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The Rice-Sneddon model for BaBiO$_3$ is a nice model Hamiltonian forconsidering the properties of polarons and bipolarons in a three-dimensionaloxide crystal. We use exact diagonalization methods on finite samples to studythe stability and properties of polarons and bipolarons. Because polarons, whenthey form, turn out to be very well-localized, we are able to convergeaccurately our calculations for two-electron bipolaron wavefunctions,accounting for the Coulomb interaction without approximation. Some of ourresults are compared with and interpreted by reference to the variationalmethod of Landau and Pekar. We calculate both electronic and vibrationalexcitations of the small polaron solutions, finding a single vibrational statelocalized with the full symmetry of the polaron, which has its energysignificantly increased. Both on-site (Hubbard) and long-range Coulombrepulsion are included in the bipolaron calculation, but due to the high degreeof localization, the long-range part has only a small influence. For areasonable on-site repulsion $U$ equal to 2 times the band width $W$, bipolaronformation is significantly suppressed; there is a large window ofelectron-phonon coupling where the polaron is stable but the bipolaron decaysinto two polarons.Comment: REVTeX, 8 pages, 7 figure

Polaron and bipolaron formation in a cubic perovskite lattice