Jahn-Teller distortion

3.4.10 · Chemistry › Coordination Chemistry

WHAT is it?

Key trigger word hai degeneracy jo unevenly occupied ho.

Ek octahedral field mein paanch d-orbitals split hote hain:

== $t_{2 g}$ == ( $d_{x y}, d_{y z}, d_{x z}$ ) — lower, ligands ke beech point karte hain.
== $e_{g}$ == ( $d_{z^{2}}, d_{x^{2} - y^{2}}$ ) — higher, ligands ki taraf point karte hain.

JT distortion tab hoti hai jab koi bhi ek set degenerate ho lekin unequally filled ho.

WHY does it happen?

Strong vs weak rule:

$e_{g}$ orbitals ligands seedhe ki taraf point karte hain → inhe uneven bhar ne se strong, observable distortion hoti hai.
$t_{2 g}$ orbitals ligands ke beech point karte hain → inhe uneven bhar ne se sirf weak, often unobservable distortion hoti hai.

HOW does the energy actually split? (Derivation from scratch)

$z$ -axis ke saath elongation consider karo: do axial ligands metal se door jaate hain.

Step 1 — $e_{g}$ set ko split karo.

$d_{z^{2}}$ ke lobes $z$ ke saath hain → repulsion kam hoti hai → energy $δ_{1} /2$ se lower hoti hai.
$d_{x^{2} - y^{2}}$ $x y$ plane mein hai → repulsion badhti hai → energy $δ_{1} /2$ se raise hoti hai.

Yeh step kyun? Kyunki perturbation ko energy ka centre ("barycentre") conserve karna hota hai: jo utna neeche jaata hai, doosra utna upar jaata hai.

Step 2 — $t_{2 g}$ set ko split karo.

$d_{x z}, d_{y z}$ (jinmein $z$ component hai) → $δ_{2} /3$ se lower hote hain.
$d_{x y}$ (purely in-plane) → $2 δ_{2} /3$ se raise hota hai (barycentre conserve karte hue).

Barycentre weighting kyun? Do orbitals neeche jaate hain, ek upar jaata hai; signed total zero hona chahiye: $- 2 (δ_{2} /3) + (2 δ_{2} /3) = 0$ . ✓

Step 3 — net stabilization. Kisi configuration ke liye, JT stabilization energy yeh hoti hai — (lowered orbitals mein electrons)×(drop) minus (raised orbitals mein electrons)×(rise):

WHICH configurations distort?

Distortion strong hoti hai jab $e_{g}$ set unevenly occupied ho:

Configuration	$e_{g}$ occupation	JT?
$d^{9}$ (Cu²⁺)	$e_{g}^{3}$ ( $t_{2 g}^{6} e_{g}^{3}$ )	Strong
high-spin $d^{4}$ (Cr²⁺, Mn³⁺)	$e_{g}^{1}$	Strong
low-spin $d^{7}$ (Co²⁺ ls, Ni³⁺)	$e_{g}^{1}$	Strong
$d^{3}$ , $d^{6}$ ls, $d^{8}$ , $d^{10}$	$e_{g}$ evenly filled (0,0,2)	None

Weak (usually ignore kiya jaata hai) jab sirf $t_{2 g}$ uneven ho: jaise $d^{1}, d^{2}, t_{2 g}$ low-spin cases.

Flashcards

Jahn–Teller theorem statement

Ek non-linear molecule jo degenerate electronic ground state mein hai, geometrically distort hota hai degeneracy hatane aur apni energy girane ke liye.

Trigger condition for JT distortion

Degenerate orbitals ke set ki uneven (asymmetric) occupation.

Which orbital set gives STRONG distortion and why

e_{g}

— iske lobes seedhe ligands ki taraf point karte hain, bond lengths se strongly couple karte hain.

Which gives only WEAK distortion

t_{2 g}

— lobes ligands ke beech point karte hain, weak coupling hoti hai.

Classic strong-JT ion

Cu²⁺ (

d^{9}

t_{2 g}^{6} e_{g}^{3}

d

counts with strong JT (high spin)

d^{9}

, high-spin

d^{4}

, low-spin

d^{7}

(uneven

e_{g}

d

counts with NO JT

d^{3}

d^{8}

d^{10}

, low-spin

d^{6}

(evenly filled sets).

Effect on Cu²⁺ octahedron geometry

z ke saath elongation → 4 short + 2 long M–L bonds.

Why elongation lowers

d_{z^{2}}

Axial ligands door jaate hain → z-pointing orbital par kam repulsion hoti hai.

Does barycentre stay fixed during splitting

Haan — split set ki total energy conserve hoti hai; rises aur drops balance hote hain.

Recall Feynman: explain to a 12-year-old

Ek metal ball imagine karo jiske around 6 marbles ki cage hai, bilkul even. Ball ke paas chhote electron "pockets" hain. Agar electrons lopsidedly baith jaate hain — ek taraf doosre se zyada — to cage uncomfortable ho jaata hai. To woh stretch karta hai: do marbles thoda aur door khisakti hain. Isse crowded electrons zyada relaxed hote hain aur poori cheez khush (lower energy). Yeh stretching hi Jahn–Teller effect hai. Yeh copper ke liye bahut hota hai, isliye copper compounds "squashed" dikhte hain.

Connections

Crystal Field Theory — $t_{2 g} / e_{g}$ splitting provide karta hai jis par JT act karta hai.
Octahedral Splitting and $\Delta_o$ — parent energy diagram.
High-spin vs Low-spin Complexes — $e_{g}$ occupation decide karta hai, isliye JT bhi.
Colour of Transition Metal Complexes — JT splitting extra absorption bands add karti hai (jaise broad Cu²⁺ band).
Stability and Distortion in $d^9$ Cu(II) — direct application.
Tetrahedral vs Octahedral Geometry — kahan distortions strong vs weak hoti hain.