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

International Tables for Crystallography (2010). Vol. B, ch. 1.2, pp. 21-22   | 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@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[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}). &(1.2.12.5a)}]

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(1.2.12.5b)]When it is substituted for t, (1.2.12.5b)[link] is the characteristic function, or Fourier transform of [P(t)] (Kendall & Stuart, 1958[link]).

The first two terms in the exponent of (1.2.12.5a)[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 (1.2.12.5a)[link] is, by analogy with the left-hand part of (1.2.12.5b)[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 &&(1.2.12.6)}]where the first two terms have been omitted. Expression (1.2.12.6)[link] is similar to (1.2.12.4)[link] except that the entire series is in the exponent. Following Schwarzenbach (1986)[link], (1.2.12.6)[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}). &(1.2.12.7)}]

This formulation, which is sometimes called the Edgeworth approximation (Zucker & Schulz, 1982[link]), clearly shows the relation to the Gram–Charlier expansion (1.2.12.4)[link], and corresponds to the probability distribution [analogous to (1.2.12.3)[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\}. &(1.2.12.8)}]

The relation between the cumulants [\kappa^{\,jkl}] and the quasi­moments [c^{\,jkl}] are apparent from comparison of (1.2.12.8)[link] and (1.2.12.4)[link]: [\eqalignno{c\hskip 2pt^{jkl} &= \kappa^{\,jkl}\cr c^{\,jklm} &= \kappa^{\,jklm}\cr c^{\,jklmn} &= \kappa^{\,jklmn}\cr c^{\,jklmnp} &= \kappa^{\,jklmnp} + 10\kappa^{\,jkl} \kappa^{mnp}. &(1.2.12.9)}]

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 (1.2.12.6)[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.

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








































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