  We employ the non-perturbative time-dependent basis function (tBF) approach
to study the scattering of the deuteron on $^{208}$Pb below the Coulomb
barrier. We obtain the bound and discretized scattering states of the
projectile, which form the basis representation of the tBF approach, by
diagonalizing a realistic Hamiltonian in a large harmonic oscillator basis. We
find that the higher-order inelastic scattering effects are noticeable for sub
barrier scatterings with the tBF method. We have successfully reproduced
experimental sub Coulomb barrier elastic cross section ratios with the tBF
approach by considering only the electric dipole ($\mathit{E1}$) component of
the Coulomb interaction between the projectile and the target during
scatterings. We find that the correction of the polarization potential to the
Rutherford trajectory is dominant in reproducing the data at very low
bombarding energies, whereas the role of internal transitions of the deuteron
projectile induced by the $\mathit{E1}$ interaction during the scattering
becomes increasingly significant at higher bombarding energies.
