Crystal field theory

Now we want to consider larger complexes with transition metals. The idea of crystal field theory is to use the properties/characteristics of the $d$ orbitals to explain the bonding patterns in transition metal complexes in a simple way (no LCAO).

The theory treats the complex as a central metal cation perturbed by the approach of negatively charged entities called the ligands, which are treated as simple point particles. In this way, the theory is physically motivated (and hence didn't appeal to chemists when it was first introduced).

In an octahedral complex, the charges approach the central metal cation along the $\pm x$, $\pm y$ and $\pm z$ axes. Now the approach of the negatively charged ligands perturbs the electrons in the $d$-orbitals of the metal and this changes the energies of these $d$ orbitals because of the Coulomb repulsion between the electrons in these orbitals and the ligands.

In the octahedral case, the electrons in the $d_{z^2}$ and $d_{x^2-y^2}$ orbitals, which have most of their amplitude along the coordinate axes, are the most strongly perturbed by the ligands, and their energy increases substantially.

Electrons in the $d_{xy}$, $d_{xz}$ and $d_{yz}$ orbitals have most of their amplitude between the coordinate axes. There is still Coulomb repulsion, but the repulsion is less than for the other two orbitals, and so these orbitals have their energies raised but to a lesser extent.

Consequently, the $d$ orbitals split into 2 groups. The lower energy orbitals are called $t_{2g}$ orbitals and correspond, in the octahedral case, to $d_{xz}$, $d_{yz}$ and $d_{xy}$. The higher energy orbitals are called $e_g$ and, in the octahedral case, corresponding to $d_{z^2}$ and $d_{x^2-y^2}$. The notation $t_{2g}$ and $e_g$ comes from group theory, so we won't go into the details of the notation. The $t_{2g}$ orbitals are triply degenerate while the $e_g$ orbitals are double degenerate.

The difference in energy between the $t_{2g}$ and $e_g$ orbitals is called the crystal field energy splitting $\Delta_0$. Let's apply the idea to a few examples.

First, consider Cr$^{3+}$, which has the electronic configuration [Ar]$3d^3$. In a coordination complex, the $d$ orbitals are pertubed so we have a diagram like:

Figure: Correlation diagram of Cr$^{3+}$

Hence, Cr$^{3+}$ are unpaired, so complexes of Cr$^{3+}$ are predicted to be paramagnetic. Note the example of [Cr(C$_2$O$_4$)$_3$] in which the C$_2$O$_4$ are chelating the Cr$^{3+}$.

As another example, consider Mn$^{3+}$, which has 4 $d$ electrons. Two configurations are possible:

Figure: Two possible correlation diagrams for Mn$^{3+}$ complexes

Here, the second configuration has a higher energy, but the cost of promoting the electron to an $e_g$ level is partially offset by the loss Coulomb repulsion between the two $d_{xy}$ electrons. The low energy configuration is a low spin configuration because there are only 2 unpaired electrons, while the high energy configuration is a high-spin configuration due to the four unpaired electrons. When $\Delta_0$ is large, the low spin configuration is favored, but when $\Delta_0$ is small, the high spin configuration is favored because the loss of Coulomb repulsion more than compensates for the cost of promotion. Hence, low-spin configurations with large $\Delta_0$ are called strong field configurations, while the high-spin configurations with small $\Delta_0$ are called weak field configurations.