The cumulant expansion

Coppens, P.

doi:10.1107/97809553602060000550

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

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

Section 1.2.12.2. The cumulant expansion

P. Coppens^a ^*

^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 $[P({\bf u})]$ 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) is the characteristic function, or Fourier transform of [P(t)] (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 $[P_{0}({\bf u})]$ .

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 $[2\pi ih]$ ), $[\eqalignno{T({\bf H}) &= \exp \left[{(2\pi i)^{3} \over 3!} \kappa\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {(2\pi i)^{4} \over 4!} \kappa\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m} + \ldots \right] T_{0}({\bf H})\cr &= \exp \left[- {4 \over 3} \pi^{3}i \kappa\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {2 \over 3} \pi^{4} \kappa\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m} + \ldots \right] T_{0}({\bf H}),\cr &&(1.2.12.6)}]$ where the first two terms have been omitted. Expression (1.2.12.6) 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 $[\eqalignno{T({\bf H}) &= \left\{1 + {(2\pi i)^{3} \over 3!} \kappa\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {(2\pi i)^{4} \over 4!} \kappa\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m} + \ldots\right.\cr &\quad + {(2\pi i)^{6} \over 6!} \left[\vphantom{{\textstyle\sum\limits_{1}^{2}}}\kappa\hskip 2pt^{jklmp} + {6! \over 2!(3!)^{2}} \kappa\hskip 2pt^{jkl}\kappa^{mnp}\right] h_{j}h_{k}h_{l}h_{m}h_{n}h_{p}\cr &\quad + \left.\vphantom{{\textstyle\sum\limits_{1}^{2}}}\hbox{higher-order terms}\right\} T_{0}({\bf H}). &(1.2.12.7)}]$

This formulation, which is sometimes called the Edgeworth approximation (Zucker & Schulz, 1982), 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)] $[\eqalignno{P({\bf u}) &= P_{0}({\bf u}) \left\{1 + {1 \over 3!} \kappa\hskip 2pt^{jkl}H_{jkl}({\bf u}) + {1 \over 4!} \kappa\hskip 2pt^{jklm}H_{jklm}({\bf u}) + \ldots\right.\cr &{\hbox to 7.25pt{}} + {1 \over 6!} \left[\vphantom{{\textstyle\sum\limits_{1}^{2}}}\kappa\hskip 2pt^{jklmnp} + 10 \kappa\hskip 2pt^{jkl} \kappa^{mnp} \vphantom{{\textstyle\sum\limits_{1}^{2}}}\right] H_{jklmnp}\cr &\quad \left.+ \hbox{ higher-order terms}\vphantom{{\textstyle\sum\limits_{1}^{2}}}\right\}. &(1.2.12.8)}]$

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

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), 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

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