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Combinatorial analysis of the energy graphs under operations of composition (Reference: Potapov LRC120)


Project Description

This PhD position is funded by the Leverhulme Centre for Functional Materials Design that have been recently organized at the University of Liverpool. This Centre is one of only 4 such centres funded in the UK and brings together chemical knowledge with state-of-the-art computer science and automated technologies to develop a new approach to revolutionise the design of functional materials at the atomic scale.

The research questions that we are going to consider within the current PhD project “Combinatorial analysis of the energy graphs under operations of composition” will be related to several areas of Theoretical Computer Science (e.g. Algorithms, Computational Models, Computational Complexity), Discrete Mathematics (e.g. Combinatorics on Graphs, Computational Geometry) and partially will cover some questions in Computational Chemistry (e.g. Crystal Structure Prediction).

The prediction of crystal structures with lowest potential energy, and therefore most stable, configuration of the atoms is still a largely unresolved challenging computational problem. The main complexity comes from combinatorial explosion in the number of possibilities and unpredictable nature of non-convex energy landscape. The current computational methods mainly fail for providing reliable low energy structures prediction for atomic structures containing more than 100-150 atoms per the unit cell, non-periodic and quasi-periodic structures and become even useless with significant increase of potentially available types of atoms for new structures.

The current variants of simulated annealing, genetic algorithms, topological modelling methods, molecular packing approaches, predictions at finite temperature and structure models by analogy have different limitations. The most crucial one is that they cannot reveal the principals of formation for the low energy structure, but instead find some plausible candidates and propose their own order for energy landscape exploration. The usual approach for minimizing the potential energy of the system with respect to the structural parameters of these models is currently based on methods which start form empirically based information on the positions of the atoms in the unit cell and typically moves atoms to nearest location, swapping atoms and changing the shape of the unit cell. A desired fully predictive method should not include experimentally known unit-cell dimensions, but may use predefined cell dimensions without the specification of the atomic contents of the unit cell (atomic coordinates). Moving forward towards the design of fully predictive methods the most essential task is to understand the properties of the energy landscape under various changes.

In this project we propose to analyse real structures by energy graph abstractions embedded into higher-dimensional spaces and representing atomic structures and potentially use it for the algorithmic design of new low energy structures. Initially will analyse/evaluate the dynamics of simpler “cost functions” (based, for example, on bond lengths, coordination numbers, etc) and later potential energy function under standard theoretical operations of composition (insertion, deletion, substitution) as well as under iterative application of formal self-modifying rewriting rules. In general we want to explore the behavior of the energy landscape function under modification of the crystal structures with adaptive graph/grammar (probabilistic or deterministic) rewriting systems for growing new structures with desired properties, exploration of the energy landscape with different 3D tessellations covering periodic, quasi-periodic and non-periodic structures.

This work can help with understanding the principals of low energy formation and the design of a more universal and principally new search procedures in contrast to existing exploration methods. The research topics on this PhD project will be within several areas of Theoretical Computer Science, Discrete Mathematics and partially cover some questions in Computational Chemistry which should also provide real life motivation for several mathematical abstractions.

Qualifications: We welcome talented and highly motivated candidates with good first degree (BSc or MSc) in Computer Science, Mathematics or closely related subject. A programming experience and/or previous research experience would be a distinct advantage though it not essential.

Please apply by completing the online postgraduate research application form here: https://www.liverpool.ac.uk/study/postgraduate-taught/applying/online/
Please ensure you quote the following reference on your application: Combinatorial analysis of the energy graphs under operations of composition (Reference: Potapov LRC120)

Funding Notes

The award is primarily available to students resident in the UK/EU and will pay full tuition fees and a maintenance grant for 3 years (£14,553 pa in 2017/18). Non-EU nationals are not eligible for this position and applications from non-EU candidates will not be considered unless you have your own funding.

Please note that this is a PhD Graduate Teaching Assistantships (GTA) and as such will have teaching commitments and contractual obligations to teaching associated with it.

References

Scott M. Woodley and Richard Catlow. Crystal structure prediction from first principles.
Nature Materials 7, 937–946 (2008) doi:10.1038/nmat2321 http://www.nature.com/articles/nmat2321

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