Armstrongs Probabilistic Electron Phase-Space Overlap Model (APEPM)

Context Assessing whether a bonding interaction is geometrically possible, that is, whether the probabilistic electron clouds of two atoms spatially overlap at a given internuclear separation, is a fundamental but often implicit aspect of molecular modeling. Conventional electronic structure methods address this question indirectly through wavefunction optimization and energetic analysis, which can obscure the underlying geometric constraints and limit scalability. Here, Armstrong’s Probabilistic Electron Phase-Space Overlap Model (APEPM) is introduced as a geometry-driven, wavefunction-free framework for evaluating geometric bond feasibility by explicitly determining whether nonzero electron cloud overlap exists for any specified atomic separation. The model provides closed-form expressions for the size and extent of the shared electron phase-space and tracks how this overlap evolves continuously under bond compression or extension. By mapping the overlap region onto radially partitioned orbital shells, APEPM identifies which atomic orbital regions are geometrically involved in the mandatory shared phase-space, yielding an interpretable description of spatial orbital involvement that is independent of energetic weighting or bond strength assignment. Methods APEPM represents bonded atoms as interacting probabilistic electron volumes characterized by effective electron cloud radii derived from atomic electron cloud extents. Geometric overlap is determined analytically using closed-form expressions for the intersection of two spheres as a function of internuclear distance, yielding overlap radius, cross-sectional area, and overlap volume, with zero overlap defining the geometric infeasibility limit. Orbital participation is evaluated by projecting the overlap region along the bond axis and identifying intersections with predefined radial orbital shells. Admissible electron participation within intersected orbitals is constrained using Armstrong’s Range Exponent Electron Configuration Model (AREECM), which assigns minimum and maximum probability bounds consistent with normalization and exclusion principles and incorporates a discrete bond-interaction constraint for each bonding interaction. All calculations are analytic and geometry-based and are implemented in Python using custom software developed for this work.