Augmented Hückel molecular orbital model of π ‐electron systems: from topology to metric . I. General theory

The π-electron molecule is planar (but the strict planarity requirement may be relaxed), with a set of electrons esiding on the (valence) molecular orbitals of the π-symmetry (πMOs). The quantum state of these electrons nfluences the geometry of the molecule, most notably the bond lengths. The Hückel molecular orbital (HMO) model provides the simplest (yet quite effective) description of the πMOs. Longuet-Higgins and Salem, and their followers, made efforts to augment this model by making its empirical parameters dependent on the bond lengths; this approach was later often referred to as the SCβ HMO model. In the present paper, we introduce the AugHMO model that is a generalization of these earlier efforts. We show that the mathematical formulation of this model necessarily implies the principle of a bond-order–bond-length (BO-BL) relationship that is universal for a given type of bond (C–C, C–N, etc.) in all π-electron molecules. A plausible assumption that the BO-BL relationship is a linear one leads to a family of workable computational methods, such as the Hückel–Su–Schrieffer–Heeger method and the Hückel–Longuet–Higgins– Salem method. Our AugHMO model is developed with the following goals in mind: (i) to offer a predictive tool for calculating the bond lengths in π-electron molecules, capable of dealing with very large systems; (ii) to provide insights into the problem of coupling between the π-electrons and the bond lengths; and (iii) to offer a simple educational model illustrating the principles of geometry optimization in quantum chemistry (involving the gradient and Hessian of the molecular energy).