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

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

## Section 1.3.4.2.1.9. 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

#### 1.3.4.2.1.9. Differential syntheses

| top | pdf |

Another particular instance of the convolution theorem is the duality between differentiation and multiplication by a monomial (Sections 1.3.2.4.2.8, 1.3.2.5.8).

In the present context, this result may be writtenin Cartesian coordinates, andin crystallographic coordinates.

A particular case of the first formula iswhereis the Laplacian of ρ.

The second formula has been used with 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 and Hessian matrix are readily obtained asand a step of Newton iteration towards the nearest stationary point of will proceed by

The modern use of Fourier transforms to speed up the computation of derivatives for model refinement will be described in Section 1.3.4.4.7.

The converse property is also useful: it relates the derivatives of the continuous transform to the moments of :For and , this identity gives the well known relation between the Hessian matrix of the transform at the origin of reciprocal space and the inertia tensor of the motif . This is a particular case of the moment-generating properties of , which will be further developed in Section 1.3.4.5.2.