Sequential Adaptive Natural Density Partitioning (AdNDP) Systematically Produces Non-N-Representable Densities

Adaptive natural density partitioning (Ad-NDP) sequentially extracts fragment natural orbitals from the one-electron density matrix. Herein, we show, with an exact and basisindependent argument, that the residual density left behind after two non-orthogonal bonding elements are extracted is not guaranteed to be N-representable. The breakdown has a precise, traceable origin: whenever two sequentially depleted orbitals carry a nonzero overlap, S12 ̸= 0, the second extraction re-subtracts density along a direction the first depletion step has already emptied, driving a diagonal element of the compressed 2 × 2 residual block negative independently of basis set, correlation treatment, or system size. At CCSD/cc-pVTZ, five benchmark multicenter systems (diborane, B−