Lecture 4: Coordination Polyhedra and Pauling's Rules

How do coordination polyhedra or environments relate to real crystals?
How do we build crystals?

The following discussion is essentially independent of whether the bonds are ionic or covalent although we will find it easiest to think about these properties as being ionic.

Closest Packing: HCP and CCP

Soccer ball demo

cation	anion
1	1	1.0	12	cuboctahedron
1	1.3	.732	8	cube
1	2.5	.414	6	octahedron
1	4.5	.225	4	tetrahedron
1	6.5	.155	3	triangular
1	>6.5	<.155	2	linear

5, 7, 9, and 10 fold coordination are possible in non-closest packed structures

Site size is proportional to CN environment, oxidation state:

Oxygen in CN=2 = 1.35 Å

Oxygen in CN=8 = 1.42 Å

Fe = 1.24 Å
Fe ++ = .74 Å
Fe +++ = .64 Å

Pauling noticed that size determines CN (Rule #1).

In the coordination polyhedron of anions about each cation, the cation-anion distance is constrained by the radius sum and the coordination number of the cation is controlled by the radius ratio.
Ex: Mg:O .72/1.36 = .53 therefore 6 C.N.

Pauling Rule #2: Electrostatic Valency Principle
Sum of the bond strengths reaching a cation from the anions must equal the charge on the cation.

calculate the electrostatic valency = charge/C.N.
all bonds of equal strength in structure? (isodesmic)
small highly charged cations: compact groups CO3, SO4 (anisodesmic)
group with exactly half the charge satisfied: silicates (mesodesmic)

Pauling Rule #3: Sharing of edges and especially faces decreases the stability of a structure.

especially a problem for high charge cations and radius ratio near lower limit for polyhedron

Pauling Rule #4: When stacking polyhedra in crystals using different cations, there is a tendency for those individual atoms with high valence and small coordination number NOT to share edges and faces.

i.e. if someone has to, it won't be those w/ high charge
problem with mineral hematite (Fe2O3)
- what is valence of Fe?
- what is the size? radius ratio?
- Fe-O electrostatic valency? 3/6 = 0.5 so we need 4 octahedra
- therefore face sharing

Rules 1-4 maximize cation-anion attraction and minimize cation-cation and anion-anion repulsion forces.

Pauling Rule #5: Principle of Parsimony
Nature is stingy. The # of essentially different types of constituents is small (due to limited # of different sites in close packed atoms).

Problems with the Rules:

radii not accurate or fixed
bonds not ionic
irregular polyhedra
ratios near border allow some cations to occur in 2 types of sites: Ca