This work addresses the question of whether low-lying excitations in conjugated polymers are comprised of free charge carriers or excitons. States are characterized as bound or unbound according to the scaling of the average particle-hole separation with system size. We critically examine other criteria commonly used to characterize states. The polymer is described by an extended Hubbard model with alternating transfer integrals. The model is solved by exact diagonalization and the density-matrix renormalization-group (DMRG) method. We demonstrate that the DMRG accurately determines excitation energies, transition dipole moments and particle-hole separations of a number of dipole-forbidden
(Ag) and dipole-allowed
(Bu) states. Within a parameter regime considered reasonable for polymers such as polyacetylene, it is found that the charge gap, often used to define the exciton binding energy, is not a good criterion by which to decide whether a state is bound or unbound. The essential nonlinear optical state
mAg is found to mark the onset of unbound excitations in the
Ag symmetry sector. In the
Bu symmetry sector, on the other hand, it is found that all low-lying states are unbound and that there is no well-defined
nBu state. That is, the
1Bu state marks the onset of unbound excitations in this sector.
The symmetry adapted density matrix renormalization group (SDMRG) technique has been an efficient method for studying low-lying eigenstates in one- and quasi-one-dimensional electronic systems. However, the SDMRG method had bottlenecks involving the construction of linearly independent symmetry adapted basis states as the symmetry matrices in the DMRG basis were not sparse. We have developed a modified algorithm to overcome this bottleneck. The new method incorporates end-to-end interchange symmetry
(C2), electron-hole symmetry
(J), and parity or spin-flip symmetry
(P) in these calculations. The one-to-one correspondence between direct-product basis states in the DMRG Hilbert space for these symmetry operations renders the symmetry matrices in the new basis with maximum sparseness, just one nonzero matrix element per row. Using methods similar to those employed in the exact diagonalization technique for Pariser-Parr-Pople (PPP) models, developed in the 1980s, it is possible to construct orthogonal SDMRG basis states while bypassing the slow step of the Gram-Schmidt orthonormalization procedure. The method together with the PPP model which incorporates long-range electronic correlations is employed to study the correlated excited-state spectra of 1,12-benzoperylene and a narrow mixed graphene nanoribbon with a chrysene molecule as the building unit, comprising both zigzag and cove-edge structures.