# Crystal field and ligand field theory

The crystal and ligand field theory are two different but mutually complementary theories of complex compounds. The crystal field theory, often abbreviated as KFT, provides a qualitative understanding and the ligand field theory allows quantitative predictions of the properties of transition metal salts or complexes . Both theories explain the structure, color and magnetism of these substances.

The crystal field and the ligand field theory have the quantum mechanical treatment of the complex center and the electrostatic description of the ligands in common.

The approach of both theories is different. The crystal field theory leads to the quantum mechanical reality of the complexes, the ligand field theory is based on the quantum mechanical reality.

Further theories are required to fully understand the complex compounds. The most important are valence bond theory (VB theory) and molecular orbital theory (MO theory). The VB theory provides an explanation for the complex geometry that is assumed by the crystal field and the ligand field theory. The MO theory enables a complete understanding of the covalent bond component in complexes.

## Structure of the complex compounds

In contrast to purely electrostatic ionic bonding, e.g. For example, in salt formation, where cations and anions electrostatically attract each other as point charges, crystal field and ligand field theory are based on a partially quantum mechanical description, which is more appropriate for most complex compounds. In the description of the crystal field and ligand field theory, the complex center is treated quantum mechanically, analogous to the quantum mechanical description of an atom, while the ligands surrounding the complex center are described purely electrostatically. Put simply, this has the following consequences:

• there is an electrostatic attraction between the central atom and the ligand.
• However, there is also an electrostatic repulsion between the central atom and the ligand, because the complex center has an electron system with an energetic shell structure determined by quantum mechanics. The electrons in the outer d orbitals have a repulsive effect on the negatively charged ligand system because of their negative charge.

The interaction of the electrostatic ligand potential and the quantum mechanical structure of the valence electrons of the central atom is fundamental for both crystal field and ligand field theory.

The approach of both theories is different:

• The crystal field theory starts from the description of the ligands as point-like charges and examines the effect of the so classically precisely defined crystal field on the quantum-mechanically determined electron system of the complex center (with the ground state as the purest representative).
• The ligand field theory is based on the detailed quantum mechanical description of the complex center (ground state plus excited states) into which the electrostatic ligand field is integrated. The ligand field is characterized in more detail by empirically determinable properties (field strength), which enables an arbitrarily precise adaptation to the observable reality.

## Crystal field theory

The crystal field theory , often abbreviated as KFT or CF theory, corresponding to the English term Crystal Field Theory , was developed from 1932 by John H. van Vleck - based on a work by Hans Bethe from 1929 - to determine the physical properties of transition metal salts that show unexpected magnetic and optical behavior.

It is a purely electrostatic model in which the anions or ligands are viewed as negative point charges whose electric field , the crystal field , influences the electrons of the outer d orbitals of the cations . A crystal is not considered as a whole, but a cation is picked out and only the influence of the closest neighbors in the crystal lattice is examined.

### Principle of the crystal field

The starting point is a classic description of the ligand system, whereby the ligands are viewed as negative point charges, whose electrostatic field, the crystal field, influences the electrons of the outer d orbitals of the complex center. The term "crystal field" expresses that in the case of the effect of the ligands on the complex center, according to the classical understanding of a crystal, only the influence of the closest neighbors is examined.

## Ligand field theory

The ligand field theory comes from Hermann Hartmann and Friedrich Ernst Ilse and was published in 1951. It allows a very precise interpretation of the spectroscopic complex properties.

### Principle of the ligand field

The starting point of the ligand field theory is the electrostatic ligand potential, which is integrated into the quantum mechanical description from the start and characterized in more detail by empirically-experimentally determinable quantities such as polarizability and field strength. That is why the ligand field theory is also referred to as a semi-empirical theory. The ligand field theory is not a MO theory because only the valence electrons of the complex center are treated quantum mechanically and not the electron system of the ligands.

## Geometry-related energy level splits

In the free complex center the d orbitals are degenerate, i.e. that is, they have the same energy. If you bring an atom into a spherically symmetrical ligand field, the degeneracy is retained, but the energy content increases due to the repulsive interaction between d electrons and ligands. In real complexes the ligand system is not spherically symmetrical, but has a special geometry that depends on the size relationships between the central atom and the ligands. As a result, the ligands destabilize some d orbitals more than the others and, from an energetic point of view, the states are split.

The type of splitting is determined by the geometry of the ligand system. The crystal field theory provides qualitative estimates in the form of energy level diagrams. This results in the often found graphic representations of the splitting of states through the geometry of the ligand system.

The ligand field theory makes it possible to calculate the magnitude of the splitting very precisely and quantitatively. The size of the splitting depends on the “strength” of the central atom and the “strength” of the ligands. The field strength parameters of the complex components are determined empirically and recorded relative to one another in spectrochemical series .

### Octahedral complexes

Representation of the squares of the D orbitals. If one approaches ligands on the coordinate axes, the energy increases in the orbital.${\ displaystyle d_ {z ^ {2}}}$${\ displaystyle d_ {x ^ {2} -y ^ {2}}}$
Energy level diagram of the d orbitals of a complex center in the octahedral field

Six point charges are arranged around the central atom in the form of an octahedron . Thereby, the orbitals and energetically raised the orbitals , and energetically lowered. There is a 2-3 split. ${\ displaystyle d_ {z ^ {2}}}$${\ displaystyle d_ {x ^ {2} -y ^ {2}}}$${\ displaystyle d_ {xy}}$${\ displaystyle d_ {yz}}$${\ displaystyle d_ {xz}}$

A transition metal central atom can potentially accommodate 3 × 2 = 6 electrons in the three favorable orbitals, but energy must be expended so that two electrons can be in one orbital. Whether the favorable orbitals are fully occupied depends on whether more energy is gained or lost, i.e. on how big the energy difference between the orbitals is.

In weak ligands , for example, one finds high-spin -configured central atoms in which the lower-energy orbitals are not fully occupied, and low-spin -configured complex centers with fully occupied orbitals in comparatively strong ligands. The two electron arrangements "high spin" and "low spin" exist in the octahedral crystal field only at d 4 , d 5 , d 6 , d 7 .

Are energy levels degenerate in an octahedral complex, i.e. This means that it cannot be determined in which orbital an electron is located, a geometric distortion occurs until this degeneracy has been eliminated. This is known as the Jahn-Teller effect .

One example is the d 1 valence electron configuration: In an octahedral field, this electron must be assigned to one of the three t 2g orbitals. Since it cannot be determined in which it is actually located, there is a distortion, which on the one hand leads to the fact that it can now be said exactly in which orbital the electron is located and on the other hand, the energy for this electron is minimized.

In this example (d 1 ), the Jahn-Teller distortion causes a compression in the z-direction. This results in a further splitting of the t 2g and e g orbitals: The orbitals with a z component ( d xz , d yz , ) are destabilized by the approach of those ligands that are on the z axis, whereas those without z-component , d xy ) are further stabilized. In this new energetic orbital sequence, one electron can be ascribed to the d xy orbital, which now represents the most stable (energetically lowest) orbital. Since the amount of stabilization and destabilization is the same, this d xy -orbital is now lower than before in the "composite" of the degenerate t 2g- theorem, which means for the electron that there is a larger one in the distorted (here: compressed) octahedron Undergoes stabilization. ${\ displaystyle d_ {z ^ {2}}}$${\ displaystyle d_ {x ^ {2} -y ^ {2}}}$

In other cases, an extension along the z-axis is also possible, as a result of which the ligands on this axis are further removed from the central atom. This is accompanied by a stabilization of all orbitals with a z component and consequently a destabilization of all orbitals without a z component.

Jahn-Teller stable complexes, i.e. those that are not subject to the Jahn-Teller distortion: d 3 , high-spin d 5 , low-spin d 6 , d 8 and d 10 . In these the electronic states are not degenerate.

### Tetrahedral complexes

Energy level diagram of the d orbitals of a complex center in the tetrahedral ligand field

Four point charges can be arranged in the form of a tetrahedron around the central transition metal. Through this geometry the orbitals , and are energetically raised and as well as energetically lowered. This gives a 3–2 split (t 2 and e). ${\ displaystyle d_ {xy}}$${\ displaystyle d_ {yz}}$${\ displaystyle d_ {xz}}$${\ displaystyle d_ {z ^ {2}}}$${\ displaystyle d_ {x ^ {2} -y ^ {2}}}$

With d 3 , d 4 , d 5 , d 6 , both configurations "low spin" and "high spin" would be expected - due to the low field splitting, however, only "high spin" complexes exist (an exception is, for example, tetrakis (1-norbornyl ) cobalt (IV) , whose norbornyl ligands cause sufficient splitting).

The ligand field splitting in the tetrahedral crystal field corresponds to 4/9 of the octahedral splitting.

### Square-planar complexes

Energy level diagram of the d orbitals of a complex center in the square-planar ligand field, starting from the octahedral. Energy values ​​in Dq are for 3d transition metals. The dz 2 orbital is usually higher in energy than the two degenerate orbitals.

Another possibility for 4 point charges is the square . The resulting split is more complicated: d x 2 - y 2 is severely disadvantaged, slightly disadvantaged is d xy , below are d yz and d xz on one level , the lowest is d z 2 (1-1-2-1 split) . (Depending on the metal, however, the order of the two lowest levels can be reversed and the d z 2 lies above the d yz and d xz , e.g. for Ni 2+ )

This geometry is often found in d 8 configurations (or 16 electron complexes) with large ligand field splitting. The d x 2 - y 2 orbital, which is energetically very high due to electrostatic repulsion to all ligands, remains unoccupied.

This splitting is typical for palladium, platinum and gold cations, since they usually lead to the typical large splitting of ligand fields. All complexes formed by these ions are diamagnetic low-spin complexes.

### Other complex geometries

The crystal and ligand field theories have also been successfully applied to interpret the effect of many other complex geometries.

Metallocenes:    Metallocenes have the split 2-1-2. The orbitals in the xy plane ( d xy and d x 2 - y 2) hardly interact with the ligands and are therefore favored. d z 2 only interacts with one part and lies in the middle. Strongly destabilized are d xz and d yz , which point completely to the rings.

## Conclusions from the splitting of energy

### colour

The colors of the transition metal salts come about through the splitting of the d orbitals described . Electrons from the d orbitals of lower energy can be excited with light into the orbitals of higher energy. Only light with a certain wavelength is absorbed , which corresponds exactly to the energy difference between the favored and the disadvantaged orbital. Since the distances are small, the absorption is in the visible range.

### Occupation of the orbitals to form high-spin or low-spin complexes

There are two ways to occupy d orbitals:

• If the splitting is small, the orbitals can be regarded as approximately degenerate. The occupation is then based on Hund's rule , i. This means that each orbital is initially occupied by a single person and the unpaired electrons all have parallel spins. More electrons must have a negative spin. The complex therefore has a high net spin and is called the high spin complex .
• If the energy splitting of the orbitals is greater than the spin pairing energy, the construction principle applies and the orbitals with less energy are initially occupied twice. This results in the lower total spin of the low-spin complexes .

If an orbital is to be filled with two electrons, spin-pairing energy must be expended. If the ligand field splitting exceeds the spin pairing energy, a "low-spin complex" can occur. That means, d-orbitals lying deeper are first filled with two electrons before d-orbitals lying higher are filled.

Strong ligands promote the splitting of the ligand field and thus the formation of low-spin complexes (see also the spectrochemical series ). Central atoms of the 5th and 6th periods tend to form low-spin complexes thanks to the larger ligand field splitting. The higher the oxidation number of the central atoms, the more pronounced the ligand field splitting and thus the preference for low-spin complexes.

### magnetism

The more unpaired electrons there are on the cation, the more paramagnetic it is. Based on the statements about the resettlement of the d orbitals, the magnetic properties of many transition metal salts could be clarified, especially the formulation of high spin and low spin configured cations explains the high paramagnetism of iron or cobalt salts with weak anions / ligands and the comparatively low paramagnetism with strong anions / ligands. If all electrons are paired, then the ion is diamagnetic .

### Thermodynamic stability

A compound is thermodynamically stable when it is itself energetically favorable and a possible product from this compound is energetically less favorable. With the crystal field theory one can use the d orbital splitting to estimate whether a compound is more favorable or more disadvantageous than its product and how big the energetic difference between them is. This makes it possible to predict whether a reaction is thermodynamically possible. These predictions hold true for most of the Ionic and Classical complexes .

### Kinetic inertness

A compound is kinetically inert if the reaction to a product is possible but very slow; that is, if the activation energy for the reaction to the product is very high. The crystal field theory enables the calculation of a substantial part of the activation energies for the reactions of transition metal complexes by considering what the possible transition states or intermediate products in the reaction might look like and how the d- orbital splitting and electron distribution at the cation changes when these transition states / intermediate products are formed . If the possible transition states are very unfavorable in terms of energy compared to the initial state, the activation energy is very high. Accordingly, the reaction hardly takes place at all. The statements of crystal field theory on the kinetics of ligand substitutions on complexes are very reliable even for non-classical complexes.

### Solid state physics

The ligand field theory is also used in solid-state physics to describe deep impurities in semiconductors - crystals .

## Scope of validity of the crystal field and ligand field theory

The crystal field theory is semi-classical and the ligand field theory is semi-empirical. Despite the restrictive prerequisites, the contribution of crystal field theory to the qualitative understanding and the contribution of ligand field theory to the quantitative derivation of the complex properties is great. The reason for this is the quantum mechanical treatment of the complex center in both theories.

The purely quantum mechanical MO theory provides a more precise picture of the complex structure because the ligands are also treated quantum mechanically, but the resulting splitting pattern is the same as in the crystal field and ligand field theory. What the crystal field or ligand field theory describes as stronger electrostatic repulsion, in the MO theory is greater splitting and raising of the antibonding orbitals (the binding orbitals are occupied by the electrons of the ligands). Only the MO theory provides an understanding of the covalent bond component in complexes with π backbonding, as they are e.g. B. occurs in carbonyl complexes.

Linus Pauling's valence bond (VB) theory provides an explanation for the complex geometry assumed by crystal field and ligand field theory.

Several theories are required to fully understand the complex compounds.

The systematic connection between all complex theories, especially the complementarity of crystal field and ligand field theory, points to the existence of a uniform field of molecular behavior that is closely linked to the spatial structure.

## literature

### Individual evidence

1. Erwin Riedel: Modern Inorganic Chemistry . WdeG, 1999, p. 237
2. ^ E. Riedel: Modern inorganic chemistry . WdeG, 1999, p. 695