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Capturing Coordination Core Topology in Metal--Organic Frameworks Using Persistent Mayer Homology

This paper introduces persistent Mayer homology (PMH) as a novel topological descriptor that captures geometric variability in local coordination cores of metal-organic frameworks and polyoxometalates, demonstrating that combining PMH with element-resolved persistent homology significantly improves the prediction of adsorption and self-diffusion properties compared to conventional methods alone.

Original authors: Caleb Simiyu Khaemba, Chun-Long Chen, Guo-Wei Wei

Published 2026-10-14

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CC BY 4.0

Original authors: Caleb Simiyu Khaemba, Chun-Long Chen, Guo-Wei Wei

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Materials science often feels like trying to understand a city by looking at a single brick. To truly grasp how a building functions, one must see how the bricks are arranged, how the streets connect them, and how the entire structure breathes. This is the central challenge when studying metal–organic frameworks, a class of crystalline materials that act like molecular sponges. These materials are built from metal ions or clusters linked together by organic molecules, forming vast, repeating networks filled with tiny pores. Because scientists can mix and match different metals and linkers, they can design these frameworks to trap specific gases, separate molecules, or catalyze chemical reactions. However, the sheer number of possible combinations is so vast that testing every single one in a lab is impossible. Researchers must rely on computer models to predict which structures will work best, but these models struggle when the internal geometry of the material is too complex to describe with simple measurements.

The difficulty lies in capturing the subtle details of how atoms arrange themselves around the metal centers. While scientists have long used mathematical tools to map the overall shape of the pores, these tools often miss the intricate, local arrangements of atoms that dictate how a gas molecule interacts with the material. A standard map might show that a room is empty, but it cannot tell you if the furniture inside is arranged in a way that blocks the door. In the world of metal–organic frameworks, this local arrangement is critical. It determines how easily a gas like nitrogen or oxygen can enter the pores, how strongly it sticks to the surface, and how fast it moves through the structure. Without a way to quantify these hidden local patterns, computer predictions of material performance remain imprecise, leaving a gap between theoretical design and real-world application.

In a recent study, researchers introduced a new mathematical approach to solve this problem, focusing specifically on the "coordination cores" of these materials. These cores are the immediate neighborhoods of atoms surrounding the metal centers, acting as the primary gatekeepers for any molecule trying to pass through. The team developed a method called persistent Mayer homology, a sophisticated way of analyzing the shape of these atomic neighborhoods. Unlike traditional methods that look only at how atoms connect to form loops or empty spaces, this new technique examines higher-order relationships between groups of atoms. It asks not just which atoms are neighbors, but how clusters of three, four, or more atoms are organized in space. By applying this method to the metal centers within the frameworks, the researchers could detect geometric variations that were previously invisible to standard analysis.

To test their idea, the scientists applied this new tool to a large collection of known metal–organic framework structures. They combined the new local analysis with an existing method that maps the overall chemical organization of the entire crystal. The goal was to see if adding this detailed view of the metal centers would improve the ability to predict how the materials would behave. They focused on eight specific properties related to how nitrogen and oxygen gases interact with the frameworks, including how much gas the material could hold and how fast the gas molecules could diffuse through it. The results were clear: the models that included the new local analysis consistently outperformed those that relied on the overall structure alone. The most successful version of the new method, which looked at relationships between groups of three atoms, provided the most accurate predictions for all eight properties.

The study revealed that the internal arrangement of atoms around the metal centers contains essential information that global measurements simply cannot capture. Even when two materials have the same overall pore size and chemical composition, differences in how the atoms are stacked around the metal cores can lead to significantly different performance. The new method successfully identified these subtle differences, allowing the computer models to distinguish between structures that would have otherwise looked identical. This finding is particularly important for a subset of these materials that incorporate polyoxometalates, which are complex clusters of metal and oxygen atoms. In these cases, the internal organization of the cluster is just as vital as the surrounding framework, and the new tool is uniquely suited to describe this complexity.

By translating these invisible geometric patterns into numbers that a computer can understand, the researchers have provided a more powerful way to screen and design new materials. The approach does not require expensive experiments or massive computational power to simulate every possible scenario; instead, it extracts the most relevant structural information directly from the atomic coordinates. The study suggests that the future of designing better gas storage or separation materials lies in paying closer attention to these local neighborhoods. While the current work focused on nitrogen and oxygen, the method is general enough to be applied to other gases and materials. The researchers noted that future work will be needed to refine the tool for even more complex chemical environments, but the current results demonstrate that looking deeper into the local structure yields a clearer picture of how these molecular sponges truly function.

Technical Summary: Capturing Coordination Core Topology in Metal–Organic Frameworks Using Persistent Mayer Homology

Problem Statement
Metal–organic frameworks (MOFs) and polyoxometalate (POM)-based MOFs offer programmable platforms for gas storage, separation, and catalysis. However, the vast chemical and structural space of these materials cannot be systematically explored through synthesis and measurement alone, necessitating high-throughput computational screening. A primary bottleneck in data-driven discovery is the representation of structural information. While adsorption and diffusion properties depend on both global pore organization and local coordination environments, existing topological descriptors often fail to capture the geometric variability of local coordination cores. Specifically, classical persistent homology (PH), governed by the boundary relation $d^2=0$ , summarizes connectivity (components, loops, cavities) but does not explicitly examine the additional boundary relations available within the higher-order simplicial organization of metal coordination cores. Consequently, distinct internal atomic arrangements with identical ordinary topological summaries may be indistinguishable, limiting the predictive accuracy of machine learning models for properties governed by these local configurations.

Methodology
The authors propose a dual-representation framework that combines Element-Resolved Persistent Homology (ERPH) with Persistent Mayer Homology (PMH) to capture both the extended framework and the inner coordination cores.

Element-Resolved Persistent Homology (ERPH):
- MOF structures are periodically expanded to approximately 64 Å along lattice directions to ensure comparable spatial coverage without rescaling.
- Atoms are partitioned into chemically defined groups (e.g., main-group metals, transition metals, carbon, donors, etc.).
- $\alpha$ -complex filtrations are constructed for each element group, and classical persistent homology is computed in dimensions $H_0, H_1, H_2$ .
- This yields 6,750 features characterizing the multiscale organization of chemically defined element groups across the extended framework.
Persistent Mayer Homology (PMH) of Coordination Cores:
- PMH generalizes classical PH by allowing higher-order differentiations $d^N=0$ for $N \ge 2$ . This introduces new boundary relations that can detect geometric changes invisible to classical PH.
- The method identifies "metal anchors" (atoms in metal groups) and defines coordination cores as periodic point clouds within 10 Å of these anchors.
- Three distinct channels are analyzed for each anchor:
  - All-heavy-atom: All non-hydrogen atoms.
  - Carbon: Carbon atoms only.
  - Potential-donor: Atoms capable of coordination (N, O, S, Se, P, halogens).
- PMH is computed at orders $N=3$ and $N=5$ for each channel. The resulting Mayer Betti curves ( $H_{k,q}$ for $q=1, \dots, N-1$ ) are summarized using pointwise maximum and arithmetic mean across distinct coordination-core classes.
- This generates 960 features for $N=3$ and 1,920 features for $N=5$ .
Predictive Modeling:
- The ERPH features are combined separately with PMH features ( $N=3$ and $N=5$ ) to form the primary models.
- A gradient-boosted decision tree (GBDT) is trained to predict eight properties: Henry constants, uptake, and self-diffusion coefficients for Nitrogen ( $N_2$ ) and Oxygen ( $O_2$ ) at 1 bar and infinite dilution.
- Performance is evaluated against ERPH alone, feature concatenation, and prediction consensus, as well as external benchmarks (descriptor-based models, MOFTransformer, PMTransformer).

Key Contributions

Introduction of PMH to MOF Characterization: The paper introduces Persistent Mayer Homology as a tool to capture the internal simplicial configurations of metal coordination cores, providing topological invariants that classical PH cannot detect.
Dual-Scale Representation: The framework successfully integrates a global description of the porous framework (ERPH) with a local description of the metal coordination environment (PMH), addressing the limitation of representations that focus solely on one scale.
Demonstration of Higher-Order Sensitivity: Through a theoretical example using a six-point Vietoris–Rips filtration, the authors demonstrate that PMH can distinguish simplicial arrangements that share identical ordinary Betti numbers ( $\beta_0, \beta_1$ ) but differ in their higher-order incidence relations.

Results
The combined representations were evaluated on a dataset of 4,744 to 5,241 MOFs derived from the CoRE MOF 2019 database.

Predictive Performance: The combination of ERPH and PMH at order $N=3$ $N = 3$ (ERPH+PMH $_{N=3}$ $_{N = 3}$ ) consistently outperformed ERPH alone and all other configurations (including $N=5$ $N = 5$ , concatenation, and consensus).
- For adsorption properties, ERPH+PMH $_{N=3}$ achieved mean $R^2$ values of 0.84 ( $N_2$ Henry), 0.88 ( $O_2$ Henry), 0.83 ( $N_2$ uptake), and 0.90 ( $O_2$ uptake).
- For self-diffusion, mean $R^2$ values ranged from 0.81 to 0.82.
- These results surpassed the descriptor-based benchmark and were competitive with or superior to transformer-based benchmarks (MOFTransformer, PMTransformer) where direct comparisons were possible.
Statistical Significance: Paired analysis confirmed that the improvement of ERPH+PMH $_{N=3}$ over ERPH alone was statistically significant ( $p \le 0.00983$ ) across all ten data splits for every property.
Channel Analysis: The "all-heavy-atom" channel was identified as the strongest individual contributor within the PMH representation, suggesting that the broader non-hydrogen geometry surrounding metal centers contains critical predictive information.

Significance and Claims
The paper claims that the inner coordination configurations captured by PMH contain essential structural information not fully represented by conventional element-resolved topological descriptors. By resolving internal simplicial relations through higher-order boundary operators, PMH provides a quantitative, interpretable topological framework for understanding local coordination environments.

The authors emphasize that this approach is particularly relevant for POM-based MOFs, where complex multimetal oxide cores are embedded within extended porous structures. The method allows for the description of internal metal-oxygen arrangements (via PMH) while retaining the organization of the surrounding pore structure (via ERPH). The authors modestly note that the current benchmark does not encode oxidation states, charge, or specific oxygen environments (terminal vs. bridging) and that future work with curated POM-specific datasets is required to fully establish the broader applicability of these findings to structurally complex inorganic cores.

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