Particle‐hole symmetry and complementary quantum systems

Abstract

AbstractIt is well known that positive and negative ions derived from the same alternant hydrocarbon have almost indistinguishable electron spectra. Effective charges, bond orders, and many other properties also correlate between the two systems. These «alternancy» properties are due to the particle‐hole symmetry, and they can be generalized far beyond alternant hydrocarbons. In general, with each quantum system one can associate a «complementary» system. The eigenstates of the complementary system are related to the eigenstates of the original system in the same way as the anionic eigenstates of an alternant hydrocarbon are related to the cationic eigenstates of this hydrocarbon. In particular, the (N + k)‐electron spectrum of the complementary system is the same, up to the uniform shift, as the (N − k)‐electron spectrum of the original system; the effective charges in the complementary system are opposite to the corresponding effetive charges in the original system; bond orders in the two systems between vertices of the same and of the opposite parity are, respectively, opposite and the same, etc. Each particle‐hole symmetry operator associates with the original system one such complementary system. There is hence a large number of the systems complementary to the same original system. In the PPP model and in various generalizations of this model, systems complementary to conjugated hydrocarbons are Möbius‐type alterations of original hydrocarbons. In the case of conjugated heterocompounds, there is a formal replacement of all heteroatoms with some other heteroatoms, with or without an additional Möbius‐type change. Such complementary systems may correspond to entirely different molecules, and they also may describe some transition states of these molecules.