Coordination Chemistry

1. Crystal Field Theory

1.1 The Crystal Field Concept

Definition 1 (Crystal Field Theory): A model in which ligands are treated as point charges (or point dipoles) that interact electrostatically with the $d$ orbitals of the central metal ion.

1.2 Octahedral Complexes

In an octahedral field, the five $d$ orbitals split into two groups:

Theorem 1 (Octahedral Crystal Field Splitting):

$\Delta_o = 10\,Dq$

• $e_g$ orbitals ($d_{z^2}$, $d_{x^2-y^2}$): Higher energy, point directly at ligands.
• $t_{2g}$ orbitals ($d_{xy}$, $d_{xz}$, $d_{yz}$): Lower energy, point between ligands.

$E(e_g) = +0.6\,\Delta_o = +6\,Dq$ $E(t_{2g}) = -0.4\,\Delta_o = -4\,Dq$

The barycenter (weighted average) is preserved: $2(+6\,Dq) + 3(-4\,Dq) = 0$.

Example 1: For $[\text{Ti(H}_2\text{O)}_6]^{3+}$ ($d^1$), the single electron occupies $t_{2g}$. The absorption at 20,300 cm$^{-1}$ gives $\Delta_o = 20,300$ cm$^{-1}$ = 243 kJ/mol.

$\blacksquare$

1.3 Tetrahedral Complexes

Theorem 2 (Tetrahedral Crystal Field Splitting):

$\Delta_t = \frac{4}{9}\Delta_o$

The splitting is inverted and smaller:

• $e$ orbitals (lower): $d_{z^2}$, $d_{x^2-y^2}$
• $t_2$ orbitals (higher): $d_{xy}$, $d_{xz}$, $d_{yz}$

$E(t_2) = +0.6\,\Delta_t, \quad E(e) = -0.4\,\Delta_t$

1.4 Square Planar Complexes

Derived from octahedral by removing the two axial ligands. The $d$-orbital energies:

$d_{x^2-y^2} > d_{xy} > d_{z^2} > d_{xz}, d_{yz}$

Square planar complexes are common for $d^8$ metals (Ni$^{2+}$, Pd$^{2+}$, Pt$^{2+}$, Au$^{3+}$).

2. Spectrochemical Series and $\Delta$

2.1 The Spectrochemical Series

Definition 2 (Spectrochemical Series): Ranking of ligands by their ability to split $d$ orbitals (weak-field to strong-field):

$\text{I}^- < \text{Br}^- < \text{Cl}^- < \text{SCN}^- < \text{F}^- < \text{OH}^- < \text{ox}^{2-} < \text{H}_2\text{O} < \text{NCS}^- < \text{NH}_3 < \text{en} < \text{bipy} < \text{NO}_2^- < \text{CN}^- < \text{CO}$

Spectrochemical series of metals:

$\text{Mn}^{2+} < \text{Ni}^{2+} < \text{Co}^{2+} < \text{Fe}^{2+} < \text{Fe}^{3+} < \text{Cr}^{3+} < \text{Co}^{3+} < \text{Rh}^{3+} < \text{Ir}^{3+} < \text{Pt}^{4+}$

Higher oxidation states and heavier metals produce larger $\Delta$.

2.2 Factors Affecting $\Delta$

$\Delta \propto \frac{Z\,q}{r^5}$

where $Z$ is the metal charge, $q$ is the ligand charge, and $r$ is the metal-ligand distance.

• Higher metal oxidation state → larger $\Delta$.
• Stronger ligand field → larger $\Delta$.
• 4d and 5d metals → larger $\Delta$ (diffuse orbitals interact more with ligands).

3. High-Spin vs Low-Spin Complexes

3.1 The Decision

Theorem 3 (High-Spin vs Low-Spin): When pairing energy $P$ is compared to $\Delta$:

• $\Delta < P$: High-spin (weak field). Electrons fill all orbitals singly before pairing.
• $\Delta > P$: Low-spin (strong field). Electrons pair in lower orbitals before occupying upper orbitals.

Only octahedral $d^4$–$d^7$ complexes have a high-spin/low-spin choice.

3.2 Electron Configurations

Example 2: $[\text{Fe(H}_2\text{O)}_6]^{2+}$ ($d^6$, $\Delta_o \approx 10,400$ cm$^{-1}$): High-spin ($\Delta_o < P$), $t_{2g}^4\,e_g^2$, 4 unpaired electrons.

$[\text{Fe(CN)}_6]^{4-}$ ($d^6$, $\Delta_o \approx 32,200$ cm$^{-1}$): Low-spin ($\Delta_o > P$), $t_{2g}^6\,e_g^0$, 0 unpaired electrons.

$\blacksquare$

4. Crystal Field Stabilization Energy (CFSE)

4.1 Calculation

Definition 3 (CFSE): The net energy lowering of a complex relative to the barycenter:

$\text{CFSE} = n_{t_{2g}}(-4\,Dq) + n_{e_g}(+6\,Dq) + n_p\,P$

where $n_{t_{2g}}$ and $n_{e_g}$ are electron counts, $P$ is the pairing energy, and $n_p$ is the number of extra electron pairs relative to the high-spin configuration.

Example 3: CFSE for $d^6$ low-spin octahedral:

$\text{CFSE} = 6(-4\,Dq) + 0(+6\,Dq) + 3P = -24\,Dq + 3P$

For $d^6$ high-spin:

$\text{CFSE} = 4(-4\,Dq) + 2(+6\,Dq) + 0P = -4\,Dq$

$\blacksquare$

4.2 CFSE and Thermodynamic Properties

CFSE contributes to:

• Lattice energies (hydrated transition metal ions).
• Hydration enthalpies (dip in the series at $d^3$, $d^8$ due to extra stabilization).
• Ligand substitution rates (low-spin $d^6$ is inert; high-spin $d^5$ is labile).

5. Ligand Field Theory

5.1 Beyond Crystal Field Theory

Definition 4 (Ligand Field Theory): An extension of CFT that includes covalent bonding (sigma and pi interactions between metal and ligand orbitals) alongside electrostatic effects.

5.2 Sigma Bonding

Ligand donor orbitals overlap with metal $d_{z^2}$, $d_{x^2-y^2}$, $s$, $p_x$, $p_y$, $p_z$ to form $\sigma$ bonding and $\sigma^*$ antibonding molecular orbitals.

5.3 Pi Bonding

Pi-donor ligands (e.g., F$^-$, O$^{2-}$, Cl$^-$):

• Donate electron density into empty metal $t_{2g}$ orbitals.
• Decrease $\Delta_o$ (weak field).
• Examples: halides, oxide, hydroxide.

Pi-acceptor ligands (e.g., CO, CN$^-$, NO):

• Accept electron density from filled metal $t_{2g}$ orbitals into empty ligand $\pi^*$ orbitals.
• Increase $\Delta_o$ (strong field).
• Examples: CO, CN$^-$, phosphines (PR$_3$).

This explains the spectrochemical series: $\pi$-acceptors > no $\pi$ interaction > $\pi$-donors.

6. The Jahn-Teller Effect

6.1 Statement

Theorem 4 (Jahn-Teller Theorem): Any nonlinear molecular system in a degenerate electronic state will undergo distortion to remove the degeneracy.

6.2 Octahedral Jahn-Teller Distortion

High-spin $d^4$ ($t_{2g}^3\,e_g^1$): One electron in $e_g$ — the complex elongates along one axis to lower the energy of the singly occupied orbital.

Low-spin $d^7$ ($t_{2g}^6\,e_g^1$): Same $e_g$ degeneracy — elongation.

$d^9$ ($t_{2g}^6\,e_g^3$): One hole in $e_g$ — strong Jahn-Teller effect (e.g., Cu$^{2+}$).

Examples:

• $[\text{Cu(H}_2\text{O)}_6]^{2+}$: Two long axial bonds (~2.4 Å) and four short equatorial bonds (~2.0 Å).
• $[\text{Mn(H}_2\text{O)}_6]^{3+}$ ($d^4$, high-spin): Elongated octahedral.

6.3 Consequences

• Splitting of $d$-orbital degeneracy leads to additional spectroscopic transitions.
• Structural distortions lower symmetry.
• $d^3$, $d^5$ (high-spin), and $d^8$ (low-spin) have no Jahn-Teller distortion (no degeneracy in $e_g$ or $t_{2g}$).

7. Magnetism

7.1 Spin-Only Formula

Theorem 5 (Spin-Only Magnetic Moment):

$\mu_{\text{eff}} = \sqrt{n(n+2)}\,\mu_B$

where $n$ is the number of unpaired electrons and $\mu_B = 9.274 \times 10^{-24}$ J/T is the Bohr magneton.

7.2 Magnetic Properties

• Diamagnetic: All electrons paired; $\mu_{\text{eff}} = 0$; repelled by magnetic field.
• Paramagnetic: Unpaired electrons; attracted to magnetic field.
• Spin crossover: Some $d^6$ complexes switch between high-spin and low-spin with temperature.

8. Stability Constants

8.1 Stepwise and Overall Formation Constants

Definition 5 (Formation Constant): For the reaction $\text{M}^{n+} + \text{L} \rightleftharpoons \text{ML}^{n+}$:

$K_1 = \frac{[\text{ML}^{n+}]}{[\text{M}^{n+}][\text{L}]}$

Overall formation constant:

$\beta_n = K_1 \cdot K_2 \cdot \ldots \cdot K_n = \frac{[\text{ML}_n^{n+}]}{[\text{M}^{n+}][\text{L}]^n}$

8.2 Chelate Effect

Theorem 6 (Chelate Effect): Multidentate ligands form more stable complexes than equivalent monodentate ligands:

$[\text{Ni(en)}_3]^{2+} \text{ (log } \beta_3 = 18.8) \gg [\text{Ni(NH}_3)_6]^{2+} \text{ (log } \beta_6 = 8.6)$

Explanation:

• Entropy: One chelate replaces several monodentate ligands, increasing the number of free particles ($\Delta S > 0$).
• Ring size: 5-membered chelate rings are most stable (en, acac). 3-membered rings are strained; 7+ membered rings are floppy.

8.3 Irving-Williams Series

Theorem 7 (Irving-Williams Series): The stability of M$^{2+}$ complexes with a given ligand:

$\text{Mn}^{2+} < \text{Fe}^{2+} < \text{Co}^{2+} < \text{Ni}^{2+} < \text{Cu}^{2+} > \text{Zn}^{2+}$

Explained by a combination of CFSE (peaks at $d^8$ Ni$^{2+}$) and Jahn-Teller effects (extra stabilization for Cu$^{2+}$, $d^9$).

9. Ligand Substitution Reactions

9.1 Inert and Labile Complexes

Definition 6 (Labile): Complexes that undergo rapid ligand substitution (half-life < 1 minute). Definition 7 (Inert): Complexes with slow ligand substitution (half-life > 1 minute).

Theorem 8: Low-spin $d^6$ complexes (e.g., $[\text{Co(CN)}_6]^{3-}$, $[\text{Cr(NH}_3)_6]^{3+}$) are inert. High-spin complexes and $d^{10}$ are labile.

9.2 Octahedral Substitution Mechanisms

S$\_\text{N}$1 (Dissociative): First, a ligand leaves, creating a 5-coordinate intermediate; then the new ligand enters.

$[\text{ML}_6] \to [\text{ML}_5] + \text{L} \to [\text{ML}_5\text{L}"]$

Rate: $v = k[\text{complex}]$ (independent of incoming ligand).

S$\_\text{N}$2 (Associative): The incoming ligand attacks to form a 7-coordinate intermediate; then a ligand leaves.

$[\text{ML}_6] + \text{L}' \to [\text{ML}_6\text{L}'] \to [\text{ML}_5\text{L}'] + \text{L}$

Rate: $v = k[\text{complex}][\text{L}']$.

9.3 Trans Effect (Square Planar)

Definition 8 (Trans Effect): In square planar complexes, some ligands labilize the ligand trans to them, accelerating its substitution.

Trans effect series:

$\text{CN}^- \approx \text{CO} \approx \text{C}_2\text{H}_4 > \text{PR}_3 > \text{H}^- > \text{SC(NH}_2)_2 > \text{CH}_3^- > \text{SCN}^- > \text{I}^- > \text{Br}^- > \text{Cl}^- > \text{py} > \text{NH}_3 > \text{OH}^- > \text{H}_2\text{O}$

This is exploited in the synthesis of square planar Pt complexes (e.g., cisplatin).

10. Electronic Spectra and Color

10.1 d–d Transitions

Definition 9 (d–d Transition): An electron is promoted from a lower-energy $d$ orbital to a higher-energy $d$ orbital, absorbing light in the visible or near-UV region.

$\Delta_o = h\nu = \frac{hc}{\lambda}$

• The absorbed wavelength determines the color (complementary color is observed).
• Selection rules: Laporte forbidden ($\Delta\ell = \pm 0$ not satisfied), but weakly allowed by vibronic coupling or low symmetry.

10.2 Orgel Diagrams

Definition 10 (Orgel Diagram): Qualitative diagrams showing the energy of $d$-orbital states as a function of $\Delta/B$ (field strength ratio).

For octahedral $d^n$, the Orgel diagram shows which transitions are spin-allowed and their approximate positions.

10.3 Charge Transfer Transitions

Definition 11 (Charge Transfer): Intense transitions involving electron transfer between metal and ligand:

• LMCT (Ligand to Metal Charge Transfer): Electron transfers from ligand to metal (e.g., $[\text{MnO}_4]^-$, purple color from O$^{2-}$ → Mn$^{7+}$).
• MLCT (Metal to Ligand Charge Transfer): Electron transfers from metal to ligand (e.g., $[\text{Ru(bpy)}_3]^{2+}$, MLCT absorption in visible).

Charge transfer transitions are much more intense ($\varepsilon \sim 10^3$–$10^5$) than d–d transitions ($\varepsilon \sim 1$–$100$).

Common Pitfalls

1. Confusing $\Delta_o$ and pairing energy $P$ units. $\Delta_o$ is in most cases in cm$^{-1}$ (wavenumbers) or kJ/mol; $P$ is in the same units. Fix: Always compare in the same units.
2. Wrong tetrahedral splitting direction. $\Delta_t$ is inverted relative to $\Delta_o$: $e$ is lower, $t_2$ is higher. Fix: Tetrahedral has fewer ligands and less direct overlap, so the splitting is smaller ($\frac{4}{9}\Delta_o$) and inverted.
3. Assuming all octahedral $d^n$ complexes can be high-spin or low-spin. Only $d^4$–$d^7$ have this choice. Fix: $d^1$–$d^3$ and $d^8$–$d^{10}$ have only one configuration regardless of field strength.
4. Ignoring the chelate effect for stability. EDTA forms extremely stable complexes not because of bond strength but because of entropy. Fix: $\beta_n$ for chelates is much larger than for monodentate analogs.
5. Wrong trans effect vs trans influence. Trans effect is a kinetic phenomenon (rate of substitution); trans influence is a thermodynamic phenomenon (bond weakening). Fix: Trans effect relates to substitution rates; trans influence relates to ground-state bond lengths.
6. Misassigning spectrochemical series positions. The spectrochemical series ranks ligands, not metals. Fix: Memorize the ligand series; also note that higher oxidation state metals produce larger $\Delta$.
7. Ignoring orbital contributions to magnetic moments. The spin-only formula works for first-row transition metals but fails for heavier metals where orbital contributions are significant. Fix: Use $\mu_{\text{eff}} = \sqrt{4S(S+1) + L(L+1)}\,\mu_B$ when orbital angular momentum is not quenched.

Summary

• CFT: $d$-orbital splitting in ligand fields; $\Delta_o$ (octahedral), $\Delta_t = \frac{4}{9}\Delta_o$ (tetrahedral).
• Spectrochemical series: I$^-$ < Cl$^-$ < F$^-$ < H$_2$O < NH$_3$ < CN$^-$ < CO.
• High-spin vs low-spin: Determined by $\Delta$ vs $P$; only $d^4$–$d^7$ octahedral.
• CFSE: Net stabilization from $d$-orbital splitting; explains hydration enthalpies.
• Jahn-Teller: Degenerate states distort; most important for $d^4$ (high-spin) and $d^9$.
• Magnetism: $\mu_{\text{eff}} = \sqrt{n(n+2)}\,\mu_B$; spin-only formula for first-row metals.
• Stability: Irving-Williams series; chelate effect (entropy-driven).
• Color: d–d transitions (weak, $\epsilon \sim 10$) and charge transfer (strong, $\epsilon \sim 10^4$).

Worked Examples

Example 1: CFSE Calculation

Problem: Calculate the crystal field stabilisation energy for [CoF6]^3- (high-spin, octahedral) and [Co(CN)6]^3- (low-spin, octahedral). Co^3+ has d^6 configuration. Delta_o for F- is 15,000 cm^-1 and for CN- is 33,000 cm^-1. Solution: High-spin [CoF6]^3-: t2g^4 eg^2. CFSE = 4(-0.4 Delta_o) + 2(0.6 Delta_o) = -1.6 + 1.2 = -0.4 Delta_o = -0.4 x 15,000 = -6,000 cm^-1 = -71.8 kJ/mol. Low-spin [Co(CN)6]^3-: t2g^6 eg^0. CFSE = 6(-0.4 Delta_o) + 0 = -2.4 Delta_o = -2.4 x 33,000 = -79,200 cm^-1 = -947.5 kJ/mol. The low-spin complex is much more stabilised.

Example 2: Isomer Counting

Problem: How many geometric isomers does [Co(NH3)2(en)2]^2+ have, and how many are optically active? (en = ethylenediamine, bidentate) Solution: The bidentate ligands occupy two coordination sites each. Possible arrangements: cis and trans for the NH3 pairs relative to each other. In the cis form, the two en ligands can be arranged as fac (with N atoms on a triangular face) or mer (with N atoms in a meridian). Total geometric isomers: 3 (cis-fac, cis-mer, trans). The cis-fac and cis-mer forms are chiral (no plane of symmetry), so there are 2 pairs of enantiomers.

Cross-References