To understand representations and character tables, it is essential to introduce three closely related ideas: the order of a group, the classes of symmetry operations, and the characters associated with representations.
Order of a group
The order of a point group, usually denoted \(h\), is the total number of symmetry operations in the group.
For example:
- \(C_{2v}\) has four operations (\(E, C_2, \sigma_v, \sigma_v'\)), so \(h=4\).
- \(C_{3v}\) has six operations (\(E, 2C_3, 3\sigma_v\)), so \(h=6\).
Classes of symmetry operations
A class is a set of symmetry operations that are related to one another by a similarity transformation. Two operations \(A\) and \(B\) belong to the same class if there exists some symmetry operation \(R\) in the group such that
\[ B = RAR^{-1}. \]
Operations in the same class are considered “equivalent” because they have the same effect on the symmetry properties of a molecule.
Example: \(C_{2v}\)
The point group \(C_{2v}\) contains the operations \(E, C_2, \sigma_v, \sigma_v'\). In this group, no two distinct operations are related by a similarity transformation.
As a result, each symmetry operation forms a class by itself:
\[ \{E\},\quad \{C_2\},\quad \{\sigma_v\},\quad \{\sigma_v'\}. \]
A group in which every operation forms its own class is called abelian. In an abelian group, all symmetry operations commute: \(AB = BA\).
The fact that \(C_{2v}\) is abelian explains why all of its irreducible representations are one-dimensional.
Example: \(C_{3v}\)
The point group \(C_{3v}\) has six symmetry operations, but they are grouped into three classes:
- Identity: \(\{E\}\)
- Rotations: \(\{C_3, C_3^2\}\) (these are related by similarity transformations)
- Reflections: \(\{\sigma_v, \sigma_v', \sigma_v''\}\)
Within each class, the operations are symmetry-equivalent even though they are distinct geometric operations.
Connection to irreducible representations
A fundamental result of group theory is that the number of irreducible representations of a group is always equal to the number of classes in the group.
- \(C_{2v}\) has four classes → four irreducible representations.
- \(C_{3v}\) has three classes → three irreducible representations.
Big idea: classes determine the structure of character tables. Understanding how symmetry operations group into classes explains why character tables have the size and form that they do.
In group theory, a character is a single number that summarizes how a symmetry operation acts within a given representation. Characters are the quantities that appear in character tables and are the primary tools used in practical symmetry analysis.
Definition of a character
For a symmetry operation \(R\) represented by a matrix \(D(R)\), the character \(\chi(R)\) is defined as the trace of the matrix:
\[ \chi(R) = \mathrm{tr}\big[D(R)\big] \]
The trace is the sum of the diagonal elements of the matrix. Because the trace is unchanged by a change of basis, characters provide a compact and basis-independent way to describe a representation.
Characters and classes
A crucial result from group theory is that all symmetry operations belonging to the same class have the same character in a given irreducible representation.
This is why character tables list one character per class, rather than one per individual symmetry operation.
Example: \(C_{2v}\)
In \(C_{2v}\), every symmetry operation forms its own class. All irreducible representations are one-dimensional, so each operation is represented by a \(1\times1\) matrix.
For example, in the \(A_1\) irreducible representation, all operations are represented by \(+1\), giving
\[ \chi^{(A_1)} = (1,\;1,\;1,\;1) \]
In the \(B_1\) representation, some operations change sign, producing
\[ \chi^{(B_1)} = (1,\;-1,\;1,\;-1). \]
In both cases, the characters completely describe how the representation behaves under the symmetry operations.
Example: \(C_{3v}\)
In \(C_{3v}\), not all irreducible representations are one-dimensional. The two-dimensional irreducible representation \(E\) is represented by \(2\times2\) matrices.
For this representation, the characters are
\[ \chi^{(E)} = (2,\;-1,\;0) \quad \text{for the classes } (E,\;2C_3,\;3\sigma_v). \]
Here, the identity operation has character 2 because the identity matrix has two diagonal ones. The mirror-plane operations have character 0 because the trace of the reflection matrices is zero.
Why characters are so useful
- Characters are much simpler than full matrices, yet retain all essential symmetry information.
- They obey orthogonality relationships guaranteed by the Great Orthogonality Theorem.
- They allow reducible representations to be decomposed into irreducible ones using simple algebra.
Big idea: a character is a compact, basis-independent summary of how a representation responds to a symmetry operation. This is why character tables are the central working tool of molecular symmetry.
A character table is a compact summary of all irreducible representations of a point group. It organizes symmetry information by listing the characters of each irreducible representation for each class of symmetry operations.
Character tables also indicate how common physical quantities—such as Cartesian coordinates and rotations—transform under the symmetry operations. This makes character tables the primary working tool in molecular symmetry.
How to read a character table
- Columns correspond to classes of symmetry operations.
- Rows correspond to irreducible representations.
- Each entry is the character (trace) of the representation matrix for that class.
- The far-right columns list common basis functions that transform according to each irreducible representation.
Character table for \(C_{2v}\)
\[ \begin{array}{c|cccc|cc} C_{2v} & E & C_2(z) & \sigma_v(xz) & \sigma_v'(yz) & \text{Linear} & \text{Rotations} \\ \hline A_1 & 1 & 1 & 1 & 1 & z & \\ A_2 & 1 & 1 & -1 & -1 & & R_z \\ B_1 & 1 & -1 & 1 & -1 & x & R_y \\ B_2 & 1 & -1 & -1 & 1 & y & R_x \\ \end{array} \]
Because \(C_{2v}\) is an abelian group, each symmetry operation forms its own class and all irreducible representations are one-dimensional.
The table shows, for example, that:
- The \(z\) coordinate transforms as \(A_1\).
- Rotations about the \(z\)-axis transform as \(A_2\).
Character table for \(C_{3v}\)
\[ \begin{array}{c|ccc|cc} C_{3v} & E & 2C_3 & 3\sigma_v & \text{Linear} & \text{Rotations} \\ \hline A_1 & 1 & 1 & 1 & z & \\ A_2 & 1 & 1 & -1 & & R_z \\ E & 2 & -1 & 0 & (x,y) & (R_x,R_y) \\ \end{array} \]
In \(C_{3v}\), symmetry operations fall into three classes, so there are three irreducible representations. The two-dimensional irreducible representation \(E\) reflects the fact that certain quantities (such as \(x\) and \(y\)) transform together.
From this table we see that:
- The \(z\) coordinate transforms as \(A_1\).
- Rotations about the principal axis transform as \(A_2\).
- The pairs \((x,y)\) and \((R_x,R_y)\) transform as \(E\).
Big idea: a character table encodes all symmetry information needed to determine how coordinates, rotations, vibrations, and orbitals transform under the operations of a point group.