Question

Explain why the \(d_{x y}, d_{x z}\), and \(d_{y z}\) orbitals lie lower in energy than the \(d_{z}^{2}\) and \(d_{x^{2}-y^{2}}\) orbitals in the presence of an octahedral arrangement of ligands about the central metal ion.

Short answer

In an octahedral arrangement of ligands, the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals lie lower in energy than the \(d_z^2\) and \(d_{x^2-y^2}\) orbitals due to differences in their electron cloud orientations. The former orbitals have their electron clouds in-between the axes, resulting in less repulsion with ligands and lower energy levels, while the latter orbitals' electron clouds align along the axes, causing greater repulsion with the ligands and higher energy levels. Overall, it is the interaction between the d orbitals and the ligands that determines the relative energy levels within the central metal ion.

Step-by-step solution

Recall the Crystal Field Theory:

According to the Crystal Field Theory, the energy levels of the d orbitals in a central metal ion are affected by the presence of ligands. In an isolated metal ion, the five d orbitals have the same energy (degenerate), but when ligands approach the central metal ion, they will cause the energy to split, resulting in the rising of some orbitals' energy and the lowering of others.

Visualizing the octahedral arrangement of ligands:

An octahedral arrangement consists of six ligands surrounding the central metal ion, positioned at equal distances along the x, y, and z-axis. There are two ligands directly opposite to each other along each axis, creating an octahedral shape.

Understand the interaction between ligands and d orbitals :

The interaction depends on the extent of the overlap between the electron clouds of the metal ion's d orbitals and the electron clouds of the ligands. If the overlap is greater, the repulsion between the ligands and the metal ion will cause an increase in the energy level of the corresponding orbital. Conversely, if the interaction is smaller, the energy level will be lower.

Analyze each d orbital's interaction with the ligands :

1. For \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\), these orbitals' electron clouds lie in-between the axes, so they don't point toward the ligands, causing less repulsion and correspondingly lower energy. 2. For \(d_z^2\) and \(d_{x^2-y^2}\), the electron clouds are aligned along the axes. The ligands directly interact with the lobes of these orbitals, causing higher repulsion between ligands and the metal ion and as a result, a higher energy level of these orbitals.

Conclude with the explanation:

Based on the analysis of the interactions between d orbitals and ligands in an octahedral arrangement, we can conclude that the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals lie lower in energy than the \(d_z^2\) and \(d_{x^2-y^2}\) orbitals. This is because the electron clouds of the former orbitals are located in-between the axes, causing less repulsion with the ligands and lower energy levels, while the electron clouds of the latter orbitals are aligned along the axes and directly interact with the approaching ligands, leading to higher repulsion and higher energy levels.

Key concepts

d Orbitals

In an isolated metal ion, the five d orbitals—namely, \(d_{xy}\), \(d_{xz}\), \(d_{yz}\), \(d_{z^2}\), and \(d_{x^2-y^2}\)—are at an equal energy level, making them degenerate. These orbitals are electron clouds where electrons are likely to be found. The shape of each d orbital dictates its properties and interactions, particularly with surrounding atoms or ions, often referred to as ligands.
With the introduction of a ligand environment, as explained by Crystal Field Theory, the once degenerate d orbitals undergo a change. This change is due to the distinct overlap between the orbitals and the ligands, leading to a split in their energy levels.
The specific orientation and shape of these orbitals allow us to predict how they will interact with ligands and, subsequently, how their energy will change. In particular, when these orbitals are in environments such as octahedral arrangements, they do not all have the same energy anymore, resulting in a crystal field splitting pattern.

Octahedral Arrangement

An octahedral arrangement is one of the most common geometric configurations in coordination chemistry. It involves a central metal ion surrounded by six ligands, positioned at equal distances along the x, y, and z-axis. This forms a geometric shape of an octahedron, where two ligands lie opposite each other along each axis.
This specific geometric formation holds significance as it dictates the interaction potential of ligands with the metal ion's d orbitals. In the case of an octahedral arrangement, we observe how some d orbitals align along the axes, directly interacting with the ligands, whereas others lie in-between, thus, less directly interacting with them. This differential interaction leads to variations in the energy levels of these orbitals—some rise in energy, and others fall.

Ligand Interaction

The nature of ligand interaction with d orbitals is central to understanding Crystal Field Theory. Ligands are molecules or ions that donate pairs of electrons to a central metal ion, forming coordination bonds. Their interaction depends heavily on which d orbitals they are engaging with.
The overlap of the ligand's electron clouds with those of the metal ion's d orbitals determines the extent of the interaction. Greater overlap means stronger repulsion, leading to a rise in the energy levels of those orbitals. Conversely, lesser overlap translates to weaker repulsion, resulting in lower energy levels. Thus, knowing which orbitals lie towards or between the axes can help predict their energy after ligand interaction.

• Ligands interacting with \(d_{z^2}\) and \(d_{x^2-y^2}\) cause high repulsion due to their alignment along the axes.
• Ligands interacting with \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) result in lower repulsion as these orbitals are oriented between the axes.

Electron Repulsion

Electron repulsion within the Crystal Field Theory framework refers to how the negative charges of ligands, which are electron-donating species, repel the electron clouds of the central metal ion's d orbitals. This repulsion alters the energy of these orbitals, depending on their orientation and positioning relative to the ligands.
In an octahedral arrangement, ligands approach from six directions. The d orbitals whose lobes are aligned along the axes, such as \(d_{z^2}\) and \(d_{x^2-y^2}\), experience significant electron repulsion. This is because the ligands directly confront these orbitals. On the other hand, the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals, which are situated between the axes directions, face less direct ligand encounters and consequently endure less electron repulsion.
By understanding and analyzing these repulsive interactions in different orbital-ligand configurations, one can predict how the energy levels of d orbitals will adjust in various chemical settings.