Question

Predict the crystal field energy-level diagram for a linear \(\mathrm{ML}_{2}\) complex that has two ligands along the \(\pm z\) axis:

Short answer

The energy-level order is: d_{z^2} > d_{xz}, d_{yz} > d_{x^2-y^2}, d_{xy}.

Step-by-step solution

Understand the Geometry

In a linear ML2 complex, the metal ion (M) is situated along the z-axis, and the two ligands (L) are symmetrically positioned along the +z and -z axes. This results in a linear geometry.

Identify the d-Orbital Splitting

The d-orbitals in a metal ion are affected by the electronic field of the ligands. In linear complexes, particularly those on the z-axis, the d-orbitals experience different levels of repulsion: 1. The d_{z^2} orbital is directly aligned with the ligand axes and thus experiences maximum repulsion, leading to higher energy. 2. The d_{x^2-y^2} and d_{xy} orbitals experience the least repulsion as they are perpendicular to the ligands, resulting in lower energy. 3. The d_{xz} and d_{yz} orbitals, while partially aligned with the ligand axes, also experience less repulsion than d_{z^2} but more than d_{x^2-y^2} and d_{xy}.

Arrange the d-Orbitals in the Energy Diagram

Based on repulsion, the d-orbitals order from highest to lowest energy in a linear ML2 complex is generally: d_{z^2} > d_{xz}, d_{yz} > d_{x^2-y^2}, d_{xy}. This order reflects the increasing distance from the direct z-axis alignment with the ligands.

Draw the Energy Diagram

Visualize the splitting by drawing an energy-level diagram: - d_{z^2} is at the highest energy level. - d_{xz} and d_{yz} are at a slightly lower energy level and are degenerate. - d_{x^2-y^2} and d_{xy} are at the lowest energy level and are also degenerate. Label these levels to show the relative position and energy.

Key concepts

d-Orbital Splitting

In a metal ion surrounded by ligands, like in a linear \(\mathrm{ML}_2\) complex, the five d-orbitals do not remain degenerate. The interaction between the d-orbitals and the ligands' electronic field causes them to split into different energy levels. In a linear \(\mathrm{ML}_2\) complex, this splitting occurs because the ligands are positioned symmetrically along the \(\pm z\) axis.
When examining the nature of d-orbital splitting:

• The \(d_{z^2}\) orbital, being directly aligned with the z-axis, faces maximum repulsion from the ligands.
• In contrast, the \(d_{x^2-y^2}\) and \(d_{xy}\) orbitals are perpendicular to the z-axis and experience minimal repulsion.
• Orbital \(d_{xz}\) and \(d_{yz}\) have intermediate repulsion as they are partially aligned with the bond axis but not as directly as \(d_{z^2}\).

As a result, there is a distinct pattern of levels where the energy ranges from high, for \(d_{z^2}\), to lower energies for \(d_{x^2-y^2}\) and \(d_{xy}\). It's essential to understand this concept of splitting, as it dictates the electronic configuration and properties of the complex.

Linear ML2 Complex

A linear \(\mathrm{ML}_2\) complex is a coordination compound where the central metal atom is positioned in a straight line with two ligands. This linear arrangement is often due to the coordination number of two, placing ligands directly opposite each other on the axis. This geometry significantly affects the electronic characteristics of the metal center.
Here are key features of linear \(\mathrm{ML}_2\) complexes:

• The ligands lie along the \(\pm z\) axis, influencing only certain d-orbitals.
• Such geometry is common in molecules with large or specific directional ligand fields.
• Its simplicity aids in understanding complex theories like crystal field theory, due to reduced steric and repulsive interactions.

Linear \(\mathrm{ML}_2\) complexes offer clarity in studying d-orbital splitting and help visualize the theoretical predictions about electronic transitions and spectroscopy.

Energy-Level Diagram

An energy-level diagram is a visual representation showing the relative energy positions of the d-orbitals in a complex. For a linear \(\mathrm{ML}_2\) complex, it highlights the sequence from the highest to lowest energy orbital based on the d-orbital interactions with the ligands.
The energy-level diagram in such complexes typically shows:

• \(d_{z^2}\) at the highest level due to direct alignment with the ligand fields.
• \(d_{xz}\) and \(d_{yz}\) at an intermediate energy level, reflecting moderate interaction with the ligands.
• Lowest energy for \(d_{x^2-y^2}\) and \(d_{xy}\), given they face the least direct electronic influence.

Drawing the energy-level diagram involves:

• Starting with a baseline of degenerate d-orbitals.
• Adjusting them according to interaction strength and alignment with the ligands.
• Creating a diagram that accurately depicts the transition and splitting patterns.

This diagram not only assists in predicting electronic transitions but also helps in correlating theoretical predictions with experimental data, like UV-Vis spectroscopy.