Real-Space and $\overrightarrow{\mathrm{k}}$-Space Electron Pairing in $\mathrm{Ba}{\mathrm{Pb}}_{1-x}{\mathrm{Bi}}_x{\mathrm{O}}_3$

A theory of the condensation of electrons, coupled strongly to optic phonons, into a lattice of real-space electron pairs is presented. Even if the on-site interaction is repulsive, real-space pairing of the Anderson form may occur because of interaction with other sites. Dilution weakens the intersite interaction and causes a phase transition that may be to a metallic state which undergoes $\overrightarrow{\mathrm{k}}$-space (BCS) pairing at low temperatures. The results are applied to $\mathrm{Ba}{\mathrm{Pb}}_{1-x}{\mathrm{Bi}}_x{\mathrm{O}}_3$.