Chapter 20: d-Metal Complexes - Electronic Structure and Properties

Electronic Structure

There are two widely used models of the electronic structure of d-metal complexes:

Crystal-Field Theory

Emerged from analysis of spectra of d-metal ions in solids. Treats ligands as point charges that create an electrostatic field around the metal ion.

Ligand-Field Theory

An application of molecular orbital theory. Provides a more complete description and accounts for a wider range of properties.

The striking colours of many d-metal complexes were a mystery to Werner when he elucidated their structures. The origin of these colours was clarified only when the description of electronic structure in terms of orbitals was applied to the problem (1930-1960).

20.1 Crystal-Field Theory

Key Points: In crystal-field theory, a ligand lone pair is modelled as a point negative charge that repels electrons in the d orbitals of the central metal ion. The theory concentrates on the resultant splitting of the d orbitals into groups with different energies.

(a) Octahedral Complexes

In an octahedral complex, six negative charges representing the ligands are placed at points in an octahedral array around the central metal ion. Electrons in different d orbitals interact with the ligands to different extents:

d-Orbital Splitting in Octahedral Field

d orbitals

Spherical
environment

→

e_g (d_z², d_x²-y²) +0.6Δ_O

t_2g (d_xy, d_yz, d_zx) -0.4Δ_O

Octahedral
crystal field

Δ_O ↕

The e_g orbitals (d_z² and d_x²-y²) are concentrated close to the ligands along the axes and are repelled more strongly. The t_2g orbitals (d_xy, d_yz, d_zx) are concentrated in regions between the ligands.

Ligand-Field Splitting Parameter (Δ_O)

The energy separation between the t_2g and e_g orbital sets. The subscript 'O' signifies an octahedral crystal field. The barycentre (average energy) remains unchanged.

The Spectrochemical Series

Ligands can be arranged in order of increasing energy of transitions (and therefore Δ_O):

I⁻ < Br⁻ < S²⁻ < SCN⁻ < Cl⁻ < F⁻ < OH⁻ < H₂O < NCS⁻ < NH₃ < en < NO₂⁻ < CN⁻ < CO

Weak-field ligands give rise to low-energy transitions (small Δ_O), while strong-field ligands give high-energy transitions (large Δ_O).

(b) Ligand-Field Stabilization Energies (LFSE)

Key Points: The ground-state configuration reflects the relative values of Δ_O and the pairing energy P. For octahedral 3dⁿ species with n = 4-7, high-spin and low-spin complexes occur in weak-field and strong-field cases, respectively.

The ligand-field stabilization energy (LFSE) is the additional stability relative to the barycentre:

LFSE = (-0.4x + 0.6y)Δ_O for t_2g^xe_g^y configuration

High-Spin vs Low-Spin Complexes

d⁴ Configuration

High-spin (Δ_O < P)

↑

e_g

↑

t_2g

4 unpaired, LFSE = -0.6Δ_O

Low-spin (Δ_O > P)

e_g

↑↓

↑

t_2g

2 unpaired, LFSE = -1.6Δ_O + P

d⁶ Configuration

High-spin (weak field)

↑

e_g

↑↓

↑

t_2g

4 unpaired, LFSE = -0.4Δ_O

Low-spin (strong field)

e_g

↑↓

t_2g

0 unpaired, LFSE = -2.4Δ_O + 2P

LFSE Values for Octahedral Complexes

dⁿ	Example	High-spin LFSE/Δ_O	Low-spin LFSE
d⁰	Sc³⁺	0	—
d¹	Ti³⁺	-0.4	—
d²	V³⁺	-0.8	—
d³	Cr³⁺, V²⁺	-1.2	—
d⁴	Cr²⁺, Mn³⁺	-0.6	-1.6Δ_O + P
d⁵	Mn²⁺, Fe³⁺	0	-2.0Δ_O + 2P
d⁶	Fe²⁺, Co³⁺	-0.4	-2.4Δ_O + 2P
d⁷	Co²⁺	-0.8	-1.8Δ_O + P
d⁸	Ni²⁺	-1.2	—
d⁹	Cu²⁺	-0.6	—
d¹⁰	Cu⁺, Zn²⁺	0	—

Example 20.1: Calculating the LFSE

For high-spin d⁵: Configuration t_2g³e_g² (no additional pairing)

LFSE = (3 × -0.4 + 2 × 0.6)Δ_O = 0

For low-spin d⁶: Configuration t_2g⁶ with 3 pairs of electrons

LFSE = 6 × (-0.4Δ_O) + 2P = -2.4Δ_O + 2P

What is the LFSE for both high- and low-spin d⁷ configurations?

(c) Magnetic Measurements

Key Points: Magnetic measurements determine the number of unpaired spins in a complex, hence identifying its ground-state configuration. The spin-only magnetic moment is given by μ = √[N(N+2)] μ_B.

μ = [N(N + 2)]^½ μ_B

where N is the number of unpaired electrons and μ_B is the Bohr magneton (9.274 × 10⁻²⁴ J T⁻¹).

Ion	Configuration	N	μ/μ_B (calc)	μ/μ_B (exp)
Ti³⁺	t_2g¹	1	1.73	1.7-1.8
V³⁺	t_2g²	2	2.83	2.7-2.9
Cr³⁺	t_2g³	3	3.87	3.8
Mn³⁺	t_2g³e_g¹	4	4.90	4.8-4.9
Fe³⁺	t_2g³e_g²	5	5.92	5.9

(e) Tetrahedral Complexes

Key Points: In a tetrahedral complex, the e orbitals lie below the t₂ orbitals; only the high-spin case need be considered because Δ_T ≈ (4/9)Δ_O.

d-Orbital Splitting in Tetrahedral Field

d orbitals

Spherical
environment

→

t₂ (d_xy, d_yz, d_zx) +0.4Δ_T

e (d_z², d_x²-y²) -0.6Δ_T

Tetrahedral
crystal field

Note: The splitting is inverted compared to octahedral!

(g) The Jahn-Teller Effect

Key Points: If the ground electronic configuration of a nonlinear complex is orbitally degenerate and asymmetrically filled, the complex distorts to remove the degeneracy and achieve a lower energy.

Regular Octahedron

→

Elongated (z-axis)

Common for d⁹ Cu²⁺

Jahn-Teller distortions are pronounced for:

d⁹: Cu(II) complexes - most pronounced
High-spin d⁴: Cr²⁺, Mn³⁺
Low-spin d⁷: Ni³⁺

20.2 Ligand-Field Theory

Ligand-field theory is an application of molecular orbital theory that concentrates on the d orbitals of the central metal atom. It provides a more substantial framework for understanding the origins of Δ_O.

Key Points: In ligand-field theory, the building-up principle is used with a molecular orbital energy-level diagram constructed from metal-atom orbitals and symmetry-adapted linear combinations (SALCs) of ligand orbitals.

(a) σ Bonding

In an octahedral (O_h) environment, the metal orbitals divide by symmetry:

Metal Orbital	Symmetry Label	Degeneracy
s	a_1g	1
p_x, p_y, p_z	t_1u	3
d_x²-y², d_z²	e_g	2
d_xy, d_yz, d_zx	t_2g	3

The t_2g orbitals remain nonbonding as there is no combination of ligand σ orbitals that has matching symmetry. The e_g orbitals become antibonding due to overlap with ligand SALCs.

(b) π Bonding

Key Points: π-Donor ligands decrease Δ_O whereas π-acceptor ligands increase Δ_O. This explains the spectrochemical series!

π-Donor Ligands

Filled π orbitals on ligand

Cl⁻ Br⁻ OH⁻

Δ_O DECREASES

No π Effects

No suitable π orbitals

NH₃ H⁻

σ bonding only

π-Acceptor Ligands

Empty π* orbitals on ligand

CO CN⁻ PR₃

Δ_O INCREASES

Electronic Spectra

The magnitudes of ligand-field splittings correspond to energies in the UV-visible region. The analysis of electronic spectra allows us to extract values of Δ_O and understand the electronic structure of complexes.

20.3 Electronic Spectra of Atoms

Key Points: Electron-electron repulsions result in multiple absorptions. Different microstates of the same configuration have different energies when electron repulsions are considered.

Spectroscopic Terms

Terms are labelled using Russell-Saunders coupling for light atoms (3d series):

Term Symbol: ^2S+1L_J

L value	0	1	2	3	4
Letter	S	P	D	F	G

Hund's Rules

Hund's First Rule

For a given configuration, the term with the greatest multiplicity lies lowest in energy.

This means triplet terms (S=1) are lower than singlet terms (S=0) when both are possible.

Hund's Second Rule

For terms of given multiplicity, the term with the greatest value of L lies lowest in energy.

When L is high, electrons can stay clear of one another and experience lower repulsion.

Example 20.6: Identifying the Ground Term

For d⁵ (Mn²⁺):

Maximum S = 5/2 → Multiplicity = 6 (sextet)

All electrons in different orbitals: M_L = +2 +1 +0 -1 -2 = 0 → L = 0

Ground term: ⁶S

Racah Parameters

The Racah parameters (A, B, C) describe electron-electron repulsion energies. Parameter B is most commonly used in analyzing spectra:

Ion	2+	3+
Ti	—	720 cm⁻¹
V	765 cm⁻¹	860 cm⁻¹
Cr	830 cm⁻¹	1030 cm⁻¹
Mn	960 cm⁻¹	1130 cm⁻¹
Fe	1060 cm⁻¹	—
Co	1120 cm⁻¹	—
Ni	1080 cm⁻¹	—

20.4 Electronic Spectra of Complexes

Tanabe-Sugano Diagrams

Key Points: Tanabe-Sugano diagrams show the correlation of all terms and are used to extract values of Δ_O and B from experimental spectra. Term energies (E/B) are plotted against Δ_O/B.

Tanabe-Sugano Diagram Principles

Weak Field (left)

Free ion terms dominate

High-spin configuration

Strong Field (right)

Large Δ_O/B values

Low-spin configuration

Lines representing terms of the same symmetry obey the noncrossing rule: they bend apart rather than crossing.

Example 20.8: Calculating Δ_O and B

For [Cr(NH₃)₆]³⁺ with transitions at 21,550 and 28,500 cm⁻¹:

Ratio = 28,500/21,550 = 1.32

From d³ Tanabe-Sugano diagram: Δ_O/B ≈ 33.0

Lower transition at 32.8B → B = 657 cm⁻¹

Therefore: Δ_O = 21,700 cm⁻¹

The Nephelauxetic Effect

The nephelauxetic parameter β measures the reduction of B from its free-ion value:

β = B(complex) / B(free ion)

A small β indicates large d-electron delocalization onto ligands (covalent character).

Nephelauxetic series:

Br⁻ < CN⁻ < Cl⁻ < NH₃ < H₂O < F⁻

← More covalent (softer) | More ionic (harder) →

20.5 Charge-Transfer Bands

Key Points: Charge-transfer (CT) transitions involve migration of electrons between orbitals predominantly ligand in character and orbitals predominantly metal in character. They are identified by high intensity and sensitivity to solvent polarity.

LMCT

Ligand → Metal

L → M

Favored with:

High oxidation state metals
Ligands with lone pairs
Examples: [MnO₄]⁻, [CrO₄]²⁻

MLCT

Metal → Ligand

M → L

Favored with:

Low oxidation state metals
Ligands with π* orbitals
Examples: [Ru(bpy)₃]²⁺

Trends in LMCT Energies

Oxidation State	Energy Order (increasing)
+7	[MnO₄]⁻ < [TcO₄]⁻ < [ReO₄]⁻
+6	[CrO₄]²⁻ < [MoO₄]²⁻ < [WO₄]²⁻
+5	[VO₄]³⁻ < [NbO₄]³⁻ < [TaO₄]³⁻

The trend correlates with the electrochemical series: lowest-energy transitions occur for most easily reduced metal ions.

Visible Spectrum

400 nm (violet) 550 nm (green) 700 nm (red)

CT bands typically have ε_max = 1,000 - 50,000 dm³ mol⁻¹ cm⁻¹

20.6 Selection Rules and Intensities

Key Points: The strength of an electronic transition is determined by the transition dipole moment. Selection rules determine which transitions are allowed (nonzero intensity) and which are forbidden.

Spin Selection Rule

ΔS = 0 for allowed transitions

Electromagnetic field cannot change spin orientations.

Spin-forbidden: ε < 1 dm³ mol⁻¹ cm⁻¹

Laporte Selection Rule

g ↔ u allowed; g ↔ g forbidden

In centrosymmetric molecules, parity must change.

d-d transitions: forbidden in O_h

Typical Intensities

Band Type	ε_max / dm³ mol⁻¹ cm⁻¹
Spin-forbidden	< 1
Laporte-forbidden d-d (octahedral)	20 - 100
Laporte-allowed d-d (tetrahedral)	~ 250
Symmetry-allowed (CT)	1,000 - 50,000

Relaxation mechanisms:

Spin-orbit coupling can relax spin selection rule (heavy-atom effect)
Asymmetric vibrations can relax Laporte rule (vibronic coupling)
Departure from centrosymmetry allows forbidden transitions

20.7 Luminescence

Fluorescence

Radiative decay from an excited state of the same multiplicity as the ground state. Fast (nanoseconds).

Phosphorescence

Radiative decay from a state of different multiplicity. Requires intersystem crossing. Slow (microseconds or longer).

Ruby Luminescence (Cr³⁺ in Al₂O₃)

Absorptions: ⁴T₂g ← ⁴A₂g (green) and ⁴T₁g ← ⁴A₂g (violet)

Intersystem crossing to ²E state

Red 627 nm phosphorescence: ²E → ⁴A₂g

This was used in the first laser (1960)!

Magnetism

20.8 Cooperative Magnetism

Key Points: In solids, spins on neighbouring metal centres may interact to produce cooperative magnetic behaviour such as ferromagnetism and antiferromagnetism.

🧲

Paramagnetic

↑ ↓ ↑ ↓

Random spin orientations

χ > 0, attracted to field

⬆️

Ferromagnetic

↑ ↑ ↑ ↑

Parallel alignment

Below T_C (Curie temp)

↕️

Antiferromagnetic

↑ ↓ ↑ ↓

Antiparallel alignment

Below T_N (Néel temp)

🔻

Ferrimagnetic

⬆ ↓ ⬆ ↓

Unequal antiparallel

Net moment (Fe₃O₄)

Superexchange Mechanism

Antiferromagnetic coupling occurs through intervening ligands:

↑

Metal 1

—

↑↓

O²⁻

—

↓

Metal 2

Examples: MnO (T_N = 122 K), Cr₂O₃ (T_N = 310 K)

20.9 Spin-Crossover Complexes

Key Points: When factors determining spin state are closely matched, complexes can change spin state in response to external stimuli (temperature, pressure, light).

Spin-crossover is most common for:

d⁴, d⁵, d⁶, d⁷ octahedral complexes
3d metals where Δ_O ≈ P
Fe(II) and Fe(III) complexes are most studied

Low Temperature / High Pressure LOW-SPIN

Smaller volume (e_g orbitals less populated)

Diamagnetic or fewer unpaired electrons

High Temperature / Low Pressure HIGH-SPIN

Larger volume (antibonding e_g occupied)

Paramagnetic with more unpaired electrons

Applications:

Magnetic information storage
Pressure-sensitive devices
Molecular switches
Understanding O₂ binding to haemoglobin

Chapter Summary

Crystal-Field Theory

Ligands as point charges
d-orbital splitting: Δ_O, Δ_T
High-spin vs low-spin
LFSE calculations

Ligand-Field Theory

MO approach
σ and π bonding effects
Explains spectrochemical series
Nephelauxetic effect

Electronic Spectra

Term symbols and Racah parameters
Tanabe-Sugano diagrams
CT transitions (LMCT, MLCT)
Selection rules

Magnetism

Spin-only formula
Cooperative magnetism
Superexchange
Spin-crossover complexes