Bond-valence theory is a general electrostatic valence rule

Abstract

Pauling's electrostatic valence rule assigns to each bond of a cation of charge $z$ with $n$ anion neighbors the strength $z/n$. Bond-valence theory replaces the equal split of valence among bonds by $s=\exp[(R_0-d)/B]$ in the bond length $d$ and takes the sum rule, that the valences at each atom add to its formal charge, as its first axiom. Here the bond valence is defined in the electron density. Atomic basins of a first-principles density are bounded by faces, one per bond critical point, and the electric flux of that density through a face is a charge associated with the bond. The fluxes leaving a cation sum to its basin charge by Gauss's law, so on this surface the sum rule is a theorem. Across 3680 oxides, halides, chalcogenides and pnictides the flux follows the bond-valence law site by site, with a median deviation of ten percent. The softness $B$ varies nearly an order of magnitude from stiff sulfides to softer arsenides. In all cases it is set by the electrostatic flux at atomic basin faces. $R_0$ then follows from the bond lengths and their differences. Anion valences equal those of bonded cations to within 5\% for most of the oxide, fluoride, sulfide and nitride sites. Transferability of $(R_0,B)$ between compounds fails in 96 of 106 families, suggesting that the softness is a local site property defined by the basin charge rather than a transferable constant of a given cation--anion pair. Eliminating $B$ by the sum rule gives $s=(z/n)^{(R_0-d)/(R_0-d_B)}$, Pauling's strength raised to a distance exponent at the softness-weighted mean bond length $d_B$. Thus, bond-valence theory is Pauling's rule with a distance exponent, and the electron density supplies the surface on which the two are equivalent.