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