19.3 Spectroscopic and Magnetic Properties of Coordination Compounds

Crystal Field Theory explains how metal ions interact with surrounding ligands in coordination compounds. It's the key framework for understanding why these compounds have specific colors, magnetic properties, and reactivities.

The theory focuses on how ligands split the metal's d-orbitals into different energy levels. This splitting affects electron distribution, determining whether a complex is high-spin or low-spin, which in turn shapes its magnetic behavior and the wavelengths of light it absorbs.

Crystal Field Theory and Coordination Compounds

Principles of crystal field theory

Crystal field theory (CFT) models the electronic structure of coordination compounds by treating ligands as negative point charges surrounding a central metal ion. These point charges create an electrostatic field that interacts with the metal's d orbitals, causing them to split into groups with different energy levels.

Before ligands approach, all five d orbitals on a free metal ion have the same energy (they're degenerate). Once ligands are in place, d orbitals that point directly at the ligands get pushed to higher energy (more electron-electron repulsion), while those pointing between the ligands stay at lower energy. This energy gap between the sets of orbitals is what drives the color, magnetism, and stability of the complex.

Geometry and d-orbital splitting

The geometry of a coordination compound determines how the d orbitals split.

• Octahedral complexes have six ligands arranged symmetrically around the metal ion. The d orbitals split into a higher-energy pair called $e_g$ and a lower-energy trio called $t_{2g}$. The energy gap between them is labeled $\Delta_o$.
• Tetrahedral complexes have four ligands. Here the splitting is inverted: $t_2$ orbitals sit at higher energy and $e$ orbitals at lower energy. The energy gap $\Delta_t$ is smaller, roughly $\frac{4}{9}$ of $\Delta_o$ for the same metal-ligand combination. Because $\Delta_t$ is so much smaller, tetrahedral complexes are almost always high-spin.
• Square planar complexes also have four ligands, but they lie in a plane around the metal. This geometry produces a more complex splitting pattern with a particularly large energy gap between the highest and lowest d orbitals. Square planar geometry is most common for $d^8$ metal ions (like $\text{Pt}^{2+}$ and $\text{Ni}^{2+}$ with strong-field ligands).

The Jahn-Teller effect can distort certain geometries further. When an octahedral complex has an unequal electron population in the $e_g$ orbitals (for example, a $d^9$ ion like $\text{Cu}^{2+}$), the complex distorts to remove the degeneracy, typically by elongating or compressing along one axis. This causes additional splitting of the d orbital energy levels.

Electron distribution in d orbitals

Once you know the splitting pattern, the next question is: how do the d electrons fill those orbitals? Two factors compete with each other:

• Hund's rule favors spreading electrons across orbitals to maximize unpaired spins (which costs pairing energy to violate).
• The crystal field splitting energy ($\Delta$) favors putting electrons in the lower-energy set first.

When $\Delta$ is small relative to the pairing energy, electrons spread out across all orbitals before any pairing occurs. That gives a high-spin complex with the maximum number of unpaired electrons. When $\Delta$ is large, electrons pair up in the lower set before occupying the higher set. That gives a low-spin complex with fewer unpaired electrons.

Here are some octahedral examples to see this in action:

• $d^3$ (e.g., $\text{Cr}^{3+}$): Configuration is $t_{2g}^3$ regardless of field strength. Three electrons each go into one $t_{2g}$ orbital. There's no difference between high-spin and low-spin here because you don't need to pair or promote any electrons.
• $d^6$ (e.g., $\text{Fe}^{2+}$): High-spin gives $t_{2g}^4 \, e_g^2$ (4 unpaired electrons). Low-spin gives $t_{2g}^6$ (0 unpaired electrons). This is a case where the ligand field strength makes a dramatic difference.
• $d^8$ (e.g., $\text{Ni}^{2+}$): Configuration is $t_{2g}^6 \, e_g^2$ in both cases (2 unpaired electrons). With 8 electrons, the $t_{2g}$ set fills completely and two electrons must go into $e_g$ no matter what, so high-spin and low-spin are identical in octahedral geometry.

Notice that the high-spin vs. low-spin distinction only matters for $d^4$ through $d^7$ configurations in octahedral complexes. Outside that range, there's only one possible arrangement.

Spectroscopic and Magnetic Properties

Properties of coordination compounds

Color and absorption spectra

The vivid colors of coordination compounds come from d-d transitions: an electron absorbs a photon and jumps from a lower-energy d orbital to a higher-energy one. The energy of that photon corresponds to the crystal field splitting energy ($\Delta$).

The color you see is the complement of the color absorbed. For example, if a complex absorbs red/orange light (lower energy, longer wavelength), it appears blue or green. If it absorbs violet/blue light (higher energy, shorter wavelength), it appears yellow or orange.

• Complexes with a larger $\Delta$ absorb higher-energy (shorter-wavelength) light.
• Complexes with a smaller $\Delta$ absorb lower-energy (longer-wavelength) light.

Not all d-d transitions are equally "allowed." Selection rules govern which transitions can occur:

• The spin selection rule says transitions that require a change in the total number of unpaired electrons are forbidden (though they can still happen weakly).
• The Laporte selection rule says d-to-d transitions are forbidden in centrosymmetric complexes (like perfect octahedral geometry). Vibrations in the molecule relax this rule, which is why octahedral complexes are colored but often have relatively pale, muted colors compared to charge-transfer complexes.

Magnetic properties

The magnetic behavior of a complex depends directly on how many unpaired electrons it has.

• Paramagnetic complexes have one or more unpaired electrons and are attracted into a magnetic field.
• Diamagnetic complexes have zero unpaired electrons and are very weakly repelled by a magnetic field.

You can estimate the magnetic moment using the spin-only formula:

$\mu = \sqrt{n(n+2)} \; \mu_B$

where $n$ is the number of unpaired electrons and $\mu_B$ is the Bohr magneton. For example, a high-spin $d^5$ complex (like $[\text{Fe}(\text{H}_2\text{O})_6]^{3+}$) has 5 unpaired electrons, giving $\mu = \sqrt{5(7)} = \sqrt{35} \approx 5.92 \; \mu_B$. A low-spin $d^6$ complex with 0 unpaired electrons gives $\mu = 0$, confirming it's diamagnetic.

Measuring the magnetic moment experimentally (using a Gouy balance, for instance) is one of the most direct ways to determine whether a complex is high-spin or low-spin.

Some complexes can even switch between high-spin and low-spin states depending on temperature or pressure. This phenomenon is called spin crossover.

Spectrochemical series

The spectrochemical series ranks ligands by their field strength, from weak-field to strong-field:

$\text{I}^- < \text{Br}^- < \text{Cl}^- < \text{F}^- < \text{H}_2\text{O} < \text{NH}_3 < \text{en} < \text{NO}_2^- < \text{CN}^- < \text{CO}$

Weak-field ligands (left side) produce a small $\Delta$ and tend to give high-spin complexes. Strong-field ligands (right side) produce a large $\Delta$ and tend to give low-spin complexes. You'll want to memorize this series, or at least the general trend: halides are weak, water and ammonia are moderate, and $\text{CN}^-$ and $\text{CO}$ are strong.

Advanced Theoretical Approaches

Crystal field theory works well for explaining color and magnetism, but it treats ligands as pure point charges and ignores covalent bonding entirely. Ligand field theory extends CFT by incorporating ideas from molecular orbital theory. It accounts for the overlap between metal d orbitals and ligand orbitals, giving a more complete picture of bonding, back-bonding (especially with ligands like $\text{CO}$ and $\text{CN}^-$), and why certain ligands are strong-field. For an intro course, CFT covers what you need, but knowing that ligand field theory exists helps explain cases where CFT's predictions fall short.