A Unified Operator Resolution of Quadratic Bond Orders

Quadratic bond-order indices provide compact measures of chemical covalency, but their scalar form conceals the orbital channels contributing to a bond. We present a common operator framework for closed-shell, open-shell, and correlated one-particle densities in a fixed orthonormal atomic partition. Bond indices are defined as traces of positive-semidefinite atom-pair operators, recovering the Wiberg and per-spin Mayer forms in the respective single-determinant limits. Spectral resolution, or diagonal projection onto a complete common spinless basis, yields non-negative orbital weights summing to the trace of the corresponding operator. For correlated densities, occupation-scaled cumulative spectral layers define a distinct one-particle extension, generally different from direct quadratic evaluation. The additional criterion is population consistency: in the exact spectral construction, the layer weights reconstruct the supplied density, while the associated quadratic population partition retains its orbital and atomic population marginals. These populations are stationary under the resulting Markov transition matrix without requiring an idempotent physical density. The exact construction requires no near-degeneracy threshold and is continuous with respect to the supplied density within the fixed atomic representation. Implemented in runEDDB and illustrated on a chemically diverse set of main-group, transition-metal, and actinide systems, the framework provides a bond-by-bond and symmetry-resolved description of covalency and localization.