2.1 Crystal Field Theory

Crystal Field Theory explains how ligands affect transition metal complexes' properties. It's all about the electrostatic interactions between metal ions and ligands, causing d-orbital splitting. This splitting impacts color, magnetism, and reactivity.

Understanding d-orbital splitting in octahedral and tetrahedral complexes is key. The energy difference between split orbitals determines whether a complex is high-spin or low-spin, affecting its magnetic properties and stability.

Crystal field theory principles

Fundamentals of crystal field theory

• Crystal field theory (CFT) is an electrostatic model describing the bonding and properties of transition metal complexes based on the interaction between the metal's d-orbitals and the ligands' electron pairs
• In CFT, ligands are treated as point negative charges creating an electrostatic field around the central metal ion, causing the splitting of the metal's d-orbital energies
• The splitting of d-orbital energies depends on the geometry of the complex (octahedral or tetrahedral) and the nature of the ligands (strong-field or weak-field)
• CFT helps explain and predict various properties of transition metal complexes (color, magnetic behavior, and reactivity)

Application of crystal field theory to transition metal complexes

• CFT is used to explain the electronic structure, geometry, and properties of transition metal complexes
• The theory helps in understanding the absorption spectra and colors of transition metal complexes based on the d-orbital splitting and electronic transitions
• CFT provides insights into the reactivity of transition metal complexes, such as ligand substitution reactions and redox reactions
• The model is also used to predict the magnetic properties of complexes based on the number of unpaired electrons in the split d-orbitals

d-orbital splitting in complexes

Octahedral complexes

• In octahedral complexes, six ligands approach the metal ion along the x, y, and z axes, causing the d-orbitals to split into two energy levels:
  1. Lower-energy t2g set (dxy, dxz, and dyz)
  2. Higher-energy eg set (dx²-y² and dz²)
• The energy difference between the t2g and eg sets in octahedral complexes is denoted as Δo (delta octahedral) or 10Dq
• The magnitude of Δo depends on the strength of the ligands, with strong-field ligands (CN⁻, CO) causing a larger splitting than weak-field ligands (I⁻, Br⁻)

Tetrahedral complexes

• In tetrahedral complexes, four ligands approach the metal ion along the vertices of a tetrahedron, causing the d-orbitals to split into two energy levels:
  1. Lower-energy e set (dx²-y², dz², and dxy)
  2. Higher-energy t2 set (dxz and dyz)
• The energy difference between the e and t2 sets in tetrahedral complexes is denoted as Δt (delta tetrahedral) and is approximately 4/9 of the Δo value for octahedral complexes with the same ligands
• Tetrahedral complexes generally have smaller d-orbital splitting compared to octahedral complexes due to reduced ligand-metal orbital overlap

Crystal field stabilization energy

Calculation of crystal field stabilization energy

• Crystal field stabilization energy (CFSE) is the energy difference between the electronic configuration of a transition metal ion in a complex and the electronic configuration of the same ion in a spherical field (without ligands)
• CFSE is calculated by summing the energies of the electrons in the split d-orbitals and subtracting the energy of the electrons in the spherical field
• For octahedral complexes, the CFSE is calculated using the formula: $CFSE = [-0.4n(t2g) + 0.6n(eg)] × Δo$, where $n(t2g)$ and $n(eg)$ are the number of electrons in the t2g and eg orbitals, respectively
• For tetrahedral complexes, the CFSE is calculated using the formula: $CFSE = [-0.6n(e) + 0.4n(t2)] × Δt$, where $n(e)$ and $n(t2)$ are the number of electrons in the e and t2 orbitals, respectively

Relationship between CFSE and complex stability

• Complexes with higher CFSE values are generally more stable than those with lower CFSE values
• The CFSE contributes to the overall stability of the complex, along with other factors such as the strength of the metal-ligand bonds and the size of the metal ion
• Complexes with a high CFSE tend to have slower ligand exchange rates and are less likely to undergo substitution reactions compared to complexes with a low CFSE

Geometry and magnetism of complexes

Preferred geometry based on d-orbital splitting

• The preferred geometry of a transition metal complex depends on the d-orbital splitting and the number of d-electrons
• High-spin complexes have a larger number of unpaired electrons and occur when the energy gap between the split d-orbitals (Δo or Δt) is smaller than the spin-pairing energy
• Low-spin complexes have a smaller number of unpaired electrons and occur when the energy gap between the split d-orbitals is larger than the spin-pairing energy
• Octahedral complexes with d⁴ to d⁷ electron configurations can exhibit both high-spin and low-spin states, depending on the strength of the ligands (strong-field or weak-field)

Magnetic properties of transition metal complexes

• The magnetic properties of transition metal complexes can be predicted using the spin-only formula: $μ = √[n(n+2)] μ_B$, where $μ$ is the magnetic moment, $n$ is the number of unpaired electrons, and $μ_B$ is the Bohr magneton
• Complexes with unpaired electrons are paramagnetic and are attracted to an external magnetic field, while complexes with no unpaired electrons are diamagnetic and are weakly repelled by an external magnetic field
• The magnetic moment of a complex can be experimentally determined using techniques such as the Gouy balance or the Faraday balance
• Examples of paramagnetic complexes include high-spin [Fe(H₂O)₆]²⁺ and [Co(NH₃)₆]³⁺, while examples of diamagnetic complexes include low-spin [Fe(CN)₆]⁴⁻ and [Co(NH₃)₆]³⁺