Question

Explain the crystal field splitting in octahedral complexes with the help of a diagram. (4)

Hint:

Six ligands along the axes split the five d orbitals into lower \(t_{2g}\) (between axes) and higher \(e_g\) (along axes); the gap is \(\Delta_o\), with \(t_{2g}\) at \(-0.4\Delta_o\) and \(e_g\) at \(+0.6\Delta_o\).

Answer 1

Step 1 (Starting point): In a free (gaseous) metal ion the five d orbitals (\(d_{xy}, d_{yz}, d_{zx}, d_{x^2-y^2}, d_{z^2}\)) have exactly the same energy, i.e. they are degenerate.
Step 2 (Approach of ligands): In an octahedral complex six ligands approach the metal ion along the three axes (the +x, −x, +y, −y, +z and −z directions). The lone pairs of the ligands create a negative electric field around the metal.
Step 3 (Effect on the two sets of orbitals): The two \(e_g\) orbitals \((d_{x^2-y^2}\) and \(d_{z^2})\) lie along the axes, so they point straight at the ligands and feel strong repulsion. Their energy is raised. The three \(t_{2g}\) orbitals \((d_{xy}, d_{yz}, d_{zx})\) lie between the axes, so they feel less repulsion and their energy is lowered.
Step 4 (Splitting): Thus the five degenerate d orbitals split into two sets. The energy gap between them is called the crystal field splitting energy, \(\Delta_o\) (o = octahedral).
Step 5 (Barycentre rule): The average energy stays constant (barycentre). The \(t_{2g}\) set is lowered by \(0.4\,\Delta_o\) (i.e. \(-0.4\Delta_o\) each) and the \(e_g\) set is raised by \(0.6\,\Delta_o\) each, so the centre of gravity is unchanged.
Step 6 (Diagram, described):
\[ \underbrace{d_{x^2-y^2},\; d_{z^2}}_{e_g \text{ (higher, } +0.6\Delta_o)} \]
\[ \text{---- barycentre (average energy) ----} \]
\[ \underbrace{d_{xy},\; d_{yz},\; d_{zx}}_{t_{2g} \text{ (lower, } -0.4\Delta_o)} \]
Free ion (5 degenerate d orbitals) → spherical field (all raised equally) → octahedral field (split into lower \(t_{2g}\) and higher \(e_g\)).
Step 7 (Result): The magnitude of \(\Delta_o\) decides whether the complex is high spin (weak field, small \(\Delta_o\)) or low spin (strong field, large \(\Delta_o\)), and it explains the colour and magnetic behaviour of the complex.
\[\boxed{e_g: +0.6\Delta_o,\quad t_{2g}: -0.4\Delta_o}\]