which explicitly includes the finite-range Coulomb repulsion, $V_c$, and the strong long-range EPI [@ale5; @alekor2]. The implicitly present (infinite) Hubbard $U$ prohibits double occupancy and removes the need to distinguish the fermionic spin, if we are interested in the charge rather than spin excitations. Introducing spinless fermion operators $c_{{\bf n}}$ and phonon operators $d_{{\bf m}}$, the Hamiltonian of CFM is written in the real-space representation as [@alekor2] $$\begin{aligned}
H &=&\sum_{{\bf n\neq n^{\prime }}}T({\bf n-n^{\prime }})c_{{\bf n}%
}^{\dagger }c_{{\bf n^{\prime }}}+{1\over {2}}\sum_{{\bf n\neq n^{\prime }}}V_{c}({\bf %
n-n^{\prime }})c_{{\bf n}}^{\dagger }c_{{\bf n}}c_{{\bf n^{\prime }}%
}^{\dagger }c_{{\bf n^{\prime }}}+ \\
&&\omega _{0}\sum_{{\bf n\neq m} }g({\bf m-n})({\bf e}\cdot {\bf
e}_{{\bf m-n}})c_{{\bf n}}^{\dagger }c_{{\bf n}}(d_{{\bf m}
}^{\dagger }+d_{{\bf m} })+  \nonumber \\
&&\omega _{0}\sum_{{\bf m}}\left( d_{{\bf m} }^{\dagger }d_{{\bf m}%
}+\frac{1}{2}\right),  \nonumber\end{aligned}$$ where $T({\bf n})$ is the bare hopping integral in a rigid lattice. In general, this many-body model is of considerable complexity. However, if we are interested in the non or near adiabatic limit and the strong EPI, the kinetic energy is a perturbation. Then the model can be grossly simplified using the Lang-Firsov canonical transformation [@lan] in the Wannier representation for electrons and phonons, $$S=\sum_{{\bf m\neq n} }g({\bf m-n})({\bf e}\cdot {\bf e}_{%
{\bf 