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

International Tables for Crystallography (2006). Vol. B. ch. 1.2, pp. 22-23   | 1 | 2 |

## Section 1.2.12.2. The cumulant expansion

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

#### 1.2.12.2. The cumulant expansion

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A second statistical expansion which has been used to describe the atomic probability distribution is that of Edgeworth (Kendall & Stuart, 1958 ; Johnson, 1969 ). It expresses the function as Like the moments μ of a distribution, the cumulants κ are descriptive constants. They are related to each other (in the one-dimensional case) by the identity When it is substituted for t, (1.2.12.5b) is the characteristic function, or Fourier transform of (Kendall & Stuart, 1958 ).

The first two terms in the exponent of (1.2.12.5a) can be omitted if the expansion is around the equilibrium position and the harmonic term is properly described by .

The Fourier transform of (1.2.12.5a) is, by analogy with the left-hand part of (1.2.12.5b) (with t replaced by ), where the first two terms have been omitted. Expression (1.2.12.6) is similar to (1.2.12.4) except that the entire series is in the exponent. Following Schwarzenbach (1986) , (1.2.12.6) can be developed in a Taylor series, which gives This formulation, which is sometimes called the Edgeworth approximation (Zucker & Schulz, 1982 ), clearly shows the relation to the Gram–Charlier expansion (1.2.12.4) , and corresponds to the probability distribution [analogous to (1.2.12.3) ] The relation between the cumulants and the quasimoments are apparent from comparison of (1.2.12.8) and (1.2.12.4) : The sixth- and higher-order cumulants and quasimoments differ. Thus the third-order cumulant contributes not only to the coefficient of , but also to higher-order terms of the probability distribution function. This is also the case for cumulants of higher orders. It implies that for a finite truncation of (1.2.12.6) , the probability distribution cannot be represented by a finite number of terms. This is a serious difficulty when a probability distribution is to be derived from an experimental temperature factor of the cumulant type.

### References Johnson, C. K. (1969). Addition of higher cumulants to the crystallographic structure-factor equation: a generalized treatment for thermal-motion effects. Acta Cryst. A25, 187–194.Google Scholar Kendall, M. G. & Stuart, A. (1958). The advanced theory of statistics. London: Griffin.Google Scholar Schwarzenbach, D. (1986). Private communication.Google Scholar Zucker, 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