Lattice vibrations in polyacetylene

The phonon modes for polyacetylene are calculated in the framework of the continuum model. Included are the lattice vibrations in polyacetylene containing the low-lying topological excitations corresponding to a soliton or a polaron, as well as those for the perfectly dimerized chain. Numerical solutions are obtained for the chain containing these topological excitations. The phonon modes for the chain with a soliton are compared to the domain-wall ${\phi }^4$ solutions, and it is shown that the translational or Goldstone mode for the ${\phi }^4$ model is an ‘‘exact’’ solution with ω=0. The chain containing a polaron is found to have four localized vibrational modes, one of which is a translational mode where the frequency is numerically determined to be near zero. Excitation energies including lattice vibrations for a chain containing a soliton or a polaron are also calculated.