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Doctoral Training Partnerships: a range of postgraduate training is funded by the Research Councils. For information on current funding routes, see the common terminology at https://www.ukri.org/apply-for-funding/how-we-fund-studentships/. Training grants may be to one organisation or to a consortia of research organisations. This portal will show the lead organisation only.

Doctoral Training Partnerships: a range of postgraduate training is funded by the Research Councils. For information on current funding routes, see the common terminology at https://www.ukri.org/apply-for-funding/how-we-fund-studentships/. Training grants may be to one organisation or to a consortia of research organisations. This portal will show the lead organisation only.

This project will open new horizons in materials science by developing the high-risk techniques of computational geometry that are needed for an efficient design of solid crystalline materials (crystals). Our project team combines the crucial skills in computer science, mathematics, and crystal chemistry to cause the necessary paradigm shift. Every year, pharmaceutical companies refuse many crystalline drugs during the early stages of trials because of their poor solubility in the human body. Billions of pounds and years of work could be saved if the physical properties of a crystal can be guaranteed by design. The first obstacle is the insufficiency of discrete classifications that were established for periodic crystals already in the 19th century. Now, the Cambridge Crystallographic Data Centre, the industry partner in our project, curates the world's largest collection, the Cambridge Structural Database (CSD), of more than 1.1M existing crystals in a conventional form consisting of an elementary pattern (a motif of atoms) and a linear basis generating the same underlying crystal lattice. This conventional form works in the ideal world where all measurements have infinite precision. However, even tiny atomic displacements (e.g., from measurement error) can break the symmetry of a crystal and make it incomparable with its idealised version. As a result, experimental databases keep growing by accepting near-duplicates of known materials because all available comparison tools are slow or require manually chosen parameters. Recent research from our project team revealed five pairs of suspected duplicates even in the well-curated CSD because our new invariants provably distinguish all generic crystals up to isometry preserving rigid forms of crystals. This project will tackle the unresolved challenge of making the invariants invertible in the sense that any set of invariants gives rise to a well-defined periodic crystal, like a blueprint of a new building which is sufficient for full construction. The simple case of triangles illustrates the challenges of invertibility. The list of side lengths of a triangle is an isometry invariant and can be represented as a point in the positive octant of 3-dimensional space. Not all points with positive coordinates can represent a triangle, but simple inequality conditions define those points that do represent triangles. No equivalent conditions are known for isometry invariants of periodic crystals. The second obstacle is the established paradigm of materials discovery based on trial-and-error of mixing components in the lab or on lottery-type searches, when a huge space of parameters is randomly sampled for subsequent slow optimisation without guarantees of success. What if we could locate the most promising spots in this vast space, where we can confidently find all desired crystals? The exciting and disruptive idea of inverse design is to start from a target property and test only a shortlist of potential candidates. For crystals, a key property is their thermodynamic stability, which is not universally defined for all types of crystals and is currently explored using various approximate energy functions tuned for specific compositions. The increasing complexity of energy functions makes their computation slower without reducing the search space. Imagine that the most promising crystals are peaks of mountains on a new planet: the past way to find such highest peaks is to randomly throw millions of 'bugs' that slowly move to the higher ground, and most of them become stuck on the much more numerous small hills (local maxima), rather than the few highest peaks (global maxima). Following this analogy, our radically new method is to push down the atmospheric clouds and watch the highest peaks appear on a global scale. This 'cloud pushing' will be realised by a simple geometric function whose analytically computable local extrema approximate realistic crystals.

Databases of crystal structures are essential tools for researchers working in the solid state. Initially established as repositories of experimentally determined structures, the large data sets contained within databases, such as the Cambridge Structural Database (CSD), have become the subject of research in their own right through the development of "data mining". The usefulness of such databases is, however, highly dependent on the quality of the data they contain. In the vast majority of cases the structures were obtained via X-ray diffraction (XRD). While XRD is the pre-eminent tool for establishing the three-dimensional structure of crystalline materials, there are areas where XRD studies struggle and some art is required on the part of the crystallographer to establish a correct structure. For instance, hydrogen atoms scatter X-rays very weakly, and fragments with the same (OH vs F) or very similar (Si vs Al) numbers of electrons are very hard to distinguish. In addition, any disruption of the regular ordering of a crystal creates major challenges for structure solution; diffraction is not the natural tool for understanding such "disorder". Historically XRD experts have used measures such as "R factors" to assess how well a proposed structure fits to the experimental data, but ideally independent experimental evidence would be used to verify crystal structures. We and other research groups have shown in recent years that solid-state nuclear magnetic resonance (SS-NMR) can now be used very effectively to distinguish between different possible crystalline structures. Developments in quantum chemistry (mostly notably through Density Functional Theory) allow NMR spectra to be calculated with excellent precision. Since the NMR spectrum is sensitive to very small changes in the local structure - deviations of the order of a picometre (10^-12 m) will change the spectrum measurably - even small imperfections in a crystal structure solution can be identified. Moreover different types of "disorder" e.g. due to the motion of atoms or irregular atomic positioning, have clear and distinct effects on the NMR spectrum. This proposal seeks to develop systematic approaches to the validation of crystal structures via solid-state NMR and computational chemistry. We will establish which NMR experiments are required in order to distinguish crystal structure solutions and also to "validate" a structural solution. This will involve the creation of "NMR confidence parameters" which will measure the extent to which a structure is compatible with the NMR data available, and the effectiveness of these parameters will be verified against more traditional diffraction-based tools. By taking a systematic approach, we will be able to show how NMR can be used to resolve the different types of structural ambiguity and show the value of NMR as a complement to conventional diffraction-based studies.