expressed via $E_{p}$ and $V_{ph}$ as $$g^{2}({\bf n-n^{\prime }})=\frac{1}{\omega _{0}}\left[ E_{p}-\frac{1}{2}%
V_{ph}({\bf n-n^{\prime }})\right] . \label{g2}$$

The Hamiltonian $\tilde{H}$, Eq.(\[trans\]), in zero order with respect to the hopping describes localised polarons and independent phonons, which are vibrations of ions relative to new equilibrium positions depending on the polaron occupation numbers. Importantly the phonon frequencies remain unchanged in this limit at any polaron density, $n$. At finite $\lambda$ and $n$ there is a softening of phonons $\delta \omega_0$ of the order of $\omega_0 n/\lambda^2$ [@ale2]. Interestingly the optical phonon can be mixed with a low-frequency polaronic plasmon forming a new excitation, “plasphon”, which was proposed in [@plasphon; @ale2] as an explanation of the anomalous phonon mode splitting observed in cuprates [@splitting]. The middle of the electron band is shifted down by the polaron level-shift $E_{p}$ due to the potential well created by lattice deformation.

When $V_{ph}$ exceeds $V_{c}$ the full interaction becomes negative and polarons form pairs. The real space representation allows us to elaborate more physics behind the lattice sums in $V_{ph}$ [@alekor2]. When a carrier (electron or hole) acts on an ion with a force ${\bf f}$, it displaces the ion by some vector ${\bf
x}={\bf f}/k$. Here $k$ 