Simply put, a symmetry operation is a transformation that can be applied to a molecule which results in no overall change in the overall positions of the atoms in it, except for the labels we put on atoms. There are six different types of symmetry operations we can find in certain molecules, all of which affect how that molecule will show up in vibrational spectroscopy.
The simplest symmetry operation is the identity operation, E. It does nothing to the positions of the atoms. Therefore all molecules possess this kind of symmetry. It may seem to be trivial, but it is important to include this type of symmetry in calculations, in order to be complete.
Many simple molecules possess at least one mirror plane, or plane of reflectional symmetry. As can be seen in the diagrams, ethene possesses three mirror planes - one bisecting the C=C bond without passing through any molecules, one containing both C atoms but not the H atoms, and one in the plane of the molecule, containing all the atoms.
Look at the three mirror planes. The first is a plane in the x and y axes, and can therefore be called sxy. Similarly, the second is sxz, and the third is syz. When looking for symmetry operations in a molecule, be aware of mirror planes in the plane of a flat molecule - these are easy to miss.
A molecule has an axis of rotational symmetry when it can be rotated by a certain fraction of a full rotation, to end up identical to how it started. The subscript n denotes what fraction of 360° the molecule can be rotated by - so a molecule with a C2 axis can be spun 180° to end up just how it started, such as the water molecule, H20. A molecule with a C3 axis can be rotated 120°, such as the ammonia molecule NH3. It follows that all molecules have an infinite number of C1 axes, which are effectively all the identity operation E.
As can be seen, ethene has three rotational axes, all of which are C2.
In molecules with more than one rotation axis, we must choose the principal axis. The first choice is always the Cn axis with the highest value of n. Where this is not appropriate (e.g. in ethene, with its 3 C2 axes) we choose the axis which passes through the greatest number of atoms. This is shown for ethene in the diagram.
By convention, once we have identified the principal axis of rotation we define that axis as the z-axis.
A centre of symmetry is a point through which it is possible to draw a straight line from one atom and arrive at an identical atom the same distance the other side of the point. In mathematical terms, the relationship is going from x, y, z to -x, -y, -z. This can be seen in the diagram.
An Improper Rotation-Reflection axis involves rotation followed by reflection at right angles to the axis.
This type of symmetry is often difficult to spot. It is best to get used to checking for it along Cn axes. An S2n axis will always be coincident with a Cn or C2n axis. For example, the S4 axis in allene coincides with the C2 axis.
The S1 and S2 operations do not exist. S1 is equivalent to a horizontal reflection sh. S2 is the same as an inversion, i.
Be careful when looking at multiples of Sn transformations. Any even number of operations is the same as the corresponding Cn operation, since the reflections cancel out.
Hence S32 is equivalent to C32. S3 is equivalent to sh.
In practice, you do not always have to spot every symmetry operation present in a molecule. Rather, you need to classify the molecule into its point group. From this, you can look up how many of each type of symmetry operation the molecule has. Then all you have to do is find them.
An important thing to remember about Group Theory is that although, by definition, a symmetry operation cannot change the position of the atoms in the molecule, it can alter the signs of the orbitals on the atoms - so a p orbital, for example, could end up inverted. This is what the use of Group Theory to predict IR spectra is based upon.