Crystal Field Theory: Splitting in an Octahedral Field

Crystal field theory (CFT) takes a different, purely electrostatic approach from valence bond theory: it treats each ligand as a point negative charge (or, for a neutral ligand such as $\text{NH}_3$, the negative end of a dipole) that approaches the central metal ion and electrostatically repels its d-electrons. Because the five d-orbitals of a free, isolated metal ion have different spatial orientations, they are not all repelled equally once a specific arrangement of ligands actually surrounds the metal — and this unequal repulsion is what "splits" their energies apart.

In an octahedral complex, six ligands approach the metal ion symmetrically along the six directions of the Cartesian axes: $+x$, $-x$, $+y$, $-y$, $+z$, $-z$. Two of the five d-orbitals — $d_{z^2}$ and $d_{x^2-y^2}$, together called the $\boldsymbol{e_g}$ set — have their electron density concentrated directly along these same axes, pointing straight at the incoming ligands. Electrons in these orbitals therefore experience strong, direct electrostatic repulsion from the approaching ligand charges, and their energy is raised. The other three d-orbitals — $d_{xy}$, $d_{yz}$, and $d_{xz}$, together called the $\boldsymbol{t_{2g}}$ set — have their electron density concentrated between the axes, in the gaps where no ligand approaches directly. Electrons in these orbitals experience comparatively weaker repulsion, and their energy is lowered relative to the $e_g$ set.

Taking the average (hypothetical, spherically-symmetric) energy of all five orbitals together as a reference "barycentre," the $t_{2g}$ set is lowered by $0.4\,\Delta_o$ below this barycentre and the $e_g$ set is raised by $0.6\,\Delta_o$ above it — chosen so that, if all five orbitals were equally occupied (as in a $d^0$ or $d^{10}$ or exactly half-filled-and-doubled configuration), the total energy would come out exactly the same as the unsplit, spherical-field case (three orbitals lowered by $0.4\Delta_o$ plus two orbitals raised by $0.6\Delta_o$ sums to zero net change: $3(-0.4) + 2(0.6) = -1.2 + 1.2 = 0$). The total energy gap between the $t_{2g}$ and $e_g$ sets is called the crystal field splitting energy, symbol $\boldsymbol{\Delta_o}$ (the subscript "$o$" for octahedral; it is also frequently written $10Dq$). Its size depends on the identity of the ligand (Section 5.3), the identity and charge of the metal ion, and, to a smaller extent, the metal-ligand distance. …