Pauling's Rules

Pauling’s Rules are a set of five principles developed in the late 1920s that describe how anions and cations pack together in ionic crystal structures. ^[1] They rest on a key simplifying assumption: that the bonding in many common minerals is sufficiently ionic that the ions can be treated as hard spheres of consistent sizes, packing together on purely geometric principles without the directional constraints of covalent bonding. ^[1] This approximation is justified because the resulting predictions match what is actually observed in detailed structural studies. ^[1]

The rules apply most directly to minerals in which oxygen is the dominant anion, because oxygen-cation bonds in common silicates and oxides range from about half ionic (Si-O) to strongly ionic (K-O, Na-O), making the ionic sphere model a reasonable first approximation for understanding these structures. ^[1]

Rule 1: The Coordination Principle

The coordination principle states that a coordination polyhedron of anions forms around each cation, and both the cation-anion bond length and the number of anions coordinating with the cation are controlled by the relative sizes of the cation and anion. ^[1] The coordination number (CN) is the number of anions in contact with a given cation, and the coordination polyhedron is the geometric shape defined by connecting those anion centers. Twelve-fold coordination (CN = 12) is the largest common coordination, but it occurs only when the cation is nearly the same size as the anion; as the cation shrinks relative to the anion, the number of anions that can fit around it decreases through 8, 6, 4, 3, and finally 2. ^[1]

The Radius Ratio

The relative size of cation and anion is expressed as the radius ratio (RR): ^[1]

RR = R_c / R_a

where R_c is the cation radius and R_a is the anion radius. As a general rule, a cation will bond with as many anions as can fit around it while maintaining cation-anion contact; the maximum coordination is limited by the geometry of packing. ^[1] The radius ratios defining the boundaries between coordination geometries are given in Table 4.1 of the source and summarized below:

Because O^2- has an effective ionic radius of about 1.26 Å, it is possible to predict which coordination each common cation will adopt in oxygen-bearing minerals. Si⁴⁺ and Al³⁺ are small enough for tetrahedral (4-fold) sites; Fe²⁺, Mg²⁺, and Al³⁺ fit in octahedral (6-fold) sites; Na⁺, Ca²⁺, and K⁺ favor 8-fold and 12-fold sites. ^[1]

Rule 2: The Electrostatic Valency Principle

The electrostatic valency principle dictates that for an ionic crystal to be stable, the sum of the electrostatic bond strengths radiating from all adjacent cations must precisely balance the negative charge of the central anion. ^[1] The strength of each electrostatic valence bond (evb) is defined as the cation charge divided by its coordination number: ^[1]

evb = cation charge / CN

This principle leads to three fundamentally different structural categories depending on how bond strengths are distributed across the anions.

Isodesmic Bonding

When all ionic bonds have the same strength - meaning no single bond takes more than half of any anion’s charge - the structure is called isodesmic. ^[1] In isodesmic structures, the anions tend to pack in highly symmetrical arrangements (often based on cubic or hexagonal closest packing), and the cations occupy interstitial polyhedra. Most minerals with simple anions - halides, oxides, and many sulfides - have isodesmic structures, and they tend to crystallize in the isometric, tetragonal, or hexagonal crystal systems because of this high symmetry. ^[1]

Anisodesmic Bonding

When small, highly charged cations form bonds that take more than half of the anion’s charge, the result is anisodesmic bonding and the formation of discrete anionic groups. ^[1] Calcite (CaCO₃) is the clearest example. Carbon (C⁴⁺) coordinates with three O^2- anions, giving each C-O bond an evb of 4/3 = 1.33, which is more than half of the available -2 charge on oxygen. This means oxygen is much more strongly attracted to carbon than to any other cation, and distinct CO₃^2- carbonate groups form as structural units. ^[1] The carbonates, sulfates, and phosphates are all anisodesmic groups.

Mesodesmic Bonding

Mesodesmic bonding is a special case in which the anion-cation bond takes exactly half of the anion’s charge, leaving the other half available to form an identical bond with a second cation. The silicates are the only common mesodesmic group. ^[1] Silicon (Si⁴⁺) coordinates with four O^2- in a silicon tetrahedron, giving each Si-O bond an evb of 4/4 = 1, exactly half of oxygen’s -2 charge. ^[1] This leaves each oxygen with a residual -1 charge available to bond with a second Si⁴⁺, allowing silicon tetrahedra to share oxygen atoms and link into pairs, chains, rings, sheets, and three-dimensional frameworks - the structural diversity that defines the silicate mineral class. ^[1]

Rule 3: Sharing of Polyhedral Elements I

Adjacent coordination polyhedra in an ionic structure prefer to share as few anions as possible, ideally sharing only a single corner (one anion), because this keeps positively charged cations at the greatest distance from each other. ^[1] For tetrahedra, the distance between cation centers is 2.45 R_a when only a corner is shared, but drops to 1.414 R_a when an edge (two anions) is shared, and to 1.15 R_a when a face (three anions) is shared. ^[1] Because like-charged cations repel each other, bringing them closer together requires energy and destabilizes the structure. Corner sharing is therefore the preferred and most stable configuration; edge sharing is tolerated under some conditions; face sharing is the least stable.

Rule 4: Sharing of Polyhedral Elements II

In structures that contain more than one type of cation, the highest-charged cations minimize the number of anions they share with adjacent polyhedra, keeping high-charge cations well separated from each other. ^[1] Conversely, low-charge cations tolerate more sharing of edges or faces between their coordination polyhedra. Perovskite (CaTiO₃) illustrates this rule clearly: Ti⁴⁺ occupies octahedral (6-fold) sites that share only single O^2- corners, keeping the highly charged Ti⁴⁺ cations widely spaced; Ca²⁺, which has a lower charge and a larger 12-fold coordination site, shares four O^2- on a face. ^[1]

Rule 5: The Principle of Parsimony

The number of fundamentally different structural site types in a mineral tends to be small, even in chemically complex minerals. ^[1] Cations systematically distribute themselves among no more than four (and usually fewer) distinct coordination polyhedra, allocated by size and charge. ^[1] The strongest evidence for this rule is that the chemical formulas of minerals universally express the ratio of anions to cations as small integers - a direct consequence of a structure built from a limited number of repeating site types. ^[1] If mineral structures were truly random assemblages of many different site geometries, no such simple ratio would arise.

Application: Halite as a Worked Example

The NaCl structure of halite demonstrates how Pauling’s Rules predict and constrain a real mineral structure. The ionic radii of Na⁺ and Cl^- are 1.16 Å and 1.81 Å respectively, giving a radius ratio of 0.64 - within the 0.414-0.732 range for octahedral (6-fold) coordination. ^[1] Rule 2 (electrostatic valency) requires that every octahedral site between the {111} layers of Cl^- be occupied, so that every Na⁺ bonds to six Cl^- and vice versa. This forces Na⁺ octahedra to share edges - acceptable here because Na⁺ has only a single +1 charge, making edge sharing less destabilizing under Rule 3. The structure involves only one cation site type, satisfying Rule 5. ^[1]

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