Editing Ligand field theory (section)

==Bonding==
{{Unreferenced section|date=January 2021}}

===σ-bonding (sigma bonding)===

In an octahedral complex, the molecular orbitals created by coordination can be seen as resulting from the donation of two [[electron]]s 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''<sub>''z''<sup>2</sup></sub> and ''d''<sub>''x''<sup>2</sup>&minus;''y''<sup>2</sup></sub> orbitals. The ''d''<sub>''xy''</sub>, ''d''<sub>''xz''</sub> and ''d''<sub>''yz''</sub> 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<ref>{{Cite web |date=2021-09-03 |title=10.3.1: Ligand Field Theory - Molecular Orbitals for an Octahedral Complex |url=https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_(LibreTexts)/10%3A_Coordination_Chemistry_II_-_Bonding/10.03%3A_Ligand_Field_Theory/10.3.01%3A__Ligand_Field_Theory_-_Molecular_Orbitals_for_an_Octahedral_Complex |access-date=2023-12-07 |website=Chemistry LibreTexts |language=en}}</ref>

[[Image:LFTi(III).png|center|thumb|400px|Ligand-Field scheme summarizing σ-bonding in the octahedral complex [Ti(H<sub>2</sub>O)<sub>6</sub>]<sup>3+</sup>.]]

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 representation]]s that these span are ''a<sub>1g</sub>'', ''t<sub>1u</sub>'' and ''e<sub>g</sub>''. The metal also has six valence orbitals that span these [[irreducible representation]]s - the s orbital is labeled ''a<sub>1g</sub>'', a set of three p-orbitals is labeled ''t<sub>1u</sub>'', and the ''d''<sub>''z''<sup>2</sup></sub> and ''d''<sub>''x''<sup>2</sup>&minus;''y''<sup>2</sup></sub> orbitals are labeled ''e<sub>g</sub>''.  The six σ-bonding molecular orbitals result from the combinations of ligand SALCs with metal orbitals of the same symmetry.<ref>{{Cite web |date=2021-09-03 |title=10.3.1: Ligand Field Theory - Molecular Orbitals for an Octahedral Complex |url=https://chem.libretexts.org/Bookshelves/Inorganic_Chemistry/Inorganic_Chemistry_(LibreTexts)/10%3A_Coordination_Chemistry_II_-_Bonding/10.03%3A_Ligand_Field_Theory/10.3.01%3A__Ligand_Field_Theory_-_Molecular_Orbitals_for_an_Octahedral_Complex |access-date=2023-12-07 |website=Chemistry LibreTexts |language=en}}</ref>

===π-bonding (pi 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<sup>*</sup> molecular orbitals present on the ligand.

In the usual analysis, the ''p''-orbitals of the metal are used for σ bonding (and have the wrong [[symmetry]] to overlap with the ligand p or π or π<sup>*</sup> orbitals anyway), so the π interactions take place with the appropriate metal ''d''-orbitals, i.e. ''d''<sub>''xy''</sub>, ''d''<sub>''xz''</sub> and ''d''<sub>''yz''</sub>. These are the orbitals that are non-bonding when only σ bonding takes place.
[[File:Pi backbonding orbitals.svg|thumb|220x220px|Example of π backbonding with [[Carbon monoxide|carbonyl]] (CO) ligands.]]
One important π bonding in coordination complexes is metal-to-ligand π bonding, also called [[Pi backbonding|π backbonding]]. It occurs when the [[LUMO]]s (lowest unoccupied molecular orbitals) of the ligand are anti-bonding π<sup>*</sup> orbitals. These orbitals are close in energy to the ''d''<sub>''xy''</sub>, ''d''<sub>''xz''</sub> and ''d''<sub>''yz''</sub> 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<sub>O</sub> has increased and the bond between the ligand and the metal strengthens. The ligands end up with electrons in their π<sup>*</sup> 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 ''d''<sub>''xy''</sub>, ''d''<sub>''xz''</sub> and ''d''<sub>''yz''</sub> 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]] (highest occupied molecular orbital) of the complex. For that reason, Δ<sub>O</sub> decreases when ligand-to-metal bonding occurs.

The greater stabilization 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 [[synergy|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<sub>2g</sub>'' symmetry. The ''d''<sub>''xy''</sub>, ''d''<sub>''xz''</sub> and ''d''<sub>''yz''</sub> 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<sub>2g</sub>'' symmetry.)