Crystal Field Theory (CFT): Explanation, Need and Examples

Crystal field theory is most useful in the realm of coordination chemistry for insight into the electronic structure and properties of transition metal complexes. More specifically, it revolves around the description of how the spatial arrangement of ligands about a central metal ion affects the energy levels of the d orbitals involved in the metal. It is for this reason that the theory is applied to explain phenomena such as color, magnetism, and reactivity to these complexes, which are of the essence in the procedures—both natural and synthetic.

This Story also Contains

1. Types and Aspects of Crystal Field Theory
2. Factors Affecting CFSE
3. Applications and Limitations of Crystal Field Theory
4. Limitations
5. Some Solved Examples
6. Summary

Types and Aspects of Crystal Field Theory

Octahedral field

For example, in an octahedral field when surrounded by six ligands, the d orbitals split into two different sets of energy: lower-energy t₂g consisting of dxy, dyz, dzx, and higher-energy e consisting of dx²−y² and dz². This energy distance between the two sets is referred to as the crystal field splitting energy (Δₒ). The magnitude of this splitting depends upon the nature of the ligands; strong field ligands like CN⁻ produce a more considerable splitting than weak field ligands like I⁻.

Crystal Field Splitting in Tetrahedral Field

Tetrahedral arrangement results when the ligands are arranged at the corners of a tetrahedron around the metal ion. In this field also, the d orbitals split but in the opposite manner than in octahedral complexes. dx²-y², dz² orbitals lie at a lower energy, while on the contrary, dxy, dyz, and dzx are high in energy. Normally, Δₜₑₜ, CFSE is smaller in tetrahedral as compared to octahedral complexes, so the electronic configuration and hence magnetic properties will also be different.

The crystal field theory (CFT) is an electrostatic model which considers the metal-ligand bond to be ionic arising purely from electrostatic interactions between the metal ion and the ligand. Ligands are treated as point charges in the case of anions or point dipoles in the case of neutral molecules. The five d orbitals in an isolated gaseous metal atom/ion have the same energy, i.e., they are degenerate. This degeneracy is maintained if a spherically symmetrical field of negative charges surrounds the metal atom/ion. However, when this negative field is due to ligands (either anions or the negative ends of dipolar molecules like NH₃ and H₂O) in a complex, it becomes asymmetrical and the degeneracy of the d orbitals is lifted. It results in the splitting of the d orbitals. The pattern of splitting depends upon the nature of the crystal field.

In an octahedral coordination entity with six ligands surrounding the metal atom/ion, there will be repulsion between the electrons in metal d orbitals and the electrons (or negative charges) of the ligands. Such a repulsion is more when the metal d orbital is directed towards the ligand than when it is away from the ligand. Thus, the $d_{x^2-y^2}$ and $d_{z^2}$ orbitals which are oriented towards the axes along the direction of approach of the ligand will experience more repulsion and will be raised in energy; and the $d_{x y}, d_{y z \text { and }} d_{z x}$ and orbitals which are directed between the axes will be lowered in energy relative to the average energy in the spherical crystal field. Thus, the degeneracy of the d orbitals has been removed due to ligand electron-metal electron repulsions in the octahedral complex to yield three orbitals of lower energy, $t_{2 n}$ set, and two orbitals of higher energy, e_g set. This splitting of the degenerate levels due to the presence of ligands in a definite geometry is termed as crystal field splitting and the energy separation is denoted by $\Delta_0$. Thus, the energy of the two $e_g$ orbitals will increase by $\frac{3}{5} \Delta_0$ and that of the three $t_{2 g}$ will decrease by $\frac{2}{5} \Delta_0$.

The crystal field splitting, $\Delta_0$, depends upon the field produced by the ligand and charge on the metal ion. Some ligands can produce strong fields in which case, the splitting will be large whereas others produce weak fields and consequently result in a small splitting of d orbitals. In general, ligands can be arranged in a series in the order of increasing field strength as given below:

$\mathrm{I}^{-}<\mathrm{Br}^{-}<\mathrm{SCN}^{-}<\mathrm{Cl}^{-}<\mathrm{S}^{2-}<\mathrm{F}^{-}<\mathrm{OH}^{-}<\mathrm{C}_2\mathrm{O}_4^{2-}<\mathrm{H}_2\mathrm{O}< \mathrm{NCS}^{-}<e \text { edta } ^{4-}<\mathrm{NH}_3<e \text { en }<\mathrm{CN}^{-}<\mathrm{CO}$

Such a series is termed a spectrochemical series. It is an experimentally determined series based on the absorption of light by complexes with different ligands. Let us assign electrons in the d orbitals of metal ions in octahedral coordination entities. The single d electron occupies one of the lower energy t_2g orbitals. In d² and d³ coordination entities, the d electrons occupy the t_2g orbitals singly in accordance with Hund’s rule. For d⁴ ions, two possible patterns of electron distribution arise: (i) the fourth electron could either enter the t_2g level and pair with an existing electron or (ii) it could avoid paying the price of the pairing energy by occupying the e_g level. Which of these possibilities occurs, depends on the relative magnitude of the crystal field splitting, $\Delta_0$, and the pairing energy, P (P represents the energy required for electron pairing in a single orbital). The two options are:

• If $\Delta_0<P$, the fourth electron enters one of the e_g orbitals giving the configuration $t_{2 g}^3 e_g^1$. Ligands for which ∆_o < P are known as weak field ligands and form high spin complexes.
• If $\Delta_0>P$, it becomes more energetically favorable for the fourth electron to occupy a t_2g orbital with configuration $t_{2 g}^4 e_g^0$. Ligands that produce this effect are known as strong field ligands and form low-spin complexes.

Calculations show that $d^4$ to $d^7$ to coordination entities are more stable for strong field as compared to weak field cases.

In tetrahedral coordination entity formation, the d orbital splitting is inverted and is smaller as compared to the octahedral field splitting. For the same metal, the same ligands and metal-ligand distances, it can be shown that $\Delta_t=\frac{4}{9} \Delta_0$. Consequently, the orbital splitting energies are not sufficiently large for forcing pairing, and, therefore, low spin configurations are rarely observed. The ‘g’ subscript is used for the octahedral and square planar complexes which have a center of symmetry. Since tetrahedral complexes lack symmetry, the ‘g’ subscript is not used with energy levels.

Factors Affecting CFSE

One of the key concepts involves the Crystal Field Stabilization Energy, CFSE, which describes the idea that there is a way to quantify transition metal complex stability due to the arrangement of electrons in the split d orbitals. CFSE depends on the number of electrons in the t₂g and e orbitals and the magnitude of the splitting energy, Δ. Factors affecting CFSE include the oxidation state of the metal, the nature of the ligand—strong versus weak field—and the geometry of the complex. Higher CFSE values mean high stability, thus changing the formation and reactivity of metal complexes in various chemical reactions.

1. Nature of central metal atom: As we move down the group, the CFSE increases. From 3d to 4d, there is a 30% increase in CFSE, and from 4d to 5d, there is a 50% increase in CFSE.
2. The oxidation state of the central metal atom: The oxidation state is directly proportional to CFSE.
   The CFSE of [Fe(CN)₆]^3- is greater than [Fe(CN)₆]^4-.
3. Nature of ligand: In the case of a strong field ligand, the magnitude of CFSE is high. In the case of a weak field ligand, the magnitude of CFSE is low.
4. Nature of complex: In an octahedral complex, 6 ligands approach the central metal atom and thus repulsion is higher, due to which the CFSE is higher. But, in a tetrahedral complex, 4 ligands approach the central metal atom and thus repulsion is lower, and thus the CFSE is lower.

Applications and Limitations of Crystal Field Theory

These are the various applications of crystal field theory.

• Configuration of metal ion: In weak field ligand, the crystal field splitting energy difference is very less. Thus, the pairing of electrons is done by Hund's rule. But for strong field ligands, the splitting energy difference is high due to which the pairing of electrons is not done by Hund's rule.
• Magnetic nature of complex: Based on magnetism, all the complexes can be divided into two categories, paramagnetic and diamagnetic. Paramagnetic complexes are weakly attracted by the magnetic field and have unpaired electrons, while diamagnetic complexes are weakly repelled by the magnetic field and these molecules have no unpaired electrons.
• Color in coordination compounds: The color of the complex is complementary to that which is absorbed. The complementary color is the color generated from the wavelength left over; if the green light is absorbed by the complex, it appears red. The color in the coordination compounds can be readily explained in terms of the crystal field theory. Consider, for example, the complex, which is violet in color. This is an octahedral complex where the single electron ($\left(T i^{3+}\right.$ is a $3 d^1$ system) in the metal d orbital is in the t_2g level in the ground state of the complex. The next higher state available for the electron is the empty e_g level. If light corresponding to the energy of the blue-green region is absorbed by the complex, it would excite the electron from t_2g level to the e_g level i.e. the configuration changes from $t_{2 g}^1 e_g^0$ to. Consequently, the complex appears violet in color. This phenomenon is called d-d transition and the crystal field theory attributes the color of the coordination compounds to this d-d transition of the electron.

Limitations

As general as the CFT is, it nonetheless, has its limitations. First of all, it neglects the covalent character of the metal-ligand bonds and hence cannot account for, for instance, the behavior of some ligands that seem to lie well outside of the obvious strong or weak field categories. Secondly, it does not include the contribution of s and p orbitals to the bonding and therefore at times has the tendency to paint a false picture. Very often, one is prompted to believe that a splitting is much greater than is sometimes the case because weak field ligands are being judged in a weak field.

The crystal field model is successful in explaining the formation, structures, color, and magnetic properties of coordination compounds to a large extent. However, from the assumptions that the ligands are point charges, it follows that anionic ligands should exert the greatest splitting effect. The anionic ligands actually are found at the low end of the spectrochemical series. Further, it does not take into account the covalent character of bonding between the ligand and the central atom. These are some of the weaknesses of CFT, which are explained by ligand field theory (LFT) and molecular orbital theory which are beyond the scope of the present study.

Recommended topic video on (Crystal Field Theory)

Some Solved Examples

Example 1

According to CFT the nature of the metal-ligand bond is

1)Covalent

2)Coordinate

3) (correct)Ionic

4)van der Waal

Solution

According to this Crystal field theory, metals and ligands are considered as point charges. This theory deals with the electrostatic force of attraction between the metal and the ligand and hence assumes ionic character in the Metal-Ligand bond.

Hence, the answer is the option(3).

Example 2

The magnitude of crystal field stabilization energy in an octahedral field depends on:

(a) the nature of the ligand

(b) the charge on the metal ion

(c) whether the element is in the first, second or third row of transition elements.

1)only (a) and (b) are correct

2) (correct)(a),(b) and (c) are correct

3)only (b) and (c) are correct

4)only (c) is correct

Solution

As we have learned,

Factors Affecting CFSE -

1. Nature of central metal atom: As we move down the group, the CFSE increases. From 3d to 4d, there is a 30% increase in CFSE, and from 4d to 5d, there is a 50% increase in CFSE.
2. The oxidation state of the central metal atom: The oxidation state is directly proportional to CFSE.
   The CFSE of [Fe(CN)₆]^3- is greater than [Fe(CN)₆]^4-.
3. Nature of ligand: In the case of a strong field ligand, the magnitude of CFSE is high. In the case of weak field ligands, the magnitude of CFSE is low.
4. Nature of complex: In an octahedral complex, 6 ligands approach the central metal atom and thus repulsion is higher, due to which the CFSE is higher. But, in a tetrahedral complex, 4 ligands approach the central metal atom and thus repulsion is lower, and thus the CFSE is lower.

The factor affecting the CFT energy in the octahedral field depends on the nature of the ligand, the charge on the metal ion, and whether the element is in the first, second or third row of transition elements.

Therefore, option (2) is correct.

Example 3

According to Crystal Field Theory, what happens to the degeneracy of the five d - d-orbitals when a spherically symmetric ligand field is applied to the metal atom/ion?

1)the degeneracy is destroyed

2) (correct)the orbitals are still degenerate but at a higher potential energy level as compared to an isolated metal atom/ion

3)the orbitals are still degenerate but at a lower potential energy level as compared to an isolated metal atom/ion

4)splitting of the five d-orbitals occurs into two discrete energy levels

Solution

As we have learned,

The five d orbitals in an isolated gaseous metal atom/ion have the same energy, i.e., they are degenerate. This degeneracy is maintained at a higher potential energy level if a spherically symmetrical field of negative charges surrounds the metal atom/ion.

It is to be mentioned that the potential energy of the orbitals will increase as there are some repulsions between the electrons in the ligand and the electrons present in the metal but since the field is spherically symmetrical, these repulsions will be equal and as a result, the five d- orbitals will still be degenerate but at a higher energy level.

Hence, the answer is the option (2).

Example 4

Which of the following cannot be explained by crystal field theory?

1) (correct)The order of spectrochemical series

2)Stability of metal complexes

3)Magnetic properties of transition metal complexes

4)Colour of metal complexes

Solution

Crystal field theory introduces spectrochemical series based upon the experimental value of $\Delta$ but can’t explain its order. While the other three points are explained by CFT. Especially when the CFSE increases thermodynamic stability of the complex increases.

Hence, the answer is the option (1).

Example 5

Among the following species the one which causes the highest CFSE, $\Delta$o as a ligand is :

1)CN^-

2)NH₃

3)F^-

4) (correct)CO

Solution

Strong Field Ligand -

These are the ligands that pair the electron in the inner d-orbital and make required d-orbitals empty/available.

- wherein

For example: CO, CN, etc

and,

Spectrochemical series -

$\begin{aligned} & \mathrm{I}^{-}<\mathrm{Br}^{-}<\mathrm{SCN}^{-}<\mathrm{Cl}^{-}<\mathrm{S}^{2-}<\mathrm{F}^{-}<\mathrm{OH}^{-}<\mathrm{C}_2 \mathrm{O}_4^{2-}<\mathrm{H}_2 \mathrm{O}< \\ & E D T A<\mathrm{NH}_3<\mathrm{en}<\mathrm{Cn}^{-}<\mathrm{CO}\end{aligned}$

The order of the spectrochemical series is

$\begin{aligned} & \mathrm{I}^{-}<\mathrm{Br}^{-}<\mathrm{SCN}^{-}<\mathrm{Cl}^{-}<\mathrm{S}^{2-}<\mathrm{F}^{-}<\mathrm{OH}^{-}<\mathrm{C}_2 \mathrm{O}_4^{2-}<\mathrm{H}_2 \mathrm{O}< \\ & E D T A^{4-}<\mathrm{NH}_3<\mathrm{en}<\mathrm{CN}^{-}<\mathrm{CO}\end{aligned}$

Hence, the answer is the option (4).

Summary

That is to say, the Crystal Field Theory is an underlying framework within which one may possibly understand the electronic structure of transition metal complexes, which clearly gives an explanation of how the energy levels of the d orbitals are influenced by the arrangement of the ligands. Though the splitting in octahedral and tetrahedral geometries is through crystal field splitting, we have become quite conversant with a variety of geometries that give a variation of patterns of energy in the splitting, which usher in a corresponding change in the properties of the complex. CFSE takes a different lead in defining or putting across a measure of the stability of such complexes based on the electron distribution among the split d orbitals.