Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 96-97   | 1 | 2 |

Section Cochran's Fourier method

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 Cochran's Fourier method

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Cochran (1948b[link],c[link], 1951a[link]) undertook to exploit an algebraic similarity between the right-hand side of the normal equations in the least-squares method on the one hand, and the expression for the coefficients used in Booth's differential syntheses on the other hand (see also Booth, 1948a[link]). In doing so he initiated a remarkable sequence of formal and computational developments which are still actively pursued today.

Let [\rho\llap{$-\!$}_{\rm C} ({\bf x})] be the electron-density map corresponding to the current atomic model, with structure factors [|F_{{\bf h}}^{\,\rm calc}| \exp (i\varphi_{{\bf h}}^{\rm calc})]; and let [\rho\llap{$-\!$}_{\rm O} ({\bf x})] be the map calculated from observed moduli and calculated phases, i.e. with coefficients [\{|F_{{\bf h}}|^{\rm obs} \exp (i\varphi_{{\bf h}}^{\rm calc})\}_{{\bf h}\in {\scr H}}]. If there are enough data for [\rho\llap{$-\!$}_{\rm C}] to have a resolved peak at each model atomic position [{\bf x}_{j}], then[(\nabla_{{\bf x}} \rho\llap{$-\!$}_{\rm C})({\bf x}_{j}) = {\bf 0} \quad \hbox{for each } j \in J\hbox{\semi}]while if the calculated phases [\varphi_{\bf h}^{\rm calc}] are good enough, [\rho\llap{$-\!$}_{\rm O}] will also have peaks at each [{\bf x}_{j}]:[(\nabla_{{\bf x}} \rho\llap{$-\!$}_{\rm O})({\bf x}_{j}) = {\bf 0} \quad \hbox{for each } j \in J.]It follows that[\eqalign{[\nabla_{{\bf x}} (\rho\llap{$-\!$}_{\rm C} - \rho\llap{$-\!$}_{\rm O})] ({\bf x}_{j}) &= {\textstyle\sum\limits_{{\bf h}}} (-2 \pi i{\bf h}) [(|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) \exp (i\varphi_{\bf h}^{\rm calc})]\cr &\quad \times \exp (-2 \pi i{\bf h} \cdot {\bf x}_{j})\cr &= {\bf 0} \hbox{ for each } j \in J,}]where the summation is over all reflections in [{\scr H}] or related to [{\scr H}] by space-group and Friedel symmetry (overlooking multiplicity factors!). This relation is less sensitive to series-termination errors than either of the previous two, since the spectrum of [\rho\llap{$-\!$}_{\rm O}] could have been extrapolated beyond the data in [{\scr H}] by using that of [\rho\llap{$-\!$}_{\rm C}] [as in van Reijen (1942)[link]] without changing its right-hand side.

Cochran then used the identity[{\partial F_{{\bf h}}^{\,\rm calc} \over \partial {\bf x}_{j}} = (2 \pi i{\bf h}) g_{j} ({\bf h}) \exp (2 \pi i{\bf h} \cdot {\bf x}_{j})]in the form[(-2 \pi i{\bf h}) \exp (-2 \pi i{\bf h} \cdot {\bf x}_{j}) = {1 \over g_{j} ({\bf h})} {\overline{\partial F_{{\bf h}}^{\,\rm calc}} \over \partial {\bf x}_{j}}]to rewrite the previous relation as[\eqalign{&[\nabla_{{\bf x}} (\rho\llap{$-\!$}_{\rm C} - \rho\llap{$-\!$}_{\rm O})] ({\bf x}_{j})\cr &\quad= \sum\limits_{{\bf h}} {1 \over g_{j} ({\bf h})} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) {\scr R}{e} \left[{\overline{\partial F_{{\bf h}}^{\,\rm calc}} \over \partial {\bf x}_{j}} \exp (i \varphi_{{\bf h}}^{\rm calc})\right]\cr &\quad= \sum\limits_{{\bf h}} {1 \over g_{j} ({\bf h})} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) {\partial |F_{{\bf h}}^{\,\rm calc}| \over \partial {\bf x}_{j}}\cr &\quad= {\bf 0} \quad \hbox{for each } j \in J}](the operation [{\scr Re}][] on the first line being neutral because of Friedel symmetry). This is equivalent to the vanishing of the [3 \times 1] subvector of the right-hand side of the normal equations associated to a least-squares refinement in which the weights would be[W_{{\bf h}} = {1 \over g_{j} ({\bf h})}.]Cochran concluded that, for equal-atom structures with [g_{j} ({\bf h}) = g ({\bf h})] for all j, the positions [{\bf x}_{j}] obtained by Booth's method applied to the difference map [\rho\llap{$-\!$}_{\rm O} - \rho\llap{$-\!$}_{\rm C}] are such that they minimize the residual[{\textstyle{1 \over 2} }\sum\limits_{{\bf h}} {1 \over g({\bf h})} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs})^{2}]with respect to the atomic positions. If it is desired to minimize the residual of the ordinary least-squares method, then the differential synthesis method should be applied to the weighted difference map[\sum\limits_{{\bf h}} {W_{{\bf h}} \over g({\bf h})} (|F_{{\bf h}}^{\,\rm calc}| - |F_{{\bf h}}|^{\rm obs}) \exp (i \varphi_{{\bf h}}^{\rm calc}).]He went on to show (Cochran, 1951b[link]) that the refinement of temperature factors could also be carried out by inspecting appropriate derivatives of the weighted difference map.

This Fourier method was used by Freer et al. (1976)[link] in conjunction with a stereochemical regularization procedure to refine protein structures.


Booth, A. D. (1948a). A new Fourier refinement technique. Nature (London), 161, 765–766.
Cochran, W. (1948b). The Fourier method of crystal structure analysis. Nature (London), 161, 765.
Cochran, W. (1948c). The Fourier method of crystal-structure analysis. Acta Cryst. 1, 138–142.
Cochran, W. (1951a). Some properties of the [F_{o} - F_{c}]-synthesis. Acta Cryst. 4, 408–411.
Cochran, W. (1951b). The structures of pyrimidines and purines. V. The electron distribution in adenine hydrochloride. Acta Cryst. 4, 81–92.
Freer, S. T., Alden, R. A., Levens, S. A. & Kraut, J. (1976). Refinement of five protein structures by constrained Fo − Fc Fourier methods. In Crystallographic Computing Techniques, edited by F. R. Ahmed, pp. 317–321. Copenhagen: Munksgaard.
Reijen, L. L. van (1942). Diffraction effects in Fourier syntheses and their elimination in X-ray structure investigations. Physica, 9, 461–480.

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