🔬 Selection Rules for Colour in Complex
The intensity of color (absorption) in transition metal complexes is governed by two major quantum mechanical principles, known as Selection Rules. These rules determine the probability of an electronic transition occurring when a complex absorbs a photon.
1. The Laporte (Orbital/Symmetry) Selection Rule
📜 Statement
A transition is allowed only if there is a change in the parity (symmetry) of the orbital. This is mathematically expressed as a change in the azimuthal quantum number ($\Delta l$).
- Allowed: Transitions must involve a change in parity ($\text{g} \leftrightarrow \text{u}$), i.e., $\Delta l = \pm 1$.
- Forbidden: Transitions between orbitals of the same parity are forbidden ($\text{g} \leftrightarrow \text{g}$ or $\text{u} \leftrightarrow \text{u}$), i.e., $\Delta l = 0$ or $\pm 2$.
🔑 Key Application: $d-d$ Transitions
- $s$ and $d$ orbitals have gerade ($\text{g}$) symmetry (symmetric with respect to the center of inversion).
- $p$ and $f$ orbitals have ungerade ($\text{u}$) symmetry (antisymmetric with respect to the center of inversion).
- Since $d-d$ transitions involve moving an electron from a $d$-orbital ($\text{g}$) to another $d$-orbital ($\text{g}$), they are $\text{g} \to \text{g}$. $$\text{Conclusion: } \mathbf{d-d \text{ transitions in centrosymmetric complexes are Laporte-Forbidden.}}$$
Relaxation of the Rule (Why we still see colour)
The Laporte rule is often broken, allowing weak transitions to occur (low intensity, $\epsilon \approx 1-100 \text{ L mol}^{-1} \text{ cm}^{-1}$).
- Vibronic Coupling: Molecular vibrations temporarily destroy the center of symmetry, allowing a momentary mixing of $d$ and $p$ orbitals ($\text{g}$ and $\text{u}$). This is the main reason why octahedral complexes are weakly coloured.
- Absence of $\mathbf{i}$ (Center of Inversion): Tetrahedral ($\text{T}_d$) complexes lack a center of symmetry, so $d$ and $p$ orbitals can naturally mix, making their transitions $\text{Laporte-allowed}$ (or partially allowed).
- Example: $[\text{CoCl}_4]^{2-}$ (deep blue) has an extinction coefficient ($\epsilon$) up to $\approx 600$, much higher than the octahedral $[\text{Co}(\text{H}_2\text{O})_6]^{2+}$ (pale pink), where $\epsilon \approx 10$.
2. The Spin (Multiplicity) Selection Rule
📜 Statement
An electronic transition is allowed only if the total spin quantum number ($S$) of the system does not change. This is expressed as no change in the spin multiplicity ($2S+1$).
$$\text{Allowed: } \Delta S = 0 \quad (\text{i.e., } (2S+1)_{\text{initial}} = (2S+1)_{\text{final}})$$ $$\text{Forbidden: } \Delta S \neq 0$$🔑 Key Application: High-Spin $d^5$ Complexes
- Ground State: $[\text{Mn}(\text{H}_2\text{O})_6]^{2+}$ ($\text{d}^5$ high-spin). $S = 5/2$. Multiplicity is $\mathbf{6}$ (Sextet, ${}^{6}\text{A}_{1\text{g}}$).
- Excited States: The available excited states are Quartets ($\mathbf{4}$, $S=3/2$) or Doublets ($\mathbf{2}$, $S=1/2$).
- Transition: ${}^{6}\text{A}_{1\text{g}} \to {}^{4}\text{T}_{1\text{g}}$ is $\mathbf{6} \to \mathbf{4}$, meaning $\Delta S \neq 0$. $$\text{Conclusion: } \mathbf{d-d \text{ transitions in } [\text{Mn}(\text{H}_2\text{O})_6]^{2+} \text{ are Spin-Forbidden.}}$$
Relaxation of the Rule
Spin-forbidden transitions are extremely weak ($\epsilon \approx 0.01-1 \text{ L mol}^{-1} \text{ cm}^{-1}$) but occur due to:
- Spin-Orbit Coupling (SOC): The coupling of the electron's spin and orbital motion momentarily mixes states of different multiplicities, allowing a very weak transition. SOC increases significantly for 2nd ($4d$) and 3rd ($5d$) row transition metals, making their spin-forbidden bands more intense.
3. Summary of Transition Intensity
The intensity of a colour is directly proportional to how many selection rules are obeyed. Molar absorptivity ($\epsilon$) is the measure of intensity.
| Transition Type | Laporte Rule | Spin Rule | Example | Intensity ($\epsilon \text{ L mol}^{-1} \text{ cm}^{-1}$) |
|---|---|---|---|---|
| Charge Transfer (LMCT/MLCT) | Allowed ($\text{p} \to \text{d}$) | Allowed ($\Delta S=0$) | $\text{MnO}_4^{-}$ (Deep Purple) | $10^3 - 10^6$ (Very Intense) |
| $d-d$ (Tetrahedral) | Partially Allowed | Allowed ($\Delta S=0$) | $[\text{CoCl}_4]^{2-}$ (Deep Blue) | $50 - 1000$ (Strong) |
| $d-d$ (Octahedral) | Forbidden ($\text{g} \to \text{g}$) | Allowed ($\Delta S=0$) | $[\text{Ti}(\text{H}_2\text{O})_6]^{3+}$ (Violet) | $1 - 100$ (Weak) |
| $d-d$ (Doubly Forbidden) | Forbidden ($\text{g} \to \text{g}$) | Forbidden ($\Delta S \neq 0$) | $[\text{Mn}(\text{H}_2\text{O})_6]^{2+}$ (Very Pale Pink) | $< 1$ (Extremely Weak) |
Understanding the degree of forbiddenness (Doubly Forbidden $\ll$ Laporte Forbidden $\ll$ Allowed) is essential for solving CSIR/GATE questions on colour intensity.
Read also
Selection Rule of Electronic Spectra
Charge Transfer Spectra