Editing Crystal field theory (section)

==Overview==
According to crystal field theory, the interaction between a transition metal and [[ligand]]s 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 orbital|degenerate]] [[Atomic orbital|''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 molecular geometry|octahedral]], 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, '''Δ<sub>oct</sub>''' (the [[crystal-field splitting parameter]], also commonly denoted by '''10''Dq''''' for ten times the "differential of quanta"<ref name="PenneySchlapp1932">{{cite journal|last1=Penney|first1=William G.|last2=Schlapp|first2=Robert|title=The Influence of Crystalline Fields on the Susceptibilities of Salts of Paramagnetic Ions. I. The Rare Earths, Especially Pr and Nd|journal=Physical Review|volume=41|issue=2|year=1932|pages=194–207|issn=0031-899X|doi=10.1103/PhysRev.41.194|bibcode=1932PhRv...41..194P}}</ref><ref name="SchlappPenney1932">{{cite journal|last1=Schlapp|first1=Robert|last2=Penney|first2=William G.|title=Influence of Crystalline Fields on the Susceptibilities of Salts of Paramagnetic Ions. II. The Iron Group, Especially Ni, Cr and Co|journal=Physical Review|volume=42|issue=5|year=1932|pages=666–686|issn=0031-899X|doi=10.1103/PhysRev.42.666|bibcode=1932PhRv...42..666S}}\
</ref>) where the ''d<sub>xy</sub>'', ''d<sub>xz</sub>'' and ''d<sub>yz</sub>'' orbitals will be lower in energy than the ''d''<sub>''z''<sup>2</sup></sub> and ''d''<sub>''x''<sup>2</sup>-''y''<sup>2</sup></sub>, 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 '''t<sub>2g</sub>''', and the two higher-energy orbitals as '''e<sub>g</sub>'''. These labels are based on the theory of [[molecular symmetry]]: they are the names of [[irreducible representation]]s of the [[Octahedral symmetry#Full octahedral symmetry|octahedral point group]], O<sub>h</sub>.(see the [[List of character tables for chemically important 3D point groups#Cubic groups|O<sub>h</sub> 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 '''Δ<sub>tet</sub>'''. The lower energy orbitals will be ''d''<sub>''z''<sup>2</sup></sub> and ''d''<sub>''x''<sup>2</sup>-''y''<sup>2</sup></sub>, and the higher energy orbitals will be ''d''<sub>''xy''</sub>, ''d''<sub>''xz''</sub> and ''d''<sub>''yz''</sub> - 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 molecular geometry|Square planar]] 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]]. 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 [[ligand#Examples of common ligands (by field strength)|this table]]):

[[iodide|I]]<sup>−</sup> < [[bromide|Br]]<sup>−</sup> < [[sulfide|S]]<sup>2−</sup> < [[thiocyanate|SCN]]<sup>−</sup> (S–bonded) < [[chloride|Cl]]<sup>−</sup> < [[nitrate|NO<sub>3</sub>]]<sup>−</sup> < [[azide|N<sub>3</sub>]]<sup>−</sup> < [[fluoride|F]]<sup>−</sup> < [[hydroxide|OH]]<sup>−</sup> < [[oxalate|C<sub>2</sub>O<sub>4</sub>]]<sup>2−</sup> < [[water (molecule)|H<sub>2</sub>O]] < [[isothiocyanate|NCS]]<sup>−</sup> (N–bonded) < [[acetonitrile|CH<sub>3</sub>CN]] < [[pyridine|py]] < [[ammonia|NH<sub>3</sub>]] < [[ethylenediamine|en]] < [[bipy|2,2'-bipyridine]] < [[phenanthroline|phen]] < [[nitrite|NO<sub>2</sub>]]<sup>−</sup> < [[triphenylphosphine|PPh<sub>3</sub>]] < [[cyanide|CN]]<sup>−</sup> < [[carbon monoxide|CO]].

It is useful to note that the ligands producing the most splitting are those that can engage in metal to ligand [[Pi backbonding|back-bonding]].

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 V<sup>3+</sup> complex will have a larger Δ than a V<sup>2+</sup> complex for a given set of ligands, as the difference in charge density allows the ligands to be closer to a V<sup>3+</sup> ion than to a V<sup>2+</sup> 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===
{{Main|Spin states (d electrons)}}
{{See also|Magnetochemistry}}
[[File:CFT-Low Spin Splitting Diagram-Vector.svg|thumb|right|250px|'''Low Spin''' [Fe(NO<sub>2</sub>)<sub>6</sub>]<sup>3−</sup> crystal field diagram]]

Ligands which cause a large splitting Δ of the [[Atomic orbital|''d''-orbitals]] are referred to as strong-field ligands, such as CN<sup>−</sup> 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]]. Complexes such as this are called "low spin". For example, NO<sub>2</sub><sup>−</sup> is a strong-field ligand and produces a large Δ. The octahedral ion [Fe(NO<sub>2</sub>)<sub>6</sub>]<sup>3−</sup>, which has 5 ''d''-electrons, would have the octahedral splitting diagram shown at right with all five electrons in the ''t''<sub>2''g''</sub> level. This low spin state therefore does not follow [[Hund's rule]].

[[Image:CFT-High Spin Splitting Diagram-Vector.svg|thumb|right|250px|'''High Spin''' [FeBr<sub>6</sub>]<sup>3−</sup> crystal field diagram]]

Conversely, ligands (like I<sup>−</sup> and Br<sup>−</sup>) 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<sup>−</sup> is a weak-field ligand and produces a small Δ<sub>oct</sub>. So, the ion [FeBr<sub>6</sub>]<sup>3−</sup>, 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<sub>''g''</sub> orbital at an energy cost of Δ. As noted above, e<sub>''g''</sub> refers to the
''d''<sub>''z''<sup>2</sup></sub> and ''d''<sub>''x''<sup>2</sup>-''y''<sup>2</sup></sub> which are higher in energy than the t<sub>2g</sub> 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<sub>''g''</sub>, high spin splitting occurs.

The crystal field splitting energy for tetrahedral metal complexes (four ligands) is referred to as Δ<sub>tet</sub>, and is roughly equal to 4/9Δ<sub>oct</sub> (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.