Electron Density of Delocalized Bonds from Correlated One-Particle Density Matrices

Extending bond-delocalization analysis to correlated one-particle density matrices requires separating fractional occupations from the projector structure underlying determinant-based algorithms. We define the electron density of delocalized bonds (EDDB) by combining bond-orbital projection (BOP) results for cumulative spectral projectors with occupation-gap weights. A structural bound on the raw BOP matrix ensures a positive semidefinite density layer bounded by its parent density. The construction recovers determinant limits, provides consistent spin conventions under stated conditions, and uses an orbital-pair population matrix consistent with the parent orbital populations. An analytic three-center model distinguishes this construction from direct substitution of a fractionaloccupation density into BOP. We further distinguish full-space spectral resolution from subsequent working-space reduction, and numerical occupation grouping and representability preprocessing from the exact definition. The resulting formulation extends EDDB analysis beyond single determinants while making its mathematical properties and limitations explicit. The proposed method is fully implemented in the latest release of runEDDB.