3.1 Displacement Parameter Restraints
The mean square displacement of an atom from its mean position is a commonly
determined quantity in small-molecule single-crystal refinements and will sometimes be a single parameter defining the mean square displacement of an atom from
its position (isotropic displacement), or a 3x3 covariance matrix (six independent
parameters) defining the mean square displacements of an atom in three dimensions
(anisotropic displacement).
Isotropic displacement parameters, where used, can be restrained to be equivalent
to neighbouring isotropic displacement parameters, or to a ‘U equiv ’ value when the
neighbouring atom has an anisotropic set of displacement parameters. This type of
restraint is most commonly used to provide sensible values for the isotropic displacement parameters of a hydrogen atom attached to an organic ligand or molecule.
A multiplier of 1.2–1.3 may be used to reflect the expectation that a H atom will have
a larger displacement than a heavier atom to which it is attached due to the difference
in mass.
Anisotropic displacement parameters (ADPs) often constitute approximately
two-thirds of the total number of refined parameters in a structure analysis, yet
compared to the number of geometrical restraints available, there are relatively
few displacement parameter restraints available. Of those in routine use, the
Hirshfeld rigid-bond restraint [30] is the most physically valid and useful. It is
based on the Hirshfeld criterion that the components of displacement parameters
along the direction of a bond connecting two atoms should be approximately equal.
It is also routinely applied to the direction between 1,3 connected atoms due to the
relative rigidity of bond angles in molecules.
A further relationship commonly imposed upon displacement parameters of
connected or overlapping disordered atoms is to set corresponding parameters of
the anisotropic displacement matrix to be equal. This is also often applied as a
constraint, but in both cases has the effect of making two atoms have the same or
approximately the same mean square displacements from their average positions.
This type of relationship is not physically meaningful and is rarely empirically
observed between bonded atoms, so it is used sparingly, though it is occasionally
useful for relating displacements of partially occupied atoms that are located on the
same site or very close together in the unit cell representation of the structure.
A significant development in displacement parameter restraints was the introduction of the RIGU restraint, implemented in SHELXL [31]. This introduces the
additional well-established empirical relationship between the covariances of displacement parameters perpendicular to the vector between two bonded atoms
(Fig. 4b).
Both the Hirshfeld rigid-bond restraint and the RIGU enhancement make use of a
restraint which can be readily expressed in a Cartesian coordinate system, which is
conveniently aligned with the local geometry – in this case the Z axis is aligned with
the interatomic vector. To write general expressions for restraints using similar
coordinate systems, a transformation operation T is defined which transforms from
Recent Developments in the Refinement and Analysis of Crystal Structures
57
The mean square displacement of an atom from its mean position is a commonly
determined quantity in small-molecule single-crystal refinements and will sometimes be a single parameter defining the mean square displacement of an atom from
its position (isotropic displacement), or a 3x3 covariance matrix (six independent
parameters) defining the mean square displacements of an atom in three dimensions
(anisotropic displacement).
Isotropic displacement parameters, where used, can be restrained to be equivalent
to neighbouring isotropic displacement parameters, or to a ‘U equiv ’ value when the
neighbouring atom has an anisotropic set of displacement parameters. This type of
restraint is most commonly used to provide sensible values for the isotropic displacement parameters of a hydrogen atom attached to an organic ligand or molecule.
A multiplier of 1.2–1.3 may be used to reflect the expectation that a H atom will have
a larger displacement than a heavier atom to which it is attached due to the difference
in mass.
Anisotropic displacement parameters (ADPs) often constitute approximately
two-thirds of the total number of refined parameters in a structure analysis, yet
compared to the number of geometrical restraints available, there are relatively
few displacement parameter restraints available. Of those in routine use, the
Hirshfeld rigid-bond restraint [30] is the most physically valid and useful. It is
based on the Hirshfeld criterion that the components of displacement parameters
along the direction of a bond connecting two atoms should be approximately equal.
It is also routinely applied to the direction between 1,3 connected atoms due to the
relative rigidity of bond angles in molecules.
A further relationship commonly imposed upon displacement parameters of
connected or overlapping disordered atoms is to set corresponding parameters of
the anisotropic displacement matrix to be equal. This is also often applied as a
constraint, but in both cases has the effect of making two atoms have the same or
approximately the same mean square displacements from their average positions.
This type of relationship is not physically meaningful and is rarely empirically
observed between bonded atoms, so it is used sparingly, though it is occasionally
useful for relating displacements of partially occupied atoms that are located on the
same site or very close together in the unit cell representation of the structure.
A significant development in displacement parameter restraints was the introduction of the RIGU restraint, implemented in SHELXL [31]. This introduces the
additional well-established empirical relationship between the covariances of displacement parameters perpendicular to the vector between two bonded atoms
(Fig. 4b).
Both the Hirshfeld rigid-bond restraint and the RIGU enhancement make use of a
restraint which can be readily expressed in a Cartesian coordinate system, which is
conveniently aligned with the local geometry – in this case the Z axis is aligned with
the interatomic vector. To write general expressions for restraints using similar
coordinate systems, a transformation operation T is defined which transforms from
Recent Developments in the Refinement and Analysis of Crystal Structures
57