Symmetry Elements

Symmetry elements are operations, such as rotation, reflection or inversion, that, when they are performed for a molecule or other object, give an object that is indistinguishable from the one on which the symmetry operation was performed. A perfect billiard ball (without any markings) can, for instance, spin without an observer being able to detect the movement. This is because a sphere has an infinite number of axes about which it can spin without changing its appearance. There are five common symmetry elements used to determine molecular symmetry:

5.1.1. The Identity, I

The identity is the simplest symmetry operation of all - we simply do nothing. Thus, the identity is the one symmetry element that is present in every object. It seems trivial even to consider a symmetry element that does absolutely nothing, but the identity is necessary for the accounting in symmetry treatments using group theory. It is given the symbol I (or sometimes E).

5.1.2. Proper Rotation Axes, C_n

Consider a perfect cylinder. Similarly to the billiard ball, if it were spinning about its axis, an observer would not be able to detect the movement. Put another way, we can rotate the cylinder by any angle about its axis without seeing any change. This is shown in the left-hand figure of Figure 5.1. The cylinder can be rotated to any one of an infinite number of positions about its axis without any visible change. This means that the cylindrical axis is a rotation axis with an infinite number of possible rotation angles that give the same result. Rotation axes are denoted by the symbol c with a suffix to denote their order, or the number of equivalent positions obtained by rotation about the axis. Thus, the axis of the cylinder is denoted C_¥.

If we now draw a cross on the surface of the cylinder, as also shown in Figure 5.1, we must rotate the cylinder by a full turn before it looks identical again. Of course, rotation by 360° is exactly equivalent to the identity, so that C₁ axes are not considered in molecular symmetry treatments. The third cylinder shown in Figure 5.1, however, now has two identical crosses drawn exactly at opposite sides of the surface of the cylinder. If we now rotate the cylinder by 180°, the crosses swap positions, but because they are identical the cylinder is indisinguishable from what we started with. There are thus two positions that are indistinguishable when the cylinder is rotated about its axis. The axis is therefore a two-fold proper rotation axis, or C₂. Quite generally, for a C_n axis, indistinguishable structures are obtained by rotating (360/n)° about the axis.

Figure 5.1 Three different cylinders with inifinite-fold, one-fold and two-fold proper rotation axes.

Let us now apply the same principles to a molecule. Water has a C₂ axis, as shown in Figure 5.2.

Figure 5.2 The two-fold axis of water.

In the figure, the hydrogens have been labelled H₁ and H₂, but they are of course identical in a real molecule and therefore indistinguishable. Thus, rotating the black molecule by 180° about the C₂ axis (the green line) gives the blue water, which in real life, where the hydrogens are identical and all water molecules are the same color, is identical with the starting structure.

Figure 5.3 shows the three-fold axis of ammonia.

Figure 5.3 Clockwise and anticlockwise rotations about the three-fold axis of ammonia.

This axis is exactly analogous to the two-fold axis of water except that we have three equivalent positions when we rotate the black ammonia in the direction shown by the blue arrow to give first the blue structure and, on a further rotation, a second structure in which H₂ takes the original position of H₁, H₁ that of H₃ and H₃ that of H₂ (Figure 5.4):

Figure 5.4 Three-fold rotation axis of ammonia (projected on the plane of the page).

In Figure 5.3, we can also rotate in the other (red arrow) direction about the C₃ axis. A single 120° rotation in this direction will give us the red structure, which is indistinguishable from both the black and the blue ammonia molecules. There is, however, an important difference between the C₂ axis of water and the C₃ of ammonia. If the hydrogens really were labelled, it would still be impossible to determine whether we had rotated water clockwise or anticlockwise to give the second equivalent structure. The result of the two rotations does not differ at all, and so they must be considered to be one symmetry operation (it doesn't matter how we got there, just what the result is). However, rotation in the two different directions in ammonia gives the red and blue structures, which are distinguishable if we label the hydrogens. Therefore, the C₃ axis of ammonia counts not as one symmetry element, but two. 120° rotations in different directions are not exactly equivalent, so we really have one C₃ axis for each rotation direction. We will see later that ammonia has two C₃ axes that differ only in the direction of rotation. Such axes are called degenerate and point groups that contain them are also degenerate. If a molecule belongs to a degenerate point group, it can have degenerate orbitals or vibrations, as we saw above for cyclopropane, for instance. All C_n axes with n equal to or larger than three are degenerate.

The principal axis of any molecule or object is the proper rotation axis with the highest order n. If there are several proper rotation axes with the same order, the one that runs through the most atoms is the principal axis. Knowing which axis is the principal axis is important for determining the exact designation of mirror planes.

5.1.3. Mirror Planes, s

The third type of symmetry element that we will consider is the mirror plane, usually designated s. A mirror plane can be considered to be equivalent to an infinitely thin, double-sided planar mirror within the molecule or object. The simplest example of a mirror plane is the molecular plane for any planar molecule. Reflection of the atoms in the molecular plane will leave them all unchanged. For instance, the plane of the page (the molecular plane) is a mirror plane for the orientation of water shown in Figure 5.2. Because atoms that lie in a mirror plane are not moved by reflection in that plane, the molecular plane is a mirror plane for any planar molecule. In water, however, there is a second mirror plane perpendicular to the molecular plane (Figure 5.5):

Figure 5.5 Second mirror plane in water perpendicular to plane of page.

Both mirror planes in water contain the C₂-principal axis and are therefore denoted vertical, or s_v mirror planes. There are exceptions to this rule for point groups with C₂-axes perpendicular to the principal axis. In this case, vertical mirror planes that contain one of these perpendicular C₂-axes are denoted s_v and those that bisect the angle between two such axes are denoted diagonal, or s_d mirror planes. Planar BH₃, for instance, has three s_v planes (Figure 5.6):

Figure 5.6 s_v mirrors planes in borane.

Perpendicular B₂H₄, on the other hand, has two s_d mirror planes, as shown in Figure 5.7:

Figure 5.7 s_d mirror planes in B₂H₄.

Note that the principal axis is determined by the fact that it runs through the two boron atoms. The perpendicular C₂-axes bisect the angle between the planes of the two BH₂ groups, whereas the two s_d planes are identical with these planes.

The third type of mirror plane, the horizontal, lies perpendicular to the principal axis, as shown below in Figure 5.8 for planar BH₃:

Figure 5.8 s_h mirror plane in BH₃.

Mirror planes in objects without proper rotation axes are simply denoted s.

5.1.4. Improper Rotation Axes, S_n

Improper rotation axes are simply the combination of a proper rotation axis with a s_h mirror plane. It does not matter in which order these two symmetry operations are applied, as shown in Figure 5.9 below for the S₄ axis of perpendicular B₂H₄.

Figure 5.9 Improper rotation axis in perpendicular B₂H₄.

Note that neither the C₄-axis nor the s_h mirror plane are correct symmetry elements for the molecule. Only their combination is valid. Improper rotation axes with the order 2n are often found to coincide with a proper rotation axis of order n.

5.1.5. Inversion Centres, i

Inversion centres are points in the centre of the molecule that act as infinitely small mirrors in every direction. This means that every atom is "reflected" to a position on the opposite side of the inversion centre (Figure 5.10):

Figure 5.10 Inversion centre in ethane bearing three different (labelled) substituents on each carbon atom.

Symmetry Elements
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Point Groups
Irreducible Representations and Character Tables
Degenerate Orbitals