The geometry of the tight-binding method for modelling molecular systems

Abstract. We present a geometric reformulation of the Hückel Molecular Orbital/Tight-Binding (HMO/TB) method that integrates tools from algebraic graph theory and statistical mechanics. By extending classical notions of charge density and bond order to incorporate bond orbital energies, we construct a probabilistic framework that embeds molecular graphs into Euclidean (pseudometric) space. This embedding introduces new geometric descriptors—such as inter-nodal distances, node angles, and electronic propagation efficiency—that correlate with chemical properties including bond lengths, reactivity, and molecular stability. Analytical results for linear and cyclic polyenes as well as polycyclic aromatic hydrocarbons (PAH) illustrate the utility of these descriptors and suggest a pathway toward closer integration of HMO/TB with ab initio methods and machine learning approaches in molecular modeling.

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