included in the atomic model of the crystal structure, for example, when solvent
molecules occupy a void within a crystal structure with no strong interactions to
order them relative to the rest of the structure. Extreme cases are light atom
molecular structures which contribute very little resonant scattering, but which
contain very disordered solvent with some strong resonantly scattering atoms,
such as the chlorine atoms in dichloromethane.
A popular treatment of disordered solvent regions is the SQUEEZE procedure
implemented in PLATON [18]. SQUEEZE computes a contribution to the calculated
scattering from a crystal structure model via a discrete Fourier transform of the
residual electron density from the disordered region. The process is iterative so that
the residual density estimate can benefit from the improved model. However, the
corrected model cannot normally account for resonant scattering from atoms in the
disordered region as the computed electron density has no atomic resonant scattering. Straightforward determination of absolute structure from crystals where the
strongly resonantly scattering atoms are not resolved has therefore not been possible.
Application of a resonant scattering correction to the SQUEEZE contributions
from the disordered regions, weighted by a function of resonant scattering terms of
the missing atoms, has allowed recovery of absolute structure information using both
conventional Flack x refinement and other post-refinement methods [27].
It should be noted that resonant scattering effects are not observed for most atom
types in neutron diffraction; however the dynamical refinement methods used to
model multiple scattering in electron diffraction analysis can determine the absolute
structure of light atom molecules [28].
3 Embedding Information Using Crystallographic
Restraints
Crystallographic restraints are a convenient means to introduce additional information (subsidiary conditions) when fitting crystal structure model parameters to
diffraction data using least squares [29].
For conventional crystallographic least squares, the function to be minimized is
the sum of the squares of the residuals, χ
2
¼
P
i
w i y i À f h i , x
ð
Þ
½
2 , where y i is the
observed value of the i-th data point, measured under conditions h i , and f(h i , x) is the
predicted value of the same observation given a set of model parameters, x. Each
residual is assigned a suitable weight, w i , computed from an estimate of the standard
deviation of the observation. A supplementary sum of residuals allows parameter
restraints to be included as additional weighted equations which relate simple
functions of the model parameters to the expected values of those functions:
Recent Developments in the Refinement and Analysis of Crystal Structures
55
molecules occupy a void within a crystal structure with no strong interactions to
order them relative to the rest of the structure. Extreme cases are light atom
molecular structures which contribute very little resonant scattering, but which
contain very disordered solvent with some strong resonantly scattering atoms,
such as the chlorine atoms in dichloromethane.
A popular treatment of disordered solvent regions is the SQUEEZE procedure
implemented in PLATON [18]. SQUEEZE computes a contribution to the calculated
scattering from a crystal structure model via a discrete Fourier transform of the
residual electron density from the disordered region. The process is iterative so that
the residual density estimate can benefit from the improved model. However, the
corrected model cannot normally account for resonant scattering from atoms in the
disordered region as the computed electron density has no atomic resonant scattering. Straightforward determination of absolute structure from crystals where the
strongly resonantly scattering atoms are not resolved has therefore not been possible.
Application of a resonant scattering correction to the SQUEEZE contributions
from the disordered regions, weighted by a function of resonant scattering terms of
the missing atoms, has allowed recovery of absolute structure information using both
conventional Flack x refinement and other post-refinement methods [27].
It should be noted that resonant scattering effects are not observed for most atom
types in neutron diffraction; however the dynamical refinement methods used to
model multiple scattering in electron diffraction analysis can determine the absolute
structure of light atom molecules [28].
3 Embedding Information Using Crystallographic
Restraints
Crystallographic restraints are a convenient means to introduce additional information (subsidiary conditions) when fitting crystal structure model parameters to
diffraction data using least squares [29].
For conventional crystallographic least squares, the function to be minimized is
the sum of the squares of the residuals, χ
2
¼
P
i
w i y i À f h i , x
ð
Þ
½
2 , where y i is the
observed value of the i-th data point, measured under conditions h i , and f(h i , x) is the
predicted value of the same observation given a set of model parameters, x. Each
residual is assigned a suitable weight, w i , computed from an estimate of the standard
deviation of the observation. A supplementary sum of residuals allows parameter
restraints to be included as additional weighted equations which relate simple
functions of the model parameters to the expected values of those functions:
Recent Developments in the Refinement and Analysis of Crystal Structures
55