be very heavy in HHM, where EPI is short-ranged. Actually HHM led some authors to the conclusion that the formation of itinerant small polarons and bipolarons in real materials is unlikely [@mel], and high-temperature bipolaronic superconductivity is impossible [@and2]. Nevertheless treating the onsite repulsion (Hubbard $U$) and the short-range EPI on an equal footing led several authors to the opposite conclusion with respect to bipolaron mobility even in HHM, which is generally unfavorable for coherent tunnelling. Aubry [@aub93] found along with the onsite bipolaron ($S0$) also an anisotropic pair of polarons lying on two neighboring sites (i.e. the $intersite$ bipolaron, $S1$) with classical phonons in the extreme adiabatic limit. Such bipolarons were originally hypothesized in [@alexandrov:1991; @alegap] to explain the anomalous nuclear magnetic relaxation (NMR) in cuprate superconductors. The intersite bipolaron could take a form of a “quadrisinglet” ($QS$) in 2D HHM, where the electron density at the central site is 1 and “1/4” on the four nearest neigbouring sites. In a certain region of $U$, where $QS$ is the ground state, the double-well potential barrier which usually pins polarons and bipolarons to the lattice depresses to almost zero, so that adiabatic lattice bipolarons can be rather mobile.

Mobile $S1$ 