Crystal Field Theory — Revision Notes

⚡ 30-Second Revision

CFT Basis — Electrostatic model, ligands as point charges/dipoles, no covalent bond.
d-orbital Splitting — Degeneracy lifted by ligand field.
Octahedral ($\Delta_o$) — $t_{2g}$ (3 orbitals, $-0.4\Delta_o$) lower, $e_g$ (2 orbitals, $+0.6\Delta_o$) higher.
Tetrahedral ($\Delta_t$) — $e$ (2 orbitals, $-0.6\Delta_t$) lower, $t_2$ (3 orbitals, $+0.4\Delta_t$) higher. $\Delta_t \approx \frac{4}{9}\Delta_o$.
Spectrochemical Series — Ligand field strength: $\text{I}^- < \text{Br}^- < \text{Cl}^- < \text{F}^- < \text{H}_2\text{O} < \text{NH}_3 < \text{en} < \text{CN}^- < \text{CO}$.
High Spin — Weak field ligands, $\Delta < P$, maximize unpaired electrons. (For $d^4-d^7$ octahedral).
Low Spin — Strong field ligands, $\Delta > P$, minimize unpaired electrons. (For $d^4-d^7$ octahedral).
CFSE — $[n_{t_{2g}}(-0.4\Delta_o) + n_{e_g}(+0.6\Delta_o)] + mP$.
Magnetic Moment — $\mu = \sqrt{n(n+2)}$ BM, where $n$ is unpaired electrons.
Color — d-d transitions, $E = \Delta = hc/\lambda$. Observed color is complementary to absorbed color.

2-Minute Revision

Crystal Field Theory (CFT) explains the properties of transition metal complexes by treating ligands as point charges or dipoles interacting electrostatically with the metal's d-electrons. This interaction causes the five degenerate d-orbitals to split into different energy levels.

In octahedral complexes, the $t_{2g}$ orbitals are lowered in energy, and the $e_g$ orbitals are raised, with an energy difference of $\Delta_o$. In tetrahedral complexes, the splitting is inverted, and $\Delta_t$ is roughly $4/9$ of $\Delta_o$.

The magnitude of this splitting depends on the ligand, metal oxidation state, and metal identity. Ligands are ordered by their field strength in the spectrochemical series. For $d^4$ to $d^7$ octahedral complexes, the spin state (high spin or low spin) is determined by the competition between $\Delta_o$ and pairing energy (P).

If $\Delta_o < P$, it's high spin; if $\Delta_o > P$, it's low spin. This electron configuration dictates the complex's magnetic properties (paramagnetic if unpaired electrons, diamagnetic if all paired) and its color (due to d-d electronic transitions absorbing specific wavelengths of light corresponding to $\Delta$).

CFSE quantifies the stabilization gained from this splitting.

5-Minute Revision

Crystal Field Theory (CFT) is a cornerstone for understanding transition metal complexes, moving beyond the limitations of VBT by focusing on electrostatic interactions. It postulates that ligands, acting as point charges or dipoles, create an electrostatic field around the central metal ion.

This field repels the metal's d-electrons, but not uniformly, due to the varying spatial orientations of the five d-orbitals. This differential repulsion lifts the degeneracy of the d-orbitals, causing them to split into different energy levels.

For octahedral complexes, six ligands approach along the x, y, z axes. The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals ($e_g$ set) point directly at the ligands and are raised in energy by $0.6\Delta_o$. The $d_{xy}$, $d_{yz}$, $d_{xz}$ orbitals ($t_{2g}$ set) point between the axes and are lowered in energy by $0.4\Delta_o$. The energy difference is $\Delta_o$.

For tetrahedral complexes, four ligands approach from the corners of a tetrahedron. Here, the $d_{xy}$, $d_{yz}$, $d_{xz}$ orbitals ($t_2$ set) are closer to the ligands and are raised in energy by $0.4\Delta_t$. The $d_{x^2-y^2}$ and $d_{z^2}$ orbitals ($e$ set) are lowered in energy by $0.6\Delta_t$. The splitting is inverted compared to octahedral, and $\Delta_t \approx \frac{4}{9}\Delta_o$, making tetrahedral complexes almost universally high spin.

The spectrochemical series ranks ligands by their ability to cause splitting: $\text{I}^- < \text{Br}^- < \text{Cl}^- < \text{F}^- < \text{OH}^- < \text{H}_2\text{O} < \text{NH}_3 < \text{en} < \text{NO}_2^- < \text{CN}^- < \text{CO}$. Strong field ligands (right side) cause large $\Delta$, weak field ligands (left side) cause small $\Delta$.

For $d^4, d^5, d^6, d^7$ octahedral complexes, the spin state depends on the competition between $\Delta_o$ and pairing energy (P):

High spin — If $\Delta_o < P$ (weak field), electrons occupy $e_g$ before pairing in $t_{2g}$, maximizing unpaired electrons.
Low spin — If $\Delta_o > P$ (strong field), electrons pair up in $t_{2g}$ before occupying $e_g$, minimizing unpaired electrons.

Crystal Field Stabilization Energy (CFSE) is the net energy stabilization: $\text{CFSE} = [n_{t_{2g}}(-0.4\Delta_o) + n_{e_g}(+0.6\Delta_o)] + mP$. The $mP$ term is added for extra pairs formed due to strong field splitting.

Magnetic properties are determined by the number of unpaired electrons ($n$). Paramagnetic if $n > 0$, diamagnetic if $n = 0$. Magnetic moment $\mu = \sqrt{n(n+2)}$ BM.

Color arises from d-d electronic transitions. Electrons absorb light of energy $\Delta$ to jump from lower to higher d-orbitals. The observed color is the complementary color of the absorbed light. A larger $\Delta$ means higher energy absorption, corresponding to shorter wavelength light.

Example: For $\text{[Fe(CN)}_6\text{]}^{4-}$:

1
$\text{Fe}^{2+}$ is $d^6$.
2
$\text{CN}^-$ is a strong field ligand, so $\Delta_o > P$.
3
Electron configuration: $t_{2g}^6 e_g^0$ (low spin).
4
Unpaired electrons ($n$) = 0. Complex is diamagnetic.
5
CFSE = $6(-0.4\Delta_o) + 0(+0.6\Delta_o) + 2P = -2.4\Delta_o + 2P$.

Prelims Revision Notes

Crystal Field Theory (CFT) - NEET Revision Notes

1. Core Concept:

Electrostatic model: Ligands are point charges (anions) or dipoles (neutral molecules).
No covalent bonding or orbital overlap considered.
Repulsion between metal d-electrons and ligand electrons causes d-orbital splitting.

2. d-Orbital Splitting Patterns:

Isolated Metal Ion: — All five d-orbitals ($d_{xy}, d_{yz}, d_{xz}, d_{x^2-y^2}, d_{z^2}$) are degenerate.
Octahedral Field ($ML_6$): — Ligands approach along axes.

* $e_g$ set ($d_{x^2-y^2}, d_{z^2}$) raised in energy by $0.6\Delta_o$. * $t_{2g}$ set ($d_{xy}, d_{yz}, d_{xz}$) lowered in energy by $0.4\Delta_o$. * Splitting energy: $\Delta_o$ (or $10Dq$).

Tetrahedral Field ($ML_4$): — Ligands approach between axes.

* $t_2$ set ($d_{xy}, d_{yz}, d_{xz}$) raised in energy by $0.4\Delta_t$. * $e$ set ($d_{x^2-y^2}, d_{z^2}$) lowered in energy by $0.6\Delta_t$. * Splitting energy: $\Delta_t$. Inverted splitting compared to octahedral. * Relationship: $\Delta_t \approx \frac{4}{9}\Delta_o$.

Square Planar Field ($ML_4$): — More complex splitting, generally larger $\Delta$ than octahedral.

3. Factors Affecting $\Delta$:

Ligand Nature: — Strong field ligands (e.g., $\text{CN}^-$, $\text{CO}$) cause large $\Delta$. Weak field ligands (e.g., $\text{Cl}^-$, $\text{F}^-$, $\text{H}_2\text{O}$) cause small $\Delta$.
Spectrochemical Series: — $\text{I}^- < \text{Br}^- < \text{S}^{2-} < \text{SCN}^- < \text{Cl}^- < \text{NO}_3^- < \text{F}^- < \text{OH}^- < \text{C}_2\text{O}_4^{2-} \approx \text{H}_2\text{O} < \text{NCS}^- < \text{EDTA}^{4-} < \text{NH}_3 \approx \text{py} < \text{en} < \text{NO}_2^- < \text{CN}^- < \text{CO}$.
Metal Oxidation State: — Higher oxidation state $\rightarrow$ larger $\Delta$ (e.g., $\text{Fe}^{3+}$ vs $\text{Fe}^{2+}$).
Metal Identity (Period): — $\Delta$ increases down a group ($5d > 4d > 3d$).

4. Electron Filling & Spin States (Octahedral $d^4-d^7$):

Pairing Energy (P): — Energy required to pair electrons in an orbital.
High Spin Complex: — Occurs when $\Delta_o < P$ (weak field ligands). Electrons occupy $e_g$ orbitals singly before pairing in $t_{2g}$. Maximize unpaired electrons.
Low Spin Complex: — Occurs when $\Delta_o > P$ (strong field ligands). Electrons pair up in $t_{2g}$ orbitals before occupying $e_g$. Minimize unpaired electrons.
Note: — $d^1, d^2, d^3, d^8, d^9, d^{10}$ have only one possible spin state.
Tetrahedral complexes — are almost always high spin due to small $\Delta_t$.

5. Crystal Field Stabilization Energy (CFSE):

Formula for octahedral: $\text{CFSE} = [n_{t_{2g}}(-0.4\Delta_o) + n_{e_g}(+0.6\Delta_o)] + mP$.
$mP$ term: Number of extra electron pairs formed due to strong field splitting, relative to the unsplit configuration.

6. Magnetic Properties:

Paramagnetic: — Contains unpaired electrons ($n > 0$).
Diamagnetic: — All electrons are paired ($n = 0$).
Spin-only Magnetic Moment: — $\mu = \sqrt{n(n+2)}$ Bohr Magnetons (BM).

7. Color of Complexes:

Due to d-d electronic transitions: Electrons absorb light energy equal to $\Delta$ to jump from lower to higher d-orbitals.
$E = \Delta = h\nu = hc/\lambda$.
Observed color is the complementary color of the light absorbed. Larger $\Delta \rightarrow$ higher energy absorbed $\rightarrow$ shorter wavelength absorbed.

Vyyuha Quick Recall

To remember the spectrochemical series (common ligands):

I Brought Some Cold Coffee, Now For Orange Water, Nice Eggs, And Every New Cake Comes Out.

I$^-$ < Br$^-$ < S$^{2-}$ < SCN$^-$ < Cl$^-$ < **NO}_3^-$< **F**$^-$< **OH**$^-$< **C**$_2$**O**$_4^{2-}$$\approx$**H**$_2$**O** < **NCS**$^-$< **EDTA**$^{4-}< **NH}_3 $\approx$ py < en < **NO}_2^-$< **CN**$^-$ < CO

(Note: This mnemonic covers a comprehensive list, for NEET focus on the more common ones like halides, water, ammonia, ethylenediamine, cyanide, CO.)