m-n}})c_{{\bf n}}^{\dagger }c_{{\bf n}}(d_{{\bf m}}^{\dagger }-d_{%
{\bf m}}).$$

Here we consider a particular lattice structure, where intersite lattice bipolarons tunnel already in the first order in $T({\bf
n})$. That allows us to average the transformed Hamiltonian, $\tilde{H}=\exp(S) H \exp(-S)$ over phonons to obtain $$\tilde{H}=H_{0}+H_{pert}, \label{trans}$$ where $$H_{0}=-E_{p}\sum_{{\bf n}}c_{{\bf n}}^{\dagger }c_{{\bf n}}+{1\over
{2}}\sum_{{\bf n\neq
n^{\prime }}}v({\bf n-n^{\prime }})c_{{\bf n}}^{\dagger }c_{{\bf n}}c_{{\bf %
n^{\prime }}}^{\dagger }c_{{\bf n^{\prime }}} +\omega _{0}\sum_{{\bf m}}\left( d_{{\bf m} }^{\dagger }d_{{\bf m}%
}+\frac{1}{2}\right),$$ and $$H_{pert}=\sum_{{\bf n\neq n^{\prime }}}t({\bf n-n^{\prime }})c_{{\bf n}%
}^{\dagger }c_{{\bf n^{\prime }}}.$$ is a perturbation. $E_{p}$ is the familiar polaron level shift, $$E_{p}=\omega_0 \sum_{{\bf m}\nu }g^{2}({\bf m-n})({\bf e}\cdot {\bf
e}_{{\bf m-n}})^{2}, \label{polshift}$$ which is independent of ${\bf n}$. The polaron-polaron interaction is $$v({\bf n-n^{\prime }})=V_{c}({\bf n-n^{\prime }})-V_{ph}({\bf n-n^{\prime }}%
),$$ where $$V_{ph}({\bf n-n^{\prime }}) =2\omega _{0}\sum_{\bf m}g({\bf %
m-n})g({\bf m-n^{\prime }})
({\bf e}\cdot {\bf e}_{{\bf m-n}})({\bf e}\cdot {\bf e}_{%
{\bf m-n^{\prime }}}). \label{v}$$ The transformed hopping integral is $t({\bf n-n^{\prime }})=T({\bf
n-n^{\prime }})\exp [-g^{2}({\bf n-n^{\prime }})]$ with $$\begin{aligned}
g^{2}({\bf n-n^{\prime }}) &=&\sum_{{\bf m},\nu }g({\bf m-n})({\bf e}\cdot {\bf e}_{{\bf m-n}})\times \\
&&\left[ g({\bf m-n})({\bf e}\cdot {\bf e}_{{\bf m-n}})-g({\bf m-n^{\prime }})({\bf e}\cdot {\bf e}_{{\bf m-n^{\prime }}})%
\right]  \nonumber\end{aligned}$$ at low temperatures. The mass renormalization exponent can be 