Lecture 9: Coordination and Pauling's Rules

Lecture 9: 

Coordination and Pauling’s Rules 

To understand the atomic structure of minerals, , we typically think in 

terms of spherical cations and anions held together predominantly 

by ionic bonds 

One of the properties of these ions that we would like to know is their 

RADIUS, , I.e., their size 

This sounds easy, but remember, ions are really formed from nuclei 

with probability density clouds of electrons moving around them. 

Thus, we refer to: 

Effective Electrostatic Radii: “imaginary” fixed radii of an ion 

Why imaginary? Because electrons are in probability density 

clouds and because radii actually change somewhat depending on 

bonding environment

Given that the effective electrostatic radii is imaginary, how 

do we “measure” it??? 

! Look at anion-cation bond lengths in structures with predominantly 

ionic bonding 

! Once you know the total bond distance and one ion radius, , you can 

calculate the other 

!Shannon and Prewitt (1969); Shannon (1976) used many oxide 

structures to regress average effective electrostatic radii of cations 

Example: 

Periclase (MgO) 

Mg-O length 2.11Å according to XRD 

O -2 eff = 1.32Å (by definition) 

So, Mg +2 eff = 2.11Å - 1.32Å = 0.79Å

Effective Electrostatic Radii 

Cation Radii ?? Elemental Radii (> or

Some General Rules Regarding Effective Electrostatic Radii: 

! Larger with Z within groups (going down columns) 

! Larger with increasing number 

Note: An exception is the so-called “Lanthanide Contraction”; trivalent 

lanthanides (La +3 to Lu +3 ) decrease in radius with increasing Z 

Why? Inner electron orbitals build before outer orbitals are added; 

higher nuclear charge combined with relatively weak shielding draws 

electrons into the nucleus 

! For cations with same electronic structure, radii decrease with 

increasing charge 

i.e. R eff P +5 < R eff Na + due to larger nuclear charge in P +5 

Effective Electrostatic Radii of Ions typically change somewhat with: 

COORDINATION NUMBER

COORDINATION NUMBER (C.N.): in packed ionic structures, the 

number of nearest neighbors (1st coordination shell); i.e. the 

number of anions surrounding a cation (or number of cations 

surrounding an anion) 

Example: K + 

6-fold coordination -> 1.38Å 



This increase in radius with coordination number reflects expansion of the 

cation into more available space between larger number of anions 

Regular Coordination Polyhedra: : all cation-anion distances assumed to 

be equal

Is it possible to predict coordination environments?

Pauling’s Rule #1 (‘Radius( 

Ratio Principal’): 

! The distance between cations and anions can be calculated 

from their effective electrostatic radii; coordination number 

depends on the relative radii of cations and surrounding 

anions 

! Greater radius = greater coordination number 

Note: ONLY strictly true for ionic bonding with undistorted polyhedra, 

pretending that ions are spheres (which they aren’t t ) and they are 

unpolarized 

Polarization: : distortion of ion shape; large, monovalent ions are 

most easily polarized



Radius Ratio (R.R.) = R c /R a 

R c = cation radius 

R a = anion radius 

2-fold: R.R. < 0.155 

3-fold: R.R. 0.155 - 0.225 

4-fold: R.R. 0.225 5 - 0.414 

6-fold: R.R. 0.414 - 0.732 

8-fold: R.R. 0.732 - 1.000



Example: 

R eff Na + = 1.10Å 

Cl - = 1.72Å 

R eff Cl 

C.N. of Na in NaCl? 

R.R. = 1.10/1.72 = 0.64 

6-Fold 

In 1929 Linus Pauling came up with some other useful rules…

Pauling’s Rule #2 (‘Electrostatic( 

Valency Principal’): 

! The strength of a bond (electrostatic valence) equals the ionic 

valence (charge) divided by the coordination number 

! Sum of bond valences = ionic valence 

E.V. = Z./C.N. 

Example: SiO 4 

Si +4 Each: Si-O 

O -2 

Si-O bond 

E.V. = +4/4 = 1 

O -2 

Si +4 O -2 O -2 

Each: O-Si bond 

E.V. = -2/2 = -1



Example: NaCl 

Cl - 

Each: Na-Cl bond 

E.V. = +1/6 = +1/6 

Cl - Cl - 

Na + 

Cl - 

Cl - 

Cl - 

Each: Cl-Na bond 

E.V. = -1/6 = -1/6 

Note: some problems with this approach -- irregular coordination 

polyhedra, , mixed bonding types



Isodemic: compounds with all bonds of equal strength 

Ex. Spinel 

AB 2 O 4 A = +2; B = +3 

By XRD, we know that AIV, BVI 

So, A = +2/4 = +1/2 

B = +3/6 = +1/2 

Anisodesmic: bonds of unequal strength; common in compounds with 

anionic complexes; electrostatic valence within the anionic 

complex is greater than half the anion charge 

Ex. Carbonate 

CO -2 3 C +4 is 3-fold with respect to O -2 

E.V. = +4/3 = 1 1/3 

(most strength within complex) 

Mesodesmic: bond strength is exactly half the anion charge 

Ex. Si-O 

O in silicates 

Si is 4-fold with respect to O E.V. = +4/4 = +1

Pauling’s Rule #3 

! Vertex-sharing (2 cations share 1 anion) between tetrahedra 

or octahedra is energetically stable 

! Edge-sharing (2 cations share 2 anions) between polyhedra is 

less stable; ; rare for tetrahedra, , more common for octahedra 

! Face-sharing (2 cations share 3 anions) between polyhedra is 

unstable; ; never occurs for tetrahedra; ; rare for octahedra 

Why? Electrostatic Repulsion 

Less of a problem for octahedra because cation-cation distances are 

longer and cations are typically of lower charge

Pauling’s Rule #4 

! Cations of high charge and small coordination number tend not to 

share anions with other cations 

! Why? Repulsion between cations 

Pauling’s Rule #5 (‘Principal( 

of Parsimony’) 

! The number of different components in a crystal tends to be small; 

if lots of ions are present, they tend to occupy the same structural 

position (‘sites( 

sites’)

Coordination of Common Cations and Anions: 

Most common element in crust by weight? O 

Therefore, many minerals contain O 

O -2 effective electrostatic radius: 1.27-1.34Å 

Si +4 : 4-fold 

Al +3 : 4-fold or 6-fold 

Fe +3 : 6-fold (4-fold) 

Mg +2 : 6-fold or 8-fold 

Fe +2 : 6-fold 

Mn +2 : 6-fold or 8-fold 

Na + : 8-fold 

Ca +2 : 8-fold 

K + : 12-fold

Lecture 9: Coordination and Pauling's Rules

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