single-crystal case, we require multi-model refinements which compute the scattering of, and are fitted to, different types or wavelengths of radiation.
Fitting independent models to separate data sets is equivalent to analysing the
data sets in isolation; however crystallographic constraints and restraints can be used
to define relationships and link common parameters between two or more models.
For example, in a joint X-ray and neutron study, a common set of coordinates of
non-hydrogen atoms can be used to model scattering for both sets of experimental
data, while separate sets of hydrogen atoms can be used for each radiation type.
Nevertheless, the hydrogen atom positions are not entirely independent: each pair of
hydrogen atom positions can be constrained to lie on a common vector with the atom
they are bonded to.
The approach can also be applied to modelling excited-state geometrical transformations in single crystals: a large part of the geometry of the structure is common
to both ground state and excited state, and the diffraction data from the excited state,
obtained by pump-probe methods, is often noisy and insufficient on its own to
support full refinement of the excited-state structure.
The mathematical tools required to carry out these types of analyses are embedded within many currently available packages, but demonstration of their use and
developments of protocols and tools for applying them, appropriate to the nature of
the problem, are still required in order to bring them into mainstream use.
6 Summary
This chapter details some recent developments in the refinement and analysis of
small-molecule crystal structures. The IAM model coupled with harmonic atom
displacements is the cornerstone of small-molecule X-ray analysis and will undoubtedly remain important for standard structure determination and structures for a long
time, especially for structures with atom environments for which aspherical scattering factors are not known and cannot easily be calculated. Section 1.2 shows how
alternative scattering models and probability distributions can be incorporated into a
routine refinement.
Section 1.4 highlights the potential for crystallographic studies to take advantage
of improvements in computing power, not only for speeding up analyses but
potentially to test alternative hypotheses for models, for example, in building and
comparing multiple models of disordered chemical groups and molecules.
Recent developments in absolute structure determination from resonant scattering
are outlined in Sect. 2, including details of the determination of the Flack parameter
by analysis of Friedel pairs of X-ray reflections, instead of least squares fit with the
other model parameters.
Sections 3–5 outline developments within the framework of traditional crystallographic constraints and restraints to provide more physically meaningful crystal
structure models. Applications include restraining atomic displacement parameters
and linking together models which are fitted against different sources of
64
R. I. Cooper
Fitting independent models to separate data sets is equivalent to analysing the
data sets in isolation; however crystallographic constraints and restraints can be used
to define relationships and link common parameters between two or more models.
For example, in a joint X-ray and neutron study, a common set of coordinates of
non-hydrogen atoms can be used to model scattering for both sets of experimental
data, while separate sets of hydrogen atoms can be used for each radiation type.
Nevertheless, the hydrogen atom positions are not entirely independent: each pair of
hydrogen atom positions can be constrained to lie on a common vector with the atom
they are bonded to.
The approach can also be applied to modelling excited-state geometrical transformations in single crystals: a large part of the geometry of the structure is common
to both ground state and excited state, and the diffraction data from the excited state,
obtained by pump-probe methods, is often noisy and insufficient on its own to
support full refinement of the excited-state structure.
The mathematical tools required to carry out these types of analyses are embedded within many currently available packages, but demonstration of their use and
developments of protocols and tools for applying them, appropriate to the nature of
the problem, are still required in order to bring them into mainstream use.
6 Summary
This chapter details some recent developments in the refinement and analysis of
small-molecule crystal structures. The IAM model coupled with harmonic atom
displacements is the cornerstone of small-molecule X-ray analysis and will undoubtedly remain important for standard structure determination and structures for a long
time, especially for structures with atom environments for which aspherical scattering factors are not known and cannot easily be calculated. Section 1.2 shows how
alternative scattering models and probability distributions can be incorporated into a
routine refinement.
Section 1.4 highlights the potential for crystallographic studies to take advantage
of improvements in computing power, not only for speeding up analyses but
potentially to test alternative hypotheses for models, for example, in building and
comparing multiple models of disordered chemical groups and molecules.
Recent developments in absolute structure determination from resonant scattering
are outlined in Sect. 2, including details of the determination of the Flack parameter
by analysis of Friedel pairs of X-ray reflections, instead of least squares fit with the
other model parameters.
Sections 3–5 outline developments within the framework of traditional crystallographic constraints and restraints to provide more physically meaningful crystal
structure models. Applications include restraining atomic displacement parameters
and linking together models which are fitted against different sources of
64
R. I. Cooper