Free-Electron Network Model for Conjugated Systems. I. Theory

The free-electron model for conjugated systems is consistently developed as the limiting case of a three-dimensional quantum-mechanical treatment of the π electrons in such systems. Joint conditions (for branching points) and boundary conditions (for free end points) are derived and the hermiticity of the Hamiltonian is shown. A matrix formulation of the theory is established which makes the application to large systems feasible, and at the same time leads to a close analogy with the LCAO model (LCAO MO treatment considering only nearest neighbor interactions). Quantities analogous to the quantities q (the charge in an atomic orbital) and p (the bond order) are defined, and special attention is given to alternant conjugated systems for which a population theorem, analogous to the one in LCAO theory, is valid.