Bag of Bonds is a molecular descriptor for machine learning that extends the Coulomb matrix representation by decomposing it into groups of pairwise atomic interactions (bonds), sorted within each group, and concatenated into a fixed-length feature vector. By grouping interactions by atom-pair type (C-C, C-H, C-N, C-O, etc.) and sorting within groups, Bag of Bonds achieves permutation invariance while retaining more structural information than the sorted Coulomb matrix eigenspectrum.

Why Bag of Bonds Matters in AI/ML: Bag of Bonds provides a simple yet effective molecular representation for predicting quantum chemical properties (atomization energies, HOMO-LUMO gaps, dipole moments) that respects permutation invariance while encoding pairwise atomic interaction information, serving as an important baseline in molecular ML.

• Construction — From the Coulomb matrix C (where C_ij = Z_i·Z_j/|R_i-R_j| for i≠j and C_ii = 0.5·Z_i^2.4), extract all pairwise elements, group by atom-pair type (e.g., all C-C interactions, all C-H interactions), sort each group in descending order, and pad to fixed length • Permutation invariance — Sorting within each atom-type group ensures that the representation is invariant to the ordering of atoms of the same element; grouping by type prevents mixing of chemically distinct interactions (unlike eigenvalue-based approaches) • Fixed-length output — Each atom-pair type group is padded to accommodate the maximum number of such pairs in the dataset, producing a fixed-length feature vector suitable for standard ML models (kernel ridge regression, random forests, neural networks) • Information retention — Unlike the Coulomb matrix eigenspectrum (which loses off-diagonal structure), Bag of Bonds retains individual pairwise interaction values, preserving more geometric and chemical information for property prediction • Comparison to modern methods — While superseded by GNNs and equivariant networks for most tasks, Bag of Bonds remains competitive for small datasets and provides an interpretable baseline that directly encodes physical atomic interactions

Bag of Bonds is the foundational molecular descriptor that introduced the principle of grouping atomic interactions by type for permutation-invariant molecular representation, providing a simple, interpretable, and physically motivated feature encoding that bridges raw Coulomb matrix representations and modern learned molecular embeddings in the molecular ML toolkit.