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Crystal field theory

Crystal field theory

Overview
Crystal field theory is a model that describes the electronic structure of transition metal
Transition metal
The term transition metal has two possible meanings:*The IUPAC definition states that a transition metal is "an element whose atom has an incomplete d sub-shell, or which can give rise to cations with an incomplete d sub-shell." Group 12 elements are not transition metals in this definition.*Some...

 compounds, all of which can be considered coordination complexes
Complex (chemistry)
In chemistry, a coordination complex or metal complex, is an atom or ion , bonded to a surrounding array of molecules or anions, that are in turn known as ligands or complexing agents...

. CFT successfully accounts for some magnetic properties, colours, hydration
Hydration reaction
In organic chemistry, a hydration reaction is a chemical reaction in which a hydroxyl group and a hydrogen cation are added to the two carbon atoms bonded together in the carbon-carbon double bond which makes up an alkene functional group. The reaction usually runs in a strong acidic, aqueous...

 enthalpies
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

, and spinel
Spinel
Spinel is the magnesium aluminium member of the larger spinel group of minerals. It has the formula MgAl2O4. Balas ruby is an old name for a rose-tinted variety.-Spinel group:...

 structures of transition metal complexes, but it does not attempt to describe bonding. CFT was developed by physicists Hans Bethe
Hans Bethe
Hans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...

 and John Hasbrouck van Vleck
John Hasbrouck van Vleck
John Hasbrouck Van Vleck was an American physicist and mathematician, co-awarded the 1977 Nobel Prize in Physics, for his contributions to the understanding of the behavior of electrons in magnetic solids....

 in the 1930s. CFT was subsequently combined with molecular orbital theory
Molecular orbital theory
In chemistry, molecular orbital theory is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule...

 to form the more realistic and complex ligand field theory
Ligand field theory
Ligand field theory 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 valence atomic orbitals, five d, one s, and three p orbitals...

 (LFT), which delivers insight into the process of chemical bonding in transition metal complexes.

According to CFT, the interaction between a transition metal and ligand
Ligand
In coordination chemistry, a ligand is an ion or molecule that binds to a central metal atom to form a coordination complex. The bonding between metal and ligand generally involves formal donation of one or more of the ligand's electron pairs. The nature of metal-ligand bonding can range from...

s arises from the attraction between the positively charged metal cation and negative charge on the non-bonding electrons of the ligand.
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Encyclopedia
Crystal field theory is a model that describes the electronic structure of transition metal
Transition metal
The term transition metal has two possible meanings:*The IUPAC definition states that a transition metal is "an element whose atom has an incomplete d sub-shell, or which can give rise to cations with an incomplete d sub-shell." Group 12 elements are not transition metals in this definition.*Some...

 compounds, all of which can be considered coordination complexes
Complex (chemistry)
In chemistry, a coordination complex or metal complex, is an atom or ion , bonded to a surrounding array of molecules or anions, that are in turn known as ligands or complexing agents...

. CFT successfully accounts for some magnetic properties, colours, hydration
Hydration reaction
In organic chemistry, a hydration reaction is a chemical reaction in which a hydroxyl group and a hydrogen cation are added to the two carbon atoms bonded together in the carbon-carbon double bond which makes up an alkene functional group. The reaction usually runs in a strong acidic, aqueous...

 enthalpies
Enthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...

, and spinel
Spinel
Spinel is the magnesium aluminium member of the larger spinel group of minerals. It has the formula MgAl2O4. Balas ruby is an old name for a rose-tinted variety.-Spinel group:...

 structures of transition metal complexes, but it does not attempt to describe bonding. CFT was developed by physicists Hans Bethe
Hans Bethe
Hans Albrecht Bethe was a German-American nuclear physicist, and Nobel laureate in physics for his work on the theory of stellar nucleosynthesis. A versatile theoretical physicist, Bethe also made important contributions to quantum electrodynamics, nuclear physics, solid-state physics and...

 and John Hasbrouck van Vleck
John Hasbrouck van Vleck
John Hasbrouck Van Vleck was an American physicist and mathematician, co-awarded the 1977 Nobel Prize in Physics, for his contributions to the understanding of the behavior of electrons in magnetic solids....

 in the 1930s. CFT was subsequently combined with molecular orbital theory
Molecular orbital theory
In chemistry, molecular orbital theory is a method for determining molecular structure in which electrons are not assigned to individual bonds between atoms, but are treated as moving under the influence of the nuclei in the whole molecule...

 to form the more realistic and complex ligand field theory
Ligand field theory
Ligand field theory 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 valence atomic orbitals, five d, one s, and three p orbitals...

 (LFT), which delivers insight into the process of chemical bonding in transition metal complexes.

Overview of crystal field theory analysis


According to CFT, the interaction between a transition metal and ligand
Ligand
In coordination chemistry, a ligand is an ion or molecule that binds to a central metal atom to form a coordination complex. The bonding between metal and ligand generally involves formal donation of one or more of the ligand's electron pairs. The nature of metal-ligand bonding can range from...

s arises from the attraction between the positively charged metal cation and negative charge on the non-bonding electrons of the ligand. The theory is developed by considering energy changes of the five degenerate d-orbitals
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

 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.
  • 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 d groups.


The most common type of complex is octahedral
Octahedral molecular geometry
In chemistry, octahedral molecular geometry describes the shape of compounds where in six atoms or groups of atoms or ligands are symmetrically arranged around a central atom, defining the vertices of an octahedron...

; here six ligands form 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) where the dxy, dxz and dyz orbitals will be lower in energy than the dz2 and dx2-y2, which will have higher energy, because the former group is farther from the ligands than the latter and therefore experience 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
Molecular symmetry in chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can predict or explain many of a molecule's chemical properties, such as its dipole moment...

). 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 where the lower energy orbitals will be dz2 and dx2-y2, and the higher energy orbitals will be dxy, dxz and dyz - 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 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
Ligand field theory 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 valence atomic orbitals, five d, one s, and three p orbitals...

. The spectrochemical series
Spectrochemical series
A spectrochemical series is a list of ligands ordered on ligand strength and a list of metal ions based on oxidation number, group and its identity...

 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
Iodide
An iodide ion is the ion I−. Compounds with iodine in formal oxidation state −1 are called iodides. This page is for the iodide ion and its salts. For information on organoiodides, see organohalides. In everyday life, iodide is most commonly encountered as a component of iodized salt,...

 < Br
Bromide
A bromide is a chemical compound containing bromide ion, that is bromine atom with effective charge of −1. The class name can include ionic compounds such as caesium bromide or covalent compounds such as sulfur dibromide.-Natural occurrence:...

 < S2−
Sulfide
A sulfide is an anion of sulfur in its lowest oxidation state of 2-. Sulfide is also a slightly archaic term for thioethers, a common type of organosulfur compound that are well known for their bad odors.- Properties :...

 < SCN
Thiocyanate
Thiocyanate is the anion [SCN]−. It is the conjugate base of thiocyanic acid. Common derivatives include the colourless salts potassium thiocyanate and sodium thiocyanate. Organic compounds containing the functional group SCN are also called thiocyanates...

 < Cl
Chloride
The chloride ion is formed when the element chlorine, a halogen, picks up one electron to form an anion Cl−. The salts of hydrochloric acid HCl contain chloride ions and can also be called chlorides. The chloride ion, and its salts such as sodium chloride, are very soluble in water...

 < NO3
Nitrate
The nitrate ion is a polyatomic ion with the molecular formula NO and a molecular mass of 62.0049 g/mol. It is the conjugate base of nitric acid, consisting of one central nitrogen atom surrounded by three identically-bonded oxygen atoms in a trigonal planar arrangement. The nitrate ion carries a...

 < N3
Azide
Azide is the anion with the formula N3−. It is the conjugate base of hydrazoic acid. N3− is a linear anion that is isoelectronic with CO2 and N2O. Per valence bond theory, azide can be described by several resonance structures, an important one being N−=N+=N−...

 < F
Fluoride
Fluoride is the anion F−, the reduced form of fluorine when as an ion and when bonded to another element. Both organofluorine compounds and inorganic fluorine containing compounds are called fluorides. Fluoride, like other halides, is a monovalent ion . Its compounds often have properties that are...

 < OH
Hydroxide
Hydroxide is a diatomic anion with chemical formula OH−. It consists of an oxygen and a hydrogen atom held together by a covalent bond, and carrying a negative electric charge. It is an important but usually minor constituent of water. It functions as a base, as a ligand, a nucleophile, and a...

 < C2O42−
Oxalate
Oxalate , is the dianion with formula C2O42− also written 22−. Either name is often used for derivatives, such as disodium oxalate, 2C2O42−, or an ester of oxalic acid Oxalate (IUPAC: ethanedioate), is the dianion with formula C2O42− also written (COO)22−. Either...

 < H2O < NCS
Isothiocyanate
Isothiocyanate is the chemical group –N=C=S, formed by substituting sulfur for oxygen in the isocyanate group. Many natural isothiocyanates from plants are produced by enzymatic conversion of metabolites called glucosinolates. These natural isothiocyanates, such as allyl isothiocyanate, are also...

 < CH3CN
Acetonitrile
Acetonitrile is the chemical compound with formula . This colourless liquid is the simplest organic nitrile. It is produced mainly as a byproduct of acrylonitrile manufacture...

 < py
Pyridine
Pyridine is a basic heterocyclic organic compound with the chemical formula C5H5N. It is structurally related to benzene, with one C-H group replaced by a nitrogen atom...

 < NH3
Ammonia
Ammonia is a compound of nitrogen and hydrogen with the formula . It is a colourless gas with a characteristic pungent odour. Ammonia contributes significantly to the nutritional needs of terrestrial organisms by serving as a precursor to food and fertilizers. Ammonia, either directly or...

 < en < 2,2'-bipyridine < phen
Phenanthroline
Phenanthroline is a heterocyclic organic compound. As a bidentate ligand in coordination chemistry, it forms strong complexes with most metal ions...

 < NO2
Nitrite
The nitrite ion has the chemical formula NO2−. The anion is symmetric with equal N-O bond lengths and a O-N-O bond angle of ca. 120°. On protonation the unstable weak acid nitrous acid is produced. Nitrite can be oxidised or reduced, with product somewhat dependent on the oxidizing/reducing agent...

 < PPh3
Triphenylphosphine
Triphenylphosphine is a common organophosphorus compound with the formula P3 - often abbreviated to PPh3 or Ph3P. It is widely used in the synthesis of organic and organometallic compounds. PPh3 exists as relatively air stable, colorless crystals at room temperature...

 < CN
Cyanide
A cyanide is a chemical compound that contains the cyano group, -C≡N, which consists of a carbon atom triple-bonded to a nitrogen atom. Cyanides most commonly refer to salts of the anion CN−. Most cyanides are highly toxic....

 < CO
Carbon monoxide
Carbon monoxide , also called carbonous oxide, is a colorless, odorless, and tasteless gas that is slightly lighter than air. It is highly toxic to humans and animals in higher quantities, although it is also produced in normal animal metabolism in low quantities, and is thought to have some normal...



It is useful to note that the ligands producing the most splitting are those that can engage in metal to ligand 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 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
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

 are referred to as strong-field ligands, such as CN and CO from the spectrochemical series
Spectrochemical series
A spectrochemical series is a list of ligands ordered on ligand strength and a list of metal ions based on oxidation number, group and its identity...

. 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
Aufbau principle
The Aufbau principle is used to determine the electron configuration of an atom, molecule or ion. The principle postulates a hypothetical process in which an atom is "built up" by progressively adding electrons...

. Complexes such as this are called "low spin". For example, NO2 is a strong-field ligand and produces a large Δ. The octahedral ion [Fe(NO2)6]3−, which has 5 d-electrons, would have the octahedral splitting diagram shown at right with all five electrons in the t2g level.

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 pairing occurs in accord with Hund's rule and "high spin" complexes are formed. For example, Br is a weak-field ligand and produces a small Δoct. So, the ion [FeBr6]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 eg orbital at an energy cost of Δ. As noted above, eg refers to the
dz2 and dx2-y2 which are higher in energy than the t2g in octahedral complexes. If the energy required to pair two electrons is greater than the energy cost of placing an electron in an eg, Δ, 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 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.

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
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

 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 d5 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.

Crystal Field stabilization is applicable to transition-metal complexes of all geometries
VSEPR theory
Valence shell electron pair repulsion theory is a model in chemistry used to predict the shape of individual molecules based upon the extent of electron-pair electrostatic repulsion. It is also named Gillespie–Nyholm theory after its two main developers...

. Indeed, the reason that many d8 complexes are square-planar is the very large amount of crystal field stabilization that this geometry produces with this number of electrons.

Explaining the colors of transition metal complexes


The bright colors exhibited by many coordination compounds can be explained by Crystal Field Theory. If the d-orbitals
Atomic orbital
An atomic orbital is a mathematical function that describes the wave-like behavior of either one electron or a pair of electrons in an atom. This function can be used to calculate the probability of finding any electron of an atom in any specific region around the atom's nucleus...

 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
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...

 and in the excited state
Excited state
Excitation is an elevation in energy level above an arbitrary baseline energy state. In physics there is a specific technical definition for energy level which is often associated with an atom being excited to an 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 color.

As explained above, because different ligands generate crystal fields of different strengths, different colors 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 effect
Jahn-Teller effect
The Jahn–Teller effect, sometimes also known as Jahn–Teller distortion, or the Jahn–Teller theorem, describes the geometrical distortion of non-linear molecules under certain situations. This electronic effect is named after Hermann Arthur Jahn and Edward Teller, who proved, using group theory,...

s) that also affect the energy difference between the ground and excited states.

Which colors are exhibited?



This color wheel demonstrates which color a compound will appear if it only has one absorption in the visible spectrum. For example, if the compound absorbs red light, it will appear green.

λ absorbed versus color observed

400 nm Violet absorbed, Green-yellow observed (λ 560 nm)

450 nm Blue absorbed, Yellow observed (λ 600 nm)

490 nm Blue-green absorbed, Red observed (λ 620 nm)

570 nm Yellow-green absorbed, Violet observed (λ 410 nm)

580 nm Yellow absorbed, Dark blue observed (λ 430 nm)

600 nm Orange absorbed, Blue observed (λ 450 nm)

650 nm Red absorbed, Green observed (λ 520 nm)

Crystal field splitting diagrams

Crystal field splitting diagrams
Octahedral Pentagonal bipyramidal Square antiprismatic
Square planar Square pyramidal Tetrahedral
Trigonal bipyramidal