Character table for C2h point group

Character table for C2h point group

How to construct a character table for the C2h point group?

Determine reducible and irreducible representation from the C2h character table

The correct answer and explanation is :

Okay, let’s construct the character table for the C2h point group and then explain how to use it for determining reducible and irreducible representations.

1. Determine Symmetry Elements and Classes:
The C2h point group has the following symmetry operations:

• E: The identity operation.
• C2: Rotation by 180° about the principal axis (conventionally the z-axis).
• i: Inversion through a center of symmetry.
• σh: Reflection through a horizontal mirror plane (perpendicular to the C2 axis, conventionally the xy-plane).

Since C2h is an Abelian group (all operations commute), each operation constitutes its own class. So, there are 4 classes: {E}, {C2}, {i}, {σh}.

2. Determine the Number and Dimensions of Irreducible Representations (IRs):

• The number of IRs equals the number of classes, which is 4.
• The sum of the squares of the dimensions (characters under E) must equal the order of the group (h). The order h is the total number of operations, which is 4.
• For Abelian groups, all IRs are one-dimensional (character under E is 1).
• 1² + 1² + 1² + 1² = 4. This confirms all four IRs are 1D.

3. Set up the Table Structure:
The table will have 4 rows (for the 4 IRs) and 4 columns (for the 4 operations/classes).

      C2h | E | C2 | i | σh |
----|---|----|---|----|
Γ1  | 1 |    |   |    |
Γ2  | 1 |    |   |    |
Γ3  | 1 |    |   |    |
Γ4  | 1 |    |   |    |
    

All characters under E are 1.

4. Populate Characters and Assign Mulliken Symbols:
We use the properties of IRs and Mulliken symbols (A/B based on C2 character, g/u based on i character, 1/2 if needed based on other symmetry, but not strictly needed for C2h beyond g/u).

• Ag: The totally symmetric representation always has characters of +1 for all operations. (A because C2=+1, g because i=+1).
• Au: Must be A (C2=+1), u (i=-1). To be orthogonal to Ag (1,1,1,1), the σh character must be -1. (11 + 11 + (-1)*1 + (-1)*1 = 0). Row: (1, 1, -1, -1). This transforms like the z-coordinate.
• Bg: Must be B (C2=-1), g (i=+1). To be orthogonal to Ag (1,1,1,1), the σh character must be -1. (1*1 + (-1)1 + 11 + (-1)*1 = 0). Row: (1, -1, 1, -1). This transforms like rotations Rx, Ry and quadratic products xz, yz.
• Bu: Must be B (C2=-1), u (i=-1). By orthogonality to Ag (1,1,1,1), the σh character must be +1. (1*1 + (-1)*1 + (-1)1 + 11 = 0). Row: (1, -1, -1, 1). This transforms like the x and y coordinates.

The C2h Character Table:

Explanation: Reducible and Irreducible Representations (approx. 300 words)

A character table lists the irreducible representations (IRs) of a point group. These are the fundamental, symmetry-distinct ways in which things (like atomic orbitals, vibrations, vectors) can transform under the group’s symmetry operations. For C2h, these are the four rows: Ag, Au, Bg, and Bu. Each character (the number in the table) is the trace of the transformation matrix for that operation and representation.

When we consider a larger basis set, such as the set of all atomic orbitals in a molecule or the vectors representing molecular vibrations, their transformation under symmetry operations can be described by reducible representations (RRs). An RR is essentially a linear combination of IRs. The characters of an RR for each operation are found by applying the operation to the basis set and summing the characters of the resulting transformation matrix (the trace).

The key use of the character table, along with the RR characters, is to determine which IRs make up the RR. This process is called reduction. The reduction formula is:

        ni=1h∑Rχred(R)χi(R)nRn_i = \frac{1}{h} \sum_R \chi_{red}(R) \chi_i(R) n_Rni​=h1​∑R​χred​(R)χi​(R)nR​
      

Where:

• nin_ini​ is the number of times the irreducible representation ‘i’ appears in the reducible representation.
• hhh is the order of the point group (4 for C2h).
• ∑R\sum_R∑R​ means sum over all symmetry operations R.
• χred(R)\chi_{red}(R)χred​(R) is the character of the reducible representation for operation R.
• χi(R)\chi_i(R)χi​(R) is the character of the irreducible representation ‘i’ for operation R (from the table).
• nRn_RnR​ is the number of operations in the class of R (for C2h, nR=1n_R = 1nR​=1 for all R, simplifying the formula).

By applying this formula for each IR (Ag, Au, Bg, Bu), you find out exactly how the reducible representation decomposes into its irreducible components. For example, if you had an RR with characters (3, -1, -3, 1) for (E, C2, i, σh) respectively, applying the formula would show it decomposes into Au + 2Bu. This tells you that the original basis set transforms as one Au component and two Bu components.