Quantifying the physical complexity of chemical architectures through electron-density topology

Quantifying molecular complexity is a longstanding challenge with profound implications for chemical discovery, biological organization, and materials design. Existing descriptors typically rely on two-dimensional graph-theoretic connectivity or empirical heuristic rules and therefore fail to capture physically encoded structural organization and three-dimensional electronic environments. Here, we introduce the Quantum Assembly Index ( $A_{\rho }$ ), a parameter-free descriptor that defines molecular complexity through real-space electron-density topology and its associated assembly constraints. The index is constructed from shortest-path organization within QTAIM-derived topology graphs together with ring and cage contributions extracted directly from electron-density critical points. Across a chemically diverse benchmark dataset spanning simple inorganic species, organic compounds, biomolecular motifs, coordination clusters, and extended frameworks, $A_{\rho }$ scales logically with molecular size while resolving substantial complexity differences among systems possessing identical atom counts. Correlation, symmetry, ablation, ranking, and contribution analyses provide complementary assessments of the behavior and robustness of the descriptor. Computational wall-time analysis shows that evaluation of $A_{\rho }$ itself is rapid, while the upstream electronic-structure and QTAIM calculations constitute the dominant computational cost, distinguishing its practical use from essentially cost-free SMILES-based descriptors. By grounding complexity measures within quantum-mechanical observables, this work provides a physically motivated approach for quantifying organized molecular architecture and offers an extensible framework for chemical information analysis and machine-learning applications.

Molecular geometries were optimized using the GFN2-xTB semiempirical Hamiltonian. Single-point electronic structure calculations were subsequently performed at the r $_{}^2$ SCAN-3c level of theory. Wavefunction information was converted to Molden format and analyzed within the framework of the Quantum Theory of Atoms in Molecules (QTAIM) using Multiwfn. Bond, ring, and cage critical points together with associated topological properties were extracted from the electron density. The Quantum Assembly Index was constructed using custom Python implementations of graph-theoretic shortest-path algorithms and topological assembly contributions. Additional analyses included correlation, symmetry, ablation, ranking, contribution, and representative computational wall-time assessments.