Question

Draw qualitative diagrams for the crystal field splittings in (a) a linear complex ion \(\mathrm{ML}_{2},\) (b) a trigonal-planar complex ion \(\mathrm{ML}_{3}\), and (c) a trigonal-bipyramidal complex ion \(\mathrm{ML}_{5}\).

Short answer

Linear: \( d_{z^2} \) high; Trigonal-planar: \( d_{x^2-y^2}, d_{xy} \) high; Trigonal-bipyramidal: \( d_{z^2}, d_{x^2-y^2}, d_{xy} \) high.

Step-by-step solution

Understand the Geometries

First, review the given geometries: linear for \( \mathrm{ML}_2 \), trigonal-planar for \( \mathrm{ML}_3 \), and trigonal-bipyramidal for \( \mathrm{ML}_5 \). Each has distinct arrangements and affects the splitting of d orbitals differently due to varying ligand interactions.

Linear Complex (ML₂) Splitting

In a linear complex, the ligands are oriented along the z-axis, significantly affecting the energies of the \( d_{z^2} \) orbital, which remains non-degenerate and higher in energy, while \( d_{x^2-y^2} \), \( d_{xy} \), \( d_{xz} \), and \( d_{yz} \) are degenerate and lower in energy.

Trigonal-Planar Complex (ML₃) Splitting

In a trigonal-planar configuration, the ligands are in a plane, with one ligand along each axis separated by 120°. This results in \( d_{x^2-y^2} \) and \( d_{xy} \) being destabilized and higher in energy, forming an upper energy pair, while \( d_{z^2} \), \( d_{xz} \), and \( d_{yz} \) remain lower and nearly degenerate.

Trigonal-Bipyramidal Complex (ML₅) Splitting

For a trigonal-bipyramidal geometry, there are two axial and three equatorial ligands. The \( d_{z^2} \) orbital becomes higher in energy due to overlap with axial ligands. The \( d_{x^2-y^2} \) and \( d_{xy} \) orbitals are also elevated due to equatorial ligand proximity, while \( d_{xz} \) and \( d_{yz} \) are of lower energy due to lesser overlap.

Key concepts

Linear Complex

In a linear complex, two ligands are aligned directly along the z-axis of the coordination center. This specific arrangement significantly impacts the splitting of the d orbitals. Imagine the d orbitals as energy levels that can be pushed up or down based on their interactions with ligands.

For linear complexes, the most affected is the \(d_{z^2}\) orbital, as it directly aligns with the ligands on the z-axis. This causes the \(d_{z^2}\) orbital to be elevated in energy, making it non-degenerate and singularly higher than the other d orbitals.

On the other hand, the other four orbitals \(d_{x^2-y^2}\), \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) remain relatively less affected and form a degenerate group that is lower in energy. The energy difference between these orbitals forms the basis for understanding the stability and reactivity of linear complexes.

Trigonal-Planar Complex

The trigonal-planar complex has ligands arranged in a plane around the central atom, each separated by 120°. This geometry is known for its influence on the d orbitals' splitting patterns in a unique manner.

In this setup, the \(d_{x^2-y^2}\) and \(d_{xy}\) orbitals experience increased repulsion because they align with the plane where the ligands are located. This causes these orbitals to rise in energy, forming an upper pair that is higher relative to others.

The remaining orbitals \(d_{z^2}\), \(d_{xz}\), and \(d_{yz}\) experience less interaction with the ligands, thus maintaining lower and nearly degenerate energy levels. This arrangement reflects how planar arrangement shifts orbital energies and impacts the electronic properties of the complex.

Trigonal-Bipyramidal Complex

In a trigonal-bipyramidal complex, you'll find five ligands around the central metal atom—two are positioned axially and three equatorially. This complex coordination affects the energy of the metal's d orbitals distinctively.

The \(d_{z^2}\) orbital experiences a strong upward energy shift due to its alignment with the axial ligands. Meanwhile, \(d_{x^2-y^2}\) and \(d_{xy}\) orbitals, positioned around the equatorial plane, are also relatively higher in energy, reflecting the greater electron repulsion from equatorial ligands.

Conversely, the \(d_{xz}\) and \(d_{yz}\) orbitals undergo less interaction with ligands, resulting in them having comparatively lower energies. Understanding this configuration aids in comprehending the electronic transitions and the overall stability of trigonal-bipyramidal complexes.

d Orbital Splitting

Crystal field theory offers insights into how metal ions' d orbitals react under different geometrical arrangements with ligands all around them. This interaction produces a phenomenon known as "d orbital splitting."

When ligands approach a metal ion, their electric fields perturb the energy levels of the d orbitals. Depending upon the orientation of these ligands, each geometry—linear, trigonal-planar, and trigonal-bipyramidal—alters these energy levels in characteristic ways.

By observing these splits, chemists gain valuable information about the electronic structure, predict chemical behavior, and infer properties like color and magnetism. Essentially, the splitting of d orbitals embodies the core of crystal field theory, which is crucial in understanding complex chemical interactions in coordination compounds.