# Character Tables

The symmetry properties of orbitals in polyatomic molecules are denoted by symmetry labels, which look like *a*, *a*_{1}, *b*, *e* etc. These labels indicate the behaviour of the orbitals in a molecule under the symmetry operations of the point group to which the molecule belongs. This can be illustrated by a hypothetical example:

We will consider a point group that contains only two operations, one being the identity, and say that there are two possible ways an orbital can behave under an operation. We make the restriction that the orbital can either be unchanged by an operation (represented symbolically by a 1) or that it can change sign under an operation (represented symbolically by a -1).

Now, no orbital can be changed by the identity, so the behaviour of a single orbital under *E* will always be 1. The behaviour of an orbital under the other operation of the group may be 1 or -1. There are thus two possible ways an orbital can behave under all the operations of the group. If its behaviour under the two operations (*E*,x) is (1,1) , then it is given one symmetry label, if its behaviour is (1,-1) it is given a different label.

For example, in the group C_{i}, which contains the symmetry elements *E *and i, the label *a*_{g} is given to an orbital which shows (1,1) behaviour and the label a_{u }is given to an orbital which shows (1,-1) behaviour. An s orbital would be labelled *a*_{g} , as it is spherical and appears unchanged under both the identity or an inversion. A p orbital, on the other hand, would be labeled a_{u , }as while it appears unchanged under the identity, the positive and negative lobes interchange under an inversion, making it look as though it has changed sign. Though complications arise rapidly in the consideration of more complex point groups, these basic principles remain valid.

The symmetry labels are merely a shorthand telling us the behaviour of an orbital under all the operations of a point group.

A label is assigned to an orbital by referring to the character table of the point group to which the molecule belongs. This a table which lays out the possible different behaviours under all the symmetry operations of the group, and gives the symmetry labels associated with each one. (Note that not all the permutations of behaviour are permitted. For example, no orbital can remain unchanged by a C_{4 }rotation but change sign under a coincident C_{2} rotation.)

The entries in a complete character table are obtained by a branch of mathematics known as group theory, and are known as characters, χ. They characterise the behaviour of each symmetry type under each operation. The simplest way to illustrate the features of a character table is to work through an actual example; we shall consider the C_{3v}table. (The C_{3v} group contains molecules such as ammonia, which have a pyramidal structure.)

The C_{3v} character table:

C_{3v} |
E |
2C_{3} |
3σ_{v} |
h = 6 |

A_{1} |
1 | 1 | 1 | z, z^{2}, x^{2} + y^{2} |

A_{2} |
1 | 1 | -1 | |

E | 2 | -1 | 0 | (x , y) (xy, x^{2} – y^{2}) , (xz, yz) |

The top left hand box contains the name of the point group to which the table refers.

The rest of the columns (except the last) are labeled at the top with the symmetry operations of the group, in this case *E*, C_{3} and σ_{v}. The numbers before each operation are the numbers of members of each class.

Symmetry operations belong to the same class if they are of the same type (eg rotations) __and__ can be converted into one another by a symmetry operation of the group. (In C_{3v} , the two C_{3} rotations are in the same class as they are related by a reflection in one of the mirror planes. Similarly, the three vertical planes are in the same class as each other, as they related by the C_{3} rotations.) The identity is always in a class of its own.

The top right hand box gives the order, h, of the group, which is simply the total number of operations of the group. In this example, it is 6 (1 + 2 + 3 = 6). Note this is sometimes omitted from a character table, as its calculation is trivial.

The rows under the labels summarise the symmetry properties of the orbitals. They are labeled in the left-hand column with the symmetry species, A_{1}, A_{2}, and E. (**Note E the symmetry species is utterly distinct from E the symmetry element**.)

These symmetry species label the irreducible representations of the group, which are simply the basic types of behaviour an orbital may show when subjected to the symmetry operations of the group. By convention, irreducible representations are labeled A_{1}, E etc, but the orbitals to which they apply are labeled *a*_{1} ,* e* etc.

The right-hand column of the table (which is occasionally omitted) indicates to which irreducible representations various mathematical functions belong. Thus in the group C_{3v} , the z axis (coincident with the axis of rotation) transforms as A_{1}. The bracketing together of x and y indicates that neither of them transforms as one of the irreducible representations by themselves, but that it is possible to form a linear combination of the two functions which transforms as E. Note that what is especially useful about this information, is that the p_{x}, p_{y} and p_{z} orbitals transform as x, y and z respectively. Similarly the functions which transform in the same way as the five d orbitals are given.