A die has six faces, and an octahedral complex has six corners. Paint the faces of a die and you meet every difficulty of counting isomers: the same die looks different when you turn it, and a die and its mirror image are not always the same. This lesson gives a method that counts every isomer once, a test for a mirror plane, and a check by division.
This is the first lesson of Coordination Compounds: Crystal Field Calculations and Isomer Counting in Depth, free for everyone to read.
Take a die and paint two faces red and four blue. How many different dice can you make?
Turn the die in your hand. The two red faces are either next to each other or on opposite sides. There are only two different dice, however you hold them.
An octahedron is the shape that fits inside a cube, with one corner at the middle of each face. So six faces of a die match six corners of an octahedron, and opposite faces match opposite corners. Painting a die is counting the isomers of : the two B ligands are cis (next to each other) or trans (opposite).
The trouble starts when many colours are used. It is easy to draw the same isomer twice, because a turned drawing looks new. It is also easy to miss one. We need a method that does not depend on how we hold the model.
An octahedron has three pairs of opposite corners. Every ligand has exactly one partner opposite to it. Write a complex as three trans pairs. Turning the model never changes the pairs, because turning keeps opposite corners opposite.
Two arrangements with the same three trans pairs are the same geometrical isomer.
Geometrical isomers differ in the arrangement of the ligands. A mirror image is not counted as a new geometrical isomer.
Take , with two A ligands and four different ones. The sets of pairs are of two kinds.
So has geometrical isomers.
Some of these nine have mirror images that you cannot turn into the original. Such a complex is chiral. A complex that has a plane of symmetry is achiral, and it has only one form.
For monodentate ligands there is a short rule. A mirror plane exists if either of these is true.
If neither is true, the complex is chiral. In the picture, the top three isomers have , so each has one form. The other six have no identical pair and no twin pairs, so each of them has two mirror images.
The number of stereoisomers is the number of geometrical isomers plus the number of chiral ones. For that is .
Start with the sliders as they are. Then make equal to , so the x axis carries two identical ligands, and watch the mirror plane appear.
Here is a second way to count, and a check on the first. Number the six corners and place the ligands in every possible order. For six different ligands that is orders. An octahedron can be turned in 24 ways to bring it back onto itself, so every real isomer shows up 24 times among the 720 orders.
With two identical ligands there are orders, and . That matches the count for .
The division fails when some arrangement is unchanged by a turn. For there are orders, and , which is not a whole number. The true count is 8. The achiral isomers have turns that bring them back to themselves, so they appear fewer than 24 times. Division is a fast check, but only a method that looks at each isomer, like the trans pair method, gives the exact count.
A bidentate ligand, written as an arc between its two donor atoms, is a ring. It is too short to span opposite corners, so a ring always takes two corners that are next to each other (cis). Write for a ring with identical donor atoms, such as ethane-1,2-diamine, and for a ring with different donor atoms.
Rings cut down the possible arrangements, because a ring cannot sit across a trans pair. But they also create a new kind of mirror image. Two rings that do not lie in one plane wind like the blades of a propeller. The winding can be right-handed or left-handed. These twisted forms are chiral even when no pair of ligands is identical.
Here are counts that the program below finds.
A good habit is to ask two questions in order. First, how many ways can the pairs be arranged? Second, which of those ways has a plane of symmetry? A ring lying in a plane with every other ligand on the axis through the plane keeps that plane. A twisted pair of rings does not.
The same idea works for other shapes.
A formula can hide several kinds of isomer at once. The kinds multiply when they are independent. Take the anion , where is the symmetric ring oxalate.
Ionisation or hydrate isomers are different formulas inside and outside the bracket. They add to the total instead of multiplying. Count each formula separately, then add.
The first program lists every set of trans pairs, tests the mirror rule, and prints the counts.
The second program handles rings. It places the donor atoms on the corners and throws away any arrangement with a ring across a trans pair. Then it groups the arrangements that a turn connects, and the arrangements that a turn or a mirror image connects.
Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.
Coordination Compounds: Crystal Field Calculations and Isomer Counting in Depth is a Standard course. It adds 2 more lessons, practice and tests on top of the free chapter.