Local state spectrum of end substituted polyenes

Starting from the electronic part of the Su–Schrieffer–Heeger [Phys. Rev. B 22, 1099 (1980)] Hamiltonian with a strict account to the end effects, it is shown that the π-electron spectrum of substituted polyenes is determined by solutions of the Lennard-Jones type equation, where each end group is represented by a single energy dependent parameter associated with the diagonal component of the end-group Green function referred to the binding site. The conditions of the appearance/disappearance of in-gap and out-of-band local π-electron states are found in the form of analytical relations between the conjugated bridge length, the C–C bond alternation parameter, and parameters of end-groups included in the above mentioned Green functions. These relations determine a set of critical parameters (defined explicitly), at which the local state spectrum undergoes qualitative transformations. On the basis of the obtained results we propose a certain procedure that allows us to predict the number of in-gap and out-of-band states in any substituted polyene provided the π-electronic structure of end groups and the bridge parameters are known. Throughout the discussion, several conclusions are made concerning the dependence of the electron localization on the finite-length and bond-alternation effects.