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).
