3. Beyond the Gaussian Approximation
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# 3. Beyond the Gaussian Approximation

The situation is less straightforward if the distribution function is not Gaussian. A large variety of different approximation formalisms, as well as different nomenclature for similar formulations, is found in the literature. Summaries have been given by Johnson & Levy (1974), Zucker & Schulz (1982), Coppens (1993), and Kuhs (1992). By virtue of eq. (1.4.8), one may express either pdf(u) or as a series expansion and obtain the other quantity by Fourier transformation.

The most widespread approaches are based on formalisms developed in statistics to describe non-Gaussian distributions (Johnson, 1969). They use a differential expansion of the Gausssian pdf. Two formulations are found in frequently used refinement programs, the cumulant or Edgeworth expansion

and the quasi-moment or Gram-Charlier expansion

with the Gaussian Debye-Waller factor (see sections 1.4 and 2.1) and , , ... the third, fourth, ... order (anharmonic) tensorial coefficients. There are in general ten cubic, fifteen quartic, ... terms that enter into the treatment. In statistics they are called cumulants and quasi-moments, respectively. They constitute the parameters of the refinement. Various symbols for these coefficients are scattered through the literature. Greek letters are chosen here to comply with the 's of the Gaussian case, which may thus be considered as second-order coefficients. For the same reason the factors , with N the order of the tensor, are included, also to follow standard physical notation, which uses as the scattering vector. The factors and/or the factors 1/N! (e.g., Kuhs, 1992) are sometimes omitted in the literature. For comparability of future results, it is therefore proposed that only coefficients defined as in (3.1.61) and (3.1.62) be published, and that subscripts be used to indicate the type of expansion employed.

The , , ... are dimensionless quantities. As proposed by Kuhs (1992), they may be transformed to quantities of dimension (length)N by

with to be replaced by , ... Note that this is a generalization of eq. (2.1.38). It must be stressed, however, that the , , ... are simple expansion coefficients and (in general) have no direct physical meaning. The transformation (3.1.63) thus has no such merits as in the Gaussian case, and some real-space illustrations should always be given to permit the results to be appreciated. The best way is certainly to plot the corresponding pdf, obtained by inversion of (3.1.62) or (1.4.8). Only programs that produce sections of the pdf's seem to be currently available, although a three-dimensional visualisation similar to ORTEP would be highly desirable. Another way of presenting the results is by tensor contraction (Kuhs, 1992). For even-order terms, full contraction yields an invariant scalar,

For the Gram-Charlier series, this quantity indicates flatness (for negative values) or peakedness (positive values) of the pdf. The are the components of the real-space metric tensor. Note that , i.e., (3.1.64) is an extension of eq. (2.2.53). Similarly, vector invariants may be calculated for odd-order terms,

giving the direction of maximal skewness. Partial contraction of even-order terms,

reveals the directions of flatness and peakedness.

Various discussions in the literature (see, e.g., Kuhs, 1992, and references therein) indicate that the Gram-Charlier formalism is the best choice in routine crystallographic work. In particular, it has the advantage that the reverse Fourier transformation (1.4.8) can be carried out analytically,

with Hermite polynomials, and the harmonic part of the pdf. These polynomials are tabulated by Johnson & Levy (1974) up to the fourth order and by Zucker & Schulz (1982) up to the sixth order [see also Coppens (1993)]. The use of the Gram-Charlier expansion (3.1.62) is therefore recommended, although other formalisms may sometimes be advantageous for special problems. In any case, the results should always be carefully checked, especially if higher order terms are used merely to improve the agreement of the fit. Strong and extended negative regions in the pdf indicate inadequacy of the results. One also has to remember that, with anharmonic refinements, the positions and obtained are not necessarily faithful representations of the mean and variance of the pdf, respectively. This must be borne in mind if bond distances and Gaussian displacement ellipsoids are to be derived from the refined parameters. In some situations, it may be better to use only the Gaussian approximation, even though the resulting R-factors may be higher.

Another possibility is the expansion of the so called one-particle-potential (OPP) , which in the classical limit [ ] is related to the pdf by Boltzmann statistics

with Z the partition function. The second equality is obtained by setting .

The latter approach was formulated by Dawson & Willis (1967) and Willis (1969) for cubic point groups and later generalized for any symmetry by Tanaka & Marumo (1983). The OPP is written as

with the harmonic (quadratic) OPP and and the third and fourth order coefficients, respectively, which are defined in a Cartesian system. Since application of eqs. (3.1.68) and (1.4.8) does not lead to an analytical expression for , the anharmonic part is approximated in (3.1.68) by

The final expressions for are rather lengthy and may be found in Tanaka & Marumo (1983). Refinable parameters are the and . Other formulations with simpler expressions for have been introduced by Coppens (1978), Kurki-Suonio, Merisalo & Peltonen (1979) and Scheringer (1985). None of these approaches seems to have been used much in crystallographic studies, and final recommendations must await further developments in this field. It should also be noted that the OPP approach treats each atom as an individual (Einstein-)oscillator, which is a poor approximation for tightly bound atoms in molecules.

The OPP approach is physically meaningful only for purely dynamic displacive disorder (giving, e.g., the directions of weak and strong bonds), and is limited to rather small anharmonicities through the approximation (3.1.70). Occasionally special expansions (e.g., symmetry-adapted spherical harmonics) of pdf(u) or have been used for special problems (e.g., curvilinear motion, molecular disorder); see Johnson & Levy (1974), Press & Hüller (1973) and Prandl (1981). Again, these expansions do not seem yet to have entered routine crystallographic work. It should be remembered that the classical limit , which is assumed in eq. (3.1.68), may be far from the actual situation even at room temperature.

In eqs. (3.1.61), ( 3.1.62), and many of the remaining equations in this section, the summation convention has been used. It is assumed that summation occurs over indices that are repeated, such as j, k, l and m in the terms on the right-hand side of ( 3.1.61) and (3.1.62).

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