Armstrongs Probabilistic Electron Phase-Space Overlap Model (APEPM)

A new closed-form, geometry-driven framework for determining whether a chemical bond is geometrically possible, and for analyzing the resulting electron cloud overlap and orbital participation, is introduced. Armstrong’s Probabilistic Electron Phase-Space Overlap Model (APEPM) evaluates bonding feasibility by treating nonzero spatial overlap of atomic electron clouds as a necessary geometric precondition for bonding, without explicit solution of the electronic wavefunction or use of variational quantum-mechanical methods. The model combines geometric overlap formalism adapted from Armstrong’s exclusion-zone solvent-accessible surface area (SASA) methodology with probabilistic orbital occupancy constraints defined by Armstrong’s Range Exponent Electron Configuration Model (AREECM). Bonded atoms are represented as interacting probabilistic electron volumes whose effective radii and internuclear separation define, in closed form, whether electron cloud overlap exists and, if so, the size and geometry of the shared electron phase-space. When overlap is present, its depth is mapped onto radially partitioned orbital shells, allowing direct identification of which atomic orbital regions are geometrically involved in bonding. Electron probability within these participating regions is constrained using AREECM-style minimum and maximum occupancy bounds, preserving normalization and exclusion principles while remaining fully wavefunction-free. Because all electronic quantities are explicitly tethered to bond length, APEPM provides a continuous, geometry-based description of bond feasibility, electron cloud overlap, and orbital-level spatial participation under changing structural conditions such as bond compression or extension. While not intended to replace quantum-mechanical electronic structure methods or to reproduce spectroscopic accuracy, APEPM offers a fast, interpretable, and physically constrained framework for screening bond feasibility, analyzing electron cloud overlap, and identifying geometrically involved orbital regions in large systems, dynamic environments, and reduced-order chemical modeling.