Symmetry operations

In 1996, a particular pharmaceutical company cam put out an anti-HIV drug known as ritonavir (aka norvir). In 1998, the drug was recalled because it was found that its efficacy was not as high as expected from clinical trials. In 2002, the reason became clear. The powered crystal of ritonavir, from which the pills were made, had more than one stable crystal structure, something which was unknown when the drug was first issued. The method of preparation was generating the wrong crystalline form, a form that was less soluble than the one that had been used in trials. The company reissued the drug in liquid form until the crystallization process could be revamped to give the correct form.

This example illustrates a complex and important problem that touches on medinical chemistry as well as many others of chemistry (solid state, geochemistry, protein chemistry,...), namely that of determining and characterizing the crystal structures of atoms and molecules. For us, this is a departure from what we have done up to now, which has been mostly focused on single molecules. Crystals are an example of what are generally called condensed phase systems, namely collections of molecules that exist in a particular phase, here, the solid or crystalline phase. Other familiar phases are, of course, the gas and liquid phases. A so-called fourth state of matter exists, the plasma, which can be reached only a very high temperatures.

The fact that some substances can crystallize into several stable forms is a phenomenon known as polymorphism. The most important aspect of a crystal structure is the existence of a pattern or arrangement of atoms or molecules that is then repeated over very long distances. The pattern that is repeated is known as the repeat unit (not to be confused with the unit cell, which we will discuss later). Thus, characterizing the long-range order of the crystal requires characterizing the repeat unit. The latter is accomplished by listing all of the operations that can be performed on the repeat unit, transforming it into a pattern that looks the same as the original one.

The types of operations we can perform are

1.

Rotations.

2.

Reflections through a plane.

3.

Spatial inversions through a point.

These are known as symmetry operations. In order to illustrate how the symmetry operations work, let us consider the very simple example of a cube, which is a shape with a high degree of symmetry. There are numerous operations that can be performed on a cube that make it look like the original cube.

The first is the identity transformation, i.e. a transformation that does nothing to the cube. This operation is trivial but, nevertheless, it must be included in the set of transformations. Next, there are three axes around which a rotation by 90$^{\circ}$ transforms the cube into an orientation that looks the same as the original one (see figure).

Figure: Symmetry elements for a cube

These rotations are known as $C_4$ rotations because they are rotations by $2\pi/4 = 360^{\circ}/4$, hence four such rotations bring the cube back to its original state. (In general, a $C_n$ rotation is a rotation by $2\pi/n$.) In addition, the figure shows that there are four axes about which a rotation by $2\pi/3 = 360^{\circ}/3$ transforms the cube into an orientation that looks the same. These are called $C_3$ axes. There are also 6 $C_2$ axes, 9 mirror planes and finally, one center of inversion symmetry through the center of the cube. This means that if this point were placed at the origin and all of the coordinate vectors of the vertices of the cube in this coordinate system were changed to their negative values, the cube would still look the same.

Example: What are the symmetry operations for the ammonia (NH$_3$) molecule?

Figure: Symmetry elements for an NH$_3$ molecule

First, there is the identity transformation, which leaves the molecule as it was. Then, there is one $C_3$ axis down the center of the pyramid. Finally, there are three mirror planes, each on containing just one of the NH bonds and bisecting the remaining HNH angle.