Pairing Energy
The energy required to force two unpaired electrons into a single orbital is called the pairing energy ($P$). In other words, pairing energy is the quantum electronic penalty required to place two electrons within the identical localized orbital zone.
- If the crystal field splitting parameter ($\Delta$) is small due to the presence of weak-field ligands, the relative pairing energy costs will be larger ($\Delta < P$), causing the coordination complex to adopt a high-spin configuration.
- If the crystal field splitting parameter ($\Delta$) is large due to strong-field ligands, pairing up becomes energetically favorable ($\Delta > P$), forcing the complex into a low-spin configuration.
Relation between Pairing Energy and Crystal Field Splitting Energy
When crystal field splitting is significantly large, it is energetically favorable for valence electrons to pair up in lower-energy $d$-orbitals ($t_{2g}$ in octahedral setups) before attempting to cross the gap to occupy higher-energy orbitals. This produces a low-spin structural complex. Strong-field ligands (such as $\text{CN}^-$, $\text{CO}$) produce large splitting variations.
Conversely, if the crystal field splitting parameter remains narrow, it is more favorable for incoming electrons to occupy individual $d$-orbitals singly to fully minimize electron-electron repulsion forces. This produces high-spin environments. Weak-field ligands (such as $\text{H}_2\text{O}$, $\text{F}^-$) yield tiny splitting properties.
• If $\Delta < P$, it favors the formation of high-spin complexes.
• If $\Delta = P$, high-spin and low-spin configurations exist in equilibrium.
When multiple electrons undergo pairing across target subshells, the energetic cost is evaluated as the mean pairing energy, which can be extracted via electronic absorption spectra analysis.
Generally, for $4d$ and $5d$ series transition metal complexes, the intrinsic magnitude of $\Delta$ is naturally much higher than that of the pairing energy threshold. As a consequence, **they almost exclusively form low-spin structures**.
- For $3d$ elements, a typical base value for pairing energy stands near $15,000 \text{ cm}^{-1}$.
- $3d$ complexes display variable behavior: high-spin alongside weak-field ligands and low-spin alongside strong-field ligands.
- High-valent $3d$ metal matrices (e.g., $\text{Co}^{3+}$ networks) exhibit large $\Delta$ and lean heavily low-spin.
Question 1: What is the correct relationship between pairing energy ($P$) and CFSE ($\Delta_o$) in the complex ion $[\text{Ir}(\text{H}_2\text{O})_6]^{3+}$?
A. $\Delta_o > P$
B. $\Delta_o < P$
C. $\Delta_o = P$
D. No Relation
Solution: A
Hint: Even though $\text{H}_2\text{O}$ behaves as a weak-field ligand for standard $3d$ metals, $\text{Ir}^{3+}$ belongs to the heavy $5d$ transition series. The spatial extension of $5d$ orbitals causes immense crystal field splitting ($\Delta_o$) that easily overpowers pairing energy ($P$). Therefore, $\Delta_o > P$ is always maintained.
Question 2: For $\text{Mn}^{3+}$, the pairing energy is $28,000 \text{ cm}^{-1}$ and $\Delta_o$ for $[\text{Mn}(\text{CN})_6]^{3-}$ is $38,500 \text{ cm}^{-1}$. Which of the following statements is incorrect?
A. The complex will be colored
B. The complex will be a low-spin complex
C. Net CFSE = $-33,600 \text{ cm}^{-1}$
D. The complex will be colorless
Solution: D
Since $\Delta_o > P$ ($38,500 > 28,000$), it forms a low-spin configuration. $\text{Mn}^{3+}$ is a $d^4$ ion, meaning its configuration inside this strong field organizes as $t_{2g}^4 e_g^0$.
Because there are two unpaired electrons remaining within the $t_{2g}$ subshells, the complex is paramagnetic and colored due to viable $d\text{-}d$ transitions. The mathematical calculation shows:
$$\text{CFSE} = (-0.4 \times 4) \times 38,500 + 28,000 = -33,600 \text{ cm}^{-1}$$