  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, $\alpha_{ij}(\omega)$, and third-order
polarizabilities $\gamma_{ijkl}(\omega,\omega,\omega)$ of the Hubbard and
"$U-V$" chains in an $all-trans$ polyacetylene geometry, with and without
dimerization. We rationalize the behavior of $\bar{\alpha}$ and $\bar{\gamma}$
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.
