Question

Write the multiplication table for \(\mathrm{C}_{2}\), point group.

Short answer

The multiplication table for \(\mathrm{C}_{2}\) point group is: E ∗ E = E, E * \(C_2\) = \(C_2\), \(C_2\) * E = \(C_2\), and \(C_2\) * \(C_2\) = E.

Step-by-step solution

Identify the group elements

The first step in writing the multiplication table for \(\mathrm{C}_{2}\) point group is to identify the members of the group. The \(\mathrm{C}_{2}\) point group has just two members: the identity operation (E) and the two-fold rotation (\(C_2\)).

Create an empty table

Create an empty table with two columns and two rows. Label the rows and columns with the group elements: E, \(C_2\).

Fill in the diagonal

The intersection of the same group operations (or elements) is always the operation itself. So, the diagonal of the multiplication table which goes from the top left to the bottom right should be filled with the same operations which are E and \(C_2\). Now, the result of the operation E (no change) followed by any operation is just the second operation. Similarly, the result of any operation followed by E is just the first operation. So the first row (except for the identity) and the first column (except for the identity) of the multiplication table can now be filled in.

Complete the multiplication table

Finally, to complete the table fill the last position (the intersection of \(C_2\) and \(C_2\)). As the second \(C_2\) rotation is equivalent to no rotation (It will rotate twice and end at the starting point), the intersection of \(C_2\) and \(C_2\) (i.e., \(C_2 C_2\)) gives the identity operation E. Now, the multiplication table for \(\mathrm{C}_{2}\) is completed with the four entries E, \(C_2\), \(C_2\), and E.

Key concepts

Group Elements

In the study of molecular symmetry, group elements are crucial. They represent the different transformations a molecule can undergo without altering its fundamental structure. For the \( \mathrm{C}_{2} \) point group, the group elements are quite simple yet powerful in understanding symmetry.

• Identity Operation (E): This is the most straightforward operation. It essentially means doing nothing to the object or molecule. Everything remains in its original position.
• Two-Fold Rotation (\( C_2 \)): This operation involves rotating the molecule by 180 degrees around a specific axis. After this rotation, the molecule's appearance is indistinguishable from its initial state.

By exploring these elements, students can gain an appreciation for how seemingly simple operations govern the complex symmetries of molecules.

Identity Operation

The identity operation, represented as E, is fundamental to group theory. Think of it as the equivalent of multiplying by one in arithmetic.

In the context of molecular symmetry, when the identity operation is applied, the molecule or object remains unchanged. This means that no matter how complex or simple the molecule is, performing the identity operation will always result in its initial configuration.

This concept is crucial when evaluating how different group elements interact in a multiplication table. Understanding the identity operation as the 'do nothing' step helps in predicting outcomes of any combined operations with other group elements. Without E, the group's structure would be void of this grounding element that ensures continuity in operations.

Multiplication Table

The multiplication table is a convenient way to visualize how group elements combine with one another. It's essentially a matrix that simplifies understanding of group operations in symmetry groups like \( \mathrm{C}_{2} \).

• Each row and column represents one of the group's elements, in this case: E and \( C_2 \).
• The intersection of a row and a column shows the result of the group operation. For example, the intersection of E with \( C_2 \) will result in \( C_2 \), because doing nothing first and then rotating still results in the rotation.
• As you move down the diagonal (from top left to bottom right), the identity element E appears where an element is combined with itself, such as \( C_2 C_2 \) resulting in E, as two 180-degree rotations bring the molecule back to its original state.

The multiplication table allows students to predict the outcomes of complex operations by simply observing these intersections, making symmetry operations less abstract and more tangible.

Two-Fold Rotation

The two-fold rotation, denoted as \( C_2 \), is a pivotal concept in the \( \mathrm{C}_{2} \) point group. It involves rotating a molecule by 180 degrees around an axis. Post-rotation, the molecule resembles its original form if it holds this particular symmetry.

Understanding \( C_2 \) helps with visualizing how molecules can exhibit different symmetrical properties. For example, molecules shaped like a simple line or a bisected shape often display \( C_2 \) symmetry because they appear the same when flipped by 180 degrees.

In the group multiplication table, this operation reveals an interesting aspect: \( C_2 \) applied twice results in the identity operation E. This illustrates how back-to-back 180-degree rotations realign the molecule to its starting orientation.

By mastering the concept of two-fold rotation, students can better understand how symmetry operations contribute to the overall symmetry of complex molecular structures.