# Crystal field theory (CFT)

Crystal field theory, abbreviated as CFT, unveils many fascinating properties of transition metals and metal complexes, otherwise difficult to explain. If you want to know why metal complexes exhibit beautiful, bright colors and why are they extremely stable and possess magnetic properties, then you must learn about the crystal field theory.

In this article, we will discuss everything you need to know about CFT, including its definition, postulates, benefits and limitations. So, continue reading, and happy learning!

## What is crystal field theory (CFT)?

The crystal field theory (CFT) was introduced by Hans Bethe and J.H Van Vleck around 1930. CFT is based on the principle that the forces of attraction between a metal ion and the ligands present in a transition metal complex are purely electrostatic (100 % ionic). Therefore, the metal ion and ligands are considered point charges or point dipoles in CFT.

### Definition

‘Crystal field theory explains the breaking of the degeneracy of d-orbitals of a metal ion’’.

As per CFT, the energies of d-orbitals in metal ions get strongly affected by the electric field of the approaching ligand. The interaction between the electrons of the metal cation and ligand electrons is entirely repulsive. Due to this repulsive effect of ligand electrons, the same energy/degenerate d-orbitals of central metal ion undergo crystal field splitting. A new chemical bond is formed by an electrostatic force of attraction between metal nucleus and ligands.

### Important terminology

• Crystal field splitting: The splitting of degenerate d-orbitals into two sets of different energies as a ligand approaches.
• Crystal field splitting energy (∆o): The energy difference created as a result of crystal field splitting.
• Crystal field stabilization energy (CFSE): The greater the energy difference between two groups of d-orbitals formed as a result of the crystal field splitting, the greater the stability of a metal complex. Therefore, crystal field stabilization energy is another name for crystal field splitting energy (∆o).

## Applications of CFT

Metals having coordination no. 6 can form two types of complexes depending upon the strength of the approaching ligand.

A ligand is defined as an electron-rich species (atom or ion) having lone pairs of electrons. In CFT, ligands behave as point charges (if it is an anion) or point dipoles (if the ligand is a neutral molecule). They form purely ionic bonds with the metal at the center of a transition metal complex.

A strong field ligand causes crystal field splitting such that it results in a high ∆o, so a more stable metal complex is formed. A small ∆o results as weak field ligands approach the metal.

Strong field ligands usually form low-spin diamagnetic complexes, while weak field ligands form high-spin paramagnetic complexes.

The strength of a ligand can be determined by its position in the spectrochemical series.

The oxidation state of the central metal ion also affects crystal field splitting and, thus, the stability of the complex.

The higher the metal’s oxidation state, the greater the crystal field splitting, which results in a high ∆o value.

The spectrochemical series for the metal ion is:

Now let’s come back to the main point and discuss how two different kinds of complexes are formed with coordination no.6 in the light of CFT.

### CFT of complexes having coordination no. 6

An octahedral complex is formed when six ligands approach the metal atom or ion perpendicular to the x, y and z-axis.

Example of a low-spin octahedral complex: [Co(NH3)6]3+

The electronic configuration of Co3+ is 1s2 2s2 2p6 3s2 3p6 3d6 4s0.

There are five degenerate d-orbitals in Co3+, including dz2, dx2-y2, dxy, dxz and dyz.

As NH3 molecules approach Co3+ perpendicular to dxy, dxz and dyz, it implies that dz2 and dx2-y2 lie directly in the path of the approaching ligands. Thus, these two d-orbitals experience a greater repulsive effect.

The degenerate d-orbitals split into two groups. dz2 and dx2-y2 occupy a higher energy level. It is known as the eg set. Contrarily, the dxy, dxz, and dyz atomic orbitals occupy an energy level lower than the ground state, and it is known as the t2g set. The energy difference between eg and t2g is called the crystal field splitting energy.

The t2g set is first singly filled with electrons. NH3 is comparatively a strong field ligand as per the spectrochemical series. So, the remaining three electrons of Co3+ also go into the t2g set, where the singly filled electrons are paired up.

There are no unpaired electrons in the above crystal field splitting diagram. Hence [Co(NH3)6]3+ is a diamagnetic low-spin complex as per the crystal field theory. A high ∆o value indicates that the octahedral complex formed is strong. In this case, ∆o is greater than the spin pairing energy.

In an octahedral complex, the crystal field splitting energy can be calculated as follows:

o (octa) = -0.4 (m) t2g + 0.6 (n) eg………..Equation (i)

The above equation is derived considering the fact that the total energy of the system = 1 Dq where Dq= differential of quanta implying energy.

By crystal field splitting in an octahedral complex, the energy of the t2g set is lowered by 0.4 Dq while the energy of the eg set is raised by 0.6 Dq. The decrease in energy by one set of orbitals is balanced by the increase in energy by the other set of orbitals. The total energy of the system stays conserved, i.e., 0.4 + 0.6 = 1 Dq. It is known as the barycenter rule.

In equation (i), m= electrons present in the t2g set while n= electrons present in, eg set of an octahedral complex.

Applying this formula for [Co(NH3)6]3+: m = 6, n=0

o = -0.4 (6) + 0.6 (0) = 2.4 Dq.

Example of a high-spin octahedral complex: [CoF6]3-

In the formation of [CoF6]3-, the crystal field splitting occurs in the same pattern as discussed above for [Co(NH3)6]3+. However, the difference occurs when electrons occupy the d-orbitals split into two groups.

As F is a weak field ligand so it cannot pair up all six electrons in the lower energy t2g set. Three electrons first singly fill the t2g set as per Hund’s rule. The next two electrons then occupy two orbitals in the higher energy eg set as per the Aufbau principle.  The last electron is then paired up in the first orbital of t2g, as per Pauli’s Exclusion principle, as shown below.

As there are unpaired electrons in the crystal field diagram drawn for [CoF6]3- thus it is a high-spin paramagnetic complex. This is how CFT explains magnetism in transition metal complexes.

The magnetic moment can be calculated in Bohr magnetic (BM) units by applying the formula:

Where n= number of unpaired electrons as per the crystal field diagram.

A low crystal field splitting energy (∆o) in the above diagram illustrates that [CoF6]3- is relatively less stable as it is formed by a weak field ligand.

Applying ∆o formula for [CoF6]3-: m = 4, n = 2

o = -0.4 (4) + 0.6 (2) = 0.4 Dq.

0.4 Dq < 2.4 Dq so [CoF6]3- is less stable than [Co(NH3)6]3+.

### CFT of complexes having coordination no. 4

Two different complexes are formed in which the coordination number of the central metal atom or ion is 4. These include tetrahedral and square planar complexes.

Tetrahedral complexes are usually high-spin and less stable than square planar complexes. This is because tetrahedral complexes are formed in the presence of weak field ligands. In that case, the crystal field stabilization energy of the complex is less than the energy required to pair up electrons (spin pairing energy).

#### CFT of tetrahedral complexes

A tetrahedral complex is formed when ligands approach the metal atom or ion such that its dxy, dyz and dxz orbitals experience greater repulsion than dz2 and dx2-y2. Crystal field splitting occurs such that the latter occupies a higher energy level than the former.

So in a tetrahedral complex, the t2 set lies at a higher energy than the e set, the reverse of octahedral splitting. The crystal field splitting energy (∆o) is thus determined by the formula given in equation (ii).

o (tetra) = -0.4 (m) e + 0.6 (n) t2…………Equation (ii)

Note that in this case, the subscript g is omitted from e and t2 because g is short for gerade, a German word that means even or symmetrical. As a tetrahedron lacks a definite centre of symmetry thus, the subscript g loses its relevance in a tetrahedral complex.

Example of a high-spin tetrahedral complex: [CoCl4]2-

The electronic configuration of Co2+ is 1s2 2s2 2p6 3s2 3p6 3d7 4s0.

As four Cl ions approach the central Co2+ ion, the negative charge repulsion by the approaching ligands results in crystal field splitting of the d-orbitals of Co2+, as shown below.

The first two electrons of Co2+ occupy the two orbitals lying at low energy in the e set. As Cl is a weak field ligand so it cannot immediately pair up these electrons.  Thus, the next three electrons of Co2+ singly fill the t2 set. This leaves behind two electrons which pair up the electrons present in the e set.

As unpaired electrons are present in the above crystal field diagram, thus, [CoCl4]2- is a paramagnetic high-spin complex. Its crystal field splitting energy is:

o = -0.4 (4) + 0.6 (3) = 0.2 Dq.

A low crystal field stabilization energy implies a relatively less stable complex formed in the presence of a weak field ligand.

#### CFT of square planar complexes

A square planar complex is formed when ligands approach the metal atom or ion such that the dx2-y2 orbital interacts most strongly with ligand orbitals while dz2 interacts least strongly. Therefore, crystal field splitting occurs such that dx2-y2 occupies the highest energy level, dz2 occupies the lowest energy level, while dxy, dyz and dxz are situated at intermediate energy.

Example of a low-spin square planar complex: [Ni(CN)4]2-

The electronic configuration of Ni2+ is 1s2 2s2 2p6 3s2 3p6 3d8 4s0.

The crystal field splitting diagram of 3d orbitals in Ni2+ in the presence of strong field CN ions is shown below.

An electron first singly occupies the lowest energy d-orbital, i.e., dz2. It is then paired up in the presence of the strong field CN ions. In the next step, two electrons occupy dxz and dyz and then pairing occurs. The last two electrons are paired up in dxy. In this way, there are no unpaired electrons in the d-orbitals of Ni2+ in [Ni(CN)4]2-. Thus, it is a spin-paired or low-spin diamagnetic square planar complex.

## How CFT explains the vibrant colors of metal complexes?

As per CFT, d-orbitals are split into two groups of different energy levels. Therefore, on providing energy from an external energy source such as sunlight, electrons migrate from the lower energy set to the empty higher energy d-orbitals. A wavelength of visible light complementary to the wavelength absorbed is then emitted, which gives the complex its characteristic color.

Light energy absorbed, E = hv, should be equal to the energy difference (∆o) between the two energy levels. [CoF6]3- is a green-colored octahedral complex, while [Co(NH3)6]3- is yellow in color. Conversely, the tetrahedral complex [CoCl4]2- is deeply blue in color.

Metal complexes with comparatively lower crystal field splitting energy, such as [CoCl4]2-, are generally more brightly colored than those with a higher ∆o value. This is because the latter have a greater chance of undergoing electronic transitions by absorbing low energy visible (200-800) nm radiations than the former.

## Why is the crystal field theory so important?

CFT is important because it explains the following:

• The bright colors of transition metal complexes.
• Their magnetic behaviour.
• Stability of different metal complexes.

## What is the difference between CFT and VBT?

Previously we studied the role of valence bond theory (VBT) in explaining the formation of transition metal complexes. Let us find out how these two are different from each other.

## Limitations of CFT

• CFT does not take into account the covalent bonding present in metal complexes.
• It gives no explanation of orbital overlap.
• There is no discussion of ligand electrons occupying d-orbitals of the metal atom during complex formation.
• The crystal field theory only explains the role of d-orbitals of the metal ion. It gives no credit to the contribution of other atomic orbitals, such as s, px, py, pz, etc., in chemical bonding.

## Are CFT and LFT the same?

LFT stands for the ligand field theory. CFT and LFT are often perceived as one and the same. However, that is a misconception. As we already discussed, CFT does not take into account orbital overlap during bond formation, which is the main concept behind ligand field theory. LFT is also known as the molecular orbital theory of transition metal complexes.

### CFT practice problems

Revise your concepts and practice some questions on CFT here.