The Chemistry of d- and f- Block: Crystal Field Theory
In this section we will build up our understanding in d-block complex compounds by looking the bonding in complex compounds. One of the approach is by using the crystal field theory which can derive some important properties of complexes. Then, we will see some factors that affect the crystal field splitting such as the ligands and the metal centre ion.
As we saw in previous section (the introduction) there are two point of views to describe the bonding in complexes, covalent view and ionic view. In covalent view, the complex is described by two electrons interaction and to rationalise this bonding, molecular orbital (MO) theory is required and we will not have a discussion in this part. In the other hands, in ionic view the complex is described as a positive centre ion of metal and it is surrounded by negative charged species of ligands. Then, the bonding energy comes from electrostatic interactions between anions and cations. From this assumption we can rationalise the bonding by using the crystal field (CF) theory where the energy of interaction shall be ignored largely.
To rationalise the bonding in complex compound, we can start with p-orbital as the analogue. When a charge of field is brought to the the p-orbital, all the orbitals increase its energy due to electrostatic interaction with the field of charge. Because the field is spherical, so the interaction with any points on the orbitals are the same which cause the same increased energy level. Then, the field is localised along the an axis, so the orbitals which interact directly with ligands feel greater interaction and the orbitals which do not interact directly with ligands feel less interaction. For example, when the field is localised at z-axis, pz orbital feels greater repulsion hence the energy increases with relative to the spherical field. Meanwhile, px and py orbital feel less interaction hence the energy decreases with relative to the spherical field. Therefore the energy level diagram can be drawn as shown below.
Then, we can move to d-orbital which has slightly more complicated than p-orbital as it has 5 set of orbitals. However, the same approach can be used to rationalised if we have an octahedral field. When the fields are localised along the x, y, and z axes so the orbitals which interacts directly with the ligands feel greater interaction which are dx2−y2 and dz2. In the other hands, the other orbitals (dxy, dyz, and dxz) will feel less interaction so the energy will decrease. The increasing and decreasing energy of the orbitals can be quantified so the total energy of is equal to 0. This means dx2−y2 and dz2 orbitals will increase by factor 0.6 and dxy, dyz, and dxz will decrease by factor of 0.4. The parameter that is used to describe the energy in complex stabilisation is called Δo (sometimes Δoct) which is the crystal field splitting parameter and this is a variable energy unit. Besides that, in the octahedral splitting dx2−y2 and dz2 orbitals are labelled as eg and dxy, dyz, and dxz) orbitals are labelled as t2g. Moreover, this labelling is based on the symmetry of the orbital.
Another common splitting in complex compound is the cubic field. In this field, all the field is localised at the vertices of a cube as shown in picture below.
Therefore, the splitting in cubic field is shown as:
This splitting can also be applied for tetrahedral field as it is similar with cubic field with the only difference is shown below and the splitting energy in tetrahedral field.
The Δo has bigger energy gap than Δtet and this is simply caused by the fact that in octahedral field the metal centre ion interacts directly with the ligands where in tetrahedral is not direct interaction. The relationship between Δtet is 4/9 of Δo and this generalisation can predict the behaviour of complex with this different geometries.
From this splitting, the splitting parameter can be quantified for certain complex and we will start from the octahedral field. To start with, d1 complex can be the first example by putting the electron based on the rules that govern the electron filling, so it will fill at the lowest energy level (t2g) and the the stabilisation parameter or crystal field stabilisation energy (CFSE) can be calculate as shown below.
Then, the same rule can be applied to d3. When we enter d4, the electron can fill either eg or t2g because the energy gap is small. This order is determined by how big the factor of stabilisation of the complex splitting.
If the Δo is much bigger than the pairing energy so the 4th electron would prefer to be paired up rather than filling the eg, and also vice versa. When electron is paired up the complex is called low spin complex, and when the electron fill eg the complex is called at high spin. The same rule can be applied to tetrahedral field but it will start to have low spin complex at d3 and the low spin complex in tetrahedral is so rare.
From the octahedral field we can derive the splitting in square planar field. To rationalise this field, it can be seen simply by looking the ligands at z-axis is removed into infinity. Therefore, all d-orbitals that related to z-axis will be decreased its energy and the splitting is shown below.
As mentioned earlier there are some factors that affect the splitting in complexes that can determine the spin of a complex, either high spin or low spin. One of the basic assumption of the crystal field theory is it comes from the electrostatic interaction. Therefore, as the charge on metal is greater, the splitting becomes greater as well. As a note, the oxidation number is a vague indicator of charge. Besides that, as the metal is down on the group, the splitting also becomes bigger. This is caused by bigger ionic radius which implies the d-orbital has closer interaction with ligands. Effect from ligands also determined how big the splitting is. The effect from the ligands is based on the spectrochemical series where the series is:
As we saw in previous section (the introduction) there are two point of views to describe the bonding in complexes, covalent view and ionic view. In covalent view, the complex is described by two electrons interaction and to rationalise this bonding, molecular orbital (MO) theory is required and we will not have a discussion in this part. In the other hands, in ionic view the complex is described as a positive centre ion of metal and it is surrounded by negative charged species of ligands. Then, the bonding energy comes from electrostatic interactions between anions and cations. From this assumption we can rationalise the bonding by using the crystal field (CF) theory where the energy of interaction shall be ignored largely.
To rationalise the bonding in complex compound, we can start with p-orbital as the analogue. When a charge of field is brought to the the p-orbital, all the orbitals increase its energy due to electrostatic interaction with the field of charge. Because the field is spherical, so the interaction with any points on the orbitals are the same which cause the same increased energy level. Then, the field is localised along the an axis, so the orbitals which interact directly with ligands feel greater interaction and the orbitals which do not interact directly with ligands feel less interaction. For example, when the field is localised at z-axis, pz orbital feels greater repulsion hence the energy increases with relative to the spherical field. Meanwhile, px and py orbital feel less interaction hence the energy decreases with relative to the spherical field. Therefore the energy level diagram can be drawn as shown below.
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| Splitting in p-orbitals |
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| d-orbitals |
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| octahedral field splitting |
Another common splitting in complex compound is the cubic field. In this field, all the field is localised at the vertices of a cube as shown in picture below.
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| Cubic field |
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| Cubic field splitting |
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| Tetrahedral field splitting |
From this splitting, the splitting parameter can be quantified for certain complex and we will start from the octahedral field. To start with, d1 complex can be the first example by putting the electron based on the rules that govern the electron filling, so it will fill at the lowest energy level (t2g) and the the stabilisation parameter or crystal field stabilisation energy (CFSE) can be calculate as shown below.
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| CFSE of d1 |
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| High spin (left) and low spin (right) of d4 in octahedral splitting |
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| d2 splitting ,high spin (centre) and low spin (right) of d4 in octahedral splitting |
From the octahedral field we can derive the splitting in square planar field. To rationalise this field, it can be seen simply by looking the ligands at z-axis is removed into infinity. Therefore, all d-orbitals that related to z-axis will be decreased its energy and the splitting is shown below.
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| Square planar field splitting |
As mentioned earlier there are some factors that affect the splitting in complexes that can determine the spin of a complex, either high spin or low spin. One of the basic assumption of the crystal field theory is it comes from the electrostatic interaction. Therefore, as the charge on metal is greater, the splitting becomes greater as well. As a note, the oxidation number is a vague indicator of charge. Besides that, as the metal is down on the group, the splitting also becomes bigger. This is caused by bigger ionic radius which implies the d-orbital has closer interaction with ligands. Effect from ligands also determined how big the splitting is. The effect from the ligands is based on the spectrochemical series where the series is:
I− < Br− < S2− < SCN− < Cl− < NO3− < N3− < F− < ONO− < OH− < OSO32- < ONO2− < C2O42− < H2O < NCS− < edta4− < py (pyridine) ≈ NH3 ≈ en (ethylenediamine) < SO32- < bipy (2,2'-bipyridine) < phen (1,10-phenanthroline) ≈ NO2− < PR3 < C2H4 ≈ CN− ≈ COAlthough the series is quite long, we can see that when the donating electron atom in the ligand has lower electronegativity it in strong field ligand (e.g CN-). Therefore, halides are weak field ligand where I- is the weakest ligand. The effect on this ligand is when a complex has strong field ligand, the splitting will becomes bigger.
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