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Hot Topics in Contemporary Crystallography

by Croatian Association of Crystallographers

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Šibenik, Croatia, May, 10^th to 15^th, 2014

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Croatian Association of Crystallographers

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H.B. Bürgi

The whole is more than the sum of its parts

Introduction to all three lectures

Open or download lecture slides!

The history of crystal structure determination by X-ray or neutron diffraction is the story of an unprecedented scientific success: within 100 years the method has not only become one of the most important analytical techniques, but is now so well understood that it could be coded – almost completely – into suites of computer software that do the job more or less automatically for biomolecules, newly synthesized chemical compounds and materials. It is due to this success that ‘Crystallography’ is often used as a synonym for X-ray crystal structure determination, although the field of crystallography is of course much broader. It is best seen as an intersection of mathematics, mineralogy, physics, chemistry and molecular biology.

Does all this success mean that crystallography is a mature branch of science in need only of periodically adapting experiments and their interpretations to the new technological developments, but not likely to produce important breakthroughs? Yes and no! YES, because routine crystal structure determination from Bragg reflections produces reliable structural models of nearly any molecule or chemical compound for which well-ordered crystals can be obtained. And NO, because in crystallographic research, as in other scientific research, it is becoming increasingly clear that the full potential of structure determination can only be realized by combining it with biological, physical or physicochemical experiments and interpretations. And indeed, much of today’s ground-breaking crystallographic research is done by solid-state physicists and physical chemists. Such infusions of fresh blood are necessary and healthy in a field that is 100 years old. There is still a universe to discover for all those interested not just in the ideal, motionless average structure, but rather in crystal dynamics, real crystal structures with their defects, faults and disorder, aspects which are often a prerequisite for interesting and useful materials properties. New and efficient methods are badly needed to uncover these secrets.

In this series of three lectures I will try to illustrate the information obtainable from combinations of crystal structure determinations with three different concepts and ideas from physical chemistry, namely transition state theory, vibrational motion and crystalline disorder. Some of the examples date back 40 years, others are taken from current research. However, in all of them the important results could not have been obtained without looking at one or even several standard crystal structure determinations from a physical or chemical point of view.

LECTURE 1: Structure-structure and structure-energy correlations

A summary of the ideas presented in this lecture and additional references are given in H.B. Bürgi, Faraday Discuss., 122 (2002) 41–63, entitled ‘What we can learn about fast chemical processes from slow diffraction experiments’.

- Structure-structure correlation and reaction paths. The course of chemical reactions is often represented highly schematically with the help of a reaction path diagram correlating reaction and transition state energies with a reaction coordinate summarizing the structural changes during the reaction. What is usually missing in such discussions is a detailed characterization of the reaction path in terms of the changes in interatomic distances, angles and energy. The first example of such correlations will be described in this lecture.

Several fragments X[Cd(SR)₃]Y were available from a series of service structure determinations raising the question whether there is any trend among their distances and angles. After correlating pretty much everything against everything, it became apparent that as the X-Cd distance became shorter, the Cd-Y distance became longer and the Cd(SR3) umbrella inverted. This correlation is reminiscent of Walden inversion at carbon. This idea has then been generalized to other reactions, e.g. pericyclic ring closure reactions and nucleophilic attack to carbonyl groups. These observations are interpreted in terms of a valley in an energy hyper-surface of the fragment leading up to a transition state. A perturbation by substitutents or crystal packing forces will distort the fragment of interest mostly along the valley, but only a little bit to its sides. The retrieval of fragments and their structures is now easily done from the CSD and is much simpler than in the mid 70s. A systematic search for correlations among structural parameters can be done by Principal Component Analysis.

- Structure-energy correlations. In order to also characterize the energetic dimension of a reaction path non-crystallographic information has to be included. For a series of acetals it was found that elongation of the scissile C-O bond by a mere 0.05 Å increases the reaction rate for acetal hydrolysis by a factor of 10¹⁰, corresponding to a decrease of the TS energy by ~15 kcal/mol. How can the relationship between the small structural and a huge kinetic effect be explained and and what follows for the understanding of catalysis which produces rate accelerations of similar magnitudes? Given a series of ground state (GS) structures and their associated transition state (TS) or activation energies from kinetic experiments, a simple model of an energy-reaction path diagram can be constructed and parameterized with experimental data. The model reproduces and predicts how much and why a small distortion of the GS towards the TS lowers the activation energy substantially.

- Structure correlation and Atomic Displacement Parameters (ADPs). The structure-structure correlation idea can also be used to distinguish soft from stiff types of molecular distortions and thus get qualitative information on their vibrational behavior. The idea has been applied to the sesquisiloxanes R₁₀Si₁₀O₁₅. By superimposing the Si atoms of the molecules with known crystal structures and using some symmetry arguments, the oxygen atoms are found to be distributed along the directions of the largest oxygen mean square displacements. The relationship between ADPs and molecular vibrations is discussed from a different point of view in the next lecture.

LECTURE 2: Atomic Displacement Parameters (ADPs), crystal dynamics, thermodynamics and disorder

A summary of the ideas presented in this lecture and additional references are given in H.B. Bürgi, S.C. Capelli, Helv. Chim. Acta, 86 (2003) 1625-1640, entitled ‘Getting More out of Crystal-Structure Analyses’.

All we can get from a crystal structure analysis is information on the motion of individual atoms. Nothing can be learned about the correlations of atomic motions without making assumptions. The most familiar of these is the rigid body model. It assumes that all atoms move in a correlated way by performing rigid body translational and librational oscillations. Their amplitudes, the components of the TLS tensor, are then derived with a least-squares algorithm from the observed ADPs assuming that internal molecular deformation motions contribute negligibly to the observed ADPs. But is this true? Many molecules show various degrees of flexibility that systematically fudge the TLS tensor components and everything that is derived with their help, e.g. distance corrections, libration and translation frequencies.

Often atoms are disordered over two positions which are too close to be resolved given the resolution limit of the experimental diffraction data. Instead an average position is refined. In such cases the ADPs have a contribution that is due to the disorder and does not only represent motion. How can the dynamic contributions be distinguished from the disorder ones? An important example is provided by benzene. The experimental ADPs at 15 K are equally compatible with a regular, sixfold symmetric benzene molecule and a disordered, centrosymmetric superposition of two cyclohexatriene molecules with only threefold symmetry.

The distinction between the internal and external molecular motion as well as between motion and disorder is possible from an analysis of the T-dependence of the ADPs with the help of the normal mode analysis well known from the interpretation of infrared and Raman spectroscopic data, but formulated in terms of ADPs rather than vibration frequencies. Such an analysis results in vibration frequencies, force constants and atomic displacement patterns (eigenvectors).

With the same analysis it is possible to study isotope effects on the motion in crystals. by combining translation and libration frequencies with ab initio calculations of the molecular deformation frequencies thermodynamic quantities such as c_V(T), H(T), S(T) and G(T) may be obtained and compared for different polymorphs.

LECTURE 3: Beyond Bragg diffraction, disorder and diffuse scattering

A summary of the ideas presented in this lecture and additional references are given in: Th. Weber, Chimia, 68 (2014) 60-65 entitled ‘Crystallography beyond the Bragg peaks’; H.B. Bürgi, J. Hauser, Vickie Lynch, Trans. Am. Cryst. Assoc. 41 (2010) entitled ‘Beyond Single-Crystal Structure Determination, Interpretation of 3D Disorder Diffuse Scattering’.

The models of many crystal structures make use of highly unphysical and unchemical parameters: fractional atomic site occupation factors. However, no unit cell ever contains a fraction of an atom; there is either one or none, but we don’t know which unit cells do or do not contain the atom in question. This limitation is typical of Bragg structures which only provide the average over all the many unit cells in the sample, on the order of 10¹⁵ may be, and not their full 3D arrangement, i.e. the real crystal structure. We generally call structures with fractional atomic occupancies ‘disordered’. The phenomenon of disorder is not just a nuisance. It is often responsible for whatever interesting or useful property a material may have. Alloys and high T_c superconductors are among the better known examples.

Disorder and crystal faults do affect the diffraction pattern. They produce diffuse scattering between the Bragg reflections. The thermal diffuse scattering, usually concentrated around the Braggs accounts for the fact that an x-ray photon never ever sees a perfectly periodic crystal, but an arrangement of atoms which are essentially randomly displaced from their average position due to their thermal motion. More permanent deviations from perfect periodicity include occupational and displacive disorder, i.e. cases where not all equivalent positions are occupied by the same chemical species or cases where a given chemical species can occupy several different, but mutually exclusive positions. The scattering from such crystals can be found pretty much anywhere in reciprocal space. Thus, for a detailed understanding of a crystal structure the total scattering is needed, Bragg as well as diffuse.

The intensity of diffuse scattering varies from relatively weak for lightly disordered structures to quite strong for heavily disordered materials. Measuring diffuse scattring accurately therefore requires high beam and low background intensities. Synchrotron beam lines equipped with pixel detectors are well suited for diffuse scattering experiments. The interpretation of such data, often millions of data points, requires modeling the disorder in terms of many unit cells, sometimes hundreds of thousands, differing in atomic occupations or positions. The necessary computational effort is substantial, but no longer insurmountable given the steady increase in computer performance.

The results of such efforts are experimentally determined models of disordered crystals that may serve as a structural basis for interpreting materials properties. Examples of diffuse scattering patterns and their interpretation in terms of disordered crystals will be discussed.

The workshop is generously supported by: