$%
v_{NNN}^{\prime }=\frac{V_{c}}{\sqrt{2}}-0.82\,E_{p}$, and the mass renormalization exponents as $g_{NN}^{2}=0.38(E_{p}/\omega)$, $%
g_{NNN}^{2}=0.60(E_{p}/\omega)$ and $(g^{\prime
}{}_{NNN})^{2}=0.59(E_{p}/\omega)$.

Let us now discuss different regimes of the model. At $V_{c}>1.23\,E_{p}$, no bipolarons are formed and the system is a polaronic Fermi liquid. Polarons tunnel in the [*square*]{} lattice with $t=T_{NN}\exp (-0.38E_{p}/\omega)$ and $t^{\prime
}=T_{NNN}\exp (-0.60E_{p}/\omega)$ for $NN$ and $NNN$ hoppings, respectively. Since $g_{NNN}^{2}\approx (g_{NNN}^{\prime })^{2} $ one can neglect the difference between $NNN$ hoppings within and between the octahedra. A single polaron spectrum is therefore $$E_{1}({\bf k})=-E_{p}-2t^{\prime }[\cos k_{x}+\cos k_{y}]-4t\cos
(k_{x}/2)\cos (k_{y}/2).  \label{tfifteen}$$ The polaron mass is $m^{\ast }=1/(t+2t^{\prime })$. Since in general $%
t>t^{\prime }$, the mass is mostly determined by the $NN$ hopping amplitude $t$.

If $V_{c}<1.23\,E_{p}$ then intersite $NN$ bipolarons form. The bipolarons tunnel in the plane via four resonating (degenerate) configurations $A$, $B$, $C$, and $D$, as shown in Fig.2. In the first order of the renormalized hopping integral, one should retain only these lowest energy configurations and discard all the processes that involve configurations with higher energies. The result of such a projection is the bipolaron Hamiltonian, $$\begin{aligned}
H_{b} &=&(V_{c}-3.23\,E_{p})\sum_{{\bf l}}[A_{{\bf l}}^{\dagger }A_{{\bf l}%
}+B_{{\bf l}}^{\dagger }B_{{\bf l}}+C_{{\bf l}}^{\dagger }C_{{\bf l}}+D_{%
{\bf l}}^{\dagger }D_{{\bf l}}] \\
&&-t^{\prime }\sum_{{\bf l}}[A_{{\bf 