International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2006 |
International Tables for Crystallography (2006). Vol. B. ch. 1.3, pp. 90-91
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Agarwal (1978) rederived and completed Cruickshank's results at a time when the availability of the FFT algorithm made the Fourier method of calculating the coefficients of the normal equations much more economical than the standard method, especially for macromolecules.
As obtained by Cruickshank, the modified Fourier method required a full 3D Fourier synthesis
Agarwal disposed of the latter dependence by pointing out that the multiplication involved is equivalent to a real-space convolution between the differential synthesis and , the standard electron density for atom type j (Section 1.3.4.2.1.2) smeared by the isotropic thermal agitation of that atom. Since is localized, this convolution involves only a small number of grid points. The requirement of a distinct differential synthesis for each parameter type, however, continued to hold, and created some difficulties at the FFT level because the symmetries of differential syntheses are more complex than ordinary space-group symmetries. Jack & Levitt (1978) sought to avoid the calculation of difference syntheses by using instead finite differences calculated from ordinary Fourier or difference Fourier maps.
In spite of its complication, this return to the Fourier implementation of the least-squares method led to spectacular increases in speed (Isaacs & Agarwal, 1978; Agarwal, 1980; Baker & Dodson, 1980) and quickly gained general acceptance (Dodson, 1981; Isaacs, 1982a,b, 1984).
References
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