Crystal Field Theory

Crystal field theory was developed by Hans Bethe and John Hasbrouck van Vleck in 1929. The crystal field theory describe the interaction between the positively charged metal atom and negatively charged ligands to form a metal complex. It is about d-d transition, splitting of degenerate orbitals, structure, and bonding in transition metal complexes. Crystal field theory also tells about the stability of newly formed complexes.

Crystal field theory is about coordinated complexes of transition metals. It tells us how degenerate orbitals are split up into two different sets of orbitals. What types of d-d transition are going to take place in complexes. What types of interactions are between metal atoms and ligands. It provides a way of determining stabilities, bondings, energies, and other spectroscopic properties of transition metal atoms.

Main points of crystal field theory(Assumptions)

Transition metal atom is surrounded by ligands
Ligand has lone pair of electrons
The ligands are regarded as point charges
The ligands may be either ionic i.e F^–, CN^–, Cl^–.
The ligands may be neutral molecules i.e H₂O, NH₃, CO, etc.
The negative end of the dipole is oriented toward the metal atom
There is an ionic bond between metal atoms and ligands. (purely electrostatic)
In free metal ion all five d-orbital are degenerate.
Ligands produce an electrical field that influences the energies of d-orbitals of the central metal atom
This field split up degenerate orbitals into different sets of orbitals
D orbital in the direction of ligand raised to a greater extent whereas others raised to a lesser extent.
Splitting depends upon the structure of the coordination compound.

Shapes of d orbitals

We must have an idea about the orientation of the five d orbital in space and the geometrical arrangement of ligands around the central metal atom because crystal field theory is all about it.

There are five d-orbitals. These are d_xy, d_yz, d_zx, d_x²_-y²,d_z².

These d-orbitals are divided into three different sets.

(i) Three orbitals d_xy, d_yz, d_zx are the same having four lobes each. These have high electron density lying in xy, yz, and zx planes. These lobes lie between the principal axis. These lobes are rotated around the axis about 45⁰ .

(ii) The d_x²_-y² orbital also has four lobes of high electron density along the x-axis and y-axis.

(iii) The d_z² orbital consist of two lobes along the z-axis. It has a ring of high electron density in the xy plane.

Crystal field splitting

In the free ion, the electron can occupy any one of five d-orbitals. This is due to the degeneracy of d-orbitals. But when there present a ligand then the case is not the same. Then all of five d orbitals are not at the same energy level. Rather they split into two different energy levels.

High energy orbitals
Low energy orbitals

Barycentre Rule

According to this rule “ decrease in energy of d-orbitals (one set) from barycentre must be balanced by an increase in energy of d-orbitals (another set)

Initially, the energy of five d-orbitals was raised equally but later on the energies of five d-orbitals split into two sets because of different orientations towards ligands.

Crystal field splitting in octahedral complexes

In octahedral complexes, the coordination number is six. When ligand approaches the central metal atom Spiliting will take place. all d-orbital will no longer be degenerate. They split into two sets of energy levels. One set moves to a higher energy state whereas, another set two lower energy states.

eg Orbitals (d_x²_-y²and d_z²)

D-orbitals that are in direction of approaching ligand are raised to a higher energy state. These are known as eg. These are d_x²_-y²and d_z². These orbitals lie on the axis. Ligand approaches through axis in octahedral complexes. They face greater repulsion and then move to a high energy state.

t₂g Orbitals (d_xy, d_yz, d_zx)

These orbitals lie between the axis. They face less repulsion because the ligand is approaching through the axis. These orbitals move to a lower energy state. These are referred to as t₂g orbitals d_xy, d_yz, d_zx are also known as low energy orbitals.

Crystal field splitting energy of octahedral complexes

The energy difference between the two sets of d-orbital is known as crystal field splitting energy. This is due to the approaching ligand. Total energy changes are balanced. It means that the two energy of e_g orbitals is counterbalanced by the energy of t_2g orbitals.

Symbolic representation

For octahedral complex, it is symbolically represented as 𝚫₀. The subscript ₍₀₎ is sed for octahedral complexes.

Energy calculations

𝚫₀= 10 Dq

The energy of each orbital of e_gis raised by 6 Dq. So 6 ✖ 2=12Dq or (0.6 ✖ 2=1.2𝚫₀)

The energy of each orbital of t_2gis lowed by 4 Dq. So 4 ✖ 3=12Dq or (0.4 ✖ 3=1.2𝚫₀)

Crystal field stabilization energy (CFSE)

The amount of stabilization provided by the splitting of two orbitals into two levels is called crystal field stabilization energy. In octahedral complexes field for each electron that enters t_2g is 0.4𝚫₀ or 4 Dq whereas the field for each electron that enters into e_g is 0.6𝚫₀ or 6 Dq

Filling of electrons in d orbitals

In octahedral complexes there are two sets of energy levels, electrons always prefer to be in lower energy levels.

d¹ system

In the d¹system, there is only one electron. It will be in any of three t_2g orbitals because all three are of equal energies. Their electronic configuration is t_2g

CFSE= 1(-4Dq) = -4Dq

d² system

In the d²system there present two electrons. They will be in t_2g orbitals, with parallel spins. Their filling is according to Hund’s rule. Their electronic configuration is t_2g²

CFSE= 2(-4Dq) = -8Dq

d³ system

In the d³system there present three electrons. They will be in t_2g orbitals, with parallel spins. Their filling is also according to Hund’s rule. Their electronic configuration is t_2g³

CFSE= 3(-4Dq) = -12Dq

d⁴ system

In the d⁴system there present four electrons. There are two possibilities.

All of four may occupy t_2gorbital (strong field complexes)
Three electrons occupy t_2gorbital and the fourth one will occupy e_g orbital (weak field complexes)

When there present four or more electrons then filling of electrons in t_2g or e_g is depends upon two factors

Splitting energy (𝚫₀)
Pairing energy (P)

Case 1

If splitting energy is greater than pairing energy then there will be low spin complexes. These are also known as strong field complexes.

𝚫₀ > P
In this case, the complex has less number of unpaired electrons. Because there is a large energy gap between t_2gand e_gorbitals so electrons prefer to be in t_2gorbital. The Electronic configuration is t_2g⁴

CFSE= 4(-4Dq) = -16Dq ( -16Dq+ P)

Case 2

If splitting energy is lesser than pairing energy then there will be high spin complexes. These are also known as weak field complexes.

𝚫₀ < P
In this case, the complex has a lesser number of paired electrons. Maximum electrons are unpaired. Because there is small energy gap between t_2gand e_gorbitals. The electronic configuration is t_2g³ e_g¹

CFSE= 3(-4Dq) + 1(6Dq) = -6 Dq

d⁵ system

The d⁵ system is also similar to the d⁴ system. Either all five electrons remain in t_2g orbital or two of them may occupy e_g orbitals

Case 1

When all five electrons remain in t_2g orbitals then the electronic configuration is t_2g⁵

CFSE= 5(-4Dq) = -20Dq ( -20Dq+ P)

Case 2

When three electrons are in t_2g orbitals and two electrons are in e_gorbitals then The electronic configuration is t_2g³ e_g²

CFSE= 3(-4Dq) + 2(6Dq) = 0 Dq

Note

This is the same for systems having four are more electrons. There is only one possibility in d¹, d², d³, d⁸, d⁹, d¹⁰, systems.

Filling of electrons

Crystal Field splitting of tetrahedral complexes

In tetrahedral complexes, the coordination number is four. In tetrahedral complexes, four ligands surround the metal ion by placing them on the corner of the cube. all d-orbital will no longer be degenerate. They split into two sets of energy levels. One set moves to a higher energy state whereas, another set to lower energy states.

t₂ Orbitals (d_xy, d_yz, d_zx)

These orbitals lie between the axis. These are closer to the point through which ligands are approaching. They face greater repulsion. They become unstable. These orbitals move to a higher energy state. These are referred to as t₂ orbitals. d_xy, d_yz, d_zxare also known as high energy orbitals.

e Orbitals (d_x²_-y²and d_z²)

These orbitals lie on the axis. They are a little away from the ligands approach. That’s why they face less repulsion . they become stable. These orbitals move to a lower energy state. These are known as e orbitals. These are d_x²_{– y}²and d_z².

Crystal field splitting energy of tetrahedral complexes

The energy difference between the two sets of d-orbital is known as crystal field splitting energy. This is due to the approaching ligand. Total energy changes are balanced. It means that the two energy of e orbitals is counterbalanced by the energy of t₂ orbitals.

Symbolic representation

For tetrahedral complex, it is symbolically represented as 𝚫_t. The subscript _(t) is used for tetrahedral complexes. It is also measured in terms of Dq

Crystal field stabilization energy (CFSE)

The amount of stabilization provided by the splitting of two orbitals into two levels is called crystal field stabilization energy. In tetrahedral complexes field for each electron that enters t₂ is 0.4𝚫_t or 4 Dq whereas the field for each electron that enters into e is 0.6𝚫₀ or 6 Dq

Energy calculations

𝚫_t= 10 Dq

The energy of each orbital of e is lowered by 6 Dq. So 6 ✖ 2=12Dq or (0.6 ✖ 2=1.2𝚫_t)

The energy of each orbital of t₂is raised by 4 Dq. So 4 ✖ 3=12Dq or (0.4 ✖ 3=1.2𝚫_t)

Filling of electrons

As ligands are not directly approaching any axis so there is less repulsion. Similarly, there is less energy gap between the two energy levels. There is no concept of high spin and low spin complexes. A filling of electrons is according to Hund’s rules. First electrons fill at lower energy orbitals. Then moves to higher energy orbitals. Till d⁵ system, there is no pairing of electrons. The pairing of electrons started from the d⁶ system.

Crystal field splitting of square planner and tetragonal complexes

The square planner and tetragonal complexes are known as distorted octahedral complexes. Both of these complexes are obtained from octahedral complexes. Square planner and tetragonal complexes merge into one another.

Crystal field splitting of square planer complexes

When we completely remove two of the ligand from the z-axis it leads to the formation of a square planner complex. In this case, the ligand is approaching through the x,y plane. These orbitals get destabilized whereas, z-orbital get stabilized. So form e_g orbital d_x²_{– y}² moves to a higher energy state leaving behind d_z². The d_z² gets lower than d_xy. Similarly, from t_2g orbitals, d_yzand d_zxmove down and get stabilized whereas d_xy remains at a higher state even higher than d_z².

Except for d_yzand d_zx, there are no degenerate orbitals.

Crystal field diagram of the tetragonal complex.

Tetragonal complexes are also formed by removal (not completely) two of trans ligands from the z-axis. In this case, we start withdrawing trans ligands from the z-axis. Ligands are approaching through the x-y plane. So these face greater repulsion. Degeneracy of eg orbitals is removed because of greater repulsion d_x²_{– y}² move to higher energy state as compared to d_z². Similary, from t_2g orbitals, d_yz,and d_zxmove down and get stabilized whereas d_xy is at a higher energy state.

Factors affecting crystal field splitting

There are the following factors that affect the splitting

Nature of ligands
Coordination number
Arrangement of ligand
Nature of metal atom
Charge on the metal atom
Size of ligands
Electronegativity

Applications of crystal field theory

Crystal field theory describes the interaction between ligands and metal atoms.
It explains the geometry of complexes.
Crystal field theory explains the different properties of transition metal complexes i.e,

Magantesim, reactivity, etc.

It helps to predict the color of coordination complexes.
It provides information about the lattice energy of complexes.
Crystal field theory also explains the relative stabilities of coordination complexes.
Crystal field theory provides the orbital picture (structure) of coordination complexes.
CFT explains the electronic spectra of com0plexes.

Key takes away

Concepts Berg

Who proposed the crystal field theory?

Crystal field theory was proposed by Hans Bethe and John Hasbrouck van Vleck in 1929.

What are the main points of crystal field theory?

Main points of crystal field theory

Transition Metal atom is surrounded by ligands (produce magnetic field)

Interaction between ligand and metal atom is purely electrostatic (100% ionic)

Ligands are considered as point charges

Ligands may be negative or neutral atoms (F^–, Cl^–, NH₃, H₂O, etc)

Approaching ligands cause splitting of degenerate orbitals. (d-orbitals)

What is t_2gand e_g orbital?

t₂g Orbitals (d_xy, d_yz, d_zx)

eg Orbitals (d_x²_-y²and d_z²)

d-orbitals that are in direction of approaching ligand are raised to a higher energy state. These are known as eg. These are d_x²_-y²and d_z². These orbitals lie on the axis. Ligand approaches through axis in octahedral complexes. They face greater repulsion and then move to a high energy state.

Why symbol g is not used for tetrahedral complexes?

Octahedral splitting is designated as t_2g and e_gorbitals whereas, for tetrahedral only t₂ and e orbitals. This is because tetrahedral geometry has no center of symmetry. The symbol g is used only for geometries having a center of symmetry.

What is meant by high spin complexes?

High spin complexes are given by weak ligands. In this case, there is less energy gap between the two energy levels. So electrons can easily jump to higher energy levels. High spin complexes have a large number of unpaired electrons. For example [CoF₆]³.

What is meant by low spin complexes?

low spin complexes are given by strong ligands. In this case, there is a great energy gap between the two energy levels. So electrons cannot jump easily to higher energy levels. Electrons prefer to be in lower energy levels. Low spin complexes have a large number of paired electrons. For example [Co(NH₃)₆]⁺³

What is meant by degenerate orbitals?

Degenrate orbitals are of same energy. So electrons can be in any one of them.

What is crystal field stabilization energy?

Crystal field stabilization energy is the energy gap between two sets of orbitals after splitting.

What is tetrahedral crystal field splitting?

When four ligands approach transition metal atom it cause splitting of degenerate d orbital into two energy levels this is called tetrahedral crystal field splitting.

What are the limitations of the crystal field theory?/ What are the failures of the ionic model of CFT Crystal Field Theory

Drawbacks of crystal field theory

This theory only explains sigma bonding. (no information about pi bonding)

CFT is only about d orbitals. (no information about s,p orbitals)

Cft does not provide significant information about the orbitals of ligands.

Metal ligand interactions are purely electrostatic is not true.(not always)

CFT does not provide any information about covalent bonding.

CFT can not explain the relative strength of ligands.

CFT is only about transition metals.

Why do we consider ligand as a point charge in crystal field theory?
According to CFT, there is a purely ionic interaction between the metal atom and ligands. Ligand is the source of charge in complexes. That’s why ligands are considered as point charges.

Give some applications of Crystal field theory.

Application of Crystal field theory

Crystal field theory describes the interaction between ligands and metal atoms.
It explains the geometry of complexes.
Crystal field theory explains the different properties of transition metal complexes i.e,

Magantesim, reactivity, etc.

It helps to predict the color of coordination complexes.
It provides information about the lattice energy of complexes.
Crystal field theory also explains the stabilities of coordination complexes.
Crystal field theory provides the orbital picture (structure) of coordination complexes.