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.

Strategies to simulate dephasing-assisted quantum transport on digital quantum computers

“…From these definitions, any physical system whose Hamiltonian can be cast in the required form realises an instance of the QW scheme. A straightforward analogy with equation ( 1) can be found in the tight-binding Hamiltonian used in solid-state physics [41] and the Hückel model for molecular orbitals [42], but also with the Frenkel exciton Hamiltonian used to model excitation on chromophore networks [43] and in general with model Hamiltonian built to describe charge and energy transfer in weakly interacting molecular aggregates and nanostructures [44]. In all these cases, we define a set of quantum sites {j}.…”

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“…Let us now assume that the Hamiltonian of the system formed by N QDs can be approximated by a special case of a tight-binding (TB) Hamiltonian based on the very weak superposition of wave functions for isolated QDs, as in the Hückel model [ 89 ] where is a very small perturbation over the Hamiltonian of the isolated QD located at .…”

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“…) is the density matrix of the Boltzmann thermal state [39], which is the density matrix with maximum von Neumann entropy, and Z (G) = T r (Γ (G, β)) is the partition function of the network. For other denitions of density matrices in the context of graphs/networks see, for instance [40,41]. In a network having many paths with low informational cost, the von Neumann entropy is low, indicating a good global navigability of the network.…”