Character Table For D4h Point Group

The character tablefor D4h point group presents a systematic summary of symmetry operations, irreducible representations, and their characters, serving as a foundational tool in group theory, spectroscopy, and quantum chemistry. This article walks through the essential elements of the D4h point group, explains how to construct its character table, and highlights practical applications for students and researchers alike.

Understanding the D4h Point Group

The D4h point group describes molecules with a square planar or octahedral geometry that possesses a four‑fold rotational axis (C₄), two perpendicular two‑fold axes (C₂), a horizontal mirror plane (σh), vertical mirror planes (σv), and an inversion center (i). Common examples include square planar complexes such as [Ni(CN)₄]²⁻ and planar molecules like ethylene (C₂H₄) in its eclipsed conformation. Recognizing these symmetry elements is the first step toward building the character table.

Key Symmetry Elements of D4h- C₄ – Rotation by 90° about the principal axis (z).

• C₂ – Rotations by 180° about the x and y axes (perpendicular to C₄).
• C₂′ – Rotations by 180° about the diagonal axes (x=y, x=−y).
• i – Inversion through the center of symmetry.
• σh – Horizontal mirror plane (xy plane).
• σv – Vertical mirror planes containing the C₄ axis.
• σd – Diagonal mirror planes bisecting the angles between C₂ axes.

These elements combine to give a total of 16 symmetry operations, which form the basis of the D4h character table.

Constructing the Character Table

To construct the character table, we first list all symmetry operations and then assign each to a class of operations that share the same character values across irreducible representations. The classes for D4h are:

1. E – Identity
2. 2C₄ – Two C₄ rotations (90° and 270°)
3. C₂ – One C₂ rotation about the principal axis (180°)
4. 2C₂′ – Two C₂ rotations about the x and y axes
5. 2C₂″ – Two C₂ rotations about the diagonal axes
6. i – Inversion
7. 2S₄ – Two improper rotations (S₄)
8. σh – Horizontal mirror plane
9. 2σv – Two vertical mirror planes
10. 2σd – Two diagonal mirror planes

Each class is represented by a single entry in the table, and the number in front indicates how many operations belong to that class.

Irreducible Representations

D4h contains ten irreducible representations: A₁g, A₂g, B₁g, B₂g, Eg, A₁u, A₂u, B₁u, B₂u, Eu. The “g” (gerade) and “u” (ungerade) labels indicate whether the representation is symmetric (g) or antisymmetric (u) with respect to inversion. The characters for each class are determined by how basis functions transform under the symmetry operations.

Example: The A₁g Representation- E: +1

• 2C₄: +1
• C₂: +1
• 2C₂′: +1
• 2C₂″: +1 - i: +1
• 2S₄: +1
• σh: +1
• 2σv: +1
• 2σd: +1

All characters are +1, reflecting that the A₁g representation is totally symmetric.

Example: The Eg Representation (Doubly Degenerate)

• E: +2
• 2C₄: 0
• C₂: -2
• 2C₂′: 0
• 2C₂″: 0
• i: +2
• 2S₄: 0
• σh: 2
• 2σv: 0
• 2σd: 0

The Eg representation is useful for describing vibrations or orbitals that are symmetric with respect to the horizontal plane but change sign under a 180° rotation about the principal axis.

The complete character table for D4h point group is summarized below: