6.9: Crystal Field Theory

Crystal field theory (CFT) is a bonding model that explains many important properties of transition-metal complexes, including their colors, magnetism, structures, stability, and reactivity. The central assumption of CFT is that metal–ligand interactions are purely electrostatic in nature. CFT qualitatively describes the strength of the metal-ligand interactions; based on the strength of the metal-ligand interactions, the energy of the system is altered and the degeneracy of the d-orbitals in transition metal complexes breaks. Even though the initial assumption is not strictly valid for many complexes, such as those that contain neutral ligands like CO, CFT enables chemists to explain many of the properties of transition-metal complexes with a reasonable degree of accuracy. T

Crystal Field Theory was developed by Hans Bethe and John Hasbrouck van Vleck. As the name implies, crystal field theory was developed as a way of explaining observed colors and magnetic properties of ionic crystalline solids. The colors of transition metel complexes results from the wavelength of light absorbed, which reveals something about differences in electronic energy levels in the ions of the crystalline solid. The magnetic properties depend on the presence of unpaired electrons and, thus, also depend on electronic energy levels.

Basic Concept

Consider a transition metal ion in an octahedral hole in a crystalline lattice (Figure \(\PageIndex{1}\)). In Crystal Field Theory, it is assumed that the ions are simple point charges (a simplification). The blue metal ion is a positive point charge that has six nearest neighbors, and the red ligands/counterions are negative point charges. The ions are held together by electrostatics: the negative charges of the (red) counterions are attracted to the positive charge of the (blue) metal ion. The attractive and repulsive interactions between the ions can be described by Coulomb's Law: 

\[E \propto \dfrac{q_1 q_2}{r} \nonumber \]

with

• \(E\) the bond energy between the charges and
• \(q_1\) and \(q_2\) are the charges of the interacting ions and
• \(r\) is the distance separating them.

Figure \(\PageIndex{1}\): A positively charged metal ion (blue) surrounded by six near neighbors with negative charges (red). (CC-BY-NC; Chris Schaller)

The attractive and repulsive forces between the positively charged metal ion and the surrounding ligands affect the energy of the electrons on the metal ion. To understand these interactions, we first consider a single transition metal ion; its five d-orbitals have the same energy (they are degenerate) (Figure \(\PageIndex{2}\)). When the metal ion is placed in a field of evenly distributed negative charge, there are repulsive forces between the metal ion's electrons and the surrounding negative charge. The result is the electrons on the metal increase in energy, and they all increase in energy by the same amount.

Figure \(\PageIndex{2}\): An Octahedral Arrangement of Six Negative Charges around a Metal Ion Causes the Five d Orbitals to Split into Two Sets with Different Energies. (a) Distributing a charge of −6 uniformly over a spherical surface surrounding a metal ion causes the energy of all five d orbitals to increase due to electrostatic repulsions, but the five d orbitals remain degenerate. Placing a charge of −1 at each vertex of an octahedron causes the d orbitals to split into two groups with different energies: the d_x²−y² and d_z² orbitals increase in energy, while the d_xy, d_xz, and d_yz orbitals decrease in energy. The average energy of the five d orbitals is the same as for a spherical distribution of a −6 charge, however. Attractive electrostatic interactions between the negatively charged ligands and the positively charged metal ion (far right) cause all five d orbitals to decrease in energy but does not affect the splittings of the orbitals. (b) The two e_g orbitals (left) point directly at the six negatively charged ligands, which increases their energy compared with a spherical distribution of negative charge. In contrast, the three t_2g orbitals (right) point between the negatively charged ligands, which decreases their energy compared with a spherical distribution of charge. (CC BY-SA-NC; anonymous by request)

In reality, the negative charges in a transition metal complex are not uniformly distributed around the positive metal ion. When the ligand point charges approach the metal ion, some experience more repulsion from the d-orbital electrons than others based on the geometric structure of the molecule. Due to the variation in shape and orientation of the d-orbitals, some ligands will interact more directly with d-orbitals and others will have less direct interactions. For an octahedral complex, the ligands approach the metal ion along the \(x\), \(y\), and \(z\) axes. Therefore, the electrons in the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals, which lie along these axes, experience greater repulsion (Figure \(\PageIndex{2}\)). It requires more energy to have an electron in these orbitals than it would to put an electron in one of the other orbitals due to the electron-electron repulsions. The energies of the \(d_{z^2}\) and \(d_{x^2-y^2}\) orbitals increase due to greater interactions with the ligands. By the same logic, the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals have less-direct interactions and decrease in energy. This causes a splitting in the energy levels of the d-orbitals. This is known as crystal field splitting. 

For an octahedral complex, the energy difference between the two sets of orbitals is called the octahedral field splitting, \(\Delta_o\). Because the total amount of repulsion is the same in the octahedral field and the spherical field (it is just distributed differently) the average energy level of the orbitals should be the same in the two fields. This average energy of the orbitals is called the barycenter, a term borrowed from astronomy. Because two of the orbitals are raised above the barycenter in the octahedral field and three are lowered below the barycenter, then the \(e_g\) set must be \(\frac{3}{5}\Delta_o\) (\(=0.6\Delta_o\)) above the barycenter and the \(t_{2g}\) set must be \(\frac{2}{5} \Delta_o\) (\(=0.4\Delta_o\)) below the barycenter. That means the average energy lies at the barycenter.

Thus far, we have considered only the effect of repulsive electrostatic interactions between electrons in the d orbitals and the six negatively charged ligands, which increases the total energy of the system and splits the d orbitals. Interactions between the positively charged metal ion and the ligands results in a net stabilization of the system, which decreases the energy of all five d orbitals without affecting their splitting (as shown at the far right in Figure \(\PageIndex{2}\)).

Three Complexes Explained

Description of d-Orbitals

To understand CFT, one must understand the description of the lobes:

• d_xy: lobes lie in-between the x and the y axes.
• d_xz: lobes lie in-between the x and the z axes.
• d_yz: lobes lie in-between the y and the z axes.
• d_x²-y²: lobes lie on the x and y axes.
• d_z²: there are two lobes on the z axes and there is a donut shape ring that lies on the xy plane around the other two lobes.

Figure \(\PageIndex{3}\): Spatial arrangement of ligands in the an octahedral ligand field with respect to the five d-orbitals. (LibreText)

Octahedral Complexes

In an octahedral complex, there are six ligands attached to the central transition metal. The d-orbital splits into two different levels (Figure \(\PageIndex{4}\)). The bottom three energy levels are named \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) (collectively referred to as \(t_{2g}\)). The two upper energy levels are named \(d_{x^²-y^²}\), and \(d_{z^²}\) (collectively referred to as \(e_g\)).

Figure \(\PageIndex{4}\): Spllitting of degenerate d-orbitalsdue to an on octahedral ligand field. (LibreText)

The reason they split is because of the electrostatic interactions between the electrons of the ligand and the lobes of the d-orbital. In an octahedral, the electrons are attracted to the axes. Any orbital that has a lobe on the axes moves to a higher energy level. This means that in an octahedral, the energy levels of \(e_g\) are higher (0.6∆_o) while \(t_{2g}\) is lower (0.4∆_o). The distance that the electrons have to move from \(t_{2g}\) from \(e_g\) and it dictates the energy that the complex will absorb from white light, which will determine the color. Whether the complex is paramagnetic or diamagnetic will be determined by the spin state. If there are unpaired electrons, the complex is paramagnetic; if all electrons are paired, the complex is diamagnetic.

Tetrahedral Complexes

In a tetrahedral complex, there are four ligands attached to the central metal. The d orbitals also split into two different energy levels. The top three consist of the \(d_{xy}\), \(d_{xz}\), and \(d_{yz}\) orbitals. The bottom two consist of the \(d_{x^2-y^2}\) and \(d_{z^2}\) orbitals. The reason for this is due to poor orbital overlap between the metal and the ligand orbitals. The orbitals are directed on the axes, while the ligands are not.

Figure \(\PageIndex{5}\): (a) Tetraheral ligand field surrounding a central transition metal (blue sphere). (b) Splitting of the degenerate d-orbitals (without a ligand field) due to an octahedral ligand field (left diagram) and the tetrahedral field (right diagram). (LibreText)

The difference in the splitting energy is tetrahedral splitting constant (\(\Delta_{t}\)), which less than (\(\Delta_{o}\)) for the same ligands:

\[\Delta_{t} = 0.44\,\Delta_o \label{1} \]

Consequentially, \(\Delta_{t}\) is typically smaller than the spin pairing energy, so tetrahedral complexes are usually high spin.

Square Planar Complexes

In a square planar, there are four ligands as well. However, the difference is that the electrons of the ligands are only attracted to the \(xy\) plane. Any orbital in the xy plane has a higher energy level (Figure \(\PageIndex{6}\)). There are four different energy levels for the square planar (from the highest energy level to the lowest energy level): d_x²-y², d_xy, d_z², and both d_xz and d_yz.

Figure \(\PageIndex{6}\): Splitting of the degenerate d-orbitals (without a ligand field) due to an square planar ligand field. (LibreText)

The splitting energy (from highest orbital to lowest orbital) is \(\Delta_{sp}\) and tends to be larger then \(\Delta_{o}\)

\[\Delta_{sp} = 1.74\,\Delta_o \label{2} \]

Moreover, \(\Delta_{sp}\) is also larger than the pairing energy, so the square planar complexes are usually low spin complexes.