Question

Determine the irreducible representation for the following: (a) \(\mathrm{C}_{2 v}\) point group (b) Set of \(p\) -orbitals

Short answer

The irreducible representations for \(\mathrm{C}_{2 v}\) are \(\mathrm{A}_{1}\), \(\mathrm{A}_{2}\), \(\mathrm{B}_{1}\) and \(\mathrm{B}_{2}\). For the set of \(p\)-orbitals; the orbital shapes transform under the \(\mathrm{C}_{2 v}\) point group operations such that they will correspond to one (or more) of these representations. It will depend on the specific characteristics of the \(p\)-orbitals, and how they transform under the operations of the \(\mathrm{C}_{2 v}\) group.

Step-by-step solution

Determine Irreducible representation for \(\mathrm{C}_{2 v}\) group

Irreducible representation for \(\mathrm{C}_{2 v}\) point group can be determined from its character table. For this purpose, we will directly refer to the character table for \(\mathrm{C}_{2 v}\) point group. According to character table, irreducible representations are \(\mathrm{A}_{1}\), \(\mathrm{A}_{2}\), \(\mathrm{B}_{1}\) and \(\mathrm{B}_{2}\).

Identify the set of \(p\) -orbitals

The \(p\) -orbitals are three dimensional, one each along the x, y, and z axis, namely the \(p_x\), \(p_y\), and \(p_z\). These three orbitals are treated as one entity under the transformation, and we characterize these representations using the same \(\mathrm{C}_{2 v}\) character table.

Determine the characters for the operations on \(p\) orbitals

The transformation of \(p_x\), \(p_y\), and \(p_z\) under the symmetry operations for \(\mathrm{C}_{2 v}\) will be used to determine the characters. The \(p_x\) orbital is not changed by \(E\) or \(\sigma_{x z}\), but changes sign with \(C_2\) and \(\sigma_{y z}\), character can be represented as \(\chi\)=3. Same way characters for \(p_y\) and \(p_z\) orbitals can be determined.

Reducible Representation for the set of \(p\) orbitals

Next, combine the characters for the \(p_x\), \(p_y\), and \(p_z\) orbitals under each operation into a reducible representation. This forms a reducible representation for the set of \(p\) -orbitals in the \(\mathrm{C}_{2 v}\) point group. Utilizing characters from step 3, reducible representation \(\Gamma_{red}\) can be constructed.

Determine Irreducible representations for p-orbitals

Use projection operator method to reduce the reducible representation of p-orbitals to their irreducible constituents. Each constituent will correspond to either to a \(p_x\), \(p_y\), or \(p_z\) orbital. Compare results with irreducible representations from step 1.

Key concepts

C2v point group

The \(\mathrm{C}_{2v}\) point group is one of the simplest types of symmetry groups found in molecules. It's characterized by a set of symmetry operations that leave a molecule unchanged. This group contains several operations: identity operation (\(E\)), a twofold rotation about the primary axis (\(C_2\)), and two perpendicular reflection planes (\(\sigma_{v}(xz)\) and \(\sigma_{v'}(yz)\)). Each of these operations represents a symmetry that can be found in certain molecules such as water (\(H_2O\)). This kind of symmetry is important because it helps chemists predict and explain the behavior of electrons in molecules. Knowing the \(\mathrm{C}_{2v}\) point group can help categorize molecular symmetries, which aids in further quantum mechanical calculations and the assignment of vibrational modes.

p-orbitals

The p-orbitals are a set of three degenerate orbitals found in atomic systems, each with distinct spatial orientations along the x, y, and z axes. These orbitals are known as \(p_x\), \(p_y\), and \(p_z\). They possess a dumbbell shape and are commonly associated with the second and higher periods of elements. Each p-orbital can hold up to two electrons and their orientations play a crucial role in determining molecular geometry and bonding characteristics. For instance:

• The \(p_x\) orbital lies along the x-axis,
• The \(p_y\) orbital lies along the y-axis,
• The \(p_z\) orbital lies along the z-axis.

Understanding how these orbitals interact under symmetry operations, such as those from a point group like \(\mathrm{C}_{2v}\), is fundamental for predicting chemical properties and reactions.

character table

A character table is an organized way to represent the symmetry operations and their corresponding characters for a specific point group. In the case of the \(\mathrm{C}_{2v}\) point group, the character table helps predict the behavior of molecular orbitals, vibrations, and electronic transitions under symmetry operations. The table lists:

• The symmetry operations of the group (like \(E\), \(C_2\), \(\sigma_{v}\), and \(\sigma_{v'}\)).
• The irreducible representations (like \(A_1\), \(A_2\), \(B_1\), \(B_2\)).
• The characters of these representations under each symmetry operation.

This information allows chemists to classify molecular orbitals and vibrational modes as symmetric or antisymmetric, an essential process for deeper quantum mechanical analysis and prediction.

projection operator method

The projection operator method is a mathematical tool used to break down or "reduce" a complex set of orbital interactions into simpler, irreducible representations. This method is particularly handy for analyzing how sets of atomic orbitals transform under a given point group's symmetry operations. To apply it, here are the typical steps:

• Start with a reducible representation that describes how a set of orbitals behaves under symmetry operations.
• Employ the projection operator method to mathematically decompose this reducible representation into its constituent irreducible representations.
• Compare these results with the character table to understand the symmetry properties of individual orbitals, such as \(p_x\), \(p_y\), or \(p_z\) in a \(\mathrm{C}_{2v}\) point group.

This approach simplifies the analysis of molecular symmetry, making it easier to predict molecular behavior and interaction in chemical reactions.