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Crystal field theory (CFT) describes the breaking of degeneracies of electron orbital states, usually ''d'' or ''f'' orbitals, due to a static electric field produced by a surrounding charge distribution (anion neighbors). This theory has been used to describe various spectroscopies of
transition metal In chemistry, a transition metal (or transition element) is a chemical element in the d-block of the periodic table (groups 3 to 12), though the elements of group 12 (and less often group 3) are sometimes excluded. They are the elements that can ...
coordination complexes A coordination complex consists of a central atom or ion, which is usually metallic and is called the ''coordination centre'', and a surrounding array of bound molecules or ions, that are in turn known as ''ligands'' or complexing agents. Many ...
, in particular optical spectra (colors). CFT successfully accounts for some
magnetic Magnetism is the class of physical attributes that are mediated by a magnetic field, which refers to the capacity to induce attractive and repulsive phenomena in other entities. Electric currents and the magnetic moments of elementary particles ...
properties, colors,
hydration Hydration may refer to: * Hydrate, a substance that contains water * Hydration enthalpy, energy released through hydrating a substance * Hydration reaction, a chemical addition reaction where a hydroxyl group and proton are added to a compound * ...
enthalpies, and
spinel Spinel () is the magnesium/aluminium member of the larger spinel group of minerals. It has the formula in the cubic crystal system. Its name comes from the Latin word , which means ''spine'' in reference to its pointed crystals. Properties S ...
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 In chemistry, molecular orbital theory (MO theory or MOT) is a method for describing the electronic structure of molecules using quantum mechanics. It was proposed early in the 20th century. In molecular orbital theory, electrons in a molec ...
to form the more realistic and complex
ligand field theory Ligand field theory (LFT) describes the bonding, orbital arrangement, and other characteristics of coordination complexes. It represents an application of molecular orbital theory to transition metal complexes. A transition metal ion has nine val ...
(LFT), which delivers insight into the process of
chemical bonding A chemical bond is a lasting attraction between atoms or ions that enables the formation of molecules and crystals. The bond may result from the electrostatic force between oppositely charged ions as in ionic bonds, or through the sharing of ...
in transition metal complexes.


Overview of crystal field theory

According to crystal field theory, the interaction between a transition metal and ligands arises from the attraction between the positively charged metal cation and the 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. This splitting is affected by the following factors: *the nature of the metal ion. *the metal's oxidation state. A higher oxidation state leads to a larger splitting relative to the spherical field. *the arrangement of the ligands around the metal ion. *the coordination number of the metal (i.e. tetrahedral, octahedral...) *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 ''d'' groups. The most common type of complex is
octahedral In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
, in which six ligands form the vertices of an octahedron around the metal ion. In octahedral symmetry the ''d''-orbitals split into two sets with an energy difference, Δoct (the
crystal-field splitting parameter A spectrochemical series is a list of ligands ordered by ligand "strength", and a list of metal ions based on oxidation number, group and element. For a metal ion, the ligands modify the difference in energy Δ between the d orbitals, called the ...
, also commonly denoted by 10''Dq'' for ten times the "differential of quanta"\ ) where the ''dxy'', ''dxz'' and ''dyz'' orbitals will be lower in energy than the ''d''''z''2 and ''d''''x''2-''y''2, which will have higher energy, because the former group is farther from the ligands than the latter and therefore experiences less repulsion. The three lower-energy orbitals are collectively referred to as t2g, and the two higher-energy orbitals as eg. These labels are based on the theory of
molecular symmetry Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain ...
: they are the names of irreducible representations of the octahedral point group, Oh.(see the Oh character table) Typical orbital energy diagrams are given below in the section High-spin and low-spin. 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. The lower energy orbitals will be ''d''''z''2 and ''d''''x''2-''y''2, and the higher energy orbitals will be ''d''''xy'', ''d''''xz'' and ''d''''yz'' - 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 planar The square planar molecular geometry in chemistry describes the stereochemistry (spatial arrangement of atoms) that is adopted by certain chemical compounds. As the name suggests, molecules of this geometry have their atoms positioned at the corne ...
and other complex geometries can also be described by CFT. 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. 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 Ligand field theory (LFT) describes the bonding, orbital arrangement, and other characteristics of coordination complexes. It represents an application of molecular orbital theory to transition metal complexes. A transition metal ion has nine val ...
. 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 (S–bonded) < Cl < NO3 < N3 < F < OH < C2O42− < H2O < NCS (N–bonded) < CH3CN < py < NH3 < en < 2,2'-bipyridine < phen < NO2 < PPh3 < CN < CO. It is useful to note that the ligands producing the most splitting are those that can engage in metal to ligand
back-bonding In chemistry, π backbonding, also called π backdonation, is when electrons move from an atomic orbital on one atom to an appropriate symmetry antibonding orbital on a ''π-acceptor ligand''. It is especially common in the organometallic chem ...
. The oxidation state of the metal 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.


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 spectrochemical series. 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 The aufbau principle , from the German ''Aufbauprinzip'' (building-up principle), also called the aufbau rule, states that in the ground state of an atom or ion, electrons fill subshells of the lowest available energy, then they fill subshells o ...
. Complexes such as this are called "low spin". For example, NO2 is a strong-field ligand and produces a large Δ. The octahedral ion e(NO2)6sup>3−, which has 5 ''d''-electrons, would have the octahedral splitting diagram shown at right with all five electrons in the ''t''2''g'' level. This low spin state therefore does not follow
Hund's rule Hund's rule of maximum multiplicity is a rule based on observation of atomic spectra, which is used to predict the ground state of an atom or molecule with one or more open electronic shells. The rule states that for a given electron configuration ...
. 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 in accord with Hund's rule, and "high spin" complexes are formed before any pairing occurs. For example, Br is a weak-field ligand and produces a small Δoct. So, the ion eBr6sup>3−, again with five ''d''-electrons, would have an octahedral splitting diagram where all five orbitals are singly occupied. In order for low spin splitting to occur, the energy cost of placing an electron into an already singly occupied orbital must be less than the cost of placing the additional electron into an e''g'' orbital at an energy cost of Δ. As noted above, e''g'' refers to the ''d''''z''2 and ''d''''x''2-''y''2 which are higher in energy than the t2g in octahedral complexes. If the energy required to pair two electrons is greater than Δ, the energy cost of placing an electron in an e''g'', high spin splitting occurs. The crystal field splitting energy for tetrahedral metal complexes (four ligands) is referred to as Δtet, and is roughly equal to 4/9Δoct (for the same metal and same ligands). 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 magnetic properties of co-ordination 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.


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 (as described above), 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. For example, 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. 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. As examples, consider the two ''d''5 configurations shown further up the page. The low-spin (top) example has five electrons in the ''t2g'' orbitals, so the total CFSE is 5 x 2/5 Δoct = 2Δoct. In the high-spin (lower) example, the CFSE is (3 x 2/5 Δoct) - (2 x 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.


Optical properties

The optical properties (details of absorption and emission spectra) of many
coordination complex A coordination complex consists of a central atom or ion, which is usually metallic and is called the ''coordination centre'', and a surrounding array of bound molecules or ions, that are in turn known as ''ligands'' or complexing agents. Many ...
es can be explained by Crystal Field Theory. Often, however, the deeper colors of metal complexes arise from more intense charge-transfer excitations.G. L. Miessler and D. A. Tarr “Inorganic Chemistry” 2nd Ed. (Prentice Hall 1999), p.379 .


Geometries and crystal field splitting diagrams


See also

* Schottky anomaly — low temperature spike in
heat capacity Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K). Heat capacity i ...
seen in materials containing high-spin magnetic impurities, often due to crystal field splitting *
Ligand field theory Ligand field theory (LFT) describes the bonding, orbital arrangement, and other characteristics of coordination complexes. It represents an application of molecular orbital theory to transition metal complexes. A transition metal ion has nine val ...
*
Molecular orbital theory In chemistry, molecular orbital theory (MO theory or MOT) is a method for describing the electronic structure of molecules using quantum mechanics. It was proposed early in the 20th century. In molecular orbital theory, electrons in a molec ...


References


Further reading

* * * * * *


External links


Crystal-field Theory, Tight-binding Method, and Jahn-Teller Effect
in E. Pavarini, E. Koch, F. Anders, and M. Jarrell (eds.): Correlated Electrons: From Models to Materials, Jülich 2012,
Crystal field theory (draft article)
on Citizendium.org {{Organometallics Condensed matter physics Inorganic chemistry Chemical bonding Coordination chemistry Transition metals de:Kristallfeld- und Ligandenfeldtheorie#Kristallfeldtheorie