Each symmetry operation can be represented by a transformation matrix.
You have seen what happens when a molecule is subjected to the symmetry operation that corresponds to any of the symmetry elements of the point group to which the molecule belongs. The molecule is simply transformed into itself. But the properties of the molecule in which we are interested are not necessarily so simply affected.
All properties, or motions, of a molecule, obtained perhaps as eigenfunctions of the corresponding operator, are related to the symmetry of the molecule. Let us illustrate this by exploring how the overall translational and rotational motions of any C2v molecule, the H2O molecule for example, change when the various symmetry operations of the C2v group are applied.
Let the overall translational motions of the H2O molecules be represented by the x, y, and zvectors. Some of the symmetry operations, those of the E and σ'v symmetry elements, leave x unchanged. Others, those of the C2 and the σv symmetry elements, change the direction, or sign of x. If the new translational vectors are indicated by primes, you can see that the effects of the symmetry operations on, for example, x are given by the set of +1, -1, +1, -1 and the effect onz by the set of entries +1, -1, +1, -1.
Now let us see how the rotations of the molecule about the x, y, and z axe are affected by the symmetry operations. We can do so by drawing curly arrows to represent the motions that constitute these rotations. Inspection of the effect of the symmetry operations shows that the same as two of those found when we used the vectors that represent translational motions as our basis. The effect on Rz, as illustrated and leads to a new, fourth set of +1 and -1 terms.
The four different types of symmetry behaviour that have been discovered are collected in each row represents a symmetry species. Each symmetry species is given an identifying label. We use the axis of rotation, i.e. a species for species that is symmetric with respect to the axis of rotation, i.e. a species for which +1 is the entry under the symbol for the rotation operation. We use the symbol B to indicate a symmetric species that is antisymmetric, and has a -1 entry, for this rotation operation. Here we use an additional subscript labels, choosing the subscript 1 for the more symmetric species and 2 for the less symmetric species.
The H2O molecule, or the C2v, point group, provides a simple, and special, example. In this case the translation and rotation vectors can be chosen so that the symmetry operations change each vector into itself or into its opposite. The effect of the operations change each vector into itself or into opposite. The effect of the operations on each of these vectors is represented by a +1 ora -1. The symmetry species of the C2v point group consists of sets containing +1 and -1 terms.
Transformation matrices: for some point groups the basis vectors that we use to study the effects of the symmetry operations become mixed as a result of these operations. Consider the three overall translation vectors of the NH3 molecule of the C3v point group. These and the symmetry elements of this group are nothing new enters when we consider the effects of the symmetry operations on the z vector. This vector is unchanged by each and every symmetry operation. Thus a set of +1 is shows how the z translation vector is transformed.
Now consider the effect of a C3v rotation, i.e. rotation by 1/3 revolution on the x and y vectors. The results have now the new position of x, that is, the vector of x' is related to the original vectors by
x' = -1/2x - √3/2y
The new vector y' that is produced from the original vector y is given by
y' = +√3/2x - 1/2y
The net effect of the operation C2 on the set of vectors x and y can be shown by the matrix equation
x' -1/2 - √3/2 x
y' √3/2 -1/2 y