On chirality of toroidal embeddings of polyhedral graphs

We investigate properties of spatial graphs on the standard torus. It is known that nontrivial embeddings of planar graphs in the torus contain a nontrivial knot or a non-split link due to [2, 3]. Building on this and using the chirality of torus knots and links [9, 10], we prove that the nontrivial embeddings of simple 3-connected planar graphs in the standard torus are chiral. For the case that the spatial graph contains a nontrivial knot, the statement was shown by Castle et al. [5]. We give an alternative proof using minors instead of the Euler characteristic. To prove the case in which the graph embedding contains a nonsplit link, we show the chirality of Hopf ladders with at least three rungs, thus generalizing a theorem of Simon [12]

Some topics in topological graph theory motivated by chemistry

Topological graph theory is a field of geometric topology. The mathematical objects of interest are embeddings of graphs in 3-space. The image is a so called spatial graphs. A spatial graph can be seen as a generalised knot. In addition to the resulting richer structure, questions about spatial graphs can also be motivated from other natural sciences. In particular, there are many applications to chemistry since molecules can be modelled as graphs embedded in R3. This text consists of two parts. Both cover pure mathematical problems which are motivated by questions from synthetic chemistry. The aim is to find materials with new chemical/physical properties. The structural richness of entangled, catenated and knotted structures has long been a target for synthetic chemistry. The first part investigates the behaviour of entanglements in spatial graphs that are not caused by knotted or linked subgraphs with respect to the surfaces the spatial graphs embed in. We show that all nontrivial embeddings of abstractly planar graphs on the torus contain either a nontrivial knot or a nonsplit link. It follows that ravels do not embed on the torus which was conjectured by Castle, Evans and Hyde in 2008. Our results provide general insight into properties of molecules that are synthesised on a torus. The second part predicts the topologically possible braided structures of 1-dimensional coordination polymers. Given the common way of synthesising via self-assembly, these coordination polymers can be modelled by pure braids with n rigidly congruent strands up to chirality. We discuss the properties and symmetries of 1-dimensional coordination polymers with up to five strands. This project is part of a collaboration with Prof D. M. Proserpio, Dr I. A. Baburin and Dr F. D.-H. Lau.Open Acces

Energy-based descriptors for photo-catalytically active metal-organic framework discovery

Metal–organic frameworks (MOFs) consist of metal nodes that are connected by organic linkers. They are thus highly chemically tunable materials given the broad range of potential linkers and nodes that can be chosen for their synthesis. Their tunability has recently sparked interest in the development of new MOF photo-catalysts for energy-related applications such as hydrogen (H2) evolution and CO2 reduction. The sheer number of potentially synthesizable MOFs requires defining descriptors that allow prediction of their performance with this aim. Herein we propose a systematic computational protocol to determine two energy-based descriptors that are directly related to the performance of a MOF as a photocatalyst. These descriptors assess the UV-vis light absorption capability and the band energy alignment with respect to redox processes and/or co-catalyst energy levels. High-throughput screening based on cost-effective computations of these features is envisioned to aid the discovery of new promising photoactive systems

Accurate Characterization of the Pore Volume in Microporous Crystalline Materials

Pore volume is one of the main properties for the characterization of microporous crystals. It is experimentally measurable, and it can also be obtained from the refined unit cell by a number of computational techniques. In this work, we assess the accuracy and the discrepancies between the different computational methods which are commonly used for this purpose, i.e, geometric, helium, and probe center pore volumes, by studying a database of more than 5000 frameworks. We developed a new technique to fully characterize the internal void of a microporous material and to compute the probe-accessible and-occupiable pore volume. We show that, unlike the other definitions of pore volume, the occupiable pore volume can be directly related to the experimentally measured pore volumes from nitrogen isotherms

SELFIES and the future of molecular string representations

Artificial intelligence (AI) and machine learning (ML) are expanding in popularity for broad applications to challenging tasks in chemistry and materials science. Examples include the prediction of properties, the discovery of new reaction pathways, or the design of new molecules. The machine needs to read and write fluently in a chemical language for each of these tasks. Strings are a common tool to represent molecular graphs, and the most popular molecular string representation, Smiles, has powered cheminformatics since the late 1980s. However, in the context of AI and ML in chemistry, Smiles has several shortcomings—most pertinently, most combinations of symbols lead to invalid results with no valid chemical interpretation. To overcome this issue, a new language for molecules was introduced in 2020 that guarantees 100% robustness: SELF-referencing embedded string (Selfies). Selfies has since simplified and enabled numerous new applications in chemistry. In this perspective, we look to the future and discuss molecular string representations, along with their respective opportunities and challenges. We propose 16 concrete future projects for robust molecular representations. These involve the extension toward new chemical domains, exciting questions at the interface of AI and robust languages, and interpretability for both humans and machines. We hope that these proposals will inspire several follow-up works exploiting the full potential of molecular string representations for the future of AI in chemistry and materials science

Cutting Materials in Half: A Graph Theory Approach for Generating Crystal Surfaces and Its Prediction of 2D Zeolites

© 2018 American Chemical Society. Scientific interest in two-dimensional (2D) materials, ranging from graphene and other single layer materials to atomically thin crystals, is quickly increasing for a large variety of technological applications. While in silico design approaches have made a large impact in the study of 3D crystals, algorithms designed to discover atomically thin 2D materials from their parent 3D materials are by comparison more sparse. We hypothesize that determining how to cut a 3D material in half (i.e., which Miller surface is formed) by severing a minimal number of bonds or a minimal amount of total bond energy per unit area can yield insight into preferred crystal faces. We answer this question by implementing a graph theory technique to mathematically formalize the enumeration of minimum cut surfaces of crystals. While the algorithm is generally applicable to different classes of materials, we focus on zeolitic materials due to their diverse structural topology and because 2D zeolites have promising catalytic and separation performance compared to their 3D counterparts. We report here a simple descriptor based only on structural information that predicts whether a zeolite is likely to be synthesizable in the 2D form and correctly identifies the expressed surface in known layered 2D zeolites. The discovery of this descriptor allows us to highlight other zeolites that may also be synthesized in the 2D form that have not been experimentally realized yet. Finally, our method is general since the mathematical formalism can be applied to find the minimum cut surfaces of other crystallographic materials such as metal-organic frameworks, covalent-organic frameworks, zeolitic-imidazolate frameworks, metal oxides, etc

High-Throughput Screening Approach for Nanoporous Materials Genome Using Topological Data Analysis: Application to Zeolites

The materials genome initiative has led to the creation of a large (over a million) database of different classes of nanoporous materials. As the number of hypothetical materials that can, in principle, be experimentally synthesized is infinite, a bottleneck in the use of these databases for the discovery of novel materials is the lack of efficient computational tools to analyze them. Current approaches use brute-force molecular simulations to generate thermodynamic data needed to predict the performance of these materials in different applications, but this approach is limited to the analysis of tens of thousands of structures due to computational intractability. As such, it is conceivable and even likely that the best nanoporous materials for any given application have yet to be discovered both experimentally and theoretically. In this article, we seek a computational approach to tackle this issue by transitioning away from brute-force characterization to high-throughput screening methods based on big-data analysis, using the zeolite database as an example. For identifying and comparing zeolites, we used a topological data analysis-based descriptor (TD) recognizing pore shapes. For methane storage and carbon capture applications, our analyses seeking pairs of highly similar zeolites discovered good correlations between performance properties of a seed zeolite and the corresponding pair, which demonstrates the capability of TD to predict performance properties. It was also shown that when some top zeolites are known, TD can be used to detect other high-performing materials as their neighbors with high probability. Finally, we performed high-throughput screening of zeolites based on TD. For methane storage (or carbon capture) applications, the promising sets from our screenings contained high-percentages of top-performing zeolites: 45% (or 23%) of the top 1% zeolites in the entire set. This result shows that our screening approach using TD is highly efficient in finding high-performing materials. We expect that this approach could easily be extended to other applications by simply adjusting one parameter, the size of the target gas molecule