Ligand field theory
Overview
 
Ligand field theory describes the bonding, orbital arrangement, and other characteristics of coordination complexes. It represents an application of 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 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...

 complexes. A transition metal ion has nine valence atomic orbital
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...

s, five (n)d, one (n+1)s, and three (n+1)p orbitals. These orbitals are of appropriate energy to form bonding interaction with 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. The LFT analysis is highly dependent on the geometry of the complex, but most explanations begin by describing 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...

 complexes, where six ligands coordinate to the metal.
The molecular orbitals created by coordination can be seen as resulting from the donation of two electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s by each of six σ-donor ligands to the d-orbitals on the metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...

.
Discussions
Encyclopedia
Ligand field theory describes the bonding, orbital arrangement, and other characteristics of coordination complexes. It represents an application of 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 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...

 complexes. A transition metal ion has nine valence atomic orbital
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...

s, five (n)d, one (n+1)s, and three (n+1)p orbitals. These orbitals are of appropriate energy to form bonding interaction with 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. The LFT analysis is highly dependent on the geometry of the complex, but most explanations begin by describing 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...

 complexes, where six ligands coordinate to the metal.

σ-Bonding

The molecular orbitals created by coordination can be seen as resulting from the donation of two electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

s by each of six σ-donor ligands to the d-orbitals on the metal
Metal
A metal , is an element, compound, or alloy that is a good conductor of both electricity and heat. Metals are usually malleable and shiny, that is they reflect most of incident light...

. In octahedral complexes, ligands approach along the x-, y- and z-axes, so their σ-symmetry orbitals form bonding and anti-bonding combinations with the dz2 and dx2y2 orbitals. The dxy, dxz and dyz orbitals remain non-bonding orbitals. Some weak bonding (and anti-bonding) interactions with the s and p orbitals of the metal also occur, to make a total of 6 bonding (and 6 anti-bonding) molecular orbitals.

In 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...

 terms, the six lone pair orbitals from the ligands (one from each ligand) form six symmetry adapted linear combinations (SALCs) of orbitals, also sometimes called ligand group orbitals (LGOs). The irreducible representations that these span are a1g, t1u and eg. The metal also has six valence orbitals that span these irreducible representations - the s orbital is labeled a1g, a set of three p-orbitals is labeled t1u, and the dz2 and dx2y2 orbitals are labeled eg. The six σ-bonding molecular orbitals result from the combinations of ligand SALC's with metal orbitals of the same symmetry.

π-bonding

π bonding in octahedral complexes occurs in two ways: via any ligand p-orbitals that are not being used in σ bonding, and via any π or π* molecular orbitals present on the ligand.

The p-orbitals of the metal are used for σ bonding (and are the wrong symmetry
Symmetry
Symmetry generally conveys two primary meanings. The first is an imprecise sense of harmonious or aesthetically pleasing proportionality and balance; such that it reflects beauty or perfection...

 to overlap with the ligand p or π or π* orbitals anyway), so the π interactions take place with the appropriate metal d-orbitals, i.e. dxy, dxz and dyz. These are the orbitals that are non-bonding when only σ bonding takes place.

One important π bonding in coordination complexes is metal-to-ligand π bonding, also called π backbonding. It occurs when the LUMO
Lumo
Lumo is a 2007 documentary film about twenty-year-old Lumo Sinai, a woman who fell victim to "Africa's First World War." While returning home one day, Lumo and another woman were gang-raped by a group of soldiers fighting for control of the Democratic Republic of the Congo during the 1994 Rwandan...

s of the ligand are anti-bonding π* orbitals. These orbitals are close in energy to the dxy, dxz and dyz orbitals, with which they combine to form bonding orbitals (i.e. orbitals of lower energy than the aforementioned set of d-orbitals). The corresponding anti-bonding orbitals are higher in energy than the anti-bonding orbitals from σ bonding so, after the new π bonding orbitals are filled with electrons from the metal d-orbitals, ΔO has increased and the bond between the ligand and the metal strengthens. The ligands end up with electrons in their π* molecular orbital, so the corresponding π bond within the ligand weakens.

The other form of coordination π bonding is ligand-to-metal bonding. This situation arises when the π-symmetry p or π orbitals on the ligands are filled. They combine with the dxy, dxz and dyz orbitals on the metal and donate electrons to the resulting π-symmetry bonding orbital between them and the metal. The metal-ligand bond is somewhat strengthened by this interaction, but the complementary anti-bonding molecular orbital from ligand-to-metal bonding is not higher in energy than the anti-bonding molecular orbital from the σ bonding. It is filled with electrons from the metal d-orbitals, however, becoming the HOMO
Homo
Homo may refer to:*the Greek prefix ὅμο-, meaning "the same"*the Latin for man, human being*Homo, the taxonomical genus including modern humans...

 of the complex. For that reason, ΔO decreases when ligand-to-metal bonding occurs.

The greater stabilisation that results from metal-to-ligand bonding is caused by the donation of negative charge away from the metal ion, towards the ligands. This allows the metal to accept the σ bonds more easily. The combination of ligand-to-metal σ-bonding and metal-to-ligand
π-bonding is a synergic
Synergy
Synergy may be defined as two or more things functioning together to produce a result not independently obtainable.The term synergy comes from the Greek word from , , meaning "working together".-Definitions and usages:...

 effect, as each enhances the other.

As each of the six ligands has two orbitals of π-symmetry, there are twelve in total. The symmetry adapted linear combinations of these fall into four triply degenerate irreducible representations, one of which is of t2g symmetry. The dxy, dxz and dyz orbitals on the metal also have this symmetry, and so the π-bonds formed between a central metal and six ligands also have it (as these π-bonds are just formed by the overlap of two sets of orbitals with t2g symmetry.)

High and low spin and the spectrochemical series

The six bonding molecular orbitals that are formed are "filled" with the electrons from the ligands, and electrons from the d-orbitals of the metal ion occupy the non-bonding and, in some cases, anti-bonding MO's. The energy
Energy
In physics, energy is an indirectly observed quantity. It is often understood as the ability a physical system has to do work on other physical systems...

 difference between the latter two types of MO's is called ΔO (O stands for octahedral) and is determined by the nature of the π-interaction between the ligand orbitals with the d-orbitals on the central atom. As described above, π-donor ligands lead to a small ΔO and are called weak- or low-field ligands, whereas π-acceptor ligands lead to a large value of ΔO and are called strong- or high-field ligands. Ligands that are neither π-donor nor π-acceptor give a value of ΔO somewhere in-between.

The size of ΔO determines the electronic structure of the d4 - d7 ions. In complexes of metals with these d-electron configurations, the non-bonding and anti-bonding molecular orbitals can be filled in two ways: one in which as many electrons as possible are put in the non-bonding orbitals before filling the anti-bonding orbitals, and one in which as many unpaired electrons as possible are put in. The former case is called low-spin, while the latter is called high-spin. A small ΔO can be overcome by the energetic gain from not pairing the electrons, leading to high-spin. When ΔO is large, however, the spin-pairing energy becomes negligible by comparison and a low-spin state arises.

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. It can be seen that the low-field ligands are all π-donors (such as I-), the high field ligands are π-acceptors (such as CN- and CO), and ligands such as H2O and NH3, which are neither, are in the middle.

I < Br < S2 < SCN < Cl < NO3 < N3 < F < OH < C2O42 < H2O < NCS < CH3CN < py (pyridine
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 < en (ethylenediamine) < bipy (2,2'-bipyridine
2,2'-Bipyridine
2,2'-Bipyridine is a organic compound with the formula . This colorless solid, commonly abbreviated bipy or bpy , is an important isomer of the bipyridine family. It is a bidentate chelating ligand, forming complexes with many transition metals...

) < phen (1,10-phenanthroline
Phenanthroline
Phenanthroline is a heterocyclic organic compound. As a bidentate ligand in coordination chemistry, it forms strong complexes with most metal ions...

) < NO2 < PPh3 < CN < CO

History

Ligand field theory was developed during the 1930s and 1940s as an alternative to crystal field theory
Crystal field theory
Crystal field theory 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,...

 (CFT). CFT describes certain properties of 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...

 but is based on a model that emphasizes electrostatic interactions between ligand electrons with the d-electrons on the metal. CFT does not describe bonding. Ligand Field Theory, in a sense, combined CFT and the then emerging molecular orbital theory
Molecular orbital
In chemistry, a molecular orbital is a mathematical function describing the wave-like behavior of an electron in a molecule. This function can be used to calculate chemical and physical properties such as the probability of finding an electron in any specific region. The term "orbital" was first...

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