Electron density of delocalized bonds revisited: the bond-orbital projection algorithm

Abstract The electron density of delocalized bonds (EDDBs) is a nonlocal molecular density layer representing electrons that participate effectively in multicenter bond delocalization. Its numerical realization, the bond-orbital projection (BOP) method, has evolved over the past decade into an efficient framework for analyzing resonance and delocalization in large, topologically complex molecules. Here, we provide the first unified, implementation-level formulation of the current BOP algorithm and document its realization in the runEDDB program. Conceptually, BOP extends the familiar superposition principle from atomic orbitals to bond orbitals: neighboring localized bond orbitals are tested for their ability to form phase-coherent conjugating or anticonjugating combinations, and the efficiency of this superposition defines their contribution to bond delocalization. We derive the complete computational workflow from the projector algebra of an idempotent one-particle density matrix through two- and three-center bond-order-orbital projections to the final EDDB density. Robustness and performance are demonstrated for a 360-atom porphyrin nanoring. Across three parent basis sets, the full natural atomic orbital (NAO), natural minimal basis (NMB), and natural valence basis (NVB) representations recover essentially the same $$\pi $$ π -bond-delocalization pattern. For the largest density, NMB retains 97.5% of the total EDDB population while reducing the BOP-kernel time by a factor of 194; removal of the core space has only a marginal additional effect. These results establish compact valence representations as an efficient basis for routine EDDB analyses of large molecular systems.