  We derive an effective polaronic interaction Hamiltonian, {\it exact to
second order in perturbation}, for the spinless one-dimensional Holstein model.
The small parameter is given by the ratio of the hopping term ($t$) to the
polaronic energy ($g^2 \omega_0$) in all the region of validity for our
perturbation; however, the exception being the regime of extreme
anti-adiabaticity ($t/\omega_0 \le 0.1$) and small electron-phonon coupling ($g
< 1$) where the small parameter is $t/\omega_0$. We map our polaronic
Hamiltonian onto a next-to-nearest-neighbor interaction anisotropic Heisenberg
spin model. By studying the mass gap and the power-law exponent of the
spin-spin correlation function for our Heisenberg spin model, we analyze the
Luttinger liquid to charge-density-wave transition at half-filling in the
effective polaronic Hamiltonian. We calculate the structure factor at all
fillings and find that the spin-spin correlation length decreases as one
deviates from half-filling. We also extend our derivation of polaronic
Hamiltonian to $d$-dimensions.
