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