Tables for
Volume B
Reciprocal space
Edited by U. Shmueli

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

Section 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: 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[link]; Johnson, 1969[link]). It expresses the function [P({\bf u})] as [\eqalignno{P({\bf u}) &= \exp \left(\kappa\hskip 2pt^{j}D_{j} + {1 \over 2!} \kappa\hskip 2pt^{jk}D_{j}D_{k} - {1 \over 3!} \kappa\hskip 2pt^{jkl}D_{j}D_{k}D_{l}\right.\cr &\quad\left. + {1 \over 4!} \kappa\hskip 2pt^{jklm}D_{j}D_{k}D_{l}D_{m} - \ldots\right)P_{0}({\bf u}). &(}]

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 [{\exp\left\{\kappa_{1}t + {\kappa_{2}t^{2} \over 2!} + \ldots {\kappa_{r}t^{r} \over r!} + \ldots\right\} = 1 + \mu_{1}t + {\mu_{2}t^{2} \over 2!} + \ldots + {\mu_{r}t^{r} \over r!}.} \eqno(] When it is substituted for t, ([link] is the characteristic function, or Fourier transform of [P(t)] (Kendall & Stuart, 1958[link]).

The first two terms in the exponent of ([link] can be omitted if the expansion is around the equilibrium position and the harmonic term is properly described by [P_{0}({\bf u})].

The Fourier transform of ([link] is, by analogy with the left-hand part of ([link] (with t replaced by [2\pi ih]), [\eqalignno{T({\bf H}) &= \exp \left[{(2\pi i)^{3} \over 3!} \kappa\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {(2\pi i)^{4} \over 4!} \kappa\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m} + \ldots \right] T_{0}({\bf H})\cr &= \exp \left[- {4 \over 3} \pi^{3}i \kappa\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {2 \over 3} \pi^{4} \kappa\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m} + \ldots \right] T_{0}({\bf H}),\cr &&(}] where the first two terms have been omitted. Expression ([link] is similar to ([link] except that the entire series is in the exponent. Following Schwarzenbach (1986)[link], ([link] can be developed in a Taylor series, which gives [\eqalignno{T({\bf H}) &= \left\{1 + {(2\pi i)^{3} \over 3!} \kappa\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {(2\pi i)^{4} \over 4!} \kappa\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m} + \ldots\right.\cr &\quad + {(2\pi i)^{6} \over 6!} \left[\vphantom{{\textstyle\sum\limits_{1}^{2}}}\kappa\hskip 2pt^{jklmp} + {6! \over 2!(3!)^{2}} \kappa\hskip 2pt^{jkl}\kappa^{mnp}\right] h_{j}h_{k}h_{l}h_{m}h_{n}h_{p}\cr &\quad + \left.\vphantom{{\textstyle\sum\limits_{1}^{2}}}\hbox{higher-order terms}\right\} T_{0}({\bf H}). &(}]

This formulation, which is sometimes called the Edgeworth approximation (Zucker & Schulz, 1982[link]), clearly shows the relation to the Gram–Charlier expansion ([link], and corresponds to the probability distribution [analogous to ([link]] [\eqalignno{P({\bf u}) &= P_{0}({\bf u}) \left\{1 + {1 \over 3!} \kappa\hskip 2pt^{jkl}H_{jkl}({\bf u}) + {1 \over 4!} \kappa\hskip 2pt^{jklm}H_{jklm}({\bf u}) + \ldots\right.\cr &{\hbox to 7.25pt{}} + {1 \over 6!} \left[\vphantom{{\textstyle\sum\limits_{1}^{2}}}\kappa\hskip 2pt^{jklmnp} + 10 \kappa\hskip 2pt^{jkl} \kappa^{mnp} \vphantom{{\textstyle\sum\limits_{1}^{2}}}\right] H_{jklmnp}\cr &\quad \left.+ \hbox{ higher-order terms}\vphantom{{\textstyle\sum\limits_{1}^{2}}}\right\}. &(}]

The relation between the cumulants [\kappa\hskip 2pt^{jkl}] and the quasimoments [c\hskip 2pt^{jkl}] are apparent from comparison of ([link] and ([link]: [\eqalignno{c\hskip 2pt^{jkl} &= \kappa\hskip 2pt^{jkl}\cr c\hskip 2pt^{jklm} &= \kappa\hskip 2pt^{jklm}\cr c\hskip 2pt^{jklmn} &= \kappa\hskip 2pt^{jklmn}\cr c\hskip 2pt^{jklmnp} &= \kappa\hskip 2pt^{jklmnp} + 10\kappa\hskip 2pt^{jkl} \kappa^{mnp}. &(}]

The sixth- and higher-order cumulants and quasimoments differ. Thus the third-order cumulant [\kappa\hskip 2pt^{jkl}] contributes not only to the coefficient of [H_{jkl}], 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 ([link], 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.


First citationJohnson, 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
First citationKendall, M. G. & Stuart, A. (1958). The advanced theory of statistics. London: Griffin.Google Scholar
First citationSchwarzenbach, D. (1986). Private communication.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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