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

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

Section 1.2.12.1. The Gram–Charlier 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

1.2.12.1. The Gram–Charlier expansion

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The three-dimensional Gram–Charlier expansion, introduced into thermal-motion treatment by Johnson & Levy (1974)[link], is an expansion of a function in terms of the zero and higher derivatives of a normal distribution (Kendall & Stuart, 1958[link]). If [D_{j}] is the operator [\hbox{d/d}u\hskip 2pt^{j}], [\eqalignno{P({\bf u}) &= [1 - c\hskip 2pt^{j}D_{j} + {1 \over 2!} c\hskip 2pt^{jk}D_{j}D_{k} - {1 \over 3!} c\hskip 2pt^{jkl}D_{j}D_{k}D_{l} + \ldots\cr &\quad + (-1)^{r} {c^{\alpha_{1}} \ldots c^{\alpha_{r}} \over r!} D_{\alpha_{1}} D_{\alpha_{r}}] P_{0}({\bf u}), &(1.2.12.1)}] where [P_{0}({\bf u})] is the harmonic distribution, [\alpha_{1} = 1, 2] or 3, and the operator [D_{\alpha_{1}} \ldots D_{\alpha_{r}}] is the rth partial derivative [\partial^{r}/(\partial u^{\alpha 1} \ldots \partial u^{\alpha r})]. Summation is again implied over repeated indices.

The differential operators D may be eliminated by the use of three-dimensional Hermite polynomials [H_{\alpha_{1}\ldots \alpha_{2}}] defined, by analogy with the one-dimensional Hermite polynomials, by the expression [{D_{\alpha_{1}}\ldots D_{\alpha_{r}} \exp (- {\textstyle {1 \over 2}} \sigma_{jk}^{-1} u\hskip 2pt^{j}u^{k}) = (-1)^{r} H_{\alpha_{1}\ldots \alpha_{r}} ({\bf u}) \exp (- {\textstyle{1 \over 2}} \sigma_{jk}^{-1} u\hskip 2pt^{j}u^{k}),} \eqno(1.2.12.2)] which gives [\eqalignno{P({\bf u}) &= \left[1 + {1 \over 3!} c\hskip 2pt^{jkl}H_{jkl}({\bf u}) + {1 \over 4!} c\hskip 2pt^{jklm}H_{jklm} ({\bf u}) + {1 \over 5!} c\hskip 2pt^{jklmn}H_{jklmn} ({\bf u})\right.\cr &\quad\left. + {1 \over 6!} c\hskip 2pt^{jklmnp}H_{jklmnp} ({\bf u}) + \ldots\right] P_{0}({\bf u}), &(1.2.12.3)}] where the first and second terms have been omitted since they are equivalent to a shift of the mean and a modification of the harmonic term only. The permutations of [j, k, l \ldots] here, and in the following sections, include all combinations which produce different terms.

The coefficients c, defined by (1.2.12.1)[link] and (1.2.12.2)[link], are known as the quasimoments of the frequency function [P(\bf u)] (Kutznetsov et al., 1960[link]). They are related in a simple manner to the moments of the function (Kendall & Stuart, 1958[link]) and are invariant to permutation of indices. There are 10, 15, 21 and 28 components of c for orders 3, 4, 5 and 6, respectively. The multivariate Hermite polynomials are functions of the elements of [\sigma_{jk}^{-1}] and of [u^{k}], and are given in Table 1.2.12.1[link] for orders [\leq 6] (IT IV, 1974[link]; Zucker & Schulz, 1982[link]).

Table 1.2.12.1| top | pdf |
Some Hermite polynomials (Johnson & Levy, 1974[link]; Zucker & Schulz, 1982[link])

H(u) = 1
Hj(u) = wj
Hjk(u) = wjwkpjk
Hjkl(u) = wjwkwl − (wjpkl + wkplj + wlpjk) = wjwkwl3w( jpkl)
Hjklm(u) = wjwkwlwm6w( jwkplm) + 3pj( kplm)
Hjklmn(u) = wjwkwlwmwn10w( lwmwnpjk) + 15w( npjkplm)
Hjklmnp(u) = wjwkwlwmwnwp − 15w( jwkwlwmpjk) + 45w( jwkplmpnp) − 15pj( kplmpnp)
where [w_{j}\equiv p{_{jk}}u^{k} \hbox{ and } p_{jk}] are the elements of [\sigma^{-1}], defined in expression (1.2.10.2)[link] [link]. Indices between brackets indicate that the term is to be averaged over all permutations which produce distinct terms, keeping in mind that [p_{jk} = p_{kj} \hbox{ and } w_{j}w_{k} = w_{k}w_{j}] as illustrated for [H_{jkl}].

The Fourier transform of (1.2.12.3)[link] is given by [\eqalignno{T({\bf H}) &= \left[1 - {4 \over 3} \pi^{3}ic\hskip 2pt^{jkl}h_{j}h_{k}h_{l} + {2 \over 3} \pi^{4}c\hskip 2pt^{jklm}h_{j}h_{k}h_{l}h_{m}\right.\cr &\quad + {4 \over 15} \pi^{5}ic\hskip 2pt^{jklmn}h_{j}h_{k}h_{l}h_{m}h_{n} &\cr &\quad\left. - {4 \over 45} \pi^{6}c\hskip 2pt^{jklmnp}h_{j}h_{k}h_{l}h_{m}h_{n}h_{p} + \ldots\right] T_{0}({\bf H}), &(1.2.12.4)}] where [T_{0}({\bf H})] is the harmonic temperature factor. [T({\bf H})] is a power-series expansion about the harmonic temperature factor, with even and odd terms, respectively, real and imaginary.

References

First citationInternational Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google Scholar
First citationJohnson, C. K. & Levy, H. A. (1974). Thermal motion analysis using Bragg diffraction data. In International tables for X-ray crystallography (1974), Vol. IV, pp. 311–336. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google Scholar
First citationKendall, M. G. & Stuart, A. (1958). The advanced theory of statistics. London: Griffin.Google Scholar
First citationKutznetsov, P. I., Stratonovich, R. L. & Tikhonov, V. I. (1960). Theory Probab. Its Appl. (USSR), 5, 80–97.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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