9.1 Crystal Field Theory

Crystal Field Theory explains how ligands affect the d-orbitals of a metal ion in coordination compounds. Understanding this theory is essential because it connects electronic structure to observable properties like color, magnetism, and reactivity. It also lays the groundwork for more sophisticated bonding models you'll encounter later.

The central idea: when ligands approach a metal ion, they break the degeneracy of the five d-orbitals, splitting them into groups of different energy. The size of that splitting, and how electrons fill the resulting levels, determines whether a complex is high-spin or low-spin, paramagnetic or diamagnetic, and what color it appears.

Ligand Field Theory and d-Orbital Splitting

d-Orbital Splitting in Different Geometries

In a free metal ion, all five d-orbitals have the same energy (they're degenerate). Once ligands approach along defined axes, the orbitals pointing toward ligands get pushed to higher energy by electrostatic repulsion, while those pointing between ligands stay lower. The pattern of that splitting depends on the geometry.

Octahedral complexes (six ligands along $\pm x$, $\pm y$, $\pm z$):

• The d-orbitals split into a lower set, $t_{2g}$ ($d_{xy}$, $d_{xz}$, $d_{yz}$), and a higher set, $e_g$ ($d_{x^2-y^2}$, $d_{z^2}$).
• The energy gap between these sets is called $\Delta_o$ (the octahedral splitting parameter).
• The $e_g$ orbitals point directly at the ligands, so they experience more repulsion and rise in energy. The $t_{2g}$ orbitals point between ligands and are stabilized relative to the average.

Tetrahedral complexes (four ligands between the axes):

• The splitting pattern inverts: $e$ ($d_{x^2-y^2}$, $d_{z^2}$) is now the lower set, and $t_2$ ($d_{xy}$, $d_{xz}$, $d_{yz}$) is higher.
• The tetrahedral splitting energy $\Delta_t$ is smaller than $\Delta_o$ for the same metal-ligand pair. Quantitatively, $\Delta_t \approx \frac{4}{9}\Delta_o$. Because $\Delta_t$ is small, tetrahedral complexes are almost always high-spin.

Square planar complexes (four ligands in the $xy$-plane):

• You can think of this as an octahedral complex where the two axial ligands have been removed entirely. Removing them stabilizes every orbital with a $z$-component.
• This produces four distinct energy levels (lowest to highest): $d_{xz} \approx d_{yz} < d_{z^2} < d_{xy} < d_{x^2-y^2}$.
• Square planar geometry is especially common for $d^8$ metal ions (e.g., $\text{Pt}^{2+}$, $\text{Ni}^{2+}$ with strong-field ligands, $\text{Pd}^{2+}$, $\text{Au}^{3+}$) because the large splitting makes it energetically favorable to leave the highest orbital empty.

Ligand Field Strength and the Spectrochemical Series

Not all ligands split d-orbitals by the same amount. The spectrochemical series ranks ligands from weakest to strongest field:

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

A few patterns to notice:

• Halides and other $\pi$-donors tend to be weak-field. They donate electron density into the metal's $t_{2g}$ orbitals through $\pi$-bonding, which reduces the effective splitting.
• $\pi$-acceptor ligands like $CO$ and $CN^-$ are strong-field. They withdraw electron density from the metal's $t_{2g}$ orbitals into their own empty $\pi^*$ orbitals, stabilizing $t_{2g}$ and increasing $\Delta$.
• $\sigma$-only donors like $NH_3$ and $H_2O$ fall in the middle. They neither donate into nor accept from the $t_{2g}$ set, so splitting is moderate.

Beyond the ligand itself, two other factors affect the magnitude of $\Delta$:

• Oxidation state of the metal: Higher oxidation states produce larger splitting. A $\text{Co}^{3+}$ complex has a larger $\Delta_o$ than an analogous $\text{Co}^{2+}$ complex because the higher charge draws ligands closer and increases repulsion with d-electrons.
• Period of the metal: Splitting increases going down a group (3d < 4d < 5d). The d-orbitals become more diffuse, overlapping more with ligand orbitals. This is why second- and third-row transition metal complexes are almost always low-spin.

Spin States and Crystal Field Stabilization Energy

High-Spin and Low-Spin Complexes

Once d-orbitals split, electrons must fill them according to two competing energy costs:

1. $\Delta$ (the splitting energy): the cost of placing an electron in the higher set of orbitals.
2. $P$ (the pairing energy): the cost of putting a second electron into an orbital that's already occupied (electron-electron repulsion + loss of exchange energy).

The spin state depends on which cost is larger:

• High-spin ($P > \Delta$): Electrons spread across all d-orbitals before any pairing occurs, maximizing unpaired electrons. This happens with weak-field ligands that produce a small $\Delta$.
• Low-spin ($\Delta > P$): Electrons fill the lower-energy set completely before occupying the upper set, minimizing unpaired electrons. This happens with strong-field ligands that produce a large $\Delta$.

The high-spin vs. low-spin distinction matters most for octahedral $d^4$ through $d^7$ configurations. For $d^1$ through $d^3$ and $d^8$ through $d^{10}$, there's only one possible filling order regardless of $\Delta$.

For tetrahedral complexes, $\Delta_t$ is so small that the pairing energy almost always wins, making them nearly always high-spin.

Crystal Field Stabilization Energy and Magnetic Properties

Crystal field stabilization energy (CFSE) quantifies how much stability a complex gains from the d-orbital splitting compared to a hypothetical spherical field.

To calculate CFSE for an octahedral complex:

1. Place the electrons into $t_{2g}$ and $e_g$ orbitals according to the spin state (high or low).
2. Each electron in $t_{2g}$ contributes $-0.4\Delta_o$ (stabilization).
3. Each electron in $e_g$ contributes $+0.6\Delta_o$ (destabilization).
4. Sum the contributions. If electrons are paired in the low-spin configuration, add the pairing energy $P$ for each forced pair.

$\text{CFSE} = (-0.4\Delta_o)(n_{t_{2g}}) + (+0.6\Delta_o)(n_{e_g})$

For example, a high-spin $d^5$ octahedral complex (like $[\text{Mn}(\text{H}_2\text{O})_6]^{2+}$) has three electrons in $t_{2g}$ and two in $e_g$:

$\text{CFSE} = 3(-0.4\Delta_o) + 2(+0.6\Delta_o) = -1.2\Delta_o + 1.2\Delta_o = 0$

That zero CFSE for high-spin $d^5$ is worth remembering. It explains why $\text{Mn}^{2+}$ complexes tend to be particularly labile and weakly colored.

Magnetic properties follow directly from the electron configuration:

• Paramagnetic: has unpaired electrons, attracted into a magnetic field. The magnetic moment can be estimated using the spin-only formula: $\mu = \sqrt{n(n+2)}$ BM, where $n$ is the number of unpaired electrons.
• Diamagnetic: all electrons paired, weakly repelled by a magnetic field.

Measuring the magnetic moment experimentally is one of the most direct ways to determine whether a complex is high-spin or low-spin.

CFSE also has broader chemical consequences:

• Thermodynamic stability: Complexes with larger CFSE tend to be more stable. The "double-humped" trend in hydration enthalpies across the first-row transition metals reflects CFSE contributions.
• Kinetic lability: Octahedral $d^3$ and low-spin $d^6$ complexes (like $\text{Cr}^{3+}$ and low-spin $\text{Co}^{3+}$) have maximum CFSE and tend to be kinetically inert.
• Redox potentials: CFSE differences between oxidation states influence how easily a metal center is oxidized or reduced.

Consequences of Crystal Field Theory

Jahn-Teller Distortion and Structural Effects

The Jahn-Teller theorem states that any non-linear molecule with a degenerate electronic ground state will distort to remove that degeneracy and lower its energy. In coordination chemistry, this matters most for octahedral complexes with unequal occupation of the $e_g$ orbitals.

The classic case is $\text{Cu}^{2+}$ ($d^9$): the $e_g$ set has three electrons (two in one orbital, one in the other). This uneven occupation is degenerate, so the complex distorts. The most common distortion is tetragonal elongation, where the two axial bonds stretch while the four equatorial bonds contract slightly.

Configurations that show strong Jahn-Teller effects:

• $d^9$ (e.g., $\text{Cu}^{2+}$): one electron unevenly in $e_g$
• High-spin $d^4$ (e.g., $\text{Cr}^{2+}$, $\text{Mn}^{3+}$): one electron in $e_g$
• Low-spin $d^7$: unevenly filled $e_g$

Unequal filling of the $t_{2g}$ set can also cause Jahn-Teller distortion, but the effect is much weaker because $t_{2g}$ orbitals don't point directly at ligands.

Structural consequences of Jahn-Teller distortion include:

• Two distinct bond lengths in what would otherwise be a regular octahedron
• Lowered symmetry (from $O_h$ to $D_{4h}$)
• Altered spectroscopic and reactivity patterns (e.g., broadened or split absorption bands)

Color and Spectroscopic Properties of Coordination Compounds

The colors of coordination compounds arise from d-d transitions: absorption of visible light promotes an electron from a lower-energy d-orbital to a higher-energy one. The energy of the absorbed photon corresponds to $\Delta$, and the color you see is the complementary color of what's absorbed.

For example, $[\text{Ti}(\text{H}_2\text{O})_6]^{3+}$ ($d^1$) absorbs in the yellow-green region (~500 nm), so it appears purple.

Factors that influence the color:

• Ligand identity: Changing ligands changes $\Delta$, shifting the absorption wavelength. Replacing $\text{H}_2\text{O}$ with $\text{NH}_3$ around $\text{Co}^{3+}$ shifts the absorption because $\text{NH}_3$ is a stronger-field ligand.
• Metal ion and oxidation state: Different metals and charges give different $\Delta$ values.
• Geometry: Tetrahedral complexes have smaller $\Delta_t$, so their absorptions shift to lower energy (longer wavelength) compared to octahedral analogs.

UV-visible spectroscopy is the primary tool for studying these transitions. The absorption maximum gives $\Delta$ directly (in wavenumbers or energy units), and the molar absorptivity gives information about the "allowedness" of the transition.

A note on selection rules: d-d transitions are formally forbidden by the Laporte rule (no change in parity for centrosymmetric molecules) and sometimes by the spin selection rule ($\Delta S = 0$). They occur anyway because of vibronic coupling (molecular vibrations temporarily break the center of symmetry) and spin-orbit coupling. This is why d-d transitions tend to have low molar absorptivities, and why transition metal complex colors are often pale compared to organic dyes.