🔮 Coordination Chemistry & Crystal Field Theory

Coordination chemistry combines transition metal d-orbital physics with ligand coordination field theory. In competitive exams (IIT-JEE Advanced & Olympiads), coordination chemistry is the single highest-weightage topic in inorganic chemistry, encompassing stereoisomer counting, CFSE calculations, spin-only magnetic moments, and synergic $\pi$-back-bonding.

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1. 🏛️ Werner's Coordination Theory

1. Primary Valency: Ionizable, corresponds to the oxidation state of the central metal, satisfied by anions.
2. Secondary Valency: Non-ionizable, corresponds to the coordination number, directional in space, satisfied by neutral molecules or anions.

Example: Precipitation with excess $AgN{O}_3$:

• $[Co(N{H}_3{)}_6]C{l}_3\to 3\text{ moles of }AgCl\downarrow$ per mole of complex (3 ionizable $C{l}^{-}$).
• $[Co(N{H}_3{)}_{5}Cl]C{l}_2\to 2\text{ moles of }AgCl\downarrow$.
• $[Co(N{H}_3{)}_{4}C{l}_2]Cl\to 1\text{ mole of }AgCl\downarrow$.
• $[Co(N{H}_3{)}_{3}C{l}_3]\to 0\text{ moles of }AgCl\downarrow$ (neutral complex, non-conducting).

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2. 🔀 Isomerism in Coordination Complexes

2.1 Structural Isomerism

1. Ionization Isomerism: Exchange of ions inside and outside coordination sphere ($[Co(N{H}_3{)}_{5}Br]S{O}_4$ gives white $BaS{O}_4\downarrow$ with $BaC{l}_2$, while $[Co(N{H}_3{)}_{5}S{O}_4]Br$ gives pale yellow $AgBr\downarrow$ with $AgN{O}_3$).
2. Hydrate / Solvate Isomerism: Difference in number of water molecules as ligands vs lattice water ($[Cr(H_{2}O{)}_6]C{l}_3$ violet, $[Cr(H_{2}O{)}_{5}Cl]C{l}_2\cdot H_{2}O$ grey-green).
3. Linkage Isomerism: Occurs with ambidentate ligands ($N{O}_2^{-}/ON{O}^{-}$, $SC{N}^{-}/NC{S}^{-}$, $C{N}^{-}/N{C}^{-}$).
4. Coordination Isomerism: Exchange of ligands between cationic and anionic complex entities ($[Co(N{H}_3{)}_6][Cr(CN{)}_6]$ and $[Cr(N{H}_3{)}_6][Co(CN{)}_6]$).

2.2 Stereoisomerism (Geometrical & Optical)

Complex Formula	Geometrical Isomers (GI)	Enantiomeric Pairs (Optical)	Total Stereoisomers
$[M{a}_2{b}_2]$ (Square Planar)	2 (cis, trans)	0 (all planar complexes have $\sigma$ plane)	2
$[Mabcd]$ (Square Planar)	3	0	3
$[M{a}_4{b}_2]$ (Octahedral)	2 (cis, trans)	0 (cis has plane, trans has inversion center)	2
$[M{a}_3{b}_3]$ (Octahedral)	2 (facial / fac, meridional / mer)	0	2
$[M(AA{)}_3]$ (e.g. $[Co(en{)}_3{]}^{3+}$)	1	1 pair ($d$ and $l$ enantiomers)	2 (Chiral, optically active!)
$[M(AA{)}_2{b}_2]$ (e.g. $[Co(en{)}_{2}C{l}_2{]}^{+}$)	2 (cis, trans)	cis is optically active (1 pair $d/l$); trans is achiral	3 (2 cis enantiomers + 1 trans)

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3. 💎 Crystal Field Theory (CFT)

CFT treats ligands as point negative charges that split the degenerate $d$-orbitals of the central metal ion via electrostatic repulsion.

3.1 Octahedral Field Splitting (${\mathrm{\Delta }}_o$)

The 5 degenerate $d$-orbitals split into two sets:

• $e_g$ set ($d_{x^2-y^2},d_{z^2}$): Point directly along the axes $⟹$ destabilized by $+0.6{\mathrm{\Delta }}_o$ ($+\frac{3}{5}{\mathrm{\Delta }}_o$).
• $t_{2g}$ set ($d_{xy},d_{yz},d_{xz}$): Point in between the axes $⟹$ stabilized by $-0.4{\mathrm{\Delta }}_o$ ($-\frac{2}{5}{\mathrm{\Delta }}_o$).

$${\text{CFSE}}_{\mathbf{ext}\mathbf{oct}}\mathbf{=}(\mathbf{-}\mathbf{0.4}{\mathbf{n}}_{{\mathbf{t}}_{\mathbf{2}\mathbf{g}}}\mathbf{+}\mathbf{0.6}{\mathbf{n}}_{{\mathbf{e}}_{\mathbf{g}}}){\mathbf{\Delta }}_{\mathbf{o}}\mathbf{+}\mathbf{m}\mathbf{P}$$

where $P$ is the electron pairing energy and $m$ is the number of newly paired electron pairs.

3.2 High-Spin vs Low-Spin Complexes

• Strong Field Ligands (${\mathrm{\Delta }}_o>P$): Pairing occurs in $t_{2g}$ before filling $e_g$ $⟹$ Low-Spin / Inner Orbital Complexes ($d^{2}s{p}^3$).
• Weak Field Ligands (${\mathrm{\Delta }}_o<P$): Electrons occupy $e_g$ singly before pairing $⟹$ High-Spin / Outer Orbital Complexes ($s{p}^3{d}^2$).

$$\text{Spectrochemical Series: }{I}^{-}<B{r}^{-}<S^{2-}<SC{N}^{-}<C{l}^{-}<N{O}_3^{-}<F^{-}<O{H}^{-}<C_2{O}_4^{2-}<H_{2}O<NC{S}^{-}<EDT{A}^{4-}<N{H}_3<en<N{O}_2^{-}<PP{h}_3<C{N}^{-}<CO$$

3.3 Tetrahedral Field Splitting (${\mathrm{\Delta }}_t$)

Tetrahedral splitting is inverted ($e$ lower, $t_2$ higher) and significantly smaller:

$${\mathbf{\Delta }}_{\mathbf{t}}\mathbf{=}\frac{\mathbf{4}}{\mathbf{9}}{\mathbf{\Delta }}_{\mathbf{o}}$$

Because ${\mathrm{\Delta }}_t<P$ always, tetrahedral complexes are ALMOST ALWAYS high-spin.

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4. 🧲 Synergic Bonding in Metal Carbonyls

In transition metal carbonyls ($M(CO{)}_n$, e.g. $Ni(CO{)}_4,Fe(CO{)}_5,Cr(CO{)}_6$):

1. $\sigma$-bond: Ligand $CO$ donates lone pair from carbon into empty hybrid metal orbital ($CO\to M$).
2. $\pi$-back-bond: Metal filled $d$-orbital back-donates electron density into empty ${\pi }^{\ast }$ antibonding orbital of $CO$ ($M\to CO$).

Consequences of Synergic Bonding:

$$\mathbf{M}\mathbf{-}\mathbf{C}\text{ bond order increases (shorter, stronger }\mathbf{M}\mathbf{-}\mathbf{C}\text{ bond)}$$$$\mathbf{C}\mathbf{-}\mathbf{O}\text{ bond order decreases (longer, weaker }\mathbf{C}\mathbf{-}\mathbf{O}\text{ bond, lower IR vibrational frequency }{\nu }_{\mathbf{CO}}\mathbf{)}$$

Comparison of $C-O$ bond strength / stretching frequency ${\nu }_{CO}$:

$$[Mn(CO{)}_6{]}^{+}>[Cr(CO{)}_6]>[V(CO{)}_6{]}^{-}>[Ti(CO{)}_6{]}^{2-}$$

(Higher negative charge on metal $⟹$ more back-bonding $⟹$ weaker $C-O$ bond $⟹$ lowest ${\nu }_{CO}$ frequency).