Pauling’s Rules

In 1929 Linus Pauling [J. Am. Chem. Soc. 51, 1010 (1929)] stated five principles which could be used to determine the structures of complex ionic/covalent crystals.  These ideas have since been refined and quantified but they remain an amazingly useful guide to understanding the crystal structures.  They serve as the outline for a discussion of structure determining forces in crystals where ionic interactions dominate.

(1) A coordinated polyhedron of anions is formed about each cation, the cation-anion distance determined by the sum of ionic radii and the coordination number by the radius ratio.

Although many different sets of ionic radii have been created over the years, almost everyone currently uses the ionic radii set first derived by Shannon and Prewitt (R.D. Shannon and C.T. Prewitt, Acta Cryst. B25, 925-945 (1969)) and later refined by Shannon (R.D. Shannon, Acta Cryst. A32, 751-767 (1976)).  They actually derived two sets of ionic radii, one set is called the crystal radii (CR) was derived to best approximate experimentally determined ionic radii, the second set called ionic radii (IR) have values which approximate the cation and anion sizes found in the older tables of ionic radii.  The two sets are related by the following equations:

for cations        IR = CR - 14 pm
for anions         IR = CR + 14 pm

Below I outline several trends in the crystal radii values of Shannon.

(A) As we move down a group in the periodic table, the CR increases. The exception to this is going from the 4d to 5d series among the transition metals (due to the lanthanide contraction).

For example, consider the crystal radii (in angstroms) of the following elements in 6-fold coordination.

          Al³⁺       0.675
          Ga³⁺      0.760
          In³⁺       0.940
          Tl³⁺       1.025

          Ti⁴⁺       0.745
          Zr⁴⁺       0.86
          Hf⁴⁺       0.85

(B) As you increase the oxidation state, cations get smaller.

For example, (once again in 6-fold coordination)

  Mn²⁺      0.810
 Mn³⁺      0.785
 Mn⁴⁺      0.670

 Ti²⁺        1.000
 Ti³⁺        0.810
 Ti⁴⁺        0.745

(C) As you move right across the periodic table the CR radii decreases (all other things being equal).

For example, (6 coordinate radii)

 La³⁺       1.172
 Nd³⁺      1.123
 Gd³⁺      1.078
 Lu³⁺      1.001

(D) As you increase the coordination number (CN), the CR increases.

For example, consider the CR of Sr²⁺

CN               CR
 6                  1.32
 8                  1.40
 9                  1.45
10                 1.50
12                 1.58

(2) The bond valence of each ion should be approximately equal to its oxidation state.
 

The idea of bond valence has long been applied to organic molecules. Each atom is given a valence of 1 for each bond it forms.  For example in formaldehyde the valences are:

                                        Atom          Valence
                                             H                    1
                                             O                    2
                                             C                    4

while in ethene the valences are:

                                             H                     1
                                             C                     4

Looking at valences in this way is a straightforward process in organic compounds.  The valence of each bond takes an integer value, with the bond order (i.e. double bond, single bond, etc.) correlated to the distance between atoms.  The beauty of this method is that it is simple (to the extent where organic chemists take it for granted) yet very powerful (it allows one to quickly and simply generate connectivities of organic molecules).

The catch when trying to extend this concept to inorganic molecules is that bonds can no longer be assumed to adopt integral valences (i.e. to say that Al when octahedrally coordinated has a valence of 6, while when tetrahedrally coordinated has a valence of 4 is not particularly useful, to be used predictively we need to assume the valence of aluminum is a constant value).

Pauling took this idea and extended bond valence concept to ionic/covalent inorganic compounds. He stated that the valence of an ion (V_i, equal to the oxidation state of the ion) is equal to a sum of the valences of its bonds (s_ij).

V_i = S s_ij

The best way to illustrate this concept is to do some examples.

Example 1-TiO₂ (Rutile)

Each Ti has an oxidation state of +4, and is coordinated to 6 oxygens.

4 = 6 (s_ij)
s_ij = 2/3

If we then consider the bond valence of oxygen, which is coordinated by 3 Ti atoms

Vo = 3 (s_ij) = 3 (-2/3) = -2

Each bond has a valence of s_ij with respect to the cation and -s_ij with respect to the anion

Thus Pauling's valence rule is satisfied, because each ion has a valence equal to its oxidation state.

Example 2 - GaAs (Sphalerite)

Both Ga and As are 4-coordinated, their formal oxidation states are +3 and -3 respectively

V_Ga = S s_ij
3 = 4 (s_ij)
s_ij = ¾

V_As = 4 (-3/4)
V_As = -3

Example 3 - SrTiO₃ (Perovskite)

Where the oxidation states and CN are as follows:
 

V_Ti = S s_ij
4 = 6 s (_Ti-O)
s (_Ti-O) = 2/3

V_Sr = S (_Sr-O)
2 = 12 s (_Sr-O)
s (_Sr-O) = 1/6

Vo = 2 s (_Ti-O) + 4 s (_Sr-O) = 2(2/3) + 4(1/6) = 2

Note that the bond valence concept can be applied to any compound where bonds only occur between atoms of the opposite formal charge (i.e. cations & anions), regardless of the degree of covalency/ionicity. It cannot be used for metals and compounds where cation-cation or anion-anion bonds are formed (i.e. BaO₂, FeS₂, elemental compounds, compounds with metal-metal bonds, etc.).

This formalism is useful for understanding and/or predicting structures, however, it is limited by the fact that the valence state of each cation must be arbitrarily assigned.

The power of the bond valence approach has been greatly enhanced, primarily by David Brown and later by Mike O’Keeffe, by correlating the valence of a bond s_ij with the bond distance d_ij.

This is done empirically through the relationship

s_ij = exp [(R_ij – d_ij) / b]

where b is taken to be a universal constant = 0.37 and R_ij is determined empirically from structures where bond distances and ideal valences are accurately known.

Thus tables of R_ij values for given bonding pairs (i.e. Nb-O, Cr-N, Mg-F, etc.) have been calculated, just as tables of ionic radii are available.

How is the bond valence concept put to use ?

A) To check experimentally determined structures for correctness, or bonding instabilities
B) To predict new structures
C) To locate light atoms such as hydrogen or Li ion, which are hard to find experimentally
D) To determine ordering of ions which are hard to differentiate experimentally, such as Al³⁺ and Si⁴⁺, or O^2- and F^-

(3) The presence of shared edges, and particularly shared faces decreases the stability of a structure. This is particularly true for cations with large valences and small CN.

This rule follows from a consideration of the Coulombic interactions in a crystal.  To maximize such interactions we want to maximize the cation-anion interactions, which are attractive, and minimize the anion-anion and cation-cation interactions, which are repulsive.  The cation-anion interactions are maximized by increasing the coordination number and decreasing the cation-anion distance. However, we know that if we bring ions to close together electron-electron repulsions become increasingly unfavorable.  Thus optimal cation-anion distances are dictated either by ionic radii or quantitative use of the bond valence rule, which are elucidated in Pauling's first two rules.

The basic concept behind this rule is to minimize the cation-cation interactions.  To illustrate this consider the cation-cation distances in both tetrahedra and octahedra which share common corners, edges and faces, as a function of the cation-anion distance (M-X)
 

From this table we see that the cation-cation distance decreases, which causes the Coulomb repulsion to increase, as the
 

• degree of sharing increases (corner < edge < face) .
• CN decreases (cubic < octahedral < tetrahedral)
• cation oxidation state increases (this leads to a stronger Coulomb repulsion)

(4) In a crystal containing different cations those with large valence and small CN tend not to share polyhedron elements with each other.

The reasoning for this rule can be understood through application of the bond valence rule (#2). I give an illustration of this below.

Consider CaWO₄ which adopts the Scheelite structure, the coordination about tungsten is tetrahedral while the calcium ion is 8 coordinate.  The valence of each W-O bond is

V_W = S s_ij
6 = 4 s (_W-O)
s (_W-O) = 1.5

If we now were to propose a structure where two WO4 tetrahedra shared corners, the valence of the shared oxygen would be

V_O = 2 s (_W-O) = 2(1.5) = 3

Which is clearly in violation of the second rule.  Therefore, the WO₄ tetrahedra in scheelite are isolated from each other.

(5) The number of chemically different coordination environments for a given ion in a crystal tends to be small.

What Pauling is saying here is that for a given atom type (i.e. Nb, Cu, O) all ions of that type in the crystal should have the similar coordination spheres.  Thus all other things being equal we don't expect to see a structure with aluminum ions in both octahedral and tetrahedral coordination.  They are expected to adopt the coordination which is most favorable for aluminum.

This rule can be understood in several ways.  The concept of maximizing symmetry whenever possible comes up again and again in nature. Another way to look at it is to say that once we find the optimal chemical environment for an ion, if possible all ions of that type should have the same environment.

This rule is sometimes called the rule of parsimony.