International
Tables for Crystallography Volume B Reciprocal space Edited by U. Shmueli © International Union of Crystallography 2010 
International Tables for Crystallography (2010). Vol. B, ch. 1.2, pp. 2022
Section 1.2.12. Treatment of anharmonicity ^{a}Department of Chemistry, Natural Sciences & Mathematics Complex, State University of New York at Buffalo, Buffalo, New York 14260–3000, USA 
The probability distribution (1.2.10.2) is valid in the case of rectilinear harmonic motion. If the deviations from Gaussian shape are not too large, distributions may be used which are expansions with the Gaussian distribution as the leading term. Three such distributions are discussed in the following sections.
The threedimensional Gram–Charlier expansion, introduced into thermalmotion 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 threedimensional Hermite polynomials defined, by analogy with the onedimensional 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).

The Fourier transform of (1.2.12.3) is given by where is the harmonic temperature factor. is a powerseries expansion about the harmonic temperature factor, with even and odd terms, respectively, real and imaginary.
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 as
Like the moments μ of a distribution, the cumulants κ are descriptive constants. They are related to each other (in the onedimensional case) by the identity When it is substituted for t, (1.2.12.5b) is the characteristic function, or Fourier transform of (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 .
The Fourier transform of (1.2.12.5a) is, by analogy with the lefthand part of (1.2.12.5b) (with t replaced by ), 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
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)]
The relation between the cumulants and the quasimoments are apparent from comparison of (1.2.12.8) and (1.2.12.4):
The sixth and higherorder cumulants and quasimoments differ. Thus the thirdorder cumulant contributes not only to the coefficient of , but also to higherorder 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.
When an atom is considered as an independent oscillator vibrating in a potential well , its distribution may be described by Boltzmann statistics. with N, the normalization constant, defined by . The classical expression (1.2.12.10) is valid in the hightemperature limit for which .
Following Dawson (1967) and Willis (1969), the potential function may be expanded in terms of increasing order of products of the contravariant displacement coordinates: The equilibrium condition gives . Substitution into (1.2.12.10) leads to an expression which may be simplified by the assumption that the leading term is the harmonic component represented by : in which etc. and the normalization factor N depends on the level of truncation.
The probability distribution is related to the spherical harmonic expansion. The ten products of the displacement parameters , for example, are linear combinations of the seven octapoles and three dipoles (Coppens, 1980). The thermal probability distribution and the aspherical atom description can be separated only because the latter is essentially confined to the valence shell, while the former applies to all electrons which follow the nuclear motion in the atomic scattering model.
The Fourier transform of the OPP distribution, in a general coordinate system, is (Johnson, 1970a; Scheringer, 1985a) where is the harmonic temperature factor and G represents the Hermite polynomials in reciprocal space.
If the OPP temperature factor is expanded in the coordinate system which diagonalizes , simpler expressions are obtained in which the Hermite polynomials are replaced by products of the displacement coordinates (Dawson et al., 1967; Coppens, 1980; Tanaka & Marumo, 1983).
The relative merits of the Gram–Charlier and Edgeworth expansions have been discussed by Zucker & Schulz (1982), Kuhs (1983), and by Scheringer (1985b). In general, the Gram–Charlier expression is found to be preferable because it gives a better fit in the cases tested, and because its truncation is equivalent in real and reciprocal space. The latter is also true for the oneparticle potential model, which is mathematically related to the Gram–Charlier expansion by the interchange of the real and reciprocalspace expressions. The terms of the OPP model have a specific physical meaning. The model allows prediction of the temperature dependence of the temperature factor (Willis, 1969; Coppens, 1980), provided the potential function itself can be assumed to be temperature independent.
It has recently been shown that the Edgeworth expansion (1.2.12.5a) always has negative regions (Scheringer, 1985b). This implies that it is not a realistic description of a vibrating atom.
References
International Tables for Xray Crystallography (1974). Vol. IV. Birmingham: Kynoch Press.Coppens, P. (1980). Thermal smearing and chemical bonding. In Electron and Magnetization Densities in Molecules and Solids, edited by P. J. Becker, pp. 521–544. New York: Plenum.
Dawson, B. (1967). A general structure factor formalism for interpreting accurate Xray and neutron diffraction data. Proc. R. Soc. London Ser. A, 248, 235–288.
Dawson, B., Hurley, A. C. & Maslen, V. W. (1967). Anharmonic vibration in fluoritestructures. Proc. R. Soc. London Ser. A, 298, 289–306.
Johnson, C. K. (1969). Addition of higher cumulants to the crystallographic structurefactor equation: a generalized treatment for thermalmotion effects. Acta Cryst. A25, 187–194.
Johnson, C. K. (1970a). Series expansion models for thermal motion. ACA Program and Abstracts, 1970 Winter Meeting, Tulane University, p. 60.
Johnson, C. K. & Levy, H. A. (1974). Thermal motion analysis using Bragg diffraction data. In International Tables for Xray Crystallography (1974), Vol. IV, pp. 311–336. Birmingham: Kynoch Press.
Kendall, M. G. & Stuart, A. (1958). The Advanced Theory of Statistics. London: Griffin.
Kuhs, W. F. (1983). Statistical description of multimodal atomic probability structures. Acta Cryst. A39, 148–158.
Kutznetsov, P. I., Stratonovich, R. L. & Tikhonov, V. I. (1960). Theory Probab. Its Appl. (USSR), 5, 80–97.
Scheringer, C. (1985a). A general expression for the anharmonic temperature factor in the isolatedatompotential approach. Acta Cryst. A41, 73–79.
Scheringer, C. (1985b). A deficiency of the cumulant expansion of the anharmonic temperature factor. Acta Cryst. A41, 79–81.
Schwarzenbach, D. (1986). Private communication.
Tanaka, K. & Marumo, F. (1983). Willis formalism of anharmonic temperature factors for a general potential and its application in the leastsquares method. Acta Cryst. A39, 631–641.
Willis, B. T. M. (1969). Lattice vibrations and the accurate determination of structure factors for the elastic scattering of Xrays and neutrons. Acta Cryst. A25, 277–300.
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.