transform of a Gaussian distribution which expresses the time- and space-dependent
harmonic displacements of an atom from its mean position; and, finally, exp2πi
[(hx j + ky j + lz j )] is the Fourier transform of a delta function at the position of atom j.
The structure factor expression conveniently separates all of the contributions
which makes replacing parts or extending the calculation quite straightforward. In
order to replace f j to produce models for fitting aspherical scattering density, as
described above, only one multiplicative term of the expression needs to be changed.
Note that any part of the structure factor expression that is not symmetrical about the
origin, such as aspherical electron density, results in a complex number, and the
overall structure factor computed for every observation is the product of these
complex contributions.
Additional terms can be multiplied onto the existing structure factor equation to
model scattering due to non-rectilinear motion of atoms. This corresponds to an
additional convolution operation, for example, a function describing an arc segment
can elegantly extend the existing atomic model to account for a nonatomic,
non-rectilinear distribution. Examples of this approach include convolutions of
isotropically distributed atomic density with a line, spherical shell or ring function
allowing refinement of continuously disordered atom models [7]; the hindered rotor
which convolves a ‘normal’ isotropic IAM atom with a function that distributes it
around a ring with an adjustable hindering potential allowing modelling of scattering
from groups with strong in-plane libration, e.g. coordinated cyclopentadienyl rings,
benzene and CF 3 groups [8]; and directly derived skewed anisotropic and curvilinear
distributions [9]. The first two cases are implemented in CRYSTALS [10] and have
proved useful for modelling disordered solvent molecules or substituent groups,
including spherical rotational disorder of tetramethylammonium ions in a series of
tetrachlorometallate salts [11], the disordered equatorial density of spinning PF 6
À
anions [12] and disordered guest solvent in a molecular cavitand [13].
1.3 Linear Algebra Description
Least squares optimization is an appropriate method to find the best fit of a crystallographic model to a set of measured data. The fit is computed between observed and
computed structure factor magnitude squared, or sometimes observed and computed
structure factor magnitude. Assuming suitable weighting of observations, the same
best fit is calculated from both approaches.
The least squares method and its matrix algebra notation is described here and is
useful to describe subsequent concepts.
The aim of a least squares refinement is to find the minimum value of χ
2 , the sum
of the squares of the differences between diffraction measurements under conditions
h i and their values predicted by the current model:
Recent Developments in the Refinement and Analysis of Crystal Structures
49
harmonic displacements of an atom from its mean position; and, finally, exp2πi
[(hx j + ky j + lz j )] is the Fourier transform of a delta function at the position of atom j.
The structure factor expression conveniently separates all of the contributions
which makes replacing parts or extending the calculation quite straightforward. In
order to replace f j to produce models for fitting aspherical scattering density, as
described above, only one multiplicative term of the expression needs to be changed.
Note that any part of the structure factor expression that is not symmetrical about the
origin, such as aspherical electron density, results in a complex number, and the
overall structure factor computed for every observation is the product of these
complex contributions.
Additional terms can be multiplied onto the existing structure factor equation to
model scattering due to non-rectilinear motion of atoms. This corresponds to an
additional convolution operation, for example, a function describing an arc segment
can elegantly extend the existing atomic model to account for a nonatomic,
non-rectilinear distribution. Examples of this approach include convolutions of
isotropically distributed atomic density with a line, spherical shell or ring function
allowing refinement of continuously disordered atom models [7]; the hindered rotor
which convolves a ‘normal’ isotropic IAM atom with a function that distributes it
around a ring with an adjustable hindering potential allowing modelling of scattering
from groups with strong in-plane libration, e.g. coordinated cyclopentadienyl rings,
benzene and CF 3 groups [8]; and directly derived skewed anisotropic and curvilinear
distributions [9]. The first two cases are implemented in CRYSTALS [10] and have
proved useful for modelling disordered solvent molecules or substituent groups,
including spherical rotational disorder of tetramethylammonium ions in a series of
tetrachlorometallate salts [11], the disordered equatorial density of spinning PF 6
À
anions [12] and disordered guest solvent in a molecular cavitand [13].
1.3 Linear Algebra Description
Least squares optimization is an appropriate method to find the best fit of a crystallographic model to a set of measured data. The fit is computed between observed and
computed structure factor magnitude squared, or sometimes observed and computed
structure factor magnitude. Assuming suitable weighting of observations, the same
best fit is calculated from both approaches.
The least squares method and its matrix algebra notation is described here and is
useful to describe subsequent concepts.
The aim of a least squares refinement is to find the minimum value of χ
2 , the sum
of the squares of the differences between diffraction measurements under conditions
h i and their values predicted by the current model:
Recent Developments in the Refinement and Analysis of Crystal Structures
49