Crystal Field Theory
Crystal field theory (CFT) is a model that describes the breaking of degeneracies of electronic orbital states, usually d or f-orbitals, due to a static electric field produced by a surrounding charge distribution (anion neighbours)
This theory has been used to describe various spectroscopies of transition metal coordination complexes, in particular optical spectra (colours).
CFT successfully accounts for some magnetic properties, colours, hydration enthalpies, and spinel structures of transition metal complexes, but it does not attempt to describe bonding.
CFT was developed by physicists Hans Bethe and John Hasbrouck van Vleck in the 1930s.
CFT was subsequently combined with molecular orbital theory to form the more realistic and complex ligand field theory (LFT), which delivers insight into the process of chemical bonding in transition metal complexes.

Overview of crystal field theory analysis

According to CFT, the interaction between a transition metal and ligands arises from the attraction between the positively charged metal cation and negative charge on the non-bonding electrons of the ligand.
The theory is developed by considering energy changes of the five degenerate d-orbitals upon being surrounded by an array of point charges consisting of the ligands.
As a ligand approaches the metal ion, the electrons from the ligand will be closer to some of the d-orbitals and farther away from others causing a loss of degeneracy.
The electrons in the d-orbitals and those in the ligand repel each other due to repulsion between like charges.
Thus the d-electrons closer to the ligands will have a higher energy than those further away which results in the d-orbitals splitting in energy represented by Δ .

This stability of a complexion is affected by the following factors:
In the d-block elements(transition) as we move from 3d to 4d to 5d elements ,the Δ (crystal field splitting energy) increases.
The explaination is that in larger ions ,the d orbitals are larger and more diffuse and extend farther from the nucleus in the direction of ligands.This produces a larger repulsion between the ligands and the orbitals that point at them.
Therefore , the second and third transition series have greater tendency to form low spin complexes and have high Δ.
Example :
Complexes of Ni2+ and Pt2+ with same ligands.
Pt2+ complex has larger crystal field splitting Δ, than the Ni2+ complex.
Oxidation State Of Metal Ion.
The oxidation state of the metal also contributes to the size of ? between the high and low energy levels.
As the oxidation state increases for a given metal, the magnitude of Δ increases
Example :
A V3+ complex will have a larger Δ than a V2+ complex for a given set of ligands, as the difference in charge density allows the ligands to be closer to a V3+ ion than to a V2+ ion.
The smaller distance between the ligand and the metal ion results in a larger Δ, because the ligand and metal electrons are closer together and therefore repel more.

Arrangement Of Ligands Around The Metal Ion In Octahedral Crystal Field Splitting :

The most common type of complex is octahedral; here six ligands form an octahedron around the metal ion.
The Octahedral Geometry
In octahedral symmetry the d-orbitals split into two sets with an energy difference, Δoct (the crystal-field splitting parameter) where the dxy, dxz and dyz orbitals will be lower in energy because group is farther from the ligands than the dz2 and dx2 - y2 group which will have higher energy, therefore experience less repulsion. The three lower energy orbitals are collectively referred to as t2g, the dxy, dxz and dyz orbitals The two higher-energy orbitals as eg.dz2 and dx2 - y2
Octahedral Splitting
The three lower-energy orbitals are collectively referred to as t2g, dxy, dxz and dyz.
The two higher-energy orbitals as eg. (These labels are based on the theory of molecular symmetry). dz2, and dx2 - y2

Tetrahedral Crystal Field Splitting

Tetrahedral complexes are the second most common type; here four ligands form a tetrahedron around the metal ion.
In a tetrahedral crystal field splitting the d-orbitals again split into two groups, with an energy difference of Δtet where
Tetrahedral Splitting
The lower energy orbitals will be dz2, dx2 and dx2 - y2.
The higher energy orbitals will be dxy, dxz and dyz opposite to the octahedral case.
Furthermore, since the ligand electrons in tetrahedral symmetry are not oriented directly towards the d-orbitals, the energy splitting will be lower than in the octahedral case.

Square Crystal Field Splitting

The square planar geometry may be considered to be derived from octahedral geometry.
In a square planar complex the ligands along the z-axis are completely removed.
As a result of these distortions, there is a net lowering of energy.
In this case, the three lower orbitalsd drops even lower in energy.
The dx2 - y2 orbital being the first higher energy level.
The dxy orbital is the second higher energy level.
Whereas the dz2, dxz, dyzorbitals become more stable energy levels.

Square Planar Splitting
In general, the size of the splitting in a square planar complex, is 1.3 times greater than for octahedral complexes with the same metal and ligands.
This distortion to square planar complexes is known as Jahn-Teller effect especially prevalent for d8 configurations and elements in the 4th and 5th periods such as: Rh (I), Ir (I), Pt(II), Pd(III), and Au (III).

Jahn-Teller effect :

The Jahn-Teller effect, sometimes also known as Jahn\96Teller distortion, or the Jahn-Teller theorem,
It describes the geometrical distortion of non-linear molecules under certain situations.
This electronic effect is named after Hermann Arthur Jahn and Edward Teller, who proved, using group theory, that orbital non-linear spatially degenerate molecules cannot be stable.
Jahn-Teller splitting of eg level in [Cu(H2O)6]2+.
The theorem essentially states that any non-linear molecule with a spatially degenerate electronic ground state will undergo a geometrical distortion that removes that degeneracy, because the distortion lowers the overall energy of the complex.
The Jahn-Teller effect is responsible for the tetragonal distortion of the hexaaquacopper(II) complex ion, [Cu(OH2)6]2+, which might otherwise possess octahedral geometry. The two axial Cu-O distances are 238 pm, whereas the four equatorial Cu-O distances are ~195 pm.
The tetragonal distortion of the hexa aqua copper(II) complex ion, [Cu(OH2)6]2+.
In octahedral complexes, the Jahn-Teller effect is most pronounced when an odd number of electrons occupy the eg orbitals; i.e., in d9, low-spin

The expected effects for octahedral coordination are given in the following table:

Jahn-Teller effect
Number of d electrons High spin Low spin
1 w w
2 w w
3
4 s w
5 w
6 w
7 w s
8
9 s s
10

Nickel (II) four-coordinate complexes are usually tetrahedral unless there is a very strong ligand fields such as in [Ni(CN)4]2-, which is square planar.
The size of the gap Δ between the two or more sets of orbitals depends on several factors, including the ligands and geometry of the complex.

Nature of Ligands :

Some ligands always produce a small value of Δ, while others always give a large splitting. The reasons behind this can be explained by ligand field theory.
The arrangement of ligands in order of their crystal field splitting energy(Δ ) values is known as spectro chemical series
The spectro chemical series
I- < Br- < S2- < SCN- < Cl- < NO3- < N3- < F- < OH- < C2O42- < H2O < NCS- < CH3CN < py < NH3 < en < 2,2' - bipyridine < phen < NO2- < PPh3 < CN- < CO
crystal field splitting energy(Δ) value of I- is small.
crystal field splitting energy(Δ) value of CO is large.
In the spectrochemical series the ligands before the H2O are called as Weak field ligands.
The ligands after the H2O are called as Strong field ligands.
It is useful to note that the ligands producing the most splitting are those that can engage in metal to ligand back-bonding.
The smaller distance between the ligand and the metal ion results in a larger Δ, because the ligand and metal electrons are closer together and therefore repel more.

High-spin and low-spin Complexes :

If the crystal field splitting energy Δ,is more or less than the energy required for electron pairing in an orbital(P) the complex will be a High-spin and low-spin Complex.
High-spin Complexes :
If the crystal field splitting energy Δ,is more than the energy required for electron pairing in an orbital(P) the complex will be a High-spin Complex.
Weak field Ligands tend to form High-spin Complexes.
Ligands(like I- and Br-) which cause a small splitting Δ of the d-orbitals are referred to as weak-field ligands
Example:
[FeBr6]3-
Br- is a weak-field ligand and produces a small Δ.
So, the ion [FeBr6]3-, again with five d-electrons, would have an octahedral splitting diagram where all five orbitals are singly occupied.
In this case, it is easier to put electrons into the higher energy set of orbitals than it is to put two into the same low-energy orbital, because two electrons in the same orbital repel each other. So, one electron is put into each of the five d-orbitals before any pairing occurs in accord with Hund's rule.
Therefore, the Complex forms a "high spin" complex.

Low-spin Complexes :
If the crystal field splitting energy Δ,is less than the energy required for electron pairing in an orbital(P) the complex will be a low-spin Complex.
Strong field Ligands tend to form low-spin Complexes.
Ligands which cause a large splitting Δ of the d-orbitals are referred to as strong-field ligands, such as CN- and CO from the spectrochemical series.
Example:
[Fe(NO2)6]3-
NO2- is a strong-field ligand and produces a large Δ.
Fe(NO2)6]3- has 5 d-electrons, would have the octahedral splitting with all five electrons in the t2g level.
In complexes with these ligands, it is unfavourable to put electrons into the high energy orbitals. Therefore, the lower energy orbitals are completely filled before population of the upper sets starts according to the Aufbau principle.
Therefore, the Complex forms a low-spin Complex.

Effect on Magnetism :

Low-spin complexes contain more paired electrons, since the splitting energy is larger than the pairing energy.
These complexes, such as [Fe(CN)6]3-, are more often diamagnetic or weakly paramagnetic.
High-spin complexes usually contain more unpaired electrons, since the pairing energy is larger than the splitting energy.
With more unpaired electrons, high-spin complexes are often paramagnetic.
The unpaired electrons in paramagnetic compounds create tiny magnetic fields, similar to the domains in ferromagnetic materials.
The more unpaired electrons (often the higher-spin the complex), the more strongly paramagnetic a coordination complex is, we can predict paramagnetism, and its relative strength by determining whether a compound is a weak field ligand or a strong field ligand.
Once we have determined whether a compound has a weak or strong field ligand, we can predict its magnetic properties

Crystal field splitting energy. (Δ)
Therefore, the energy required to pair two electrons is typically higher than the energy required for placing electrons in the higher energy orbitals.
Thus, tetrahedral complexes are usually high-spin.
The use of these splitting diagrams can aid in the prediction of the magnetic properties of coordination compounds.
A compound that has unpaired electrons in its splitting diagram will be paramagnetic and will be attracted by magnetic fields.
A compound that lacks unpaired electrons in its splitting diagram will be diamagnetic and will be weakly repelled by a magnetic field.
The spin state of the complex also affects an atom's ionic radius.

The following are some of the High spin and Low spin complexes :

d4
Octahedral high-spin: 4 unpaired electrons, paramagnetic, substitutionally labile. Includes Cr 2+ ionic radius 80 pm, Mn3+ ionic radius 64.5 pm.
Octahedral low-spin: 2 unpaired electrons, paramagnetic, substutionally inert. Includes Cr2+ ionic radius 73 pm, Mn3+ ionic radius 58 pm.

d5
Octahedral high-spin: 5 unpaired electrons, paramagnetic, substitutionally labile. Includes Fe3+ ionic radius 64.5 pm.
Octahedral low-spin: 1 unpaired electron, paramagnetic, substitutionally inert. Includes Fe3+ ionic radius 55 pm.

d6
Octahedral high-spin: 4 unpaired electrons, paramagnetic, substitutionally labile. Includes Fe2+ ionic radius 78 pm, Co3+ ionic radius 61 pm.
Octahedral low-spin: no unpaired electrons, diamagnetic, substitutionally inert. Includes Fe2+ ionic radius 62 pm, Co3+ ionic radius 54.5 pm, Ni4+ ionic radius 48 pm.

d7
Octahedral high-spin: 3 unpaired electrons, paramagnetic, substitutionally labile. Includes Co2+ ionic radius 74.5 pm, Ni3+ ionic radius 60 pm.
Octahedral low-spin: 1 unpaired electron, paramagnetic, substitutionally labile. Includes Co2+ ionic radius 65 pm, Ni3+ ionic radius 56 pm.

d8
Octahedral high-spin:2 unpaired electrons, paramagnetic, substitutionally labile. Includes Ni2+ ionic radius 69 pm.
Square planar low-spin: no unpaired electrons, diamagnetic, substitutionally inert. Includes Ni2+ ionic radius 49 pm.

Crystal field stabilization energy

The crystal field stabilization energy (CFSE) is the stability that results from placing a transition metal ion in the crystal field generated by a set of ligands.
It arises due to the fact that when the d-orbitals are split in a ligand field some of them become lower in energy than before with respect to a spherical field known as the barycenter in which all five d-orbitals are degenerate.
In an octahedral case, the t2g set becomes lower in energy than the orbitals in the barycenter.
As a result of this, if there are any electrons occupying these orbitals, the metal ion is more stable in the ligand field relative to the barycenter by an amount known as the CFSE. Conversely, the eg orbitals (in the octahedral case) are higher in energy than in the barycenter, so putting electrons in these reduces the amount of CFSE.
Octahedral crystal field stabilization energy
If the splitting of the d-orbitals in an octahedral field is Δoct,the three t2g orbitals are stabilized relative to the barycenter by 2/5 Δoct, and the eg orbitals are destabilized by 3/5 Δoct.
The low-spin (top)
example has five electrons in the t2g orbitals,
so the total CFSE is 5 × 2/5 Δoct = 2Δoct.
In the high-spin (lower)
example, the CFSE is (3 × 2/5 Δoct) - (2 × 3/5 Δoct) = 0
In this case, the stabilization generated by the electrons in the lower orbitals is canceled out by the destabilizing effect of the electrons in the upper orbitals.
Crystal Field stabilization is applicable to transition-metal complexes of all geometries.
Indeed, the reason that many d8 complexes are square-planar is the very large amount of crystal field stabilization that this geometry produces with this number of electrons.
Drawback in the Crystal Field Theory
The crystal field theory could explain the formation, structure, optical and magnetic properties of the coordination compounds quite satisfactorily.
However, the CFT could not explain the following :
The existence of covalent bonding in some transition metal complexes.
The order of ligands in the spectrochemical series.
On the basis of the assumption that the ligands are point charges, the anionic ligands should exert greater crystal field splitting effect.
However, most of the anionic ligands are found at the low-end of the spectrochemical series, i.e., these are considered weak field ligands.
The OH- ion although lies below H2O and NH3 in the spectrochemical series but produces a greater crystal field splitting.

Ligand Field Theory

Ligand Field Theory Along With Molecular Orbital Theory
As the Ligand Field Theory applies molecular orbital theory and symmetry concerns to transition metal complexes.
The principles of Ligand Field Theory are similar to those for Molecular Orbital Theory.
An alternative approach to understanding the bonding of transition metal complexes is Ligand Field Theory.
Crystal Field Theory is a simple model which explains the spectra, thermochemical and magnetic data of many complexes. It's main flaw is that it treats the ligands as point charges or dipoles, and fails to consider the orbitals of the ligands.
In octahedral symmetry, group theory can be used to determine the shapes and orientation of the orbitals on the metal and the ligands.
Examination of the symmetry table for OH shows that the orbitals on the metal have the following attributes.
Group theory can be used to determine the combination of ligand orbitals which have the same symmetry properties as the metal orbitals. The results of these Symmetry Adapted Linear Combinations (SALC) are provided below.
The t2g set (dxy, dyz, dxz) does not have any electron density along the bond axes, so these orbitals do not participate in sigma bonding, but will be involved with π bonding.
A molecular orbital diagram which estimates the energies of the bonding (show above) antibonding and non-bonding orbitals is shown in adjacent figure.
Since there is a large disparity in energy between the ligand orbitals and the metal orbitals, the lower lying molecular orbitals in the diagram are essentially ligand orbitals.
That is, the electrons of the ligand lone pairs fill the lower levels (eg, t1u, and a1g).
The d electrons on the metal will fill the t2g (non-bonding) and eg (anti bonding) molecular orbitals.
The split between the HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) corresponds to the Δo splitting in crystal field theory.
In crystal field theory, the electrons in these orbitals are viewed as entirely on the metal atom or ion, whereas in ligand field theory the electrons are, to some extent, on the ligands, too.
Pi (π) Bonding Pi bonding is not considered by crystal field theory, but is addressed in ligand field theory.
The orbitals on the metal which were not used for sigma bonding (the t2g set: dxy, dyz, dxz) have the same symmetry properties as combinations of the p orbitals on the ligands.
If the energy of the metal and ligand orbitals are comparable, the pi bonding orbitals formed will be significantly lower in energy than the atomic orbitals on either the metal or ligand
Likewise, the anti bonding pi orbitals will be much higher in energy. If the orbitals are very different in energy, only slight mixing will occur.
An example of pi overlap is shown in adjacent figure.
The effect on the molecular orbital diagram is as follows.
The gap between the t2g and eg set will change, because the t2g set is involved in bonding, so there is not a bonding t2g set, and an anti bonding t2g set of orbitals.
The gap, represented as Δo becomes the gap between the t2g set of anti bonding orbitals and the eg set of orbitals. As a result, the size of Δo diminishes.
The above molecular orbital diagram is for ligands which have pi antibonding orbitals too high in energy to interact with the metal orbitals.
The net effect for these pi donor ligands is to decrease the size of Δo compared to ligands which only act as sigma donors.
Ligands may have empty pi anti bonding orbitals higher in energy and with the same symmetry as the t2g orbitals of the metal.
These ligands orbitals interact with the t2g orbitals of the metal creating a bonding orbital which is slightly lower in energy than the t2g set of the metal, and an anti bonding set of orbitals which are much greater in energy than the eg set of the complex.
The net result is that the size of the splitting, Δo, increases, since the energy of the t2g bonding orbitals drops a bit.
The net result is that pi acceptor ligands (such as CO and N2), with empty anti bonding orbitals available to accept electrons from the metal, increase the size of Δo.
The spectro chemical series can be reconsidered with the possiblity of pi bonding in mind.
It shows that the order (with some notable exceptions) goes as follows:
strong π donor (small Δo) < weak π donor < no π effects (intermediate Δo) < π acceptor (large Δo)
The trend can be illustrated with the following ligands as examples:
I- < Br- < Cl- < F- < H2O < NH3 < PR3 < CO, CN-
Symmetry table for octahedral (Oh) geometry
Metal orbital Symmetry label Degeneracy
s a1g non-degenerate
px, py, pz t1u triply degenerate
dxy, dyz, dxz t2g triply degenerate
dx2 - y2, dz2v eg doubly degenerate

M.O diagrams

dπ - pπ bonding

M.O diagrams of π - donating and π - acceptor bondings