Struct Bond (2020) 185: 43–68
https://doi.org/10.1007/430_2020_76
# Springer Nature Switzerland AG 2020
Published online: 24 October 2020
Recent Developments in the Refinement
and Analysis of Crystal Structures
Richard I. Cooper
Contents
1 Introduction and Background of Crystal Structure Refinement and Analysis . . . . . . . . . . . . . . 44
1.1 Refinement and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.2 Practical Elements of Refinement: The IAM Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.3 Linear Algebra Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
1.4 Computing Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2 Determination of Absolute Configuration During Crystal Structure Refinement . . . . . . . . . . . 52
2.1 Highly Disordered Resonant Scatterers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Embedding Information Using Crystallographic Restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3.1 Displacement Parameter Restraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Validation of Structure Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.1 Leverage Analysis for Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.2 Validation with DFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5 Horizons: Analysis of Multiple Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Abstract Crystal structure refinement and analysis is a powerful method for determination of crystal structures and finds widespread application in determination of
structures of crystals of small molecules and frameworks at atomic resolution. The
independent atom model is used to describe atomic scattering for routine use, while
more accurate aspherical scattering factors are increasingly available. The structure
factor is presented as the Fourier transform of convolutions of scattering and
probability densities in the crystal structure to clarify how aspherical scattering
factors and alternative displacement probabilities can be introduced into refinement
methods. Non-linear least squares fitting of the crystal structure parameters in the
structure factor equations is described using matrix algebra notation which enables
simple derivation of the extensions required for discussion of crystallographic
restraints and leverage analysis. Finally, combined analysis of multiple single-crystal
R. I. Cooper (*)
Chemical Crystallography, Department of Chemistry, University of Oxford, Oxford, UK
e-mail: richard.cooper@chem.ox.ac.uk