  Probing extended polyene systems with energy in excess of the bright state
($1^1B_u^+$/$S_2$) band edge generates triplets via singlet fission. This
process is not thought to involve the $2^1A_g^-$/$S_1$ state, suggesting that
other states play a role. Using density matrix renormalisation group (DMRG)
calculations of the Pariser-Parr-Pople-Peierls Hamiltonian, we investigate
candidate states that could be involved in singlet fission. We find that the
relaxed $1^1B_u^-$, and $3^1A_g^-$ singlet states and $1^5A_g^-$ quintet state
lie below the $S_2$ state. The $1^1B_u^-$, $3^1A_g^-$ and $1^5A_g^-$ states are
all thought to have triplet-triplet character, which is confirmed by our
calculations of bond dimerization, spin-spin correlation and wavefunction
overlap with products of triplet states. We thus show that there is a family of
singlet excitations(i.e., $2^1A_g^-$, $1^1B_u^-$, $3^1A_g^-$, $\cdots$),
composed of both triplet-pair and electron-hole character, which are
fundamentally the same excitation, but have different center-of-mass energies.
The lowest energy member of this family, the $2^1A_g^-$ state, cannot undergo
singlet fission. But higher energy members (e.g., the $3^1A_g^-$) state, owing
to their increased kinetic energy and reduced electron-lattice relaxation, can
undergo singlet fission for certain chain lengths.
