We present density matrix renormalisation group calculations of the Pariser- Parr-Pople-Peierls model of linear polyenes within the adiabatic approximation. We calculate the vertical and relaxed transition energies, and relaxed geometries for various excitations on long chains. The triplet (3Bu+) and even- parity singlet (2Ag+) states have a 2-soliton and 4-soliton form, respectively, both with large relaxation energies. The dipole-allowed (1Bu-) state forms an exciton-polaron and has a very small relaxation energy. The relaxed energy of the 2Ag+ state lies below that of the 1Bu- state. We observe an attraction between the soliton-antisoliton pairs in the 2Ag+ state. The calculated excitation energies agree well with the observed values for polyene oligomers; the agreement with polyacetylene thin films is less good, and we comment on the possible sources of the discrepencies. The photoinduced absorption is interpreted. The spin-spin correlation function shows that the unpaired spins coincide with the geometrical soliton positions. We study the roles of electron-electron interactions and electron-lattice coupling in determining the excitation energies and soliton structures. The electronic interactions play the key role in determining the ground state dimerisation and the excited state transition energies.
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