  We investigate the effect of static electron-phonon coupling, on real-time
dynamics of spin and charge transport in $\pi$-conjugated polyene chains. The
polyene chain is modeled by the Pariser-Parr-Pople Hamiltonian with dimerized
nearest-neighbor parameter $t_{0}(1+\delta)$ for short bonds and
$t_{0}(1-\delta)$ for long bonds, and long-range electron-electron
interactions. We follow the time evolution of the spin and charge using
time-dependent density matrix renormalization group technique, when a hole is
injected at one end of the chain in its ground state. We find that spin and
charge dynamics followed through spin and charge velocities, depend both on
chain length and extent of dimerization, $\delta$. Analysis of the results
requires focusing on physical quantities such as average spin and charge
polarizations, particularly in the large dimerization limit. In the
dimerization range 0.0 $\le$ $\delta$ $\le$ 0.15, spin-charge dynamics is found
to have a well defined behavior, with spin-charge separation (measured as the
ratio of charge velocity to spin velocity) as well as, the total amount of
charge and spin transported in a given time, along the chain, decreasing as
dimerization increases. However, in the range 0.3 $\le$ $\delta$ $\le$ 0.5, it
is observed that the dynamics of spin and charge transport becomes complicated.
It is observed that for large $\delta$ values, spin-charge separation is
suppressed and the injected hole fails to travel the entire length of the
chain.
