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Pauling’s rules for oxide-based minerals: A re-examination based on quantum mechanical constraints and modern applications of bond-valence theory to Earth materials

  • Gerald V. Gibbs , Frank C. Hawthorne and Gordon E. Brown
Published/Copyright: July 2, 2022
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American Mineralogist
From the journal American Mineralogist Volume 107 Issue 7

Abstract

Since their introduction in 1929, 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 solids; amorphous materials such as silicate glasses and melts; nanomaterials, poorly crystalline solids; aqueous cation and anion complexes; and sorption complexes at mineral-aqueous solution interfaces. The predictive power of these simple yet powerful rules was challenged recently by George et al. (2020), 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. They also found that Pauling’s first rule is satisfied for only 66% of the coordination environments tested and concluded that no simple rule linking ionic radius to coordination environment will be predictive due to the variable quality of univalent radii.

We address these concerns and discuss quantum mechanical calculations that complement Pauling’s rules, particularly his first (radius sum and radius ratio rule) and second (electrostatic valence rule) rules. 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 (rc). Electron density at the bond critical point is a quantum mechanical observable that correlates well with Pauling bond strength. Moreover, a metal atom in a polyhedron has as many bonded radii as it has bonded interactions, resulting in metal and O atoms that may not be spherical. O atoms, for example, are not spherical in many oxide-based crystal structures. Instead, the electron density of a bonded oxygen is often highly distorted or polarized, 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. Bonded radii determined for metal atoms match the Shannon (1976) radii for more electropositive atoms, but the match decreases systematically as the electronegativities of the M atoms increase. As a result, significant departures from the radius ratio rule in the analysis by George et al. (2020) is not surprising. We offer a modified, more fundamental version of Pauling’s first rule and demonstrate that the second rule has a one-to-one connection between the electron density accumulated between the bonded atoms at the bond critical point 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) developed an equation and a set of parameters to describe the relation between bond length and bond strength, now redefined as bond valence to avoid confusion with Pauling bond-strength. Brown (1980) used the valence-sum rule, together with the path rule and the valence-matching principle, as the three axioms of bond-valence theory (BVT), a powerful method for understanding many otherwise elusive aspects of crystals and also their participation in dynamic processes. We show how a priori bond-valence calculations can predict unstrained bond-lengths and how bond-valence mapping can locate low-Z atoms in a crystal structure (e.g., Li) or examine possible diffusion pathways for atoms through crystal structures.

In addition, we briefly discuss Pauling’s third, fourth, and fifth rules, the first two of which concern the sharing of polyhedron elements (edges and faces) and the common instability associated with structures in which a polyhedron shares an edge or face with another polyhedron and contains high-valence cations. The olivine [α-(MgxFe1–x)2SiO4] crystal structure is used to illustrate the distortions from hexagonal close-packing of O atoms caused by metal-metal repulsion across shared polyhedron edges.

We conclude by discussing several applications of BVT to Earth materials, including the use of BVT to: (1) locate H+ ions in crystal structures, including the location of protons in the crystal structures of nominally anhydrous minerals in Earth’s mantle; (2) determine how strongly bonded (usually anionic) structural units interact with weakly bonded (usually cationic) interstitial complexes in complex uranyl-oxide and uranyl-oxysalt minerals using the valence-matching principle; (3) calculate Lewis acid strengths of cations and Lewis base strengths of anions; (4) determine how (H2O) groups can function as bond-valence transformers by dividing one bond into two bonds of half the bond valence; (5) help characterize products of sorption reactions of aqueous cations (e.g., Co2+ and Pb2+) and oxyanions [e.g., selenate (Se6+O4)2− and selenite (Se4+O3)2−] at mineral-aqueous solution interfaces and the important role of protons in these reactions; and (6) help characterize the local coordination environments of highly charged cations (e.g., Zr4+, Ti4+, U4+, U5+, and U6+) in silicate glasses and melts.

Keywords: Pauling’s rules; bonded radii; bonded interactions; bond paths; bond valence theory; bond valence analysis of crystal structures; bond valence analysis of water in nominally anhydrous mantle minerals; bond valence analysis of proton, cation, and oxyanion sorption complexes; application of bond-valence theory to uranyl-oxide and uranyl-oxysalt minerals; bond-valence analysis of proton, cation, and oxyanion adsorption at mineral-aqueous solution interfaces; bond-valence and XAFS analysis of the local coordination environments of highly charged cations in silicate glasses

Acknowledgments

The preparation of this review was stimulated by the recent paper of George et al. (2020), which evaluated the success of Pauling’s five rules, both singly and in combination, in predicting stable atom and polyhedron arrangements in over 5000 oxide-based crystal structures in the Inorganic Crystal Structure Data Base.

The Three Amigos, who have known each other professionally for the past 50+ years, decided to write this review with the hope that it might help students in a broad range of scientific disciplines, as well as their mentors, understand the major contribution made by Linus Pauling when he introduced his rules in 1929, and by David Brown, who introduced BVT in 1980. We were all fortunate to get to know both Linus Pauling and David Brown—the two stars of this review. G.B. thanks his co-author and Ph.D. mentor (G.V.G.) for introducing him to the principles of crystal chemistry, X‑ray crystallography, and structural chemistry, as well as for his unfailing guidance and friendship over the past 55 years. G.B. also acknowledges Linus Pauling for writing The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, first published in 1939. The third edition of this book, published in 1960, has had a major influence on our understanding of chemical bonding in Earth materials. In addition, G.B. thanks his many excellent graduate students, postdocs, and faculty colleagues at Stanford over the past 47 years, particularly John Bargar (SLAC National Accelerator Laboratory), the late Steven Towle, Peggy O’Day (U.C. Merced), and the late George Parks (Stanford University) for the stimulating and fruitful collaborations we had involving adsorption reactions of aqueous cations and oxyanions at mineral-aqueous solution interfaces. G.B. also thanks former postdoc Professor Francois Farges (Muséum National d’Histoire Naturelle, Paris), who was responsible for much of the work on our joint studies of the local structural environments of high-valence cations in aluminosilicate and titanosilicate glasses and melts. Finally, G.B. thanks the U.S. National Science Foundation (Chemistry and Geosciences Directorates) and the U.S. Department of Energy, Basic Energy Sciences and Biological and Environmental Research for their generous support of his research program over the past 45 years and their support of the two main synchrotron light sources (the Stanford Synchrotron Radiation Lightsource at SLAC National Accelerator Laboratory and the Advanced Photon Source at Argonne National Laboratory) where most of G.B.’s synchrotron-based studies of sorption complexes at mineral-water interfaces and his studies of structure-property relations in silicate glasses and melts were done. G.V.G. owes a debt to the late Richard Bader for tutoring him about the Feynman force exerted on the nuclei and the Ehrenfest force exerted on the electrons of the bonded atoms. Another of us (F.C.H.) was a graduate student from 1968 to 1973 at McMaster University, working in the Materials Research Institute where he was exposed on a daily basis to David Brown, Bob Shannon, and other scientific luminaries at coffee breaks and lunch. The result was a friendship between a geologist, a physicist, and a chemist that has lasted until the present time, with innumerable emails concerning the physics and chemistry of chemical bonding and its influence on the properties and behavior of solids. F.C.H. takes pleasure in acknowledging the enormous influence that David and Bob have had on his knowledge and understanding of crystals.

Finally, we thank several anonymous reviewers as well as American Mineralogist Editor Don Baker for careful reviews of our manuscript.

  1. Manuscript handled by Jie Xu,

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Received: 2020-12-22
Accepted: 2021-07-29
Published Online: 2022-07-02
Published in Print: 2022-07-26

© 2022 Mineralogical Society of America

Articles in the same Issue

  1. Highlights and Breakthroughs
  2. Mineral evolution heralds a new era for mineralogy
  3. MSA Review
  4. Pauling’s rules for oxide-based minerals: A re-examination based on quantum mechanical constraints and modern applications of bond-valence theory to Earth materials
  5. A cotunnite-type new high-pressure phase of Fe2S
  6. Density determination of liquid iron-nickel-sulfur at high pressure
  7. On the paragenetic modes of minerals: A mineral evolution perspective
  8. Lumping and splitting: Toward a classification of mineral natural kinds
  9. Thermal expansion of minerals in the amphibole supergroup
  10. A multi-faceted experimental study on the dynamic behavior of MgSiO3 glass in the Earth’s deep interior
  11. Origin of β-cristobalite in Libyan Desert Glass: The hottest naturally occurring silica polymorph?
  12. Time-resolved Raman and luminescence spectroscopy of synthetic REE-doped hydroxylapatites and natural apatites
  13. Reexamination of the structure of opal-A: A combined study of synchrotron X-ray diffraction and pair distribution function analysis
  14. A first-principles study of water in wadsleyite and ringwoodite: Implication for the 520 km discontinuity
  15. Inclusions in calcite phantom crystals suggest role of clay minerals in dolomite formation
  16. Crystal-chemical reinvestigation of probertite, CaNa[B5O7(OH)4]·3H2O, a mineral commodity of boron
  17. Crystal structure determination of orthorhombic variscite2O and its derivative AlPO4 structure at high temperature
  18. Transformation of Fe-bearing minerals from Dongsheng sandstone-type uranium deposit, Ordos Basin, north-central China: Implications for ore genesis
  19. Vaterite in a decrepitated diamond-bearing inclusion in zircon from a stromatic migmatite in the Chinese Sulu ultrahigh-pressure metamorphic belt
  20. Oxygen diffusion in garnet: Experimental calibration and implications for timescales of metamorphic processes and retention of primary O isotopic signatures
  21. Oxidation state of iron and Fe-Mg partitioning between olivine and basaltic martian melts
https://doi.org/10.2138/am-2021-7938
Keywords for this article
Pauling’s rules; bonded radii; bonded interactions; bond paths; bond valence theory; bond valence analysis of crystal structures; bond valence analysis of water in nominally anhydrous mantle minerals; bond valence analysis of proton, cation, and oxyanion sorption complexes; application of bond-valence theory to uranyl-oxide and uranyl-oxysalt minerals; bond-valence analysis of proton, cation, and oxyanion adsorption at mineral-aqueous solution interfaces; bond-valence and XAFS analysis of the local coordination environments of highly charged cations in silicate glasses

Articles in the same Issue

  1. Highlights and Breakthroughs
  2. Mineral evolution heralds a new era for mineralogy
  3. MSA Review
  4. Pauling’s rules for oxide-based minerals: A re-examination based on quantum mechanical constraints and modern applications of bond-valence theory to Earth materials
  5. A cotunnite-type new high-pressure phase of Fe2S
  6. Density determination of liquid iron-nickel-sulfur at high pressure
  7. On the paragenetic modes of minerals: A mineral evolution perspective
  8. Lumping and splitting: Toward a classification of mineral natural kinds
  9. Thermal expansion of minerals in the amphibole supergroup
  10. A multi-faceted experimental study on the dynamic behavior of MgSiO3 glass in the Earth’s deep interior
  11. Origin of β-cristobalite in Libyan Desert Glass: The hottest naturally occurring silica polymorph?
  12. Time-resolved Raman and luminescence spectroscopy of synthetic REE-doped hydroxylapatites and natural apatites
  13. Reexamination of the structure of opal-A: A combined study of synchrotron X-ray diffraction and pair distribution function analysis
  14. A first-principles study of water in wadsleyite and ringwoodite: Implication for the 520 km discontinuity
  15. Inclusions in calcite phantom crystals suggest role of clay minerals in dolomite formation
  16. Crystal-chemical reinvestigation of probertite, CaNa[B5O7(OH)4]·3H2O, a mineral commodity of boron
  17. Crystal structure determination of orthorhombic variscite2O and its derivative AlPO4 structure at high temperature
  18. Transformation of Fe-bearing minerals from Dongsheng sandstone-type uranium deposit, Ordos Basin, north-central China: Implications for ore genesis
  19. Vaterite in a decrepitated diamond-bearing inclusion in zircon from a stromatic migmatite in the Chinese Sulu ultrahigh-pressure metamorphic belt
  20. Oxygen diffusion in garnet: Experimental calibration and implications for timescales of metamorphic processes and retention of primary O isotopic signatures
  21. Oxidation state of iron and Fe-Mg partitioning between olivine and basaltic martian melts
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