Ligand field theory

Bonding

σ-bonding (sigma bonding)

In an octahedral complex, 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 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 labeled a_1g, a set of three p-orbitals is labeled t_1u, and the d_z² and d_x²−y² orbitals are labeled e_g. The six σ-bonding molecular orbitals result from the combinations of ligand SALCs with metal orbitals of the same symmetry.

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^* 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^* 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.

One important π bonding in coordination complexes is metal-to-ligand π bonding, also called π backbonding. It occurs when the LUMOs (lowest unoccupied molecular orbitals) 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 situation arises 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 (highest occupied molecular orbital) of the complex. For that reason, Δ_O 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 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.)

Crystal Field Splitting in Octahedral and Tetrahedral Complexes

Crystal field theory (CFT) describes how the presence of surrounding ligands affects the energy of the d orbitals in a transition metal ion. In an isolated metal ion, the five d orbitals are degenerate meaning they are equal in energy. However, when ligands approach the metal center, electrostatic interactions between the ligand electron pairs and the metal d orbitals cause degeneracy, which results the splitting of energy levels.

Octahedral Complex

In an octahedral complex, six ligands approach the metal ion along the x, y, and z axes. The d orbitals that point directly along these axes, namely d_x²_−y² and d_z² experience greater repulsion from the ligands and became higher in energy. These two orbitals form the e_g set. In contrast, the three orbitals placed between the axes (d_xy, d_xz and d_yz) experience less repulsion and are lower in energy, forming the t_2g set. The energy difference between these two groups is called the crystal field splitting energy (Δo).

The magnitude of Δo depends on several factors, including the identity of the metal ion, its oxidation state, and the nature of the ligands. Strong-field ligands (such as CN⁻ or CO) produce a large splitting, while weak-field ligands (such as I⁻ or Br⁻) result in a smaller splitting.

Tetrahedral Complex

In tetrahedral complexes, four ligands approach the metal ion between the coordinate axes rather than directly along them. As a result, the pattern of orbital splitting is reversed compared to the octahedral case. The orbitals d_xy, d_xz and d_yz now experience greater interaction with the ligands and are increased in energy forming t₂ set, while d_x²_−y² and d_z² are lower in energy forming the e_g set. Tetrahedral complexes lack a center of inversion (inversion center/symmetry). The labels g(gerade) and u (ungerade)refer to the parity of orbitals when inverted through a central point, which is only possible in centrosymmetric systems like octahedral complexes. The splitting energy in tetrahedral complexes is denoted as Δ_t and it is typically smaller than the octahedral splitting.

Δ_t= (4/9) Δo

Another important concept associated with crystal field splitting is the distinction between high-spin and low-spin configurations. When the splitting energy (Δ) is small (weak-field ligands), electrons occupy higher-energy orbitals before pairing, resulting in a high-spin configuration with more unpaired electrons. Conversely, when (Δ) is large (strong-field ligands), electrons occupy the lower-energy orbitals first, leading to a low-spin configuration with fewer unpaired electrons.

Crystal field splitting is an important phenomenon identifying the physical properties of coordination compounds. It explains their color, as electronic transitions between split d orbitals absorb visible light, and their magnetic properties, which depend on the number of unpaired electrons. Thus, crystal field theory provides a useful framework for understanding and predicting the behavior of transition metal complexes in chemistry.

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 MOs. The energy difference between the latter two types of MOs is called Δ_O (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 Δ_O and are called weak- or low-field ligands, whereas π-acceptor ligands lead to a large value of Δ_O and are called strong- or high-field ligands. Ligands that are neither π-donor nor π-acceptor give a value of Δ_O somewhere in-between.

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 by comparison and a low-spin state arises.

The spectrochemical series 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) < NH₃ < en (ethylenediamine) < bipy (2,2'-bipyridine) < phen (1,10-phenanthroline) < NO₂⁻ < PPh₃ < CN⁻ < CO