Since Lewis proposed the electron pair model of the covalent bond in 1916and Pauling established the quantum mechanical theory of the chemical bondin 1939, covalent, ionic, metallic, hydrogen, and van der Waals bonds havebeen described by separate theoretical frameworks. Whether a singlecontinuous variable can unify them has remained an open question. Here weshow that the undertaking tightness S, defined as S = (1 − d̂₁·d̂₂) · f(ΔS),rigorously unifies all five bond types as distinct intervals on a singlecontinuous spectrum. Starting from the mathematical foundations of ConstraintNetwork dynamics, we prove the existence, uniqueness, and continuity of S(Theorems 1–2), establish the dynamical convergence mechanism by which Sattains its steady-state value, derive the analytical functionalrelationships between the S-value and bond energy and bond length(Theorems 3–4), show that the continuous spectrum of S unifies all fivechemical bond types (Theorem 5), and demonstrate that the geometry ofpolyatomic molecules is rigorously determined by the angles between thedirection vectors of the Φ-open-branches of the central node (Theorem 6).The predicted values of all theorems are consistent with experimental datafrom the NIST database within chemical accuracy. On this basis, we constructa Molecular Periodic Table—starting from a single mathematical framework,one reads out the number and directionality of Φ-open-branches for eachnuclear node, arranges all possible undertaking networks according to theΦ-complementarity rule, and directly outputs the bond energy, bond length,bond angle, and reactivity for each combination.
Menggang Yu (Thu,) studied this question.