  Treated traditionally by the Ehrenfest approximation, dynamics of a
one-dimensional molecular crystal model with off-diagonal exciton-phonon
coupling is investigated in this work using the Dirac-Frenkel time-dependent
variational principle with the multi-D$_2$ {\it Ansatz}. It is shown that the
Ehrenfest method is equivalent to our variational method with the single D$_2$
{\it Ansatz}, and with the multi-D$_2$ {\it Ansatz}, the accuracy of our
simulated dynamics is significantly enhanced in comparison with the
semi-classical Ehrenfest dynamics. The multi-D$_2$ {\it Ansatz} is able to
capture numerically accurate exciton momentum probability and help clarify the
relation between the exciton momentum redistribution and the exciton energy
relaxation. The results demonstrate that the exciton momentum distributions in
the steady state are determined by a combination of the transfer integral and
the off-diagonal coupling strength, independent of the excitonic initial
conditions. We also probe the effect of the transfer integral and the
off-diagonal coupling on exciton transport in both real and reciprocal space
representations. Finally, the variational method with importance sampling is
employed to investigate temperature effects on exciton transport using the
multi-$\rm D_2$ {\it Ansatz}, and it is demonstrated that the variational
approach is valid in both low and high temperature regimes.
