Why Ruby Is Red and Emerald Is Green

Two complexes of the same metal can differ completely — one violet and strongly magnetic, another yellow and non-magnetic. Bonding theories explain these differences by looking at how ligands interact with the metal's d orbitals.

This part covers valence bond theory, crystal field splitting, colour and magnetism from crystal field theory, and bonding in metal carbonyls with the uses of complexes.

**Valence bond theory says the metal's empty orbitals hybridise to accept electron pairs from ligands, so the hybridisation fixes the shape — $s{p}^3$ tetrahedral, $ds{p}^2$ square planar, $d^{2}s{p}^3$ or $s{p}^3{d}^2$ octahedral — and any unpaired electrons left behind decide the magnetism.

Common cases:

- $d^{2}s{p}^3$ — octahedral inner-orbital** complex, using $(n-1)d$ orbitals
- **$s{p}^3{d}^2$ — octahedral outer-orbital** complex, using $nd$ orbitals
- **$ds{p}^2$ — square planar
- $s{p}^3$ — tetrahedral

Worked example 1.** In $[{\text{Co(NH}}_3{)}_6{]}^{3+}$, Co${}^{3+}$ is $3d^6$. Its six electrons pair up in three $3d$ orbitals, leaving two free for $d^{2}s{p}^3$ hybridisation: octahedral, inner-orbital and diamagnetic.

Worked example 2. In $[{\text{CoF}}_6{]}^{3-}$ the electrons stay unpaired, so outer $4d$ orbitals are used — $s{p}^3{d}^2$, outer-orbital and paramagnetic with $4$ unpaired electrons:

$$\mu =\sqrt{4\times 6}\approx 4.90\text{ BM}$$

Worked example 3. In $[{\text{Ni(CN)}}_4{]}^{2-}$, Ni${}^{2+}$ is $3d^8$; pairing frees one $3d$ orbital for $ds{p}^2$, so it is square planar and diamagnetic.

An everyday example. A simple magnetic balance in a chemistry laboratory pulls a paramagnetic complex into the field and pushes a diamagnetic one slightly out, confirming these predictions.

The substance. Valence bond theory cannot predict in advance whether electrons will pair — it explains the magnetism only once it has been measured.

**In crystal field theory, ligands act as point charges that raise the energies of the metal's d orbitals unequally: in an octahedral field they split into a lower $t_{2g}$ set and a higher $e_g$ set separated by ${\Delta }_o$, and the complex is low spin when ${\Delta }_o$ exceeds the pairing energy $P$ and high spin when it is smaller.

Octahedral splitting:

- $t_{2g}$** ($d_{xy}$, $d_{yz}$, $d_{xz}$) — lowered by $0.4{\Delta }_o$
- **$e_g$** ($d_{x^2-y^2}$, $d_{z^2}$) — raised by $0.6{\Delta }_o$, since they point straight at the ligands

Tetrahedral splitting is reversed and smaller, ${\Delta }_t=\frac{4}{9}{\Delta }_o$, so tetrahedral complexes are almost always high spin.

Spectrochemical series, from weak to strong field:

$${\text{I}}^{-}<{\text{Br}}^{-}<{\text{Cl}}^{-}<{\text{F}}^{-}<{\text{OH}}^{-}<{\text{H}}_2\text{O}<{\text{NH}}_3<\text{en}<{\text{CN}}^{-}<\text{CO}$$

Crystal field stabilisation energy for $x$ electrons in $t_{2g}$ and $y$ in $e_g$ is $(-0.4x+0.6y){\Delta }_o$, plus $P$ for each extra pair forced together.

Worked example. For a $d^4$ ion:

- Weak field (${\Delta }_o<P$): $t_{2g}^3{e}_g^1$, CFSE $=-1.2+0.6=-0.6{\Delta }_o$, with $4$ unpaired electrons
- Strong field (${\Delta }_o>P$): $t_{2g}^4{e}_g^0$, CFSE $=-1.6{\Delta }_o+P$, with $2$ unpaired electrons

An everyday example. Blue cobalt chloride paper turns pink in moist air as water ligands replace chloride and the complex changes from tetrahedral to octahedral, altering the splitting.

The substance. **Strong-field ligands such as CN${}^{-}$ and CO force electrons to pair**, which is why cyanide complexes are so often low spin.

**A complex is coloured because an electron in the lower d set absorbs light of energy $\Delta$ to jump to the higher set, so we see the complementary colour, and its magnetism follows from the number of unpaired electrons left after filling the split orbitals.

Colour:**

- $[{\text{Ti(H}}_2{\text{O)}}_6{]}^{3+}$, a $d^1$ ion, absorbs light around $498$ nm and looks violet
- Stronger-field ligands raise ${\Delta }_o$ and shift absorption to shorter wavelength: $[{\text{Ni(H}}_2{\text{O)}}_6{]}^{2+}$ is green, but $[{\text{Ni(en)}}_3{]}^{2+}$ is violet

Worked example. Light of wavelength $498$ nm carries energy

$$E=\frac{hc}{\lambda }=\frac{6.63\times 10^{-34}\times 3\times 10^8}{498\times 10^{-9}}\approx 4.0\times 10^{-19}\text{ J}$$

which is ${\Delta }_o$ for one ion, or about $4.0\times 10^{-19}\times 6.02\times 10^{23}\approx 240$ kJ mol${}^{-1}$.

Magnetism. $[{\text{Fe(CN)}}_6{]}^{3-}$ is low spin, $t_{2g}^5$, with $1$ unpaired electron and $\mu \approx 1.73$ BM; $[{\text{FeF}}_6{]}^{3-}$ is high spin, $t_{2g}^3{e}_g^2$, with $5$ and $\mu \approx 5.92$ BM.

Limitations:

- Valence bond theory — cannot explain colour or tell weak ligands from strong ones
- Crystal field theory — treats ligands as point charges, so it cannot explain why neutral water is a stronger ligand than the F${}^{-}$ ion, and it ignores covalent bonding

An everyday example. Blue copper sulphate crystals turn white when heated, as the water ligands responsible for the splitting are driven off.

The substance. **Neutral water outranking charged F${}^{-}$ in ligand strength** shows that bonding in complexes is partly covalent.

**In metal carbonyls, CO donates an electron pair from carbon into an empty metal orbital to form a $\sigma$ bond, while the metal donates electrons from a filled d orbital back into CO's empty ${\pi }^{\ast }$ orbital to form a $\pi$ bond; the two effects reinforce each other, giving synergic bonding.

Examples and shapes:**

- $[{\text{Ni(CO)}}_4]$ — tetrahedral
- $[{\text{Fe(CO)}}_5]$ — trigonal bipyramidal
- $[{\text{Cr(CO)}}_6]$ — octahedral

Why synergic. The $\sigma$ donation makes the metal electron-rich, which strengthens back-donation; back-donation removes that extra charge, allowing still more $\sigma$ donation.

Worked example. Counting valence electrons around the metal, each CO donates $2$:

$${\text{Fe(CO)}}_5:8+5\times 2=18,{\text{Ni(CO)}}_4:10+4\times 2=18,{\text{Cr(CO)}}_6:6+6\times 2=18$$

Importance of coordination compounds:

- Qualitative analysis — dimethylglyoxime gives a red precipitate with Ni${}^{2+}$; water hardness is measured with EDTA
- Extraction of metals — silver and gold dissolve as cyanide complexes; nickel is purified through $[{\text{Ni(CO)}}_4]$
- Biological systems — haemoglobin contains iron, chlorophyll magnesium and **vitamin B${}_{12}$ cobalt

An everyday example. Blood is red because of an iron complex, haemoglobin, which carries oxygen from the lungs to every cell.

The substance. Carbon monoxide is poisonous because it binds haemoglobin's iron far more strongly than oxygen does** — the same back-bonding ability seen in carbonyls.