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

Abstract

Local coordination cores are critically important in metal-organic frameworks (MOFs) and polyoxometalates (POMs), as they govern the fundamental chemical and structural properties of these materials. Existing methods, including element-resolved persistent homology (ERPH), have limited ability to translate local coordination cores into quantitative, scalable, and interpretable information. Here, we introduce persistent Mayer homology (PMH) to enable the detection of the geometric variability invisible by classical topology. By focusing on coordination cores formed by periodic metal-centered heavy-atom, carbon, and donor channels, PMH distinguishes different arrangements of the local coordination environment, linker framework, and potential coordinating atoms. We examine explainable PMH of orders $N=3$ and $N=5$ and separately combine each with ERPH, which characterizes the multiscale organization of chemically defined element groups across the extended framework, to predict eight nitrogen- and oxygen-related adsorption and self-diffusion properties. The resulting coordination-core PMH representations consistently improve predictive performance over ERPH alone, with the third-order Mayer representation providing the strongest overall performance. These results demonstrate that the inner coordination configurations captured by PMH contain essential structural information that is not fully represented by conventional element-resolved topological descriptors. This finding provides a quantitative interpretable topological framework for understanding local coordination environments and may facilitate the rational design of MOFs and POM-based materials.