Problem 28 Consider the following \(A B_{n}... [FREE SOLUTION]

Chapter 3: Problem 28

Consider the following $A B_{n}$ molecules and determine the symmetries and degeneracies of the $s, p,$ and $d$ orbitals on $A$ in each. a. $A B_{8}$ (cube) b. $A B_{4}$ (square plane) c. $A B_{3}$ (trigonal pyramid) d. $A B_{3}$ (trigonal plane) e. $A B_{3}(T \text { -shape })$ f. $A B_{4}(T \text { rectangular plane })$

Short Answer

Expert verified

Identify the point group and use symmetry tables to determine orbital symmetries.

Step by step solution

Identify the Point Group of Each Molecule

First, determine the point group for each molecular structure, as symmetry properties depend on these groups.- For $A B_8$ in a cube structure, the point group is $O_h$.- For $A B_4$ in a square planar structure, the point group is $D_{4h}$.- For $A B_3$ in a trigonal pyramid, the point group is $C_{3v}$.- For $A B_3$ in a trigonal plane, the point group is $D_{3h}$.- For $A B_3$ in a T-shape, the point group is $C_{2v}$.- For $A B_4$ in a T-rectangular plane, the point group is $C_{2v}$.

Determine Symmetries and Degeneracies for Each Molecule's Orbitals

For each point group, use character tables to determine the symmetries and degeneracies of the orbitals involved.### (a) $AB_8$ (Cube)- In $O_h$, the $s$ orbital is invariant; therefore, it transforms as $A_{1g}$.- The $p$ orbitals transform as the triply-degenerate representation $T_{1u}$.- The $d$ orbitals transform as $E_g$ (doubly-degenerate) and $T_{2g}$ (triply-degenerate).### (b) $AB_4$ (Square Plane)- In $D_{4h}$, the $s$ orbital transforms as $A_{1g}$.- The $p$ orbitals transform as $E_u$ (degenerate in the plane) and $A_{2u}$ (perpendicular to plane).- The $d$ orbitals transform as $E_g$ (in-plane degeneracy) and $B_{2g}$, $B_{1g}$ (non-degenerate, out-of-plane).### (c) $AB_3$ (Trigonal Pyramid)- In $C_{3v}$, the $s$ orbital is $A_1$.- The $p$ orbitals transform as $A_1$ (in-plane component) and $E$ (two-fold degeneracy).- The $d$ orbitals transform as $A_1$ and $E$.### (d) $AB_3$ (Trigonal Plane)- In $D_{3h}$, the $s$ orbital is $A_1'$.- The $p$ orbitals transform as $E''$ and $A_2''$.- The $d$ orbitals transform as $E'$ and $A_1'$, $A_2'', etc.### (e) $AB_3$ (T-shape)- In $C_{2v}$, the $s$ orbital is $A_1$.- The $p$ orbitals transform as $B_2$, $B_1$, $A_1$.- The $d$ orbitals transform as $A_1$, $B_1$, $B_2$, $A_2$.### (f) $AB_4$ (T-rectangular plane)- In $C_{2v}$, the $s$ orbital is also $A_1$.- Similarly, the $p$ orbitals are $B_2$, $B_1$, $A_1$.- And the $d$ orbitals are $A_1$, $B_1$, $B_2$, $A_2$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Point Group Theory

Understanding point group theory is essential for studying molecular symmetry. Point groups classify molecules based on their symmetrical properties. Each point group represents a set of symmetry operations like rotations and reflections that a molecule can undergo without changing its appearance. This classification helps in predicting molecular behavior, particularly in spectroscopy and crystallography.

Let's take some examples!

The point group for a molecule with a cubic structure like $AB_8$ is $O_h$, which contains the highest level of symmetry.
For a square planar molecule, such as $AB_4$, the point group is $D_{4h}$, representing symmetry in both vertical and horizontal planes.
$AB_3$ can form various shapes, like trigonal pyramid $C_{3v}$ or planar $D_{3h}$, each defining different symmetrical operations.
Complexes such as the T-shape $AB_3$ and T-rectangular $AB_4$ fit the $C_{2v}$ point group, with specific symmetry elements.

Learning the symmetry properties associated with point groups aids in determining the orbitals' behavior鈥攈ow they interact, split, or remain the same during these operations.

Orbital Symmetry

Orbital symmetry relates to how atomic or molecular orbitals behave under the symmetry operations of the point group. Each orbital can be assigned a symmetry label, which helps in determining how they overlap and form bonds or how they split in an external field.

In a simple example, the $s$ orbitals in our molecules remain invariant under any symmetry operation, leading to a simple designation. For instance:

In $AB_8$ configured in the $O_h$ group, the $s$ orbital transforms as $A_{1g}$.
In $AB_4$, a square planar arrangement under $D_{4h}$, the $s$ is also an $A_{1g}$ orbital.

The behavior of $p$ and $d$ orbitals varies significantly:

$p$ orbitals, being directional, have different symmetries based on their axis alignment.
$d$ orbitals exhibit more complex symmetry transformations, often showing multiple degeneracies.

Understanding orbital symmetry helps predict molecular shapes and reactivity, proving crucial in fields like coordination chemistry and quantum chemistry.

Character Tables

Character tables are like cheat sheets for point group symmetries. They provide a systematic way to understand how different orbitals behave under symmetry operations. Each table contains representations that denote how orbitals transform, which is invaluable in predicting molecular vibrations, electronic transitions, and more.

A typical character table presents:

Symmetry operations specific to the point group.
Irreducible representations, showing how orbital symmetries change under these operations.

In practice:

For a cubic molecule $AB_8\under )O_h$, refer to its character table to find the symmetries of $T_{1u}$ for $p$ and $T_{2g}$ for some $d$ orbitals.
Square planar molecules like $AB_4$ using $D_{4h}$ feature 脽 identifying symmetrical orbitals $A_{1g}$ for $s$ and $E_g$ for degenerate orbitals.

How to use character tables: Mastering them allows chemists to predict physical properties and observe how molecular geometries can guide reactions or spectral data analysis.

Degeneracy of Orbitals

Degeneracy refers to orbitals having the same energy level, which is a direct outcome of symmetry. Degenerate orbitals are typically seen in high symmetry environments where multiple orbitals transform equally under the point group's operations. Degeneracy is essential in determining electronic configurations, spectroscopy, and molecular interactions.

Examples include:

The orbitals in a cube-shaped $AB_8$ have high degeneracy; $p$ orbitals transform as the triply-degenerate $T_{1u}$, while $d$ orbitals split into doubly-degenerate $E_g$ and triply-degenerate $T_{2g}$.
In $AB_4$, the square planar structure: in-plane $p$ orbitals hold degeneracy $E_u$, while out-of-plane orbitals are non-degenerate $A_{2u}$.

Why does it matter? Degenerate orbitals indicate symmetrical electron distribution, affecting bonding, chemical reactivity, and interpretation of spectroscopic data.

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Short Answer

Step by step solution

Identify the Point Group of Each Molecule

Determine Symmetries and Degeneracies for Each Molecule's Orbitals

Key Concepts

Point Group Theory

Orbital Symmetry

Character Tables

Degeneracy of Orbitals

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