Ligand field theory

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Ligand field theory was developed during the 1930s and 1940s as an expansion of the electrostatic crystal field theory, which offered a good description of the electronic structure of metal ions in coordination complexes but was not able to provide a proper explanation for their bonding. It was created by combining crystal field theory with molecular orbital theory.

Contents 1 Ligand field theory in octahedral complexes 1.1 σ-bonding 1.2 High and low spin 1.3 π-bonding 2 Ligand field stabilisation energy

[edit] Ligand field theory in octahedral complexes

Like crystal field theory, ligand field theory is most easily understood by picturing ligands approaching a central metal ion and visualising the resulting orbital overlap. In an octahedral complex, six ligands coordinate to the central atom.

[edit] σ-bonding

The molecular orbitals created by coordination can be seen as resulting from the donation of two electrons by each of six σ-donor ligands to the d-orbitals on the metal. 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 d_z² and d_x²−y² orbitals. The d_xy, d_xz and d_yz 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 total.

In molecular symmetry 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 a_1g, t_1u and e_g. The metal also has six valence orbitals that span these irreducible representations - the s orbital is a_1g, a set of three p-orbitals is t_1u, and the d_z² and d_x²−y² orbitals are e_g. It is the 6 combinations of ligand SALCs with metal orbitals of the same symmetry that makes the 6 σ-bonding molecular orbitals.

[edit] High and low spin

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 anti-bonding molecular orbitals. The energy difference between the latter two types of molecular orbitals is called Δ_O (O stands for octahedral) and is determined by strength of the interaction of the ligand orbitals with the d-orbitals on the central atom, as described in crystal field theory. Ligands that interact strongly with the metal orbitals are called strong-field ligands and cause a relatively large Δ_O. Weakly interacting ligands are called weak-field ligands: they cause a relatively small energy gap between the non-bonding and anti-bonding molecular orbitals (and hence a small Δ_O) in the complex.

The size of Δ_O 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 Δ_O can be overcome by the energetic gain from not pairing the electrons, leading to high-spin. When Δ_O is large, however, the spin-pairing energy becomes negligible and a low-spin state arises.

[edit] 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 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. d_xy, d_xz and d_yz. These are the orbitals that are non-bonding when only σ bonding takes place.

The most important π bonding in coordination complexes is metal-to-ligand π bonding, also called π backbonding. It occurs when the LUMOs of the ligand are anti-bonding π^* orbitals. These orbitals are close in energy to the d_xy, d_xz and d_yz 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, Δ_O 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 happens when the π-symmetry p or π orbitals on the ligands are filled. They combine with the d_xy, d_xz and d_yz 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 of the complex. For that reason, Δ_O 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 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 t_2g symmetry. The d_xy, d_xz and d_yz 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 t_2g symmetry.)

[edit] Ligand field stabilisation energy

To determine the stabilisation of d-electrons that follows from the bonding of ligands, a quantity known as Ligand Field Stabilisation Energy (LFSE) was introduced. The LFSE is given by the equation:

LFSE = E_s − E_e

In which E_s is the relative energy (in terms of Δ_O) if the splitting of d-orbitals is taken into account, and E_e is the relative energy (again, in terms of Δ_O) if the d-electron are spread uniformly among the orbitals (which means 1/5 of all available d-electrons is put in all orbitals). For example, take the simple case where only one d-electron is available:

E_s equals zero, as the one electron is put in one of the three degenerate, lower-lying orbitals. E_e, however, equals 2/5Δ_O: 3 times 1/5 electron in the lower orbitals, plus 2 times 1/5 electron in the orbitals with energy Δ_O. The LFSE therfore equal −2/5Δ_O in this case.

For determination of E_s, the high or low spin character of the complex under consideration must be taken into account.