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

International Tables for Crystallography (2006). Vol. B. ch. 1.2, p. 23   | 1 | 2 |

Section 1.2.12.4. Relative merits of the three expansions

P. Coppensa*

aDepartment of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260-3000, USA
Correspondence e-mail: coppens@acsu.buffalo.edu

1.2.12.4. Relative merits of the three expansions

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The relative merits of the Gram–Charlier and Edgeworth expansions have been discussed by Zucker & Schulz (1982)[link], Kuhs (1983)[link], and by Scheringer (1985b)[link]. In general, the Gram–Charlier expression is found to be preferable because it gives a better fit in the cases tested, and because its truncation is equivalent in real and reciprocal space. The latter is also true for the one-particle potential model, which is mathematically related to the Gram–Charlier expansion by the interchange of the real- and reciprocal-space expressions. The terms of the OPP model have a specific physical meaning. The model allows prediction of the temperature dependence of the temperature factor (Willis, 1969[link]; Coppens, 1980[link]), provided the potential function itself can be assumed to be temperature independent.

It has recently been shown that the Edgeworth expansion (1.2.12.5a)[link] always has negative regions (Scheringer, 1985b[link]). This implies that it is not a realistic description of a vibrating atom.

References

First citationCoppens, P. (1980). Thermal smearing and chemical bonding. In Electron and magnetization densities in molecules and solids, edited by P. J. Becker, pp. 521–544. New York: Plenum.Google Scholar
First citationKuhs, W. F. (1983). Statistical description of multimodal atomic probability structures. Acta Cryst. A39, 148–158.Google Scholar
First citationScheringer, C. (1985a). A general expression for the anharmonic temperature factor in the isolated-atom-potential approach. Acta Cryst. A41, 73–79.Google Scholar
First citationScheringer, C. (1985b). A deficiency of the cumulant expansion of the anharmonic temperature factor. Acta Cryst. A41, 79–81.Google Scholar
First citationWillis, B. T. M. (1969). Lattice vibrations and the accurate determination of structure factors for the elastic scattering of X-rays and neutrons. Acta Cryst. A25, 277–300.Google Scholar
First citationZucker, U. H. & Schulz, H. (1982). Statistical approaches for the treatment of anharmonic motion in crystals. I. A comparison of the most frequently used formalisms of anharmonic thermal vibrations. Acta Cryst. A38, 563–568.Google Scholar








































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