Molecular orbitals and molecular motions belong to certain symmetry species of the point group of the molecule.
Examples of the special ways in which vectors or functions can be affected by symmetry operations are illustrated here. All wave functions solutions, or eigenfunctions, for an atom or a molecule transform according to one or another of the special symmetry species of a point group. We thus have a very powerful guide to the form of any vector or function that describes the properties or behaviour of a symmetric molecule. Each vector or function must transform according to one of the symmetry species of the point group to which the molecule belongs.
Typically, in dealing with ,molecular properties, we proceed from simple and easily pictured or easily described functions or vectors associated with the atoms of a molecule. We use these to build up functions or vectors appropriate to the whole molecule. Thus to describe the translational, rotational and vibrational motion of a molecule, we might start with the three Cartesian displacement coordinates of each atom of the molecule. To describe the translational, rotational and vibrational motion of a molecule, we often adopt a linear combination of atomic orbitals(LCAO) approach.
Now we begin the steps that let us use easy to deal with vectors or functions to deduce the symmetry of molecular vectors or functions.
Characters of transformation matrices: suppose you were to construct transformation matrices, n the basis of a set of vectors or functions. Suppose also that there existed other vectors or functions which were linear combinations of the first set of vectors or functions. You would find that the sum of the diagonal elements of the transformation matrix that represents any symmetry operation would be the same fr any basis vectors or functions. (The transformation matrices themselves would be different for different basis vectors or functions.)
The sum of the diagonal elements of a transformation matrix of a representation is known as the character of the matrix. Thus, the characters of the transformation matrices that represent a group are the same for all basis vectors or functions that are or could be formed each other by linear combinations.
We generally would need large matrices to show the effect of each symmetry operation on the molecule. For example, if we use the three Cartesian displacement coordinates on each atom of an n-atom molecule as our basis, we generally need matrices of order 3n to describe the effects of the operations. If we use bond orbitals as a basis, we generally need transformation matrices with an order equal to the number of bonds. These large matrices can be converted, or reduced, to sets of smaller matrices by forming linear combinations of the original basis vectors. The original sets of large matrices constitute a reducible representation. The smallest matrix representations obtained by appropriate linear combinations of the basis vectors are called irreducible representations. The characters of the reducible representation are the same as the sum of the characters of the irreducible representations that are obtained from the original representation.
The use of characters rather than the transformation matrices themselves brings a great simplicity and elegance to the use of symmetry. First we introduced the tables used to display these characters, and we investigate some of the special properties of the characters of the irreducible representation matrices.