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Recent Developments in the Refinement and Analysis of Crystal Structures Richard I. Cooper

Contents 1 Introduction and Background of Crystal Structure Refinement and Analysis 1.1 Refinement and Analysis 1.2 Practical Elements of Refinement: The IAM Model 1.3 Linear Algebra Description 1.4 Computing Power 2 Determination of Absolute Configuration During Crystal Structure Refinement 2.1 Highly Disordered Resonant Scatterers 3 Embedding Information Using Crystallographic Restraints 3.1 Displacement Parameter Restraints 4 Validation of Structure Refinements 4.1 Leverage Analysis for Validation 4.2 Validation with DFT 5 Horizons: Analysis of Multiple Experiments 6 Summary References

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: [email protected]

R. I. Cooper

experiments is discussed highlighting the potential of refinement tools to extract useful information from joint X-ray and neutron data and from mixed ground-state and excited-state X-ray data from pump-probe experiments. Keywords Atomic displacement parameters · Crystal structure refinement · Least squares · Leverage analysis · Restraints · Structure factor · Validation

Abbreviations
ADP DFT Fc2 FFT Fo2 FT IAM r.m.s. SCF TAAM TLS u(x) λ σ2(Fo2)

Convolution operator Anisotropic displacement parameter Density functional theory Calculated structure factor squared FAST Fourier transform Observed structure factor squared Fourier transform Independent atom model Root mean square Self-consistent field Transferable aspherical atom model Translation-libration-screw Estimated uncertainty of a model parameter, x Wavelength Estimated variance of the observed structure factor

1 Introduction and Background of Crystal Structure Refinement and Analysis X-ray crystal structures have been determined from a diverse range of materials, spanning metals and minerals, through small-molecule organic and metal-organic compounds, including covalent and metal-organic extended frameworks, to crystallized proteins and even virus particles. T