8.3 Crystal field theory and magnetic properties

Crystal Field Theory

Principles of crystal field theory

Crystal Field Theory (CFT) is an electrostatic model that explains the electronic structure and properties of transition metal coordination compounds. Rather than treating metal-ligand bonds as covalent, CFT models ligands as point negative charges (or dipoles) that interact with the d-orbitals of the central metal ion.

The core idea: not all five d-orbitals experience the same repulsion from surrounding ligands. Depending on the geometry of the complex (octahedral, tetrahedral, square planar, etc.), certain d-orbitals point directly at ligands while others point between them. This unequal repulsion causes the d-orbitals to split into groups of different energy.

The magnitude of that splitting depends on both the identity of the metal ion and the nature of the ligands. From this single concept, CFT predicts several observable properties:

• Color and absorption spectra (which wavelengths of light the complex absorbs)
• Magnetic properties (how many unpaired electrons the complex has)
• Relative stability and reactivity of different complexes

D-orbital splitting in complexes

In a free metal ion, all five d-orbitals have the same energy. Once ligands surround the ion, that degeneracy breaks.

Octahedral complexes (six ligands arranged along the x, y, and z axes):

• The $e_g$ set ($d_{x^2-y^2}$ and $d_{z^2}$) sits at higher energy because these orbitals point directly at the ligands, experiencing greater electrostatic repulsion.
• The $t_{2g}$ set ($d_{xy}$, $d_{xz}$, $d_{yz}$) sits at lower energy because these orbitals point between the ligands, experiencing less repulsion.
• The energy gap between the two sets is called $\Delta_o$ (the octahedral crystal field splitting energy).

Tetrahedral complexes (four ligands positioned between the axes):

The splitting pattern flips relative to octahedral geometry:

• The $t_2$ set ($d_{xy}$, $d_{xz}$, $d_{yz}$) is now higher in energy because in tetrahedral geometry these orbitals are closer to the ligand positions.
• The $e$ set ($d_{x^2-y^2}$ and $d_{z^2}$) is lower in energy because these orbitals point away from the tetrahedral ligand positions.
• The energy gap is called $\Delta_t$, and it's significantly smaller than $\Delta_o$: roughly $\Delta_t \approx \frac{4}{9}\Delta_o$.

Because $\Delta_t$ is so much smaller, tetrahedral complexes are almost always high-spin. The splitting is rarely large enough to force electron pairing.

Electronic configuration of coordination compounds

How electrons fill the split d-orbitals depends on a competition between two energies: the crystal field splitting energy ($\Delta$) and the pairing energy ($P$, the energy cost of putting two electrons in the same orbital).

High-spin complexes form when $\Delta < P$. It's energetically cheaper to place electrons in the higher-energy orbitals than to pair them in the lower set. This maximizes the number of unpaired electrons.

Low-spin complexes form when $\Delta > P$. The splitting is large enough that electrons pair up in the lower-energy orbitals before occupying the higher set. This minimizes unpaired electrons.

For octahedral complexes, the high-spin vs. low-spin distinction matters only for $d^4$ through $d^7$ configurations. Outside that range, the electron count gives the same arrangement regardless of $\Delta$.

Magnetic properties follow directly from the number of unpaired electrons:

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

The spin-only magnetic moment provides a quantitative prediction:

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

where $n$ is the number of unpaired electrons and $\mu_B$ is the Bohr magneton. For example, a complex with 4 unpaired electrons has $\mu = \sqrt{4(6)} = \sqrt{24} \approx 4.90 \; \mu_B$.

Spectroscopic Properties

UV-visible spectra of coordination compounds

The colors of coordination compounds come from electronic transitions between split d-orbitals. When white light passes through a solution of a complex, photons whose energy matches $\Delta$ are absorbed, promoting an electron from the lower set to the higher set. The color you see is the complement of the absorbed wavelength. For instance, a complex that absorbs orange light appears blue.

The spectrochemical series ranks ligands by their ability to split d-orbitals, from weak field to strong field:

$I^- < Br^- < Cl^- < F^- < OH^- < H_2O < NH_3 < en < NO_2^- < CN^- < CO$

• Weak field ligands (like $I^-$, $Br^-$) produce a small $\Delta$, so the complex absorbs lower-energy (longer-wavelength) light.
• Strong field ligands (like $CN^-$, $CO$) produce a large $\Delta$, so the complex absorbs higher-energy (shorter-wavelength) light.

This is also why ligand identity determines spin state: strong field ligands tend to create low-spin complexes, while weak field ligands tend to create high-spin complexes.

Selection rules govern which transitions are "allowed" (intense) vs. "forbidden" (weak):

1. Laporte rule: Transitions between orbitals of the same parity are forbidden (g → g or u → u). In octahedral complexes, d-d transitions are formally Laporte-forbidden since all d-orbitals have g symmetry. However, molecular vibrations temporarily break the center of symmetry (vibronic coupling), allowing these transitions to occur with weak to moderate intensity.
2. Spin selection rule: Transitions that require a change in spin multiplicity are forbidden. Spin-allowed transitions (no change in total spin) produce noticeably more intense absorption bands than spin-forbidden ones. This is why $Mn^{2+}$ complexes ($d^5$ high-spin, where all d-d transitions are spin-forbidden) tend to have very pale colors.