Electronic States of a Disordered Polymer

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Abstract
A method introduced by Schmidt to give the vibrational frequency spectrum of an isotopic disordered one‐dimensional lattice is generalized and applied to an infinite one‐dimensional disordered molecular crystal with nearest‐neighbor interactions. Such a model is applicable to a polymer having several molecular species arranged randomly along the polymer axis. Equations are derived which may be solved numerically to give the density of states for the following types of polymer: (1) the general disordered polymer having a random arrangement of several molecular species, the coupling constants between two species being determined by the species being coupled; (2) the glassy polymer having only one molecular species but with the coupling constants distributed randomly according to a continuous probability law. A numerical calculation of the density of states is made for the isotopic disordered polymer both with and without positional correlation between the molecular species and for the glassy polymer. When the energies of the two species in the diatomic isotopic disordered polymer differ by more than the nearest‐neighbor coupling constant, the density of states is quite structured. Several of the peaks in this structure are associated with clusters of one or two molecules of one species in a lattice of molecules of the other species. Structure also appears in the density of states for the glassy polymer when there is a finite probability of small values for the nearest‐neighbor coupling integrals.

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