2.4 Electronic Spectra of Transition Metal Complexes

Electronic Absorption Spectra of Transition Metal Complexes

Origin and Dependence of Electronic Absorption Spectra

Electronic absorption spectra of transition metal complexes arise from electronic transitions between split d orbitals. The number, energy, and intensity of absorption bands depend on three factors: the electronic configuration of the metal ion (e.g., $d^6$ for $\text{Fe}^{2+}$), the geometry of the complex (octahedral, tetrahedral, etc.), and the nature of the ligands (strong-field vs. weak-field).

Beyond d-d transitions, charge-transfer (CT) transitions also appear in electronic spectra. These are Laporte-allowed because they involve a change in parity (the electron moves between orbitals of different symmetry character), making them far more intense than d-d transitions.

• LMCT (ligand-to-metal charge transfer): electron density shifts from a ligand-based orbital to a metal-based orbital
• MLCT (metal-to-ligand charge transfer): electron density shifts from a metal-based orbital to a ligand-based orbital (common with $\pi$-acceptor ligands like CO or bpy)

Selection Rules and Intensity of Electronic Transitions

Two selection rules govern whether an electronic transition is formally allowed or forbidden:

1. Laporte selection rule: Transitions between states of the same parity are forbidden. In a centrosymmetric complex (like octahedral), this means $g \rightarrow g$ transitions (such as d-d, since all d orbitals are gerade) are forbidden. Tetrahedral complexes lack a center of inversion, so the Laporte rule does not strictly apply to them.

2. Spin selection rule: Transitions that involve a change in total spin multiplicity ($\Delta S \neq 0$) are forbidden. Only transitions between states of the same spin multiplicity are formally allowed.

In practice, "forbidden" transitions still occur but with reduced intensity. Several mechanisms relax these rules:

• Vibronic coupling temporarily distorts octahedral symmetry through molecular vibrations, partially relaxing the Laporte rule. This is why octahedral d-d bands are weak but not absent (typical $\varepsilon \sim 5-50 \text{ L mol}^{-1}\text{cm}^{-1}$).
• Spin-orbit coupling mixes states of different spin multiplicity, partially relaxing the spin selection rule. This effect is stronger for heavier metals (larger spin-orbit coupling constants). For example, the spin-forbidden $^4\text{A}_2 \rightarrow {}^2\text{E}$ transition in $[\text{CoCl}_4]^{2-}$ gains weak intensity through spin-orbit coupling.
• d-p mixing in non-centrosymmetric environments (like tetrahedral) introduces ungerade character into the d orbitals, further relaxing the Laporte rule.

The rough intensity hierarchy to remember:

Electronic Transitions in Octahedral vs. Tetrahedral Complexes

Octahedral Complexes

In octahedral complexes, the d orbitals split into a lower-energy $t_{2g}$ set ($d_{xy}$, $d_{xz}$, $d_{yz}$) and a higher-energy $e_g$ set ($d_{x^2-y^2}$, $d_{z^2}$), separated by an energy gap $\Delta_o$.

The d-d transitions are all Laporte-forbidden ($g \rightarrow g$), so they tend to be weak. The specific transitions observed depend on the $d^n$ configuration and the term symbols of the ground and excited states:

• $[\text{Ni}(\text{H}_2\text{O})_6]^{2+}$ ($d^8$): The green color arises from three spin-allowed transitions. The lowest-energy band corresponds to $^3\text{A}_{2g} \rightarrow {}^3\text{T}_{2g}$, which directly equals $\Delta_o$ in energy for a $d^8$ system.
• $[\text{Co}(\text{CN})_6]^{3-}$ ($d^6$, low-spin): The $^1\text{A}_{1g} \rightarrow {}^1\text{T}_{1g}$ transition appears at relatively short wavelength because $\text{CN}^-$ is a strong-field ligand producing a large $\Delta_o$.

Strong-field complexes with $\pi$-acceptor ligands often show intense CT bands in addition to the d-d bands, which can dominate the visible spectrum and determine the observed color.

Tetrahedral Complexes

In tetrahedral complexes, the splitting is inverted: the lower set is $e$ ($d_{z^2}$, $d_{x^2-y^2}$) and the upper set is $t_2$ ($d_{xy}$, $d_{xz}$, $d_{yz}$), separated by $\Delta_t$.

The d-d transitions are $e \rightarrow t_2$. Because tetrahedral complexes lack a center of inversion, the Laporte rule does not apply, and d-p mixing is significant. This makes tetrahedral d-d bands noticeably more intense than octahedral ones.

• $[\text{CoBr}_4]^{2-}$ ($d^7$): The deep blue color arises from the $^4\text{A}_2 \rightarrow {}^4\text{T}_1(\text{P})$ transition. The intensity is high enough ($\varepsilon \sim 100-600$) that even dilute solutions show strong color.

Crystal Field Splitting Energy and Wavelength of Absorbed Light

Relationship between Crystal Field Splitting Energy and Wavelength

The crystal field splitting energy ($\Delta$) is the energy gap between the split d orbital sets. The wavelength of light absorbed in a d-d transition is related to the transition energy by:

$E = \frac{hc}{\lambda}$

where $h$ is Planck's constant, $c$ is the speed of light, and $\lambda$ is the wavelength. A larger $\Delta$ means higher-energy (shorter-wavelength) light is absorbed, while a smaller $\Delta$ means lower-energy (longer-wavelength) light is absorbed.

Note that for multi-electron systems, the energy of an absorption band does not always equal $\Delta$ directly. Interelectronic repulsion shifts the transition energies. For $d^1$ and $d^9$ systems the lowest-energy band does correspond to $\Delta$, but for other configurations you need Tanabe-Sugano diagrams or explicit term-energy expressions to extract $\Delta$ properly.

Spectrochemical Series and Crystal Field Splitting Energy

The spectrochemical series ranks ligands by their ability to split the d orbitals:

$\text{I}^- < \text{Br}^- < \text{Cl}^- < \text{F}^- < \text{H}_2\text{O} < \text{NH}_3 < \text{en} < \text{NO}_2^- < \text{CN}^- < \text{CO}$

Ligands higher in the series produce larger $\Delta$ values and shift absorption bands to shorter wavelengths (higher energy). This trend reflects a combination of $\sigma$-donor strength and $\pi$-bonding character:

• Weak-field ligands (halides) are often $\pi$-donors, which decrease $\Delta$
• Strong-field ligands ($\text{CN}^-$, CO) are $\pi$-acceptors, which increase $\Delta$

Crystal Field Parameters from Electronic Spectra

Determining Crystal Field Splitting Energy ($\Delta$)

You can extract $\Delta$ from the electronic spectrum by identifying the appropriate d-d transition energies:

1. Record the absorption spectrum and identify the d-d bands (usually in the visible/near-IR region, with low to moderate $\varepsilon$).
2. For simple cases ($d^1$, $d^9$, or $d^8$ octahedral), the lowest-energy spin-allowed band gives $\Delta$ directly: $\Delta_o = hc/\lambda$.
3. For other configurations, use a Tanabe-Sugano diagram for the appropriate $d^n$ case. Plot the ratios of observed band energies and match them to the diagram to read off $\Delta/B$.
4. For tetrahedral complexes, $\Delta_t \approx \frac{4}{9}\Delta_o$ for the same metal-ligand combination. This smaller splitting is why tetrahedral complexes almost always adopt high-spin configurations.

• Example: For $[\text{Ni}(\text{H}_2\text{O})_6]^{2+}$ ($d^8$), the $^3\text{A}_{2g} \rightarrow {}^3\text{T}_{2g}$ band appears near 8,500 cm$^{-1}$, giving $\Delta_o$ directly.
• Example: $\Delta_t$ for $[\text{CoBr}_4]^{2-}$ is roughly $\frac{4}{9}$ of the value for a hypothetical $[\text{CoBr}_6]^{4-}$ octahedral complex.

Racah Parameters and the Nephelauxetic Effect

The Racah parameters $B$ and $C$ quantify electron-electron repulsion within the d orbitals. $B$ measures the repulsion between electrons in different d orbitals (related to the Coulomb and exchange integrals), while $C$ captures additional repulsion relevant to states of different spin multiplicity. Both parameters are needed to fully describe the term energies, but $B$ is the one most commonly extracted from spectra.

To determine $B$ from a spectrum:

1. Measure the energies of at least two spin-allowed transitions.
2. Use the Tanabe-Sugano diagram or the appropriate energy expressions to solve for both $\Delta$ and $B$ simultaneously.
3. Compare the fitted $B$ value (often written $B'$ for the complex) to the free-ion value $B_0$.

The nephelauxetic ratio is defined as:

$\beta = \frac{B'}{B_0}$

A value of $\beta < 1$ (which is almost always the case in complexes) indicates that electron-electron repulsion is reduced in the complex compared to the free ion. This happens because the d electron cloud expands onto the ligands, reducing the effective repulsion. The more covalent the metal-ligand bond, the smaller $\beta$ becomes.

The nephelauxetic series ranks ligands by how much they reduce $B$:

$\text{F}^- < \text{H}_2\text{O} < \text{NH}_3 < \text{en} < \text{NCS}^- < \text{Cl}^- < \text{Br}^- < \text{CN}^- < \text{I}^-$

Notice this ordering differs from the spectrochemical series. A ligand can be a strong-field splitter but have varying degrees of covalency.