Character theory

A character table contains information about the irreducible representations of a finite group. In chemistry, one can make with their help statements about properties of molecules based on the associated point group.

The actual character table of a group is a square table with complex numbers as entries. The rows correspond to irreducible representations of the columns in the conjugacy. The table entry for the representation and conjugacy class is the value of belonging to nature, evaluated on any element of.

Application in chemistry

Includes one now from the symmetry elements or with the aid of Schoenflies scheme to the point group of a molecule, one can conclude with the help of the character table to certain properties of the substance.

Example

• Character table of the point group

The first designation is the point group, in the first row contains the symmetry elements of R that are contained in it. If a symmetry element n times before, then you write. The number of symmetry elements is the order of the group is, in the first column are the irreducible representations. The following columns are the character (in this case -1 and 1). In the last two columns are the bases of the irreducible representations, or orbitals which transform as an irreducible representation. To say, for example, the rotation around the z-axis is transformed as.

Rotations and vibrations

• The information and refer to molecular rotations in the x-, y - and z -direction, which transform as the irreducible representations. For example, transformed with a molecule of the point group of the rotation about the z-axis as is.

The natural vibrations of the molecule also transform as one of the irreducible representations of the point group of the molecule.

Orbitals

The symmetry of the basis orbitals of a molecule can also be assigned to an irreducible representation of the point group. If a character in a particular representation and a certain element of symmetry, for example, the character " 1", then the sign of the wave function does not change when using this symmetry element. If it is "-1" then it changes.

Example

A molecule belongs to the point group (see the character table above). At its base rate, the orbital, which lies on the x- axis and how transformed belong. Reflection in the xz mirror plane does not change the orbital, it is mapped to itself, the character is " 1". Mirrors are the x - orbital contrast to the yz plane, the sign of the wave function changes, the character is so "-1", as can be seen from the character table.

Reducible and irreducible representations, ausreduzieren

An irreducible representation possesses only and as invariant subspaces. All other subspaces mix. A reducible representation is divided into several subspaces.

If a representation is completely reducible, it can be regarded as a direct sum of irreducible representations. Not every reducible representation is completely reducible.

When completely reducible representations of the shares of the irreducible representations in a reducible representation can be determined by rates or the following formula:

The frequency of an element must be n ( so ) are taken into account. h is the order of the group, the character of each irreducible representation and the character of the reducible representation.