PhD Dissertation Proposal Defense: Alexandra Camero Bejarano, Crystal Network Comparison
Content
Speaker
Description
The molecular structure of a crystal comprises a set of atoms, or a motif, arranged infinitely on a lattice. Crystallographic Information Files (CIFs) are commonly used to store crystal data. Determining whether the information in two CIFs represents the same crystal is an open problem. In particular, this happens because there are infinitely many ways to finitely describe a single crystal. In crystallography databases, this means queries looking for "equivalent" CIFs can return incomplete or incorrect answers. Available comparison tools are based only on crystal point sets. In contrast, this thesis will use topological information captured by bonds to compare crystal networks. The aim is to contribute theoretical algorithms and software for crystallographic applications.
The anticipated final deliverable is a web tool that takes CIF inputs to determine if the crystal networks they describe are the same. The tool models crystals as periodic frameworks (i.e., graphs with fixed edge lengths) where atoms appear as vertices and bonds as edges. Two crystals are equivalent if their periodic frameworks are isomorphic and their crystal point sets are isometric. The software will be validated using real data from the Crystallography Open Database. Several theoretical questions must be addressed prior to developing the tool.
The main theoretical contribution is expected to be an algorithm for comparing periodic frameworks through Marked Quotient Graphs (MQGs). An MQG is a finite descriptor of a periodic structure which includes a subset of representative vertices and edges. Edge annotations record periodicity information by describing connections between lattice cells. There are infinitely many choices of MQGs for a periodic framework. As unmarked graphs, two MQGs that represent the same periodic structure may not be isomorphic.
The first part of this thesis focuses on 2D periodic frameworks. We developed visual tools for constructing the periodic structure of a given MQG, and generating distinct MQGs describing the same structures. These pedagogical resources will help explain relevant mathematical concepts to chemists. We have also made progress on the 2D comparison problem. The simpler problem can elucidate the more complex 3D case. The primary remaining goal is to extend our work to 3D.
Advisor
Ileana Streinu