Crystal field theory

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Crystal field theory (CFT) is a model that describes the electronic structure of transition metal compounds, all of which can be considered coordination complexes. CFT successfully accounts for some magnetic properties, colours, hydration enthalpies, and spinel structures of transition metal complexes. It does not attempt to describe bonding. CFT was developed by physicists Hans Bethe and John Hasbrouck van Vleck. CFT was subsequently combined with molecular orbital theory to form ligand field theory (LFT), which delivers insight into the process of chemical bonding in transition metal complexes. CFT is considered to be a stepping stone to appreciating LFT.

In CFT, the metal complex is treated as a "free ion". The ligands are treated as point charges. It is assumed that the orbitals on the metal and the ligands do not overlap. In the more refined LFT model, metal-ligand bonding is quantified using an empirical constant called the Racah parameter. Overall the transition from CFT to LFT is one from electrostatic model to a more realistic (and more complicated) polar-covalent model for metal-ligand interactions.

Contents 1 Overview of the CFT analysis 1.1 High-spin and low-spin 2 Crystal Field Stabilisation Energy 3 Explaining the Colours of Transition Metal Complexes 3.1 Which colours are exhibited? 4 References 5 See also

[edit] Overview of the CFT 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. It is typical to consider how the d-orbitals on a metal will be affected upon their being surrounded by an octahedral array of six point charges. In such an octahedral geometry, some d orbitals are more strongly affected than others. Thus, the d-orbitals that are aligned with the two Cartesian axes (d_x²−y², d_z²) will be more strongly affected than the other three d-orbitals (d_xy, d_xz, d_yz). The resulting energy difference between these two sets of d-orbitals is given the symbol Δ. This crystal field splitting (i.e. the size of Δ) depends on the following 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. The reasons behind this can be explained by ligand field theory. The spectrochemical series is an empirically-derived list of ligands ordered by the size of the splitting Δ that they produce (small Δ to large Δ; see also this table):

I⁻ < Br⁻ < S²⁻ < SCN⁻ < Cl⁻ < NO₃⁻ < N₃⁻ < F⁻ < OH⁻ < C₂O₄²⁻ < H₂O < NCS⁻ < CH₃CN < py < NH₃ < en < 2,2'-bipyridine < phen) < NO₂⁻ < PPh₃ < CN⁻ < CO

The metal's oxidation state also 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 (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.

[edit] 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

Conversely, ligands (like I⁻ and Br⁻) which cause a small splitting Δ of the d orbitals are referred to as weak-field ligands. 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 orbital gets two — Hund's rule is obeyed, and we get high spin complexes. For example, Br⁻ is a weak-field ligand and produces a small Δ. So, the ion [FeBr₆]³⁻, again with 5 d-electrons, would have an octahedral splitting diagram that looks like

In order for Low Spin Splitting to occur, the Pairing Energy (P), that is, the energy required for electrons to pair up and obey Hund's Rule, must be less than Δ. If it is greater than Δ, high spin splitting occurs. Pairing Energy (P) = energy required to overcome repulsions (between paired electrons) + sacrificed Exchange Stabilization.

The Crystal Field Splitting Energy for tetrahedral metal complexes (4 ligands) is roughly equal to 4/9 octahedral Δ. Therefore, in practice, P > Δ for tetrahedral complexes, and high spin will occur preferentially.

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.

[edit] Crystal Field Stabilisation Energy

The crystal field stabilisation 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 (as described above), some of them become lower in energy than before. For example, in an octahedral case, the t_2g set become lower in energy than in the free ion, and as a result of this, if there are any electrons occupying these orbitals, the metal ion is more stable in the crystal field than outside it - the amount of stability is the CFSE. Conversely, the e_g orbitals (in the octahedral case) are higher in energy than in the free ion, so putting electrons in these reduces the amount of CFSE.

The energy level of the d-orbitals in the free ion is called the barycentre. If the splitting of the d-orbitals in an octahedral field is Δ_oct, the three t_2g orbitals are stabilised relative to the barycentre by ²/₅ Δ_oct, and the e_g orbitals are destabilised by ³/₅ Δ_oct. As examples, consider the two d⁵ configurations shown further up the page. The low-spin (top) example has five electrons in the t_2g orbitals, so the total CFSE is 5 x ²/₅ Δ_oct = 2Δ_oct. In the high-spin (lower) example, the CFSE is (3 x ²/₅ Δ_oct) - (2 x ³/₅ Δ_oct) = 0 - in this case, the stabilisation generated by the electrons in the lower orbitals is cancelled out by the destabilising effect of the electrons in the upper orbitals.

Crystal Field stabilisation 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 stabilisation that this geometry produces with this number of electrons.

[edit] Explaining the Colours of Transition Metal Complexes

The bright colours exhibited by many coordination compounds can be explained by Crystal Field Theory. If the d orbitals of such a complex have been split into two sets as described above, when the molecule absorbs a photon of visible light one or more electrons may momentarily jump from the lower energy d orbitals to the higher energy ones to transiently create an excited state atom. The difference in energy between the atom in the ground state and in the excited state is equal to the energy of the absorbed photon, and related inversely to the wavelength of the light. Because only certain wavelengths (λ) of light are absorbed - those matching exactly the energy difference - the compounds appears the appropriate complementary colour.

As explained above, because different ligands generate crystal fields of different strengths, different colours can be seen. For a given metal ion, weaker field ligands create a complex with a smaller Δ, which will absorb light of longer λ and thus lower frequency ν. Conversely, stronger field ligands create a larger Δ, absorb light of shorter λ, and thus higher ν. It is, though, rarely the case that the energy of the photon absorbed corresponds exactly to the size of the gap Δ; there are other things (such as electron-electron repulsion and Jahn-Teller effects) that also affect the energy difference between the ground and excited states.

[edit] Which colours are exhibited?

This colour wheel demonstrates which colour a compound will appear if it only has one absoprtion in the visible spectrum. For example, 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)

[edit] References

• Zumdahl, Steven S. Chemical Principles Fifth Edition. Boston: Houghton Mifflin Company, 2005. 550-551,957-964.
• Silberberg, Martin S. Chemistry: The Molecular Nature of Matter and Change, Fourth Edition. New York: McGraw Hill Company, 2006. 1028-1034.
• D. F. Shriver and P. W. Atkins Inorganic Chemistry 3rd edition, Oxford University Press, 2001. Pages: 227-236.

[edit] See also

• Ligand field theory
• Geometrical frustrated magnets