crystal field theory

Crystal field theory is used to describe the electronic structure of transition metal complexes. It is successful in describing the magnetic properties, colors, hydration enthalpies and spinel structures of transition metal complexes, but it cannot provide an adequate description of bonding. Crystal field theory was developed by the physicists Hans Bethe and John Hasbrouck van Vleck. It was combined with molecular orbital theory to form ligand field theory, which delivers insight into the process of chemical bonding in transition metal complexes.

In crystal field theory the metal ion is assumed to be free in gas form, the ligands are assumed to behave like point charges and it is assumed that the orbitals of the metal and the ligands do not interact. A refined form of crystal field theory called ligand field theory takes an empirical constant called the Racah parameter in the calculations to make up for covalent effects.

The bonding between a transition metal and the ligands is due to the attraction between the positively charged metal ion and the electrons of the ligand. Crystal field theory describes how the ligands affect the d electrons and split them in to higher and lower (in terms of energy) groups - the energy difference between the two sets is given the symbol Δ. This crystal field splitting (i.e. the size of Δ) depends on several factors:

• the nature of the metal ion.
• the metal's oxidation state. A higher oxidation state leads to a larger splitting.
• the arrangement of the ligands around the metal ion.
• the nature of the ligands surrounding the metal ion. The stronger the effect of the ligands then the greater the difference between the high and low energy 3d groups.

The most common type of complex is octahedral; here six ligands form an octahedron around the metal ion. The ligands point directly at the metal d-orbitals and cause a large splitting. Tetrahedral complexes are the second most common type; here four ligands form a tetrahedron around the metal ion, and since in this case the ligands' electrons aren't oriented directly towards the d-orbitals the energy splitting will be lower than in the octahedral case. Square planar complexes are mostly formed by transition metals in groups 10 and 11. Crystal field theory works best for period 4 transition metals.

Transition metals form ions with partly filled d-orbitals. There are 5 d-orbitals which each can contain two electrons. These five d-orbitals are degenerate - they have the same energy - when there are no ligands around the metal. When 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. The electrons in the d-orbitals and the electrons in the ligand repel each other (because they're both negatively charged), and so d-electrons closer to the ligands will have a higher energy than ones further away because they feel more repulsion. Thus, the d-orbitals will split in energy. What determines the way that the orbitals split is the orientation of the ligands with respect to the metal d orbitals. If there are six ligands there will most likely be one along each axis, so the complex will have octahedral symmetry. The d_xy, d_xz and d_yz orbitals will be lower energy than the d_z² and d_x²-y², which will have higher energy, because the former group are further from the ligands than the latter. In a tetrahedral crystal field splitting, the lower energy orbitals will be d_z² and d_x²-y², and the higher energy orbitals will be d_xy, d_xz and d_yz - the opposite way round to the octahedral case.

The size of the gap Δ between the two sets of orbitals depends on several factors, including the ligands. Some ligands always produce a small value of Δ, and some always give a large value. They can be ordered in a spectrochemical series devised by Japanese chemist R. Tsuchida (small Δ to large Δ, see also this table):

I^- <>- <>2- <>- <>- <>3^- <>3^- <>- <>- <>2O₄^2- <>2O <>- <>3CN < href="http://en.wikipedia.org/wiki/Pyridine" title="Pyridine">pyridine) <>3 < href="http://en.wikipedia.org/wiki/Ethylenediamine" title="Ethylenediamine">ethylenediamine) < href="http://en.wikipedia.org/wiki/2%2C2%27-bipyridine" title="2,2'-bipyridine">2,2'-bipyridine) < href="http://en.wikipedia.org/wiki/Phenanthroline" title="Phenanthroline">phenanthroline) <>2^- <>3 <>- <>

High-spin and low-spin

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 list above. In complexes with these ligands, because the upper set of orbitals is very high in energy it is unfavourable to put electrons into them. Therefore, the lower set of orbitals is completely filled before population of the upper set starts - the Aufbau rule is obeyed. Complexes such as this are called low spin. For example, NO₂^- is a strong-field ligand and produces a large Δ. The ion [Fe(NO₂)₆]^3-, which has 5 d-electrons, would have an octahedral splitting diagram that looks like

Explaining the Colours of Transition Metals

The bright, diverse colours exhibited by coordination compounds can be explained by Crystal Field Theory. The colour seen depends upon the difference in energy Δ of the split d orbitals in the crystal field. As explained above, different ligands generate crystal fields of different strengths, so different colours are seen.

When a molecule absorbs visible light, one or more electrons may momentarily jump from the lower energy d orbitals to the higher energy ones. The difference in energies of the two orbitals Δ is equal to the energy of the absorbed photon, and related inversely to the wavelength of the light.

Thus, Δ = E_photon = hν = hc/λ.

Because only certain wavelengths (λ) of light are absorbed - those matching exactly the energy Δ - the complementary colours are visible in the compound.

Weaker field ligands with smaller Δ absorb light of longer λ and thus lower ν. Similarly, stronger field ligands with larger Δ absorb light of shorter λ and thus higher ν.

This colour wheel demonstrates which colours will be observed in the compounds. If the compound absorbs red light, it will appear green.

λ absorbed versus colour observed
400nm Violet absorbed, Green-yellow observed (λ 560nm)
450nm Blue absorbed, Yellow observed (λ 600nm)
490nm Blue-green absorbed, Red observed (λ 620nm)
570nm Yellow-green absorbed, Violet observed (λ 410nm)
580nm Yellow absorbed, Dark blue observed (λ 430nm)
600nm Orange absorbed, Blue observed (λ 450nm)
650nm Red absorbed, Green observed (λ 520nm)