is the ion’s force constant. The total energy of the carrier-ion pair is $-{\bf f}%
^{2}/(2k)$. This is precisely the summand in Eq.(\[polshift\]) expressed via dimensionless coupling constants. Now consider two carriers interacting with the [*same*]{} ion. The ion displacement is ${\bf x}=({\bf f}%
_{1}+{\bf f}_{2})/k$ and the energy is $-{\bf f}_{1}^{2}/(2k)-{\bf f}%
_{2}^{2}/(2k)-({\bf f}_{1}\cdot {\bf f}_{2})/k$. Here the last term should be interpreted as an ion-mediated interaction between the two carriers. It depends on the scalar product of ${\bf f}_{1}$ and ${\bf f}_{2}$ and consequently on the relative positions of the carriers with respect to the ion. If the ion is an isotropic harmonic oscillator, then the following simple rule applies. If the angle $\phi $ between ${\bf f}_{1}$ and ${\bf f}_{2}$ is less than $\pi /2$ the polaron-polaron interaction will be attractive, if otherwise it will be repulsive. In general, some ions will generate attraction, and some repulsion between polarons.

The overall sign and magnitude of the interaction is given by the lattice sum in Eq.(\[v\]). One should note that according to Eq.(\[g2\]) an attractive EPI reduces the polaron mass (and consequently the bipolaron mass), while repulsive EPI enhances the mass. Thus, the long-range EPI serves a double 