(CN of cation) x (stoichiometric coefficient of cation) = (CN of anion) x (stoich. coeff. of anion)

So, for example, in the rutile (TiO₂) structure, the cations are octahedral, so

6 x 1 = (CN of oxygen) x 2, and therefore CN of oxygen must be 3.
 

1. TiO2. This is the "rutile" structure; there are at least two other known structures for TiO2! Make sure you can see the octahedral Ti and three-coordinate O.  Push for spacefilling  Back to ball-and-stick  Highlight one Ti4+ ion (gray) and the octahedron of six O2– ions (red) surrounding it

 

2. ReO3. This is the "starting structure" for the perovskite family. Since Re is a transition metal, you might look for it to be octahedral; that means the oxygen coordination number must be 2. In this picture, I haven't shown any complete Re octahedra. But you can see the basic "box" structure, with Re (pink) at the corners and O (red) in the middle of the edges. Note also the BIG HOLES in the centers of these cubes! (You can see the holes even in the spacefilling model. Does Nature abhor a vacuum or not???)  Push for spacefilling  Back to ball-and-stick

 

3. PbTiO3. This is an important perovskite (ABO3) structure, in which the holes in the ReO3 structure are filled by larger cations.  Push for spacefilling  Back to ball-and-stick  Things to note about PbTiO3:  (a) It has the basic structure of ReO3, with big Pb2+ ions filling the holes. BUT...  (b) The boxes are not symmetrical any more. The vertical Ti­O distances are not all the same: you can think of the Ti atoms as all being shifted in one direction. (This loss of symmetry is very important for possible practical applications of PbTiO3, such as new types of computer memory.)

 

4. The "1-2-3" superconductor YBa2Cu3O7. Push for spacefilling  Back to ball-and-stick  Things to note about "1-2-3":  (a) It has the basic structure of ReO3, with Ba2+ (yellow) and Y3+ (pink) ions filling the holes between the Cux+ (brown) and O2– (red). BUT...  (b) The boxes are not all the same. Partly this is because some have Ba2+ (yellow) in the center and some have Y3+ (pink). The boxes with Y3+ in the center are smaller, as you might expect because Y3+ is smaller than Ba2+.  (c) Someof the O atoms are "missing"! That is, some edges have no O atoms in places where the ReO3 and PbTiO3 structures would have O atoms. This is required by the stoichiometry of the compound, YBa2Cu3O7 (otherwise it would be YBa2Cu3O9). It also allows some of the Cu ions to be square-planar rather than octahedral.  (d) What is the oxidation state of Cu in this structure?

 

5. Al2O3, corundum (as in ruby and sapphire). Push for spacefilling  Back to ball-and-stick  This structure contains oxide ions in an approx. hexagonal close-packed arrangement.  The gray Al3+ ions are in octahedral sites, but because of the formula, only 2/3 of the cation sites are occupied.

6. Last example: FeCr2O4, a spinel. Push for spacefilling  Back to ball-and-stick  This is a complicated structure, so please concentrate on the following:  (a) the overall cubic symmetry;  (b) the yellow Fe2+ ions, in tetrahedral sites surrounded by 4 O atoms (red); and  (c) the gray Cr3+ ions, in octahedral sites.