A simplified derivation

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. 91 | 1 | 2 |

Section 1.3.4.4.7.8. A simplified derivation

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

1.3.4.4.7.8. A simplified derivation

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A very simple derivation of the previous results will now be given, which suggests the possibility of many generalizations.

The weighted difference map $[D({\bf x})]$ has coefficients $[D_{{\bf h}}]$ which are the gradients of the global residual with respect to each $[F_{{\bf h}}^{\rm calc}]$ : $[D_{{\bf h}} = {\partial R \over \partial A_{{\bf h}}^{\rm calc}} + i {\partial R \over \partial B_{{\bf h}}^{\rm calc}}.]$ By the chain rule, a variation of each $[F_{{\bf h}}^{\rm calc}]$ by $[\delta F_{{\bf h}}^{\rm calc}]$ will result in a variation of R by $[\delta R]$ with $[\delta R = \sum\limits_{{\bf h}} \left[{\partial R \over \partial A_{{\bf h}}^{\rm calc}} \delta A_{{\bf h}}^{\rm calc} + {\partial R \over \partial B_{{\bf h}}^{\rm calc}} \delta B_{{\bf h}}^{\rm calc}\right] = {\scr Re} \sum\limits_{{\bf h}} [\overline{D_{{\bf h}}} \delta F_{{\bf h}}^{\rm calc}].]$ The $[{\scr Re}]$ operation is superfluous because of Friedel symmetry, so that $[\delta R]$ may be simply written in terms of the Hermitian scalar product in $[\ell^{2}({\bb Z}^{3})]$ : $[\delta R = ({\bf D}, \delta {\bf F}^{\rm calc}).]$ If $[\rho\llap{$-\!$}^{\rm calc}]$ is the transform of $[\delta {\bf F}^{\rm calc}]$ , we have also by Parseval's theorem $[\delta R = (D, \delta \rho\llap{$-\!$}^{\rm calc}).]$ We may therefore write $[D ({\bf x}) = {\partial R \over \partial \rho\llap{$-\!$}^{\rm calc} ({\bf x})},]$ which states that $[D({\bf x})]$ is the functional derivative of R with respect to $[\rho\llap{$-\!$}^{\rm calc}]$ .

The right-hand side of the normal equations has $[\partial R/\partial u_{p}]$ for its pth element, and this may be written $[{\partial R \over \partial u_{p}} = \int_{{\bb R}^{3}/{\bb Z}^{3}} {\partial R \over \partial \rho\llap{$-\!$}^{\rm calc}({\bf x})} {\partial \rho\llap{$-\!$}^{\rm calc}({\bf x}) \over \partial u_{p}} \hbox{d}^{2}{\bf x} = \left(D, {\partial \rho\llap{$-\!$}^{\rm calc} \over \partial u_{p}}\right).]$ If $[u_{p}]$ belongs to atom j, then $[{\partial \rho\llap{$-\!$}^{\rm calc} \over \partial u_{p}} = {\partial (\tau_{{\bf x}_{j}} \sigma_{j}) \over \partial u_{p}} = \tau_{{\bf x}_{j}} \left({\partial \sigma\llap{$-$}_{j} \over \partial u_{p}}\right)\hbox{;}]$ hence $[{\partial R \over \partial u_{p}} = \left(D, \tau_{{\bf x}_{j}} \left({\partial \sigma\llap{$-$}_{j} \over \partial u_{p}}\right)\right).]$ By the identity of Section 1.3.2.4.3.5, this is identical to Lifchitz's expression $[(D * \partial \sigma\llap{$-$}_{j}/\partial u_{p})({\bf x}_{j})]$ . The present derivation in terms of scalar products [see Brünger (1989) for another presentation of it] is conceptually simpler, since it invokes only the chain rule [other uses of which have been reviewed by Lunin (1985)] and Parseval's theorem; economy of computation is obviously related to the good localization of $[\partial \rho\llap{$-\!$}^{\rm calc}/\partial u_{p}]$ compared to $[\partial {F}^{\rm calc}/\partial u_{p}]$ . Convolutions, whose meaning is less clear, are no longer involved; they were a legacy of having first gone over to reciprocal space via differential syntheses in the 1940s.

Cast in this form, the calculation of derivatives by FFT methods appears as a particular instance of the procedure described in connection with variational techniques (Section 1.3.4.4.6) to calculate the coefficients of local quadratic models in a search subspace; this is far from surprising since varying the electron density through a variation of the parameters of an atomic model is a particular case of the `free' variations considered by the variational approach. The latter procedure would accommodate in a very natural fashion the joint consideration of an energetic (Jack & Levitt, 1978; Brünger et al., 1987; Brünger, 1988; Brünger et al., 1989; Kuriyan et al., 1989) or stereochemical (Konnert, 1976; Sussman et al., 1977; Konnert & Hendrickson, 1980; Hendrickson & Konnert, 1980; Tronrud et al., 1987) restraint function (which would play the role of S) and of the crystallographic residual (which would be C). It would even have over the latter the superiority of affording a genuine second-order approximation, albeit only in a subspace, hence the ability of detecting negative curvature and the resulting bifurcation behaviour (Bricogne, 1984). Current methods are unable to do this because they use only first-order models, and this is known to degrade severely the overall efficiency of the refinement process.

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 91