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^- < Br^- < S^2- < SCN^- < Cl^- < NO₃^- < N₃^- < F^- < OH^- < C₂O₄^2- < H₂O < NCS^- < CH₃CN < py (pyridine) < NH₃ < en (ethylenediamine) < bipy (2,2'-bipyridine) < phen (1,10-phenanthroline) < NO₂^- < PPh₃ < CN^- < CO

High-spin and low-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, while a compound that lacks unpaired electrons in its splitting diagram will be diamagnetic and will be weakly repelled by a magnetic field.

The metal's oxidation state contributes to the size of Δ between the high and low energy levels. As the oxidation state increases for a given metal, the magnitude of Δ increases. A V³⁺ complex will have a larger Δ than a V²⁺ complex. 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.

References

• Zumdahl, Steven S. Chemical Principles Fifth Edition. Boston: Houghton Mifflin Company, 2005. 550-551,957-964.

• D. F. Shriver and P. W. Atkins Inorganic Chemistry 3rd edition, Oxford University Press, 2001. Pages: 227-236.

Kristallfeldtheorie | 結晶場理論