Calculation of localized vibrational modes around a soliton in trans-polyacetylene using the molecular orbital method

Vibrational frequencies of polyacetylene with a charged soliton are calculated for finite chain lengths using the molecular orbital method. A translational mode, an amplitude oscillation mode, a third localized mode and a staggered mode are obtained as localized modes. It is found that the frequency of the translational mode is inversely proportional to the square of the chain length, and approaches zero when the length becomes infinite. This can be explained if the soliton is regarded as a quantum particle in a box. A type of electron-hole symmetry holds, even with the Coulomb effect, between occupied orbitals of the cation and unoccupied orbitals of the anion and between unoccupied orbitals of the former and occupied orbitals of the latter.