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powder XRD pattern. As discussed in Sect.
2.5
, some implementations of the
direct-space strategy (e.g. [
35
,
36
]) have employed figures-of-merit based on
integrated peak intensities
I
(H) extracted from the experimental powder XRD
pattern in the profile fitting procedure (i.e. the intensity data (5) discussed in
Sect.
2.5
), rather than comparison of the complete powder XRD profile by means
of
R
wp
. Specific details are discussed in the papers cited.
The aim of the direct-space strategy is to find the trial crystal structure that
corresponds to lowest R-factor, and is equivalent to exploring a hypersurface
R
(
)
to find the global minimum, where
G
represents the set of variables that define the
structure (discussed in more detail below). In principle, any technique for global
optimization may be used to find the lowest point on the
R
(
G
) hypersurface, and
much success has been achieved using Monte Carlo/simulated annealing [
2
,
35
-
57
]
and genetic algorithm [
58
-
75
] techniques in this field. In addition, grid search
[
76
-
80
] and differential evolution [
81
,
82
] techniques have also been employed.
We now consider the way in which trial structures are defined within the context
of direct-space structure solution calculations for molecular solids. In principle, the
set (
G
) of structural variables could be taken to comprise the coordinates of each
individual atom within the asymmetric unit, but this approach discards any prior
knowledge of molecular geometry and corresponds to the maximal number of
structural variables (3
N
variables for
N
atoms in the asymmetric unit). Instead, it
is advantageous to use directly all information on molecular geometry that is
already known reliably [in the study of molecular materials, the identity of the
molecule is generally known before starting the structure solution calculation, and
if ambiguities remain concerning the atomic connectivity (e.g. tautomeric form),
other techniques such as solid-state NMR may be useful to resolve these
ambiguities before starting the structure solution calculation]. Thus, it is common
to fix bond lengths and bond angles at standard values in direct-space structure
solution calculations and to fix the geometries of well-defined structural units (e.g.
aromatic rings). In general, the only aspects of intramolecular geometry that need
to be determined are the values of some (or all) of the torsion angles that define the
molecular conformation. Under these circumstances, each trial structure in a direct-
space structure solution calculation is defined by a set (
G
) of structural variables
that represent, for each molecule in the asymmetric unit, the position of the molecule
in the unit cell (defined by the coordinates {
x
,
y
,
z
} of the centre of mass or a
selected atom), the orientation of the molecule in the unit cell (defined by rotation
angles {
G
y
,
f
,
c
} relative to the unit cell axes) and the unknown torsion angles
{
t
1
,
t
2
,
...
,
t
n
}. Thus, in general, there are 6 +
n
variables,
G
ΒΌ
{
x
,
y
,
z
,
y
,
f
,
c
,
t
1
,
t
2
,
t
n
}, for each molecule in the asymmetric unit.
We emphasize that an important feature contributing to the success of the direct-
space approach is that it makes maximal use of information on molecular geometry
that is already known reliably, independently of the powder XRD data, prior to
commencing the structure solution calculation. The traditional approach for struc-
ture solution, on the other hand, does not (in general) utilize prior knowledge of
features of molecular geometry.
...
,
