appears already in *the first order* with respect to the single-hole transfer $T_{pp^{\prime }}^{apex}$, and the bipolaron energy spectrum consists of two subbands $E^{x,y}({\bf K),}$ formed by the overlap of $%
p_{x}$ and $p_{y}$ $apex$ oxygen orbitals, respectively (here we take the lattice constant $a=1$): $$\begin{aligned}
E^{x}({\bf K}) &=&t\cos (K_{x})-t^{\prime }\cos (K_{y}), \\
E^{y}({\bf K}) &=&-t^{\prime }\cos (K_{x})+t\cos (K_{y}). \nonumber\end{aligned}$$

-0.5mm

They transform into one another under $\pi /2$ rotation. If $t,t^{\prime
}>0, $ “$x"$ bipolaron band has its minima at ${\bf K=}(\pm \pi ,0)$ and $y$-band at ${\bf K}=(0,\pm \pi )$. In these equations $t$ is the renormalized hopping integral between $p$ orbitals of the same symmetry elongated in the direction of the hopping ($pp\sigma $) and $t^{\prime }$ is the renormalized hopping integral in the perpendicular direction ($pp\pi $). Their ratio $t/t^{\prime
}=T_{pp^{\prime }}^{apex}/T^{\prime }{}_{pp^{\prime }}^{apex}=4$ as follows from the tables of hopping integrals in solids. Two different bands are not mixed because $T_{p_{x},p_{y}^{\prime
}}^{apex}=0$ for the nearest neighbors. A random potential does not mix them either, if it varies smoothly on the lattice scale. Hence, we can distinguish ‘$x$’ and ‘$y$’ bipolarons with a lighter effective mass in $x$ or $y$ direction, respectively. The apex $z$ bipolaron, 