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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 63-64   | 1 | 2 |

Section 1.3.4.2.1.3. Fourier series for the electron density and its summation

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

1.3.4.2.1.3. Fourier series for the electron density and its summation

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The convergence of the Fourier series for [\rho\llap{$-\!$}][\rho\llap{$-\!$}({\bf x}) = {\textstyle\sum\limits_{{\bf h}\in {\bb Z}^{3}}} F({\bf h}) \exp (-2\pi i {\bf h} \cdot {\bf x})]is usually examined from the classical point of view (Section 1.3.2.6.10[link]). The summation of multiple Fourier series meets with considerable difficulties, because there is no natural order in [{\bb Z}^{n}] to play the role of the natural order in [{\bb Z}] (Ash, 1976[link]). In crystallography, however, the structure factors [F({\bf h})] are often obtained within spheres [\|{\bf H}\| \leq \Delta^{-1}] for increasing resolution (decreasing Δ). Therefore, successive estimates of [\rho\llap{$-\!$}] are most naturally calculated as the corresponding partial sums (Section 1.3.2.6.10.1[link]):[S_{\Delta} (\rho\llap{$-\!$})({\bf x}) = {\textstyle\sum\limits_{\|({\bf A}^{-1})^{T} {\bf h}\| \leq \Delta^{-1}}} F({\bf h}) \exp (-2\pi i{\bf h} \cdot {\bf x}).]This may be written[S_{\Delta} (\rho\llap{$-\!$})({\bf x}) = (D_{\Delta} * \rho\llap{$-\!$})({\bf x}),]where [D_{\Delta}] is the `spherical Dirichlet kernel'[D_{\Delta}({\bf x}) = {\textstyle\sum\limits_{\|({\bf A}^{-1})^{T} {\bf h}\| \leq \Delta^{-1}}} \exp (-2\pi i{\bf h} \cdot {\bf x}).][D_{\Delta}] exhibits numerous negative ripples around its central peak. Thus the `series termination errors' incurred by using [S_{\Delta}(\rho\llap{$-\!$})] instead of [\rho\llap{$-\!$}] consist of negative ripples around each atom, and may lead to a Gibbs-like phenomenon (Section 1.3.2.6.10.1[link]) near a molecular boundary.

As in one dimension, Cesàro sums (arithmetic means of partial sums) have better convergence properties, as they lead to a convolution by a `spherical Fejér kernel' which is everywhere positive. Thus Cesàro summation will always produce positive approximations to a positive electron density. Other positive summation kernels were investigated by Pepinsky (1952)[link] and by Waser & Schomaker (1953)[link].

References

Ash, J. M. (1976). Multiple trigonometric series. In Studies in Harmonic Analysis, edited by J. M. Ash, pp. 76–96. MAA studies in mathematics, Vol. 13. The Mathematical Association of America.
Pepinsky, R. (1952). The use of positive kernels in Fourier syntheses of crystal structures. In Computing Methods and the Phase Problem in X-ray Crystal Analysis, edited by R. Pepinsky, pp. 319–338. State College: Penn. State University.
Waser, J. & Schomaker, V. (1953). The Fourier inversion of diffraction data. Rev. Mod. Phys. 25, 671–690.








































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