# (a) Jahn-Teller Theorem ## Statement The **Jahn-Teller theorem** states: > Any non-linear molecule in a degenerate electronic state will undergo a geometrical distortion that removes that degeneracy, resulting in a lowering of the overall energy. - **Deg electronic state:**als of equal energy are unevenly filled. - **Non-linear molecule:** Applies to most 3D complexes, such as octahedral or tetrahedral complexes. **In essence:** If an octahedral complex has unequal filling of the \( e_g \) or \( t_{2g} \) orbitals, it will distort to relieve this degeneracy and lower the energy. --- # (b) Application to Complex Ions ## Step 1: Electronic Configurations Let's determine the electron configurations for each complex ion, assuming *strong field/low spin* for all except where not possible. ### (i) \([Cu(OH_2)_6]^{2+}\) - Copper(II): \( Cu^{2+} \) - Ground state: [Ar] 3d\(^9\) - Octahedral field: \( t_{2g}^6 e_g^3 \) - Electron configuration: One of the \( e_g \) orbitals (\( d_{z^2} \) or \( d_{x^2-y^2} \)) is singly occupied → **degeneracy present**. ### (ii) \([Cr(NH_3)_6]^{3+}\) - Chromium(III): \( Cr^{3+} \) - Ground state: [Ar] 3d\(^3\) - Octahedral field: \( t_{2g}^3 e_g^ \) - Electron configuration: \( t_{2g} \) orbitals each have one electron, all unpaired, but no degeneracy in \( e_g \) orbitals. ### (iii) \([Re(CO)_6]\) - Rhenium(): \( Re^ \) - Ground state: [Xe] 4f\(^{14}\)5d\(^7\)6s\(^2\) (but CO is a strong field ligand; will promote electrons to low spin) - \( Re \) in \( Re(CO)_6 \): \( d^6 \) system (since each CO is neutral) - Octahedral field: \( t_{2g}^6 e_g^ \) - All lower energy \( t_{2g} \) orbitals fully filled; no degeneracy. --- ## Step 2: Apply the Jahn-Teller Theorem - **Jahn-Teller effect is most pronounced when the \( e_g \) set is unevenly occupied** (i.e., d\(^4\) high spin, d\(^7\) high spin, d\(^9\)). - Weak or absent for d\(^3\), d\(^6\) (low spin), d\(^5\) (high or low spin), d\(^8\), d\(^\). **Ranking (from most to least Jahn-Teller distortion):** 1. **[Cu(OH\(_2\))\(_6\)]\(^{2+}\) (d\(^9\))** - Strong Jahn-Teller distortion. - Occupation: \( t_{2g}^6 e_g^3 \) → unevenly filled \( e_g \). - **Distortion:** Elongation or compression along one axis (usually z), leading to tetragonal elongation. 2. **[Cr(NH\(_3\))\(_6\)]\(^{3+}\) (d\(^3\))** - No significant Jahn-Teller distortion. - Occupation: \( t_{2g}^3 e_g^ \); no degeneracy in \( e_g \). 3. **[Re(CO)\(_6\)] (d\(^6\), low spin)** - No significant Jahn-Teller distortion. - Occupation: \( t_{2g}^6 e_g^ \); all fully filled, no degeneracy. --- ## Step 3: Explanation of Distortion ### [Cu(OH\(_2\))\(_6\)]\(^{2+}\): - **Type:** Tetragonal distortion (usually elongation along the z-axis). - **Cause:** One electron occupies the \( e_g \) set unevenly (between \( d_{z^2} \) and \( d_{x^2-y^2} \)). - **Result:** Two axial bonds are longer than the four equatorial bonds. ### [Cr(NH\(_3\))\(_6\)]\(^{3+}\): - **Type:** No significant distortion. - **Cause:** No orbital degeneracy to relieve. ### [Re(CO)\(_6\)]: - **Type:** No significant distortion. - **Cause:** All \( t_{2g} \) orbitals filled, \( e_g \) empty; no degeneracy. --- ## Step 4: Diagrams ### Jahn-Teller Distortion in [Cu(OH\(_2\))\(_6\)]\(^{2+}\) ```plaintext Before distortion (ideal octahedral): O | O--Cu--O | O After distortion (elongation along z-axis): O | O--Cu--O | O (axial O's move further from Cu) ``` **[Alt text: Diagram of an octahedral complex showing equal bond lengths before distortion and longer axial bonds after Jahn-Teller distortion.]** --- # Final Ranking **Degree of deviation from ideal octahedral structure (most to least):** 1. \([Cu(OH_2)_6]^{2+}\) (**strong distortion**) 2. \([Cr(NH_3)_6]^{3+}\) (**negligible**) 3. \([Re(CO)_6]\) (**negligible**) --- **Summary Table** | Complex | d-electrons | Jahn-Teller Effect? | Expected Distortion | |----------------------|-------------|---------------------|----------------------------| | [Cu(OH2)\(_6\)]\(^{2+}\) | d\(^9\) | Strong | Tetragonal (axial elongation) | | [Cr(NH3)\(_6\)]\(^{3+}\) | d\(^3\) | None | None | | [Re(CO)\(_6\)] | d\(^6\) | None | None |