The electron density function of the Hückel (tight-binding) model

Abstract

The Hückel (tight-binding) molecular orbital (HMO) method has found many applications in the chemistry of alternant conjugated molecules, such as polycyclic aromatic hydrocarbons (PAHs), fullerenes and graphene-like molecules, as well as in solid-state physics. In this paper, we found analytical expressions for the electron density matrix of the HMO method in terms of odd-powers of its Hamiltonian. We prove that the HMO density matrix induces an embedding of a molecule into a high-dimensional Euclidean space in which the separation between the atoms scales very well with the bond lengths of PAHs. We extend our approach to describe a quasi-correlated tight-binding model, which quantifies the number of unpaired electrons and the distribution of effectively unpaired electrons. In this case, we found that the corresponding density matrices induce embedding of the molecules into high-dimensional Euclidean spheres where the separation between the atoms contains information about the spin–spin repulsion between them. Using our approach, we found an analytic expression which explains the bond length alternation in polyenes inside the HMO framework. We also found that spin–spin interaction explains the alternation of distances between pairs of atoms separated by two bonds in conjugated molecules.