Madelung Energy of the Valence Skipping Compound BaBiO$_3$

Several elements show valence skip fluctuation, for instance, Tl forms the compounds in valence states +1 and +3, and Bi forms in +3 and +5 states. This kind of fluctuation gives rise to a negative effective attractive interaction and the Kondo effect. In the compounds of valence skipping elements, the carrier doping will induce superconductivity with high critical temperature. For example, Ba$_{1-x}$K$_x$BiO$_3$ shows high $T_c$ which is unlikely from the conventional electron-phonon mechanism. The reason for the missing of some valence states in such valence skip compounds remains a mystery. We have performed the evaluation of the Madelung potential for BaBiO$_3$, and have shown for the first time that charge-ordered state is stabilized if we take into account the polarization of the oxygen charge. We argue that the effective Coulomb interaction energy $U$ may be negative evaluating the local excitation energy.