Long-range Coulomb forces and localized bonds.

The ionic model is shown to be applicable to all compounds in which the atoms carry a net charge and their electron density is spherically symmetric regardless of the covalent character of the bonding. By examining the electric field generated by an array of point charges placed at the positions of the ions in over 40 inorganic compounds, we show that the Coulomb field naturally partitions itself into localized regions (bonds) which are characterized by the electric flux that links neighbouring ions of opposite charge. This flux is identified with the bond valence, and Gauss' law with the valence-sum rule, providing a secure theoretical foundation for the bond-valence model. The localization of the Coulomb field provides an unambiguous definition of coordination number and our calculations show that, in addition to the expected primary coordination sphere, there are a number of weak bonds between cations and the anions in the second coordination sphere. Long-range Coulomb interactions are transmitted through the crystal by the application of Gauss' law at each of the intermediate atoms. Bond fluxes have also been calculated for compounds containing ions with non-spherical electron densities (e.g. cations with stereoactive lone electron pairs). In these cases the point-charge model continues to describe the distant field, but multipoles must be added to the point charges to give the correct local field.