Exact solution for the Hückel model of heteropolyenes

The Lennard-Jones equation, which determines the π electron spectrum of unsubstituted polyenes H2C-(CH)2Nd-CH2, is generalized to the case of heteropolyenes X-(CH)2Nd-Y. The closed form of the exact Hückel molecular orbitals for heteropolyenes is derived. The dependence of the local state spectrum on characteristic parameters is examined analytically for three types of heteropolyene, X-(CH)2Nd-X, X+-(CH)2Nd-X−, and X-(CH)2Nd-CH2. The criteria of the existence of local states and their position in the asymptotic limit are established for both signs of the alternation parameters. The results obtained predict strong and qualitatively different effects of the substitution on the optical response of the indicated types of heteropolyenes, and they can be used, therefore, in search of new compounds, which are optically active in the desired frequency region. The theory is also applied for the analysis of the dependence of all π electron levels in heteropolyenes on the electron site-energy perturbation caused by heteroatoms.