  We report quadruple configuration interaction calculations within the
extended Pariser$-$Parr$-$Pople Hamiltonian on the excited states of aggregates
of polyenes, crystalline acenes, and covalently linked dimers of acene
molecules. We determine the precise energy orderings and analyze the cluster
wave functions in order to arrive at a comprehensive physical understanding of
singlet fission in these diverse families of materials. Our computational
approach allows us to retain a very large number of basis states and thereby
obtain the correct relative energy orderings of one electron$-$one hole Frenkel
and charge-transfer excitons versus intra- and intermolecular two
electron$-$two hole triplet$-$triplet excited states. We show that from the
energy orderings it is possible to understand the occurrence of singlet fission
in polyene and acene crystals, as well as its near total absence in the
covalently linked acene dimers. As in the acene crystals, singlet fission in
the polyenes is a multichromophoric phenomenon, with the well-known
2$^1$A$_g^-$ playing no direct role. Intermolecular charge-transfer is
essential for singlet fission in both acenes and polyenes, but because of
subtle differences in the natures and orderings of the aggregate excited
states, the mechanisms of singlet fission are slightly different in the two
classes. We are thus able to give qualitative physical reasoning for the slower
singlet fission in the polyenes, relative to that in crystalline pentacene. Our
work also gives new insight into the complex exciton dynamics in tetracene
crystals, which has been difficult to understand theoretically. Our large-scale
many-body calculations provide us with the ability to understand the
qualitative differences in the singlet fission yields and rates between
different classes of $\pi$-conjugated materials.
