PAULING’S RULES FOR OXIDE-BASED MINERALS: A RE-EXAMINATION BASED ON QUANTUM MECHANICAL CONSTRAINTS

Pauling’s five rules have been used by scientists from many disciplines to rationalize and predict stable arrangements of atoms and coordination polyhedra in crystalline and amorphous solids; nanomaterials; aqueous complexes; and sorption complexes at mineral-aqueous solution interfaces. The predictive power of these simple rules was challenged recently by George et al. (2020, Angewan. Chemie), who performed a statistical analysis of the performance of Pauling’s five rules for about 5000 oxide crystal structures. They concluded that only 13% of the oxides satisfy the last four rules simultaneously and that the second rule has the most exceptions and that Pauling’s first rule is satisfied for only 66% of the coordination environments tested.

We address these concerns and discuss quantum mechanical calculations that complement Pauling’s rules (Gibbs, Hawthorne, Brown 2022, Am. Mineral.) We also present a more realistic view of the bonded radii of atoms, derived by determining the local minimum in the electron density distribution measured along trajectories between bonded atoms known as bond paths, i.e., the bond critical point (r_c). Electron density (ED) at r_c is a quantum mechanical observable that correlates well with Pauling bond strength. The ED of a bonded oxygen is often highly distorted, with its bonded radius decreasing systematically from ~1.38 Å when bonded to highly electropositive atoms like sodium to 0.64 Å when bonded to highly electronegative atoms like nitrogen. Significant departures from the radius ratio rule in the analysis by George et al. (2020) is not surprising. We offer a more fundamental version of Pauling’s first rule and demonstrate that the second rule has a one-to-one connection between the ED accumulated between bonded atoms at the r_c and the Pauling bond strength of the bonded interaction. Pauling’s second rule implicitly assumes that bond strength is invariant with bond length for a given pair of bonded atoms. Many studies have since shown that this is not the case, and Brown and Shannon (1973, Acta Crystallogr.) developed an equation and a set of parameters to describe the relation between bond length and bond strength. We also briefly discuss Pauling’s third, fourth, and fifth rules and conclude by discussing several applications of Bond Valence Theory to Earth materials.