Crystal Field Theory (CFT) — Δ_oct, Δ_tet, splitting diagrams

WHAT is Crystal Field Theory?

WHY do we even need it? Valence Bond Theory could not explain colour, magnetism trends, or why some complexes are high-spin and others low-spin. CFT predicts all three from one number: the splitting energy $\Delta$ .

HOW the splitting happens (derive it, don't memorise it)

The five $d$ orbitals fall into two geometric families:

Orbitals	Lobes point...
$d_{x^2-y^2},\ d_{z^2}$	along the axes ( $x,y,z$ )
$d_{xy},\ d_{yz},\ d_{xz}$	between the axes

Octahedral case ( $\Delta_{oct}$ )

Deriving the barycentre rule (first principles). When ligands approach, the average (weighted) energy of the five orbitals must stay constant — energy is conserved; we only redistribute it. This is the barycentre / centre-of-gravity rule.

Let the $t_{2g}$ set drop by $x$ and the $e_g$ set rise by $y$ , with $y - (-x) = \Delta_{oct}$ ... let's set the barycentre as zero:

$3(-x) + 2(+y) = 0 \quad\text{and}\quad x + y = \Delta_{oct}$

Why? 3 orbitals in $t_{2g}$ , 2 in $e_g$ ; total displacement from barycentre must sum to zero.

Solve: from the first, $3x = 2y$ . Substitute $x = \Delta - y$ : $3(\Delta - y) = 2y \Rightarrow 3\Delta = 5y \Rightarrow y = \tfrac{3}{5}\Delta,\quad x = \tfrac{2}{5}\Delta$

Figure — Crystal Field Theory (CFT) — Δ_oct, Δ_tet, splitting diagrams

Tetrahedral case ( $\Delta_{tet}$ )

WHY is $\Delta_{tet}$ smaller? Two compounding reasons:

Only 4 ligands (vs 6) → less total repulsion.
No orbital points directly at a ligand → the effect is weaker still.

A geometric/electrostatic calculation gives the famous factor:

Spin states: high-spin vs low-spin (octahedral)

If $\Delta_{oct} < P$ → weak field → fill all orbitals singly first → high-spin (max unpaired).
If $\Delta_{oct} > P$ → strong field → pair in $t_{2g}$ first → low-spin (min unpaired).

This only matters for $d^4$ – $d^7$ (for $d^1,d^2,d^3,d^8,d^9,d^{10}$ the filling is forced — no choice).

Colour (the 80/20 payoff)

The Spectrochemical Series (what controls $\Delta$ )

Ligands ordered by the $\Delta$ they produce (weak → strong field):

$I^- < Br^- < S^{2-} < Cl^- < F^- < OH^- < H_2O < NH_3 < en < NO_2^- < CN^- < CO$

Common Mistakes (Steel-manned)

Recall Feynman: explain to a 12-year-old

Imagine the metal ion's $d$ orbitals are five kids standing in a field, all equally happy. Now you place some grumpy ligand kids around them holding "keep away" signs (negative charges repel). The orbital-kids standing face-to-face with a grumpy kid feel uncomfortable and become "high energy" (annoyed). The ones standing in the gaps stay calmer ("low energy"). The size of the discomfort gap is $\Delta$ . If the gap is small, kids prefer to spread into all spots (high-spin). If the gap is huge, they'd rather squeeze together in the comfy low spots (low-spin). In an octahedron (6 grumpy kids on the axes) the face-to-face orbitals are $e_g$ . In a tetrahedron (4 kids in the gaps) the roles flip, and the discomfort is much smaller — only $4/9$ as big.

Active Recall — Flashcards

What model treats M–L bonds as point-charge electrostatic interactions?

Crystal Field Theory (CFT)

In octahedral fields, which d-orbital set is raised and why?

e_g

(

d_{x^2-y^2}, d_{z^2}

); they point along axes directly at ligands → more repulsion.

Energies of

t_{2g}

and

e_g

relative to barycentre?

t_{2g} = -0.4\Delta_{oct}

e_g = +0.6\Delta_{oct}

State the relation between

\Delta_{tet}

and

\Delta_{oct}

\Delta_{tet} = \tfrac{4}{9}\Delta_{oct} \approx 0.44\,\Delta_{oct}

Why are tetrahedral complexes almost always high-spin?

\Delta_{tet}

is small (4/9 of

\Delta_{oct}

), so

\Delta_{tet} < P

→ pairing never favoured.

Why no 'g' subscript in tetrahedral labels?

Tetrahedron lacks a centre of inversion (not centrosymmetric).

Condition for low-spin (octahedral)?

Strong field,

\Delta_{oct} > P

(pairing energy).

Config of low-spin

d^6

octahedral & unpaired electrons?

t_{2g}^6 e_g^0

; 0 unpaired (diamagnetic).

CFSE of high-spin

d^5

octahedral?

0

(3×(−0.4Δ) + 2×(+0.6Δ) = 0).

What causes colour in complexes?

d–d electronic transition (

t_{2g}\to e_g

); absorbed

\lambda

with

E=hc/\lambda=\Delta

; we see complementary colour.

Strongest and weakest common field ligand?

Strongest: CO (and CN⁻); weakest: I⁻.

Connections

Coordination Compounds — Nomenclature
Valence Bond Theory of Complexes
Magnetic Properties — Spin-only Formula μ = √(n(n+2))
Spectrochemical Series & Ligand Strength
Colour & d–d Transitions
Jahn–Teller Distortion
Stability of Complexes & CFSE

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, Crystal Field Theory ka core idea bilkul simple hai. Jab metal ion akela hota hai, uske paanch $d$ orbitals ki energy same hoti hai (degenerate). Lekin jaise hi uske around ligands aate hain, hum unhe point negative charges maan lete hain. Ab jo $d$ orbitals seedhe ligand ki taraf point karte hain, unke electrons ko zyada repulsion feel hota hai, to unki energy up chali jaati hai. Jo orbitals beech mein (gaps mein) hote hain, unko kam repulsion milta hai, to energy neeche. Yahi splitting hai, aur dono groups ke beech ka gap hai $\Delta$ .

Octahedral mein 6 ligand axes pe baithe hote hain, isliye $e_g$ ( $d_{x^2-y^2}, d_{z^2}$ ) upar jaate hain aur $t_{2g}$ neeche. Barycentre rule (energy conserve hoti hai) se nikalta hai: $t_{2g} = -0.4\Delta$ , $e_g = +0.6\Delta$ . Tetrahedral mein ulta ho jaata hai — kyunki wahan 4 ligand axes pe nahi, gaps mein hote hain. Aur ek important baat: $\Delta_{tet} = \frac{4}{9}\Delta_{oct}$ , yaani bahut chhota gap. Isliye tetrahedral complexes hamesha high-spin maane jaate hain, kyunki gap itna chhota hai ki pairing kabhi favourable nahi hoti.

High-spin vs low-spin ka funda yeh hai: electron ko decide karna hota hai ki upar $e_g$ mein jaaye (cost $\Delta$ ) ya neeche pair ho jaaye (cost pairing energy $P$ ). Agar $\Delta < P$ (weak field ligand jaise F⁻, H₂O) to spread out — high spin. Agar $\Delta > P$ (strong field jaise CN⁻, CO) to pair up — low spin. Isi se magnetism (kitne unpaired electrons) aur colour (d–d transition jisme $\Delta$ energy ka photon absorb hota hai, aur humein complementary colour dikhta hai) dono explain ho jaate hain. Spectrochemical series yaad rakho — yahi batati hai kaunsa ligand kitna strong field deta hai.

Crystal Field Theory (CFT) — Δ_oct, Δ_tet, splitting diagrams

WHAT is Crystal Field Theory?

HOW the splitting happens (derive it, don't memorise it)

Octahedral case ( $\Delta_{oct}$ )

Tetrahedral case ( $\Delta_{tet}$ )

Spin states: high-spin vs low-spin (octahedral)

Colour (the 80/20 payoff)

The Spectrochemical Series (what controls $\Delta$ )

Common Mistakes (Steel-manned)

Active Recall — Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Coordination Chemistry

Connections