Chapter 2: Crystal Field Theory and Electronic Structure

Why do transition metal compounds display such diverse colors? Learn crystal field theory and understand the relationship between d-orbital splitting and electronic states. From the Jahn-Teller effect and ligand field theory to real material applications, explore energy levels through Python calculations.

2.1 Fundamentals of Crystal Field Theory

What is Crystal Field Theory?

Crystal Field Theory (CFT) is a theory that explains how the energy levels of d-orbitals split when transition metal ions are surrounded by ligands. It was proposed by Hans Bethe and John Van Vleck in 1929.

Basic Concept:

1. In an isolated transition metal ion, the five d-orbitals are degenerate (same energy)
2. When ligands approach, electrostatic repulsion increases the energy of d-orbitals
3. The arrangement (symmetry) of ligands causes d-orbitals to split into different energy levels

Spatial Distribution of d-Orbitals

There are five types of d-orbitals, each with different spatial distributions:

2.2 Crystal Field Splitting in Octahedral Coordination

Symmetry of Octahedral Coordination

In octahedral coordination (O_h symmetry) where a transition metal ion is surrounded by six ligands, d-orbitals split as follows:

Crystal Field Splitting Energy (Δ_oct or 10Dq):

Energy Stabilization:

• e_g orbitals: + 0.6 Δ_oct (destabilized)
• t_2g orbitals: - 0.4 Δ_oct (stabilized)
• Energy barycenter is conserved

Visualizing Octahedral Coordination Energy Diagram

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def plot_octahedral_splitting(delta_oct=10):
    """
    Visualize d-orbital crystal field splitting in octahedral coordination

    Parameters:
    -----------
    delta_oct : float
        Crystal field splitting energy (in Dq units)
    """
    fig, ax = plt.subplots(figsize=(10, 6))

    # Energy levels
    E_barycenter = 0  # Energy barycenter
    E_eg = E_barycenter + 0.6 * delta_oct
    E_t2g = E_barycenter - 0.4 * delta_oct

    # Isolated ion (left)
    ax.hlines(0, 0, 1.5, colors='gray', linewidth=2, label='Isolated ion (5d degenerate)')
    ax.text(0.75, 0.5, '5d', ha='center', fontsize=12, fontweight='bold')

    # Spherical field (center)
    ax.hlines(E_barycenter, 2.5, 4, colors='blue', linewidth=2, label='Spherical field')
    ax.text(3.25, E_barycenter+0.5, '5d', ha='center', fontsize=12, fontweight='bold')

    # Octahedral field (right)
    ax.hlines(E_eg, 5, 7, colors='red', linewidth=3, label='e$_g$ (2 orbitals)')
    ax.text(6, E_eg+0.5, '$d_{z^2}$, $d_{x^2-y^2}$', ha='center', fontsize=11)

    ax.hlines(E_t2g, 5, 7, colors='green', linewidth=3, label='t$_{2g}$ (3 orbitals)')
    ax.text(6, E_t2g-0.8, '$d_{xy}$, $d_{xz}$, $d_{yz}$', ha='center', fontsize=11)

    # Arrow showing Δ_oct
    ax.annotate('', xy=(7.5, E_eg), xytext=(7.5, E_t2g),
                arrowprops=dict(arrowstyle='<->', color='purple', lw=2))
    ax.text(8, (E_eg + E_t2g)/2, f'$\\Delta_{{oct}}$ = {delta_oct} Dq',
            fontsize=13, fontweight='bold', color='purple')

    # Axis settings
    ax.set_xlim(-0.5, 9)
    ax.set_ylim(-5, 8)
    ax.set_ylabel('Energy (Dq)', fontsize=12)
    ax.set_xticks([0.75, 3.25, 6])
    ax.set_xticklabels(['Isolated Ion', 'Spherical Field', 'Octahedral Field'], fontsize=11)
    ax.grid(axis='y', alpha=0.3)
    ax.set_title('Crystal Field Splitting in Octahedral Coordination', fontsize=14, fontweight='bold')

    plt.tight_layout()
    plt.show()

# Execute
plot_octahedral_splitting(delta_oct=10)

2.3 Crystal Field Splitting in Tetrahedral Coordination

Characteristics of Tetrahedral Coordination

In tetrahedral coordination (T_d symmetry), with four ligands, the splitting pattern is reversed from octahedral:

• e orbitals ($d_{z^2}, d_{x^2-y^2}$): Lower energy
• t₂ orbitals ($d_{xy}, d_{xz}, d_{yz}$): Higher energy
• Splitting magnitude: $\Delta_{\text{tet}} \approx \frac{4}{9} \Delta_{\text{oct}}$ (approximately 44%)

Comparison of Octahedral and Tetrahedral

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import matplotlib.pyplot as plt
import numpy as np

def compare_octahedral_tetrahedral():
    """Compare crystal field splitting in octahedral and tetrahedral coordination"""
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 6))

    # Octahedral coordination
    delta_oct = 10
    E_eg_oct = 0.6 * delta_oct
    E_t2g_oct = -0.4 * delta_oct

    ax1.hlines(E_eg_oct, 0, 1, colors='red', linewidth=4, label='e$_g$')
    ax1.text(0.5, E_eg_oct + 0.5, 'e$_g$ (2)', ha='center', fontsize=12, fontweight='bold')

    ax1.hlines(E_t2g_oct, 0, 1, colors='green', linewidth=4, label='t$_{2g}$')
    ax1.text(0.5, E_t2g_oct - 0.7, 't$_{2g}$ (3)', ha='center', fontsize=12, fontweight='bold')

    ax1.annotate('', xy=(1.3, E_eg_oct), xytext=(1.3, E_t2g_oct),
                arrowprops=dict(arrowstyle='<->', color='purple', lw=2))
    ax1.text(1.6, (E_eg_oct + E_t2g_oct)/2, '$\\Delta_{oct}$', fontsize=13, fontweight='bold')

    ax1.set_xlim(-0.2, 2)
    ax1.set_ylim(-6, 8)
    ax1.set_ylabel('Energy (Dq)', fontsize=12)
    ax1.set_title('Octahedral Coordination (O$_h$)', fontsize=14, fontweight='bold')
    ax1.grid(axis='y', alpha=0.3)
    ax1.set_xticks([])

    # Tetrahedral coordination
    delta_tet = (4/9) * delta_oct  # Approximately 4.44 Dq
    E_t2_tet = 0.6 * delta_tet
    E_e_tet = -0.4 * delta_tet

    ax2.hlines(E_t2_tet, 0, 1, colors='green', linewidth=4, label='t$_2$')
    ax2.text(0.5, E_t2_tet + 0.5, 't$_2$ (3)', ha='center', fontsize=12, fontweight='bold')

    ax2.hlines(E_e_tet, 0, 1, colors='red', linewidth=4, label='e')
    ax2.text(0.5, E_e_tet - 0.7, 'e (2)', ha='center', fontsize=12, fontweight='bold')

    ax2.annotate('', xy=(1.3, E_t2_tet), xytext=(1.3, E_e_tet),
                arrowprops=dict(arrowstyle='<->', color='orange', lw=2))
    ax2.text(1.7, (E_t2_tet + E_e_tet)/2, f'$\\Delta_{{tet}}$\n≈ {delta_tet:.1f} Dq',
            fontsize=12, fontweight='bold')

    ax2.set_xlim(-0.2, 2.2)
    ax2.set_ylim(-6, 8)
    ax2.set_ylabel('Energy (Dq)', fontsize=12)
    ax2.set_title('Tetrahedral Coordination (T$_d$)', fontsize=14, fontweight='bold')
    ax2.grid(axis='y', alpha=0.3)
    ax2.set_xticks([])

    plt.tight_layout()
    plt.show()

compare_octahedral_tetrahedral()

2.4 Jahn-Teller Effect

Jahn-Teller Theorem

Conditions for occurrence:

• e_g or t_2g orbitals are partially occupied
• Typical examples: Cu²⁺ (d⁹), Mn³⁺ (d⁴ high spin), Cr²⁺ (d⁴ high spin)

Jahn-Teller Distortion in Cu²⁺

Cu²⁺ has d⁹ electron configuration with 3 electrons in e_g orbitals ($d_{x^2-y^2}$: 2, $d_{z^2}$: 1).

Simulation of Jahn-Teller Effect

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def jahn_teller_distortion_energy():
    """Calculate energy change due to Jahn-Teller distortion"""
    # Distortion parameter (percentage change in bond length)
    distortion = np.linspace(-0.15, 0.15, 100)  # -15% to +15%

    # Cu2+ case: d9 configuration, 3 electrons in eg orbitals
    # eg degenerate energy in ideal octahedron
    E_ideal = 0

    # Energy after distortion (simplified model)
    # For axial elongation (positive distortion)
    E_dz2 = E_ideal - 1000 * distortion**2  # d_z2 orbital stabilized
    E_dx2y2 = E_ideal + 800 * distortion**2  # d_x2-y2 orbital destabilized

    # Electronic configuration: 2 electrons in d_z2, 1 electron in d_x2-y2
    E_total = 2 * E_dz2 + 1 * E_dx2y2

    # Elastic energy (cost of structural distortion)
    E_elastic = 500 * distortion**2

    # Total energy
    E_net = E_total + E_elastic

    # Plot
    fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))

    # Orbital energies
    ax1.plot(distortion * 100, E_dz2, 'b-', linewidth=2, label='$d_{z^2}$ (2 electrons)')
    ax1.plot(distortion * 100, E_dx2y2, 'r-', linewidth=2, label='$d_{x^2-y^2}$ (1 electron)')
    ax1.axhline(0, color='gray', linestyle='--', alpha=0.5)
    ax1.axvline(0, color='gray', linestyle='--', alpha=0.5)
    ax1.set_xlabel('Axial Distortion (%)', fontsize=12)
    ax1.set_ylabel('Orbital Energy (cm$^{-1}$)', fontsize=12)
    ax1.set_title('e$_g$ Orbital Splitting', fontsize=14, fontweight='bold')
    ax1.legend(fontsize=11)
    ax1.grid(alpha=0.3)

    # Total energy
    ax2.plot(distortion * 100, E_total, 'g-', linewidth=2, label='Electronic Energy')
    ax2.plot(distortion * 100, E_elastic, 'orange', linewidth=2, label='Elastic Energy')
    ax2.plot(distortion * 100, E_net, 'purple', linewidth=3, label='Total Energy')

    # Most stable structure
    min_idx = np.argmin(E_net)
    min_distortion = distortion[min_idx]
    ax2.plot(min_distortion * 100, E_net[min_idx], 'ro', markersize=10, label='Most Stable Structure')

    ax2.axhline(0, color='gray', linestyle='--', alpha=0.5)
    ax2.axvline(0, color='gray', linestyle='--', alpha=0.5)
    ax2.set_xlabel('Axial Distortion (%)', fontsize=12)
    ax2.set_ylabel('Energy (cm$^{-1}$)', fontsize=12)
    ax2.set_title('Jahn-Teller Stabilization Energy', fontsize=14, fontweight='bold')
    ax2.legend(fontsize=10)
    ax2.grid(alpha=0.3)

    plt.tight_layout()
    plt.show()

    print(f"Most stable distortion: {min_distortion*100:.2f}%")
    print(f"Stabilization energy: {-E_net[min_idx]:.1f} cm^-1")

jahn_teller_distortion_energy()

2.5 Ligand Field Theory

Limitations of Crystal Field Theory and Ligand Field Theory

Crystal field theory treats ligands as simple point charges, but in reality there is covalent bonding between ligands and metal ions.

Spectrochemical Series

The arrangement of ligands by their crystal field splitting strength is called the spectrochemical series:

I^- < Br^- < Cl^- < F^- < OH^- < H₂O < NH₃ < en < CN^- < CO

← Weak ligand field ────── Strong ligand field →

• Weak ligands (I^-, Br^-, Cl^-): Small Δ → High spin state
• Strong ligands (CN^-, CO): Large Δ → Low spin state

Calculating High Spin and Low Spin States

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import matplotlib.pyplot as plt
import numpy as np

def calculate_spin_states(d_electrons, delta_oct):
    """
    Calculate high spin/low spin from d-electron count and crystal field splitting

    Parameters:
    -----------
    d_electrons : int
        Number of d-electrons (1-10)
    delta_oct : float
        Crystal field splitting energy (unit: cm^-1)

    Returns:
    --------
    dict : Information about spin states
    """
    # Pairing energy (energy required for electron pairing)
    P = 15000  # Typical value: approximately 15,000 cm^-1

    # High spin configuration (Hund's rule priority)
    if d_electrons <= 3:
        high_spin = d_electrons  # Sequential placement in t2g orbitals
        hs_t2g = d_electrons
        hs_eg = 0
    elif d_electrons <= 5:
        high_spin = d_electrons  # Fill t2g then eg orbitals
        hs_t2g = 3
        hs_eg = d_electrons - 3
    elif d_electrons <= 8:
        high_spin = d_electrons - 5  # Fill t2g, eg then start pairing
        hs_t2g = min(6, d_electrons)
        hs_eg = max(0, d_electrons - 6)
    else:
        high_spin = 10 - d_electrons
        hs_t2g = 6
        hs_eg = d_electrons - 6

    # Low spin configuration (when Δ_oct > P)
    if d_electrons <= 6:
        low_spin = d_electrons % 2  # Fill t2g orbitals first
        ls_t2g = min(6, d_electrons)
        ls_eg = max(0, d_electrons - 6)
    else:
        low_spin = (d_electrons - 6) % 2
        ls_t2g = 6
        ls_eg = d_electrons - 6

    # Crystal Field Stabilization Energy (CFSE)
    cfse_high = (-0.4 * hs_t2g + 0.6 * hs_eg) * delta_oct
    cfse_low = (-0.4 * ls_t2g + 0.6 * ls_eg) * delta_oct

    # Total energy considering pairing energy
    n_pairs_high = (d_electrons - high_spin) / 2
    n_pairs_low = (d_electrons - low_spin) / 2

    E_high = cfse_high + n_pairs_high * P
    E_low = cfse_low + n_pairs_low * P

    return {
        'high_spin': high_spin,
        'low_spin': low_spin,
        'E_high': E_high,
        'E_low': E_low,
        'stable_state': 'Low Spin' if E_low < E_high else 'High Spin'
    }

# Example: Fe2+ (d6)
d_electrons = 6
delta_values = np.linspace(5000, 25000, 100)

high_spin_list = []
low_spin_list = []

for delta in delta_values:
    result = calculate_spin_states(d_electrons, delta)
    high_spin_list.append(result['E_high'])
    low_spin_list.append(result['E_low'])

# Plot
plt.figure(figsize=(10, 6))
plt.plot(delta_values, high_spin_list, 'r-', linewidth=2, label='High Spin (S=2)')
plt.plot(delta_values, low_spin_list, 'b-', linewidth=2, label='Low Spin (S=0)')

# Crossover point (spin transition point)
idx_cross = np.argmin(np.abs(np.array(high_spin_list) - np.array(low_spin_list)))
delta_cross = delta_values[idx_cross]
plt.axvline(delta_cross, color='green', linestyle='--', linewidth=2, label=f'Spin Crossover: {delta_cross:.0f} cm$^{{-1}}$')

plt.xlabel('Crystal Field Splitting Δ$_{oct}$ (cm$^{-1}$)', fontsize=12)
plt.ylabel('Total Energy (cm$^{-1}$)', fontsize=12)
plt.title(f'High Spin and Low Spin States of Fe$^{{2+}}$ (d$^{6}$)', fontsize=14, fontweight='bold')
plt.legend(fontsize=11)
plt.grid(alpha=0.3)
plt.tight_layout()
plt.show()

print(f"Spin transition point: Δ_oct ≈ {delta_cross:.0f} cm^-1")

2.6 Tanabe-Sugano Diagrams

What are Tanabe-Sugano Diagrams?

Tanabe-Sugano diagrams show electronic states of transition metal ions with dⁿ electron configuration as a function of crystal field splitting strength (Δ/B).

Simplified Tanabe-Sugano Diagram for d³ Electron Configuration

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import numpy as np
import matplotlib.pyplot as plt

def plot_tanabe_sugano_d3():
    """Tanabe-Sugano diagram for d3 electron configuration (simplified version)"""
    # Range of Δ/B
    delta_over_B = np.linspace(0, 3.5, 100)

    # Simplified electronic state energies (actual is more complex)
    # 4F (ground state)
    E_4F = 0 * delta_over_B

    # 4P
    E_4P = 15 + 0 * delta_over_B

    # 2G
    E_2G = 17 + 2 * delta_over_B

    # 2H
    E_2H = 22 + 1.5 * delta_over_B

    # 2D
    E_2D = 28 + 0.8 * delta_over_B

    # Plot
    fig, ax = plt.subplots(figsize=(10, 7))

    ax.plot(delta_over_B, E_4F, 'b-', linewidth=3, label='$^4$F (ground state)')
    ax.plot(delta_over_B, E_4P, 'r-', linewidth=2, label='$^4$P')
    ax.plot(delta_over_B, E_2G, 'g-', linewidth=2, label='$^2$G')
    ax.plot(delta_over_B, E_2H, 'orange', linewidth=2, label='$^2$H')
    ax.plot(delta_over_B, E_2D, 'purple', linewidth=2, label='$^2$D')

    # Example absorption transition (Cr3+)
    delta_B_example = 2.3  # Typical value for Cr3+
    ax.axvline(delta_B_example, color='gray', linestyle='--', alpha=0.5)
    ax.text(delta_B_example + 0.1, 35, 'Cr$^{3+}$\n(ruby)', fontsize=10, color='red')

    ax.set_xlabel('Δ / B', fontsize=13, fontweight='bold')
    ax.set_ylabel('E / B', fontsize=13, fontweight='bold')
    ax.set_title('Tanabe-Sugano Diagram (d$^3$ Electron Configuration)', fontsize=15, fontweight='bold')
    ax.set_xlim(0, 3.5)
    ax.set_ylim(0, 40)
    ax.legend(fontsize=11, loc='upper left')
    ax.grid(alpha=0.3)

    plt.tight_layout()
    plt.show()

plot_tanabe_sugano_d3()

2.7 Applications of Transition Metal Compounds

Applications in Catalysis

Transition metal compounds are widely used as catalysts for redox reactions and ligand exchange reactions due to their partially occupied d-orbitals.

Applications in Magnetic Materials

Unpaired electrons in transition metal compounds are the source of magnetism.

• Ferromagnets: Fe, Co, Ni (metals), CrO₂ (magnetic tape)
• Antiferromagnets: MnO, NiO, FeO
• Ferrimagnets: Fe₃O₄ (magnetite), γ-Fe₂O₃ (maghemite)

Applications in Optical Materials

Materials utilizing light absorption by crystal field:

• Gemstones: Ruby (Cr³⁺), Emerald (Cr³⁺), Sapphire (Fe²⁺/Ti⁴⁺)
• Pigments: Prussian blue (Fe²⁺/Fe³⁺), chromium oxide green (Cr³⁺)
• Solar cells: Dye-sensitized solar cells (Ru complexes)

Visualizing the Relationship Between Crystal Field Splitting and Color

# Requirements:
# - Python 3.9+
# - matplotlib>=3.7.0
# - numpy>=1.24.0, <2.0.0

import matplotlib.pyplot as plt
import numpy as np
from matplotlib.patches import FancyBboxPatch

def crystal_field_color_chart():
    """Relationship between crystal field splitting and color for transition metal ions"""

    ions = ['Ti³⁺', 'V³⁺', 'Cr³⁺', 'Mn²⁺', 'Fe³⁺', 'Co²⁺', 'Ni²⁺', 'Cu²⁺']
    d_electrons = [1, 2, 3, 5, 5, 7, 8, 9]
    delta_oct = [20300, 18900, 17400, 21000, 14000, 9300, 8500, 12600]  # cm^-1
    colors = ['purple', 'green', 'red', 'pale pink', 'yellow', 'pink', 'green', 'blue']
    hex_colors = ['#9b59b6', '#27ae60', '#e74c3c', '#f8b3c7', '#f1c40f', '#ff69b4', '#2ecc71', '#3498db']

    fig, ax = plt.subplots(figsize=(12, 7))

    x_pos = np.arange(len(ions))
    bars = ax.bar(x_pos, delta_oct, color=hex_colors, edgecolor='black', linewidth=1.5, alpha=0.8)

    # Data labels
    for i, (bar, ion, d, col) in enumerate(zip(bars, ions, d_electrons, colors)):
        height = bar.get_height()
        ax.text(bar.get_x() + bar.get_width()/2., height + 500,
                f'{int(height)} cm⁻¹', ha='center', va='bottom', fontsize=9, fontweight='bold')
        ax.text(bar.get_x() + bar.get_width()/2., -2000,
                f'd{d}', ha='center', va='top', fontsize=10, color='gray')
        ax.text(bar.get_x() + bar.get_width()/2., height/2,
                col, ha='center', va='center', fontsize=9, fontweight='bold',
                color='white', rotation=90)

    ax.set_xticks(x_pos)
    ax.set_xticklabels(ions, fontsize=12, fontweight='bold')
    ax.set_ylabel('Crystal Field Splitting Δ$_{oct}$ (cm$^{-1}$)', fontsize=13)
    ax.set_title('Crystal Field Splitting and Observed Colors for Transition Metal Ions in Octahedral Coordination', fontsize=14, fontweight='bold')
    ax.set_ylim(-3000, 24000)
    ax.grid(axis='y', alpha=0.3)

    # Reference line for spectrochemical series
    ax.axhline(15000, color='red', linestyle='--', linewidth=1.5, alpha=0.5, label='Typical ligands (H₂O, NH₃)')

    ax.legend(fontsize=10)
    plt.tight_layout()
    plt.show()

crystal_field_color_chart()

2.8 Crystal Field Calculations Using First-Principles Methods

Calculating Crystal Field Parameters with DFT

In modern materials science, crystal field splitting energy can be directly calculated using first-principles calculations (DFT).

Generating Transition Metal Compound Structures with ASE

# Requirements:
# - Python 3.9+
# - numpy>=1.24.0, <2.0.0

from ase import Atoms
from ase.visualize import view
import numpy as np

def create_octahedral_complex(metal='Ti', ligand='O', bond_length=2.0):
    """
    Generate octahedral coordination transition metal complex

    Parameters:
    -----------
    metal : str
        Central metal ion
    ligand : str
        Ligand atom
    bond_length : float
        Metal-ligand bond length (Å)

    Returns:
    --------
    atoms : ase.Atoms
        Generated structure
    """
    # Ligand positions in octahedral coordination (±x, ±y, ±z axes)
    ligand_positions = np.array([
        [bond_length, 0, 0],
        [-bond_length, 0, 0],
        [0, bond_length, 0],
        [0, -bond_length, 0],
        [0, 0, bond_length],
        [0, 0, -bond_length]
    ])

    # Create atomic object
    symbols = [metal] + [ligand] * 6
    positions = np.vstack([[0, 0, 0], ligand_positions])

    atoms = Atoms(symbols=symbols, positions=positions)

    # Set cell size (for visualization)
    cell_size = bond_length * 3
    atoms.set_cell([cell_size, cell_size, cell_size])
    atoms.center()

    return atoms

# Generate TiO6 octahedron
complex_TiO6 = create_octahedral_complex(metal='Ti', ligand='O', bond_length=2.0)

print(f"Generated structure: {complex_TiO6.get_chemical_formula()}")
print(f"Atomic positions:\n{complex_TiO6.get_positions()}")

# Visualization (if ASE GUI available)
# view(complex_TiO6)

# Save structure to file
complex_TiO6.write('TiO6_octahedral.xyz')
print("Structure saved to TiO6_octahedral.xyz")

Exercises

Summary

In this chapter, we learned crystal field theory and ligand field theory and understood the relationship between d-orbital splitting and electronic states in transition metal compounds.

Review of Key Points

• ✅ Crystal Field Theory: Treats ligands as point charges and explains d-orbital splitting electrostatically
• ✅ Octahedral Coordination: Splits into e_g and t_2g, with Δ_oct as the splitting width
• ✅ Tetrahedral Coordination: Splitting pattern is reversed, Δ_tet ≈ 4/9 Δ_oct
• ✅ Jahn-Teller Effect: Degenerate electronic states are lifted by structural distortion
• ✅ Ligand Field Theory: More precise theory considering covalency
• ✅ Spectrochemical Series: Ordering of ligand crystal field splitting strength
• ✅ Tanabe-Sugano Diagrams: Graphical representation of dⁿ electronic states vs. Δ/B
• ✅ Applications: Catalysts, magnetic materials, optical materials, pigments

Connection to Next Chapter

In Chapter 3, we will learn how to actually calculate these concepts using first-principles calculations (DFT). Master the Hohenberg-Kohn theorem, Kohn-Sham equations, and practical workflows using Python libraries (ASE, Pymatgen).