tested limits on some statistical descriptors of the least squares fit (R-factor, goodness-of-fit, scattering density statistics), as well as checking for reasonable chemical
and physical consistency of the crystal structure model such as balanced charges,
bond lengths and angles, intermolecular interactions, coordination of ions, hydrogen
bond geometry and displacement parameter consistency.
While these checks provide a degree of confidence in the correctness of a crystal
structure determination, they are no substitute for a thorough understanding of a
crystal structure model and its fit to the data. Simple plots of the least squares
residual can reveal systematic errors and outliers in the data, which may not be
caught by a one-size-fits-all service such as checkCIF. In particular Henn and Meindl
have demonstrated that common crystallographic statistics, including the goodnessof-fit measure, may obscure systematic errors in data and have proposed alternative
measures that are more robust [41, 42] and fractal dimension plots of residual
scattering density to test for systematic biases in the data or model [43]. These
developments have been made in the context of analysis of charge density data sets,
but can equally be applied to routine IAM models to detect uncorrected data or
under-parameterized models.
Complementary to these statistical checks are some methods which are not as
simple to automate, either due to a relatively high computational cost or because the
results require some careful inspection and interpretation. Approaches of both types
are outlined below: leverage analysis for testing the appropriate application of
crystallographic restraints in refinement and application of DFT calculations to
crystal structure validation.
4.1 Leverage Analysis for Validation
A crystal structure is described by a set of model parameters, the optimal values of
which are found using least squares fitting to experimental diffraction measurements
and empirical restraints. In addition, the method yields a variance-covariance matrix
which, when appropriately scaled, gives an estimate of the variance of each parameter, usually reported as a standard uncertainty. This matrix can be used to derive
estimates of standard uncertainties of functions of the model parameters, such as
bond lengths, angles, etc., by taking into account the covariance between parameters.
The leverage of each observation is defined as the influence that it has – through
the structural model – on its own fitted value. For example, if the magnitude of one
observation, F obs , in a structure refinement is doubled, the leverage tells us what will
happen to the value of F calc . At one extreme a leverage of zero indicates that the
model cannot respond to the change in F obs in a way that will improve the fit for that
reflection. Such reflections are rare but provide a useful internal check on the quality
of the model. At the other extreme, a value of 1 indicates that the F calc will follow the
change in F obs exactly, meaning that no other observations have any influence on its
fitted value.
60
R. I. Cooper
and physical consistency of the crystal structure model such as balanced charges,
bond lengths and angles, intermolecular interactions, coordination of ions, hydrogen
bond geometry and displacement parameter consistency.
While these checks provide a degree of confidence in the correctness of a crystal
structure determination, they are no substitute for a thorough understanding of a
crystal structure model and its fit to the data. Simple plots of the least squares
residual can reveal systematic errors and outliers in the data, which may not be
caught by a one-size-fits-all service such as checkCIF. In particular Henn and Meindl
have demonstrated that common crystallographic statistics, including the goodnessof-fit measure, may obscure systematic errors in data and have proposed alternative
measures that are more robust [41, 42] and fractal dimension plots of residual
scattering density to test for systematic biases in the data or model [43]. These
developments have been made in the context of analysis of charge density data sets,
but can equally be applied to routine IAM models to detect uncorrected data or
under-parameterized models.
Complementary to these statistical checks are some methods which are not as
simple to automate, either due to a relatively high computational cost or because the
results require some careful inspection and interpretation. Approaches of both types
are outlined below: leverage analysis for testing the appropriate application of
crystallographic restraints in refinement and application of DFT calculations to
crystal structure validation.
4.1 Leverage Analysis for Validation
A crystal structure is described by a set of model parameters, the optimal values of
which are found using least squares fitting to experimental diffraction measurements
and empirical restraints. In addition, the method yields a variance-covariance matrix
which, when appropriately scaled, gives an estimate of the variance of each parameter, usually reported as a standard uncertainty. This matrix can be used to derive
estimates of standard uncertainties of functions of the model parameters, such as
bond lengths, angles, etc., by taking into account the covariance between parameters.
The leverage of each observation is defined as the influence that it has – through
the structural model – on its own fitted value. For example, if the magnitude of one
observation, F obs , in a structure refinement is doubled, the leverage tells us what will
happen to the value of F calc . At one extreme a leverage of zero indicates that the
model cannot respond to the change in F obs in a way that will improve the fit for that
reflection. Such reflections are rare but provide a useful internal check on the quality
of the model. At the other extreme, a value of 1 indicates that the F calc will follow the
change in F obs exactly, meaning that no other observations have any influence on its
fitted value.
60
R. I. Cooper