We extend the symmetrized density matrix renormalization group (SDMRG) method to compute the dynamic nonlinear optic coefficients for long chains. By computing correction vectors in the appropriate symmetry subspace we obtain the dynamic polarizabilities,
αij(ω), and third-order polarizabilities
γijkl(ω,ω,ω) of the Hubbard and "
U−V" chains in an
all−trans polyacetylene geometry, with and without dimerization. We rationalize the behavior of
αˉ and
γˉ on the basis of the low-lying excitation gaps in the system. This is the first study of the dynamics of a fermionic system within the DMRG framework.
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.
We extend the symmetrized density-matrix renormalization-group method to compute the dynamic nonlinear optical coefficients for long chains. By computing correction vectors in the appropriate symmetry subspace, we obtain the dynamic polarizabilities, ${\ensuremath{\alpha}}_{\mathrm{ij}}(\ensuremath{\omega})$, and third-order polarizabilities ${\ensuremath{\gamma}}_{\mathrm{ijkl}}(\ensuremath{\omega},\ensuremath{\omega},\ensuremath{\omega})$ of the Hubbard and ``U-V'' chains in an all transpolyacetylene geometry, with and without dimerization. We rationalize the behavior of $\overline{\ensuremath{\alpha}}$ and $\overline{\ensuremath{\gamma}}$ on the basis of the low-lying excitation gaps in the system.