Geometric Constraints Determine Chemical Bond Parameters: A Unified Scaling Theory of Bond Length, Bond Energy, and Isotope Effects

Abstract

Chemical bond parameters—bond length, bond energy, and isotope effects—are traditionally treated as element-specific empirical quantities requiring quantum chemical computation. Here we show that these parameters across the entire periodic table are governed by a single geometric scaling law rooted in coordination polyhedron topology. We introduce geometric looseness (δ), a distributed quantity characterizing deviation from regular polyhedral coordination. A unified bond length formula d = d₀(1+δ) achieves 3.12% equivalent mean error across 82 covalent bonds, outperforming the semiempirical PM7 method (4.62%) at calculator-level cost. A bond energy scaling relation E = k/d reduces covalent bond energy prediction error to 4.2% across 65 bonds and lowers ionic crystal lattice energy error from 4.44% (Born-Landé equation) to 2.73%. The 4.44% Born-Landé deviation falls within the geometric looseness range of non-regular polyhedral coordination (2.6%–6.6%), indicating classical theory's error range is the quantitative manifestation of coordination geometric looseness. A universal isotope correction formula calibrated via diamond-family elements (C/Si/Ge, ε₀=0.045) predicts isotope-induced bond length changes in quantitative agreement with experiment at six orders of magnitude lower cost than path-integral molecular dynamics. A complete 63-element quick-reference table enables instant bond parameter prediction. This framework reframes chemical bonding as a unified geometric scaling theory with immediate utility across chemistry, physics, and materials science.