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) , is an expansion of a function in terms of the zero and higher derivatives of a normal distribution (Kendall & Stuart, 1958 ). If is the operator , where is the harmonic distribution, or 3, and the operator is the rth partial derivative . Summation is again implied over repeated indices.

The differential operators D may be eliminated by the use of three-dimensional Hermite polynomials defined, by analogy with the one-dimensional Hermite polynomials, by the expression which gives 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 here, and in the following sections, include all combinations which produce different terms.

The coefficients c, defined by (1.2.12.1) and (1.2.12.2) , are known as the quasimoments of the frequency function (Kutznetsov et al., 1960 ). They are related in a simple manner to the moments of the function (Kendall & Stuart, 1958 ) 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 and of , and are given in Table 1.2.12.1 for orders (IT IV, 1974 ; Zucker & Schulz, 1982 ).

 Table 1.2.12.1| top | pdf | Some Hermite polynomials (Johnson & Levy, 1974 ; Zucker & Schulz, 1982 )
 H(u) = 1 Hj(u) = wj Hjk(u) = wjwk − pjk Hjkl(u) = wjwkwl − (wjpkl + wkplj + wlpjk) = wjwkwl − 3w( jpkl) Hjklm(u) = wjwkwlwm − 6w( jwkplm) + 3pj( kplm) Hjklmn(u) = wjwkwlwmwn − 10w( lwmwnpjk) + 15w( npjkplm) Hjklmnp(u) = wjwkwlwmwnwp − 15w( jwkwlwmpjk) + 45w( jwkplmpnp) − 15pj( kplmpnp) where are the elements of , defined in expression (1.2.10.2)  . Indices between brackets indicate that the term is to be averaged over all permutations which produce distinct terms, keeping in mind that as illustrated for .

The Fourier transform of (1.2.12.3) is given by where is the harmonic temperature factor. is a power-series expansion about the harmonic temperature factor, with even and odd terms, respectively, real and imaginary.

### References International Tables for X-ray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press. (Present distributor Kluwer Academic Publishers, Dordrecht.)Google Scholar Johnson, 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 Kendall, M. G. & Stuart, A. (1958). The advanced theory of statistics. London: Griffin.Google Scholar Kutznetsov, P. I., Stratonovich, R. L. & Tikhonov, V. I. (1960). Theory Probab. Its Appl. (USSR), 5, 80–97.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