We present a method for large-scale DFT-based screening of ion diffusion in crystalline solids. This is accomplished by extending the Ionic TuTraSt method to sample the potential energy surface by using single-point DFT calculations. To drastically reduce the number of single-point DFT calculations, symmetry, interpolation, and exclusion of high-energy regions are employed. This approach is tested on a large data set of solid-state Li-ion conductors, for which the interpolation and high-energy exclusion are optimized to balance computational efficiency and accuracy of the obtained diffusion properties. Furthermore, the developed workflow is validated by comparison with ab initio molecular dynamics (AIMD) simulations on a set of known Li-ion superconducting materials.

For a given spatial graph 𝒢⊂ℝ^3, we would like to find a closed orientable surface 𝒮 embedded in ℝ^3 in which 𝒢 is cellular embedded. However, for general 𝒢 this is not possible. We therefore define a property of spatial graphs, called leveled, to show that for leveled spatial graphs with a small number of levels, a surface 𝒮 can always be found. The argument is based on decomposing 𝒢 into spatial subgraphs that can be placed on a sphere and on cylinders attached as handles, in such a way that the resulting surface contains a cellular embedding of 𝒢. We generalize the procedure to an algorithm that, if successful, constructs 𝒮 for leveled spatial graphs with any number of levels. We conjecture that all connected leveled embeddings can be cellular embedded with the presented algorithm.

We investigate the property of a spatial graph of having a leveled embedding and characterize the abstract graphs with this property. We show that all leveled embeddings are free and we compare leveled and paneled (also known as flat) embeddings. Two new graph invariants are introduced: the level number, an invariant for graphs that admit a leveled embedding, and the Hamiltonian level number, an invariant for Hamiltonian graphs. These invariants provide a measure on how far a graph is from being planar. We study the relation between the (Hamiltonian) level number and other graph invariants that minimize the decomposition of a graph in planar subgraphs, namely the thickness and the book thickness of a graph. We characterize graphs with low level number and determine both the level number and the Hamiltonian level number of complete graphs and of complete bipartite graphs.

We present an efficient method to compute diffusion coefficients of multiparticle systems with strong interactions directly from the geometry and topology of the potential energy field of the migrating particles. The approach is tested on Li-ion diffusion in crystalline inorganic solids, predicting Li-ion diffusion coefficients within 1 order of magnitude of molecular dynamics simulations at the same level of theory while being several orders of magnitude faster. The speed and transferability of our workflow make it well-suited for extensive and efficient screening studies of crystalline solid-state ion conductor candidates and promise to serve as a platform for diffusion prediction even up to the density functional level of theory.

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.

A high-throughput screening protocol based on cost-effective computations of light absorption and photo-redox capabilities is proposed to discover promising photo-catalytically active metal–organic frameworks.

The study of organic molecular crystals under high pressure provides fundamental insight into crystal packing distortions and reveals mechanisms of phase transitions and the crystallization of polymorphs. These solid-state transformations can be monitored directly by analyzing electron charge densities that are experimentally obtained at high pressure. However, restricting the analysis to the featureless electron density does not reveal the chemical bonding nature and the existence of intermolecular interactions. This shortcoming can be resolved by the use of the DORI (density overlap region indicator) descriptor, which is capable of simultaneously detecting both covalent patterns and noncovalent interactions from electron density and its derivatives. Using the biscarbonyl[14]annulene crystal under pressure as an example, we demonstrate how DORI can be exploited on experimental electron densities to reveal and monitor changes in electronic structure patterns resulting from molecular compression. A novel approach based on a flood-fill-type algorithm is proposed for analyzing the topology of the DORI isosurface. This approach avoids the arbitrary selection of DORI isovalues and provides an intuitive way to assess how compression packing affects covalent bonding in organic solids.

For large-scale screening studies there is a need to estimate the diffusion of gas molecules in nanoporous materials more efficiently than (brute force) molecular dynamics. In particular for systems with low diffusion coefficients molecular dynamics can be prohibitively expensive. An alternative is to compute the hopping rates between adsorption sites using transition state theory. For large-scale screening this requires the automatic detection of the transition states between the adsorption sites along the different diffusion paths. Here an algorithm is presented that analyzes energy grids for the moving particles. It detects the energies at which diffusion paths are formed, together with their directions. This allows for easy identification of nondiffusive systems. For diffusive systems, it partitions the grid coordinates assigned to energy basins and transitions states, permitting a transition state theory based analysis of the diffusion. We test our method on CH4 diffusion in zeolites, using a standard kinetic Monte Carlo simulation based on the output of our grid analysis. We find that it is accurate, fast, and rigorous without limitations to the geometries of the diffusion tunnels or transition states.

How many copies of a parallelepiped are needed to ensure that for every point in the parallelepiped a copy of each other point exists, such that the distance between them equals the distance of the pair of points when the opposite sites of the parallelepiped are identified? This question is answered in Euclidean space by constructing the smallest domain that fulfills the above condition. We also describe how to obtain all primitive cells of a lattice (i.e., closures of fundamental domains) that realise the smallest number of copies needed and give them explicitly in 2D and 3D.

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.

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.

Zeolite-templated carbons (ZTCs) comprise a relatively recent material class synthesized via the chemical vapor deposition of a carbon-containing precursor on a zeolite template, followed by the removal of the template. We have developed a theoretical framework to generate a ZTC model from any given zeolite structure, which we show can successfully predict the structure of known ZTCs. We use our method to generate a library of ZTCs from all known zeolites, to establish criteria for which zeolites can produce experimentally accessible ZTCs, and to identify over 10 ZTCs that have never before been synthesized. We show that ZTCs partition space into two disjoint labyrinths that can be described by a pair of interpenetrating nets. Since such a pair of nets also describes a triply periodic minimal surface (TPMS), our results establish the relationship between ZTCs and schwarzites-carbon materials with negative Gaussian curvature that resemble TPMSs-linking the research topics and demonstrating that schwarzites should no longer be thought of as purely hypothetical materials.

We consider two metal–organic frameworks as identical if they share the same bond network respecting the atom types. An algorithm is presented that decides whether two metal–organic frameworks are the same. It is based on distinguishing structures by comparing a set of descriptors that is obtained from the bond network. We demonstrate our algorithm by analyzing the CoRe MOF database of DFT optimized structures with DDEC partial atomic charges using the program package ToposPro.

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].

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.

In most applications of nanoporous materials the pore structure is as important as the chemical composition as a determinant of performance. For example, one can alter performance in applications like carbon capture or methane storage by orders of magnitude by only modifying the pore structure. For these applications it is therefore important to identify the optimal pore geometry and use this information to find similar materials. However, the mathematical language and tools to identify materials with similar pore structures, but different composition, has been lacking. We develop a pore recognition approach to quantify similarity of pore structures and classify them using topological data analysis. This allows us to identify materials with similar pore geometries, and to screen for materials that are similar to given top-performing structures. Using methane storage as a case study, we also show that materials can be divided into topologically distinct classes requiring different optimization strategies.

In most applications of nanoporous materials the pore structure is as important as the chemical composition as a determinant of performance. For example, one can alter performance in applications like carbon capture or methane storage by orders of magnitude by only modifying the pore structure (1,2). For these applications it is therefore important to identify the optimal pore geometry and use this information to find similar materials. However, the mathematical language and tools to identify materials with similar pore structures, but different composition, has been lacking. Here we develop a pore recognition approach to quantify similarity of pore structures and classify them using topological data analysis (3,4). Our approach allows us to identify materials with similar pore geometries, and to screen for materials that are similar to given top-performing structures. Using methane storage as a case study, we also show that materials can be divided into topologically distinct classes -- and that each class requires different optimization strategies. In this work we have focused on pore space, but our topological approach can be generalised to quantify similarity of any geometric object, which, given the many different Materials Genomics initiatives (5,6), opens many interesting avenues for big-data science.

We give explicit deformations of embeddings of abstractly planar graphs that lie on the standard torus T^2 ⊂ℝ^3 and that contain neither a nontrivial knot nor a nonsplit link into the plane. It follows that ravels do not embed on the torus. Our results provide general insight into properties of molecules that are synthesized on a torus.