Crystal Field Splitting Diagrams

Why This Matters

Crystal field splitting diagrams are your window into understanding why coordination complexes behave the way they do—their colors, magnetic properties, stability, and reactivity all trace back to how d-orbitals split in different ligand environments. You're being tested on your ability to predict and explain these properties, not just draw diagrams. Exam questions will ask you to connect geometry to splitting patterns, determine spin states from ligand strength, and calculate stabilization energies.

The core concepts here—ligand field strength, orbital degeneracy, electron pairing energy, and geometric distortions—show up repeatedly in FRQs and problem sets. Don't just memorize that octahedral complexes split into $t_{2g}$ and $e_g$ sets; know why certain orbitals go up in energy (they point at ligands) and how that determines everything from color absorption to thermodynamic stability.

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Geometry and Splitting Patterns

The spatial arrangement of ligands determines which d-orbitals experience electrostatic repulsion, creating characteristic energy level patterns for each geometry.

Octahedral Complexes

• Six ligands arranged symmetrically along the x, y, and z axes create the most common coordination geometry in transition metal chemistry
• d-orbitals split into two sets: lower-energy $t_{2g}$ ($d_{xy}$, $d_{xz}$, $d_{yz}$) and higher-energy $e_g$ ($d_{z^2}$, $d_{x^2-y^2}$)—the $e_g$ orbitals point directly at ligands
• Splitting energy $\Delta_o$ determines spin state and is the reference point for comparing other geometries

Tetrahedral Complexes

• Four ligands at tetrahedral vertices create a geometry where no d-orbitals point directly at ligands, reducing overall repulsion
• Inverted splitting pattern: higher-energy $e$ set ($d_{z^2}$, $d_{x^2-y^2}$) and lower-energy $t_2$ set ($d_{xy}$, $d_{xz}$, $d_{yz}$)—opposite of octahedral
• $\Delta_t \approx \frac{4}{9}\Delta_o$ means tetrahedral complexes are almost always high-spin due to small splitting

Square Planar Complexes

• Four ligands in the xy-plane with no axial ligands, creating extreme splitting in the $d_{x^2-y^2}$ orbital
• Energy ordering from lowest to highest: $d_{xz}/d_{yz} < d_{z^2} < d_{xy} < d_{x^2-y^2}$—the $d_{x^2-y^2}$ orbital is strongly destabilized
• Favored by $d^8$ metals (Ni²⁺, Pd²⁺, Pt²⁺, Au³⁺) where the large splitting energy compensates for losing two ligands from octahedral

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Ligand Effects on Splitting

The identity of the ligand—specifically its ability to donate or accept electron density—determines the magnitude of d-orbital splitting.

Spectrochemical Series

• Ranks ligands by field strength: $I^- < Br^- < Cl^- < F^- < OH^- < H_2O < NH_3 < en < NO_2^- < CN^- < CO$
• Strong-field ligands (CN⁻, CO) are typically π-acceptors that stabilize metal electrons through back-bonding
• Weak-field ligands (halides, H₂O) are σ-donors or π-donors that don't effectively increase splitting

Strong Field vs. Weak Field Ligands

• Strong-field ligands create large $\Delta$ values, making electron pairing energetically favorable over occupying higher orbitals
• Weak-field ligands create small $\Delta$ values where the pairing energy $P$ exceeds $\Delta$, favoring unpaired electrons
• The crossover point occurs when $\Delta \approx P$—knowing typical values helps predict spin states for borderline cases

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Spin States and Electron Configuration

The competition between splitting energy ($\Delta$) and pairing energy ($P$) determines whether electrons spread out (high-spin) or pair up (low-spin).

High-Spin vs. Low-Spin Configurations

• High-spin occurs when $\Delta < P$—electrons occupy higher-energy orbitals before pairing, maximizing unpaired electrons
• Low-spin occurs when $\Delta > P$—electrons pair in lower-energy orbitals first, minimizing unpaired electrons
• Only $d^4$–$d^7$ configurations can exhibit both spin states; $d^1$–$d^3$ and $d^8$–$d^{10}$ have only one possible arrangement

Crystal Field Stabilization Energy (CFSE)

• CFSE quantifies stability as the energy difference between the split configuration and a hypothetical spherical field
• Calculated using: $CFSE = (-0.4\Delta \times n_{t_{2g}}) + (0.6\Delta \times n_{e_g}) + P \times (\text{extra pairs})$ for octahedral complexes
• Maximum CFSE occurs at $d^3$ and low-spin $d^6$ configurations, explaining the exceptional stability of Cr³⁺ and Co³⁺ complexes

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Distortions and Advanced Theory

Real complexes deviate from ideal geometries when electronic configurations create orbital degeneracy that can be relieved through structural changes.

Jahn-Teller Distortions

• Degenerate electronic states are unstable—complexes distort to remove degeneracy and lower total energy (Jahn-Teller theorem)
• Most pronounced in $d^4$ high-spin and $d^9$ octahedral complexes where $e_g$ orbitals are unevenly occupied
• Typically elongation along z-axis stabilizes $d_{z^2}$ relative to $d_{x^2-y^2}$, observable in Cu²⁺ complexes with two long axial bonds

Ligand Field Theory

• Extends crystal field theory by incorporating covalent character through metal-ligand orbital overlap
• Explains π-bonding effects: π-acceptor ligands (CO, CN⁻) increase $\Delta$ through back-donation; π-donors (Cl⁻, OH⁻) decrease $\Delta$
• Molecular orbital diagrams from LFT provide more accurate predictions for spectroscopic and magnetic properties

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Quick Reference Table

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Self-Check Questions

1. Why does $\Delta_t$ equal approximately $\frac{4}{9}\Delta_o$, and how does this explain why tetrahedral complexes are almost never low-spin?

2. For a $d^5$ metal ion, draw the electron configurations for both high-spin and low-spin octahedral complexes. Which ligands from the spectrochemical series would produce each?

3. Compare [Fe(H₂O)₆]²⁺ and [Fe(CN)₆]⁴⁻: both contain Fe²⁺, but they have different numbers of unpaired electrons. Explain this difference and calculate the CFSE for each.

4. Why do Cu²⁺ complexes commonly show Jahn-Teller distortion while Zn²⁺ complexes do not? What would you predict for Ni²⁺ in an octahedral environment?

5. An FRQ asks you to explain why Pt²⁺ forms square planar complexes while Ni²⁺ can form either tetrahedral or square planar. What factors determine the preferred geometry for $d^8$ ions?