Bonding in Coordination Complexes: Crystal Field Theory | Inorganic Chemistry PDF Download

What is Crystal Field Theory?

Crystal field theory describes

the net change in crystal energy resulting from the orientation of d orbitals of a transition metal cation inside a coordinating group of anions also called ligands.

A major feature of transition metals is their tendency to form complexes. A complex may be considered as consisting of a central metal atom or ion surrounded by a number of ligands. The interaction between these ligands with the central metal atom or ion is subject to crystal field theory.

Crystal field theory was established in 1929 treats the interaction of metal ion and ligand as a purely electrostatic phenomenon where the ligands are considered as point charges in the vicinity of the atomic orbitals of the central atom. Development and extension of crystal field theory taken into account the partly covalent nature of bonds between the ligand and metal atom mainly through the application of molecular orbital theory. 

Crystal field theory often termed as ligand field theory.

In order to understand clearly the crystal field interactions in transition metal complexes, it is necessary to have knowledge of the geometrical or spatial disposition of d orbitals. The d-orbitals are fivefold degenerate in a free gaseous metal ion. If a spherically symmetric field of negative ligand filed charge is imposed on a central metal ion, the d-orbitals will remain degenerate but followed by some changes in the energy of free ion.

A summary of the interactions is given below.

Crystal Field Splitting

Crystal field theory was proposed which described the metal-ligand bond as an ionic bond arising purely from the electrostatic interactions between the metal ions and ligands. Crystal field theory considers anions as point charges and neutral molecules as dipoles.

When transition metals are not bonded to any ligand, their d orbitals are degenerate that is they have the same energy. When they start bonding with other ligands, due to different symmetries of the d orbitals and the inductive effect of the ligands on the electrons, the d orbitals split apart and become non-degenerate.

The complexion with the greater number of unpaired electrons is known as the high spin complex, the low spin complex contains the lesser number of unpaired electrons. High spin complexes are expected with weak field ligands whereas the crystal field splitting energy is small Δ. The opposite applies to the low spin complexes in which strong field ligands cause maximum pairing of electrons in the set of three t₂ atomic orbitals due to large Δ_o.

• High spin – Maximum number of unpaired electrons.
• Low spin – Minimum number of unpaired electrons.

Example: [Co(CN)₆]^3- & [CoF₆]^3-

High Spin and Low Spin Complex

• [Co(CN)₆]^3- – Low spin complex
• [CoF₆]^3- – High spin complex

Splitting of d-Orbital Energies in an Octahedral Fields: 

In a free metal (or isolated) metal cation all the five d-orbitals are degenerate (same energy). When the ligands move towards the metal cation, two electrostatic forces operate. One is the attraction between the metal cation and the ligands and the second is the electrostatic repulsion between d -electrons of the metal cation and the lone pairs of electrons on the ligands. Greater the force of attraction between the metal cation and the ligands, ligands will be more closer to the metal ion, and hence more will be the repulsion between the metal d electrons and the lone pair of electrons. This repulsion causes to increase in energy of metal d electrons. In a spherical field all the ligands are at equal distance from each of the d orbitals. The energy of each d orbital will raise by the same amount and all the five d-orbitals will still remain degenerate. In an octahedral complex, however, all the five d orbitals do not affected to same extent. Since the two orbitals (d_x²–y² and d_z²) point directly towards the ligands and three orbitals (d_xy, d_yz, d_zx) point in between the path of approaching ligands. 

Therefore, the d_{x² – y²} and d_z² orbitals will be more strongly repelled t han the d_xy, d_yz, d_zx orbitals. Therefore, the energy of d_{x² – y²} and d_z² will be raised and that of d_xy, d_yz and d_zx orbitals which lie far away from the ligands will be decreased relative to the hypothetical energy state. 

The set of d_{x² – y²} and d_z² orbitals is referred to as eg set which is doubly degenerate and the set of d_xy, d_yz and d_zx orbital is referred to as t_2g set which is triply degenerate.
The separation of five d orbitals of metal cation into two sets of different energies is called crystal field splitting. The energy difference between two sets of orbitals which arise due to an octahedral field is called ∆_o or 10 D_q, where o in ∆_o stands for octahedral.

The total energy decrease of t_2g set is equal to the total energy increase of eg orbitals. The eg orbitals are destabilised b y +0.6 ∆_o or +6 Dq with respect to bary’s centre whereas t_2g orbitals are stabilised by -0.4 ∆_o or –4 D_q with respect to the bary,s centre. 

Crystal Field Stabilization Energy 

The energy which is released in a formation of a complex is called the crystal field stabilization energy of that complex.
For example, for a d¹ system, the ground state corresponds to the configuration t_2g¹ Wit h respect to the barycentre, there is stabilization energy of –0.4∆_o .This indicates that in a complex ion of d¹ configuration 0.4∆_o energy is released. This released energy is called the crystal field stabilization energy (CFSE) of d¹ complex.

Similarly, for or a d² ion, the ground state configuration is t2g2 and the CFSE is –0.8∆_o and for complex having d³ configuration (t_2g³) has a CFSE = –1.2∆_o.

This is calculated as follows:

CFSE = (no. of electrons in t_2g) x (–0.4∆_o) +   (no. of electrons in e_g) x (+0.6∆_o)
 CFSE = nt_2g x (–0.4∆_o) + neg x (0.6∆_o)

For d¹, electronic configuration is t_2g¹ eg⁰: nt_2g = 1, ne_g = 0

CFSE = 1 × (–0.4∆_o) + 0 × (+0.6 ∆_o)

CFSE = –0.4∆_o = 4 Dq

For d² configuration electronic configuration is t_2g² e_g⁰, nt_2g = 2, ne_g = 0

CFSE = 2× (–0.4∆_o) + 0 × (0.6 ∆o)
CFSE = –0.8∆_o = 8D_q

For d³ configuration electronic configuration is t_2g³ e_g⁰, nt_2g = 3, ne_g = 0

CFSE = (3×–0.4∆_o) + 0 = –1.2∆_o