  The Rice-Sneddon model for BaBiO$_3$ is a nice model Hamiltonian for
considering the properties of polarons and bipolarons in a three-dimensional
oxide crystal. We use exact diagonalization methods on finite samples to study
the stability and properties of polarons and bipolarons. Because polarons, when
they form, turn out to be very well-localized, we are able to converge
accurately our calculations for two-electron bipolaron wavefunctions,
accounting for the Coulomb interaction without approximation. Some of our
results are compared with and interpreted by reference to the variational
method of Landau and Pekar. We calculate both electronic and vibrational
excitations of the small polaron solutions, finding a single vibrational state
localized with the full symmetry of the polaron, which has its energy
significantly increased. Both on-site (Hubbard) and long-range Coulomb
repulsion are included in the bipolaron calculation, but due to the high degree
of localization, the long-range part has only a small influence. For a
reasonable on-site repulsion $U$ equal to 2 times the band width $W$, bipolaron
formation is significantly suppressed; there is a large window of
electron-phonon coupling where the polaron is stable but the bipolaron decays
into two polarons.
