The modified Fourier method

Cruickshank, D. W. J.

doi:10.1107/97809553602060000697

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 409 | 1 | 2 |

Section 18.5.5.2. The modified Fourier method

D. W. J. Cruickshank^a ^*^‡

^a Chemistry Department, UMIST, Manchester M60 1QD, England
Correspondence e-mail: dwj_cruickshank@email.msn.com

18.5.5.2. The modified Fourier method

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In the simplest form of the Fourier-map approach to centrosymmetric high-resolution structures, atomic positions are given by the maxima of the observed electron density. The uncertainty of such a position may be estimated as the uncertainty in the slope function (first derivative) divided by the curvature (second derivative) at the peak (Cruickshank, 1949a), i.e., $[\sigma (x) = \sigma (\hbox{slope})/(\hbox{atomic peak `curvature'}). \eqno(18.5.5.1)]$ However, atomic positions are affected by finite-series and peak-overlapping effects.

Hence, more generally, atomic positions may be determined by the requirement that the slope of the difference map at the position of atom r should be zero, or equivalently that the slopes at atom r of the observed and calculated electron densities should be equal. As a criterion this becomes the basis of the modified Fourier method (Cruickshank, 1952, 1959, 1999; Bricogne, 2001, Section 1.3.4.4.7.5 ), which, like the least-squares method, is applicable whether or not the atomic peaks are resolved and is applicable to noncentrosymmetric structures. For refinement, a set of n simultaneous linear equations are involved, analogous to the normal equations of least squares. Their right-hand sides are the slopes of the difference map at the trial atomic positions.

The diagonal elements of the matrix, for coordinate $[x_{r}]$ of an atom with Debye B value $[B_{r}]$ , are approximately equal to $[\hbox{`curvature'} = (4\pi^{2}/a^{2}V) \left[\textstyle\sum\limits_{hkl}\displaystyle (m / 2)h^{2} f_{r} \exp (-B_{r} \sin^{2}\theta / \lambda^{2})\right], \eqno(18.5.5.2)]$ where [m = 1] or 2 for acentric or centric reflections. The summation is over all independent planes and their symmetry equivalents. Strictly speaking, (18.5.5.2) is a curvature only for centrosymmetric structures.

In the modified Fourier method, $[\sigma (\hbox{slope}) = (2\pi/aV) \left[\textstyle\sum\limits_{hkl}\displaystyle h^{2} (\Delta |F|^{2})\right]^{1/2}. \eqno(18.5.5.3)]$ This is simply an estimate of the r.m.s. uncertainty at a general position (Cruickshank & Rollett, 1953) in the slope of the difference map, i.e., the r.m.s. uncertainty on the right-hand side of the modified Fourier method.

$[\sigma (x)]$ is then given by (18.5.5.1), using (18.5.5.3) and (18.5.5.2).

References

Bricogne, G. (2001). Fourier transforms in crystallography: theory, algorithms, and applications. In International tables for crystallography, Vol. B, edited by U. Shmueli, ch. 1.3. Dordrecht: Kluwer Academic Publishers.Google Scholar

Cruickshank, D. W. J. (1949a). The accuracy of electron-density maps in X-ray analysis with special reference to dibenzyl. Acta Cryst. 2, 65–82.Google Scholar

Cruickshank, D. W. J. (1952). On the relations between Fourier and least-squares methods of structure determination. Acta Cryst. 5, 511–518.Google Scholar

Cruickshank, D. W. J. (1959). Statistics. In International tables for X-ray crystallography, Vol. 2, edited by J. S. Kasper & K. Lonsdale, pp. 84–98. Birmingham: Kynoch Press.Google Scholar

Cruickshank, D. W. J. (1999). Remarks about protein structure precision. Acta Cryst. D55, 583–601.Google Scholar

Cruickshank, D. W. J. & Rollett, J. S. (1953). Electron-density errors at special positions. Acta Cryst. 6, 705–707.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 18.5, p. 409