2}^{m_{2}} h_{3}^{m_{3}} F ({\bf h})]in crystallographic coordinates.

A particular case of the first formula is[-4 \pi^{2} {\textstyle\sum\limits_{{\bf H} \in \Lambda^{*}}} \|{\bf H}\|^{2} F ({\bf A}^{T} {\bf H}) \exp (-2 \pi i {\bf H} \cdot {\bf X}) = \Delta \rho ({\bf X}),]where[\Delta \rho = \sum\limits_{j = 1}^{3} {\partial^{2} \rho \over \partial X_{j}^{2}}]is the Laplacian of ρ.

The second formula has been used with [|{\bf m}| = 1] or 2 to compute `differential syntheses' and refine the location of maxima (or other stationary points) in electron-density maps. Indeed, the values at x of the gradient vector [\nabla \rho\llap{$-\!$}] and Hessian matrix [(\nabla \nabla^{T}) \rho\llap{$-\!$}] are readily obtained as[\eqalign{(\nabla \rho\llap{$-\!$}) ({\bf x}) &= {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} (-2 \pi i {\bf h}) F ({\bf h}) \exp (-2 \pi i {\bf h} \cdot {\bf x}), \cr [(\nabla \nabla^{T}) \rho\llap{$-\!$}] ({\bf x}) &= {\textstyle\sum\limits_{{\bf h} \in {\bb Z}^{3}}} (-4 \pi^{2} {\bf hh}^{T}) F ({\bf h}) \exp (-2 \pi i {\bf h} \cdot {\bf x}),}]and a step of Newton iteration towards the nearest stationary point of [\rho\llap{$-\!$}] will proceed by[{\bf x} \,\longmapsto\, {\bf x} - \{[(\nabla \nabla^{T}) \rho\llap{$-\!$}] ({\bf x})\}^{-1} (\nabla \rho\llap{$-\!$}) ({\bf x}).]

The modern use of Fourier transforms to speed up the computation of derivatives for model refinement will be described in Section[link].

The converse property is also useful: it relates the derivatives of the continuous transform [\bar{\scr F}[\rho^{0}]] to the moments of [\rho^{0}]:[{\partial^{m_{1} + m_{2} + m_{3}} \bar{\scr F}[\rho^{0}] \over \partial X_{1}^{m_{1}} \partial X_{2}^{m_{2}} \partial X_{3}^{m_{3}}} ({\bf H}) = \bar{\scr F}[(2 \pi i)^{m_{1} + m_{2} + m_{3}} X_{1}^{m_{1}} X_{2}^{m_{2}} X_{3}^{m_{3}} \rho_{{\bf x}}^{0}] ({\bf H}).]For [|{\bf m}| = 2] and [{\bf H} = {\bf 0}], this identity gives the well known relation between the Hessian matrix of the transform [\bar{\scr F}[\rho^{0}]] at the origin of reciprocal space and the inertia tensor of the motif [\rho^{0}]. This is a particular case of the moment-generating properties of [\bar{\scr F}], which will be further developed in Section[link].

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