Is doped BaBiO$_3$ a conventional superconductor?

We report density functional calculations based on local density approximation (LDA) of the properties of doped barium bismuthates. Using linear-response approach developed in the framework of the linear muffin-tin-orbital method the phonon spectrum of the Ba$_{0.6}$K$_{0.4}$BiO$% _3$ system is calculated and is compared with the results of the neutron diffraction measurements. The effect of doping in the calculation is modelled by the virtual crystal and mass approximations. The electron-phonon coupling constant $\lambda $ is then evaluated for a grid of phonon wave-vectors using the change in the potential due to phonon distortion found self-consistently. A large coupling of the electrons to the bond-stretching oxygen vibrations and especially to the breathing-like vibrations is established. Also, a strongly anharmonic potential well is found for tilting-like motions of the oxygen octahedra. This mode is not coupled to the electrons to linear order in the displacements, therefore an anharmonic contribution to $\lambda $ is estimated using frozen-phonon method. Our total (harmonic plus anharmonic) $\lambda $ is found to be 0.34. This is too small to explain high-temperature superconductivity in Ba$_{0.6}$K$_{0.4}$BiO% $_3$ within the standard mechanism. Finally, based on standard LDA and LDA+U like calculations, a number of properties of pure BaBiO$_3$ such as tilting of the octahedra, breathing distortion, charge disproportionation and semiconducting energy gap value is evaluated and discussed in connection with the negative $U$ extended Hubbard model frequently applied to this compound.