Ligand field theory

Ligand field theory(LFT) describes the bonding, orbital arrangement, and other characteristics ofcoordination complexes.^[1]^[2]^[3]^[4]It represents an application ofmolecular orbital theorytotransition metalcomplexes. A transition metal ion has nine valenceatomic orbitals- consisting of fivend, one (n+1)s, and three (n+1)p orbitals. These orbitals have the appropriate energy to form bonding interactions withligands.The LFT analysis is highly dependent on the geometry of the complex, but most explanations begin by describingoctahedralcomplexes, where six ligands coordinate with the metal. Other complexes can be described with reference tocrystal field theory.^[5]Inverted ligand field theory(ILFT) elaborates on LFT by breaking assumptions made about relative metal and ligand orbital energies.

History

Ligand field theory resulted from combining the principles laid out in molecular orbital theory andcrystal field theory,which describe the loss of degeneracy of metal d orbitals in transition metal complexes.John Stanley GriffithandLeslie Orgel^[6]championed ligand field theory as a more accurate description of such complexes, although the theory originated in the 1930s with the work on magnetism byJohn Hasbrouck Van Vleck.Griffith and Orgel used the electrostatic principles established in crystal field theory to describe transition metal ions in solution and used molecular orbital theory to explain the differences in metal-ligand interactions, thereby explaining such observations as crystal field stabilization and visible spectra of transition metal complexes. In their paper, they proposed that the chief cause of color differences in transition metal complexes in solution is the incomplete d orbital subshells.^[6]That is, the unoccupied d orbitals of transition metals participate in bonding, which influences the colors they absorb in solution. In ligand field theory, the various d orbitals are affected differently when surrounded by a field of neighboring ligands and are raised or lowered in energy based on the strength of their interaction with the ligands.^[6]

Bonding

σ-bonding (sigma bonding)

In an octahedral complex, the molecular orbitals created by coordination can be seen as resulting from the donation of twoelectronsby each of six σ-donor ligands to thed-orbitals on themetal.In octahedral complexes, ligands approach along thex-,y- andz-axes, so their σ-symmetry orbitals form bonding and anti-bonding combinations with thed_z²andd_x²−y²orbitals. Thed_xy,d_xzandd_yzorbitals remain non-bonding orbitals. Some weak bonding (and anti-bonding) interactions with thesandporbitals of the metal also occur, to make a total of 6 bonding (and 6 anti-bonding) molecular orbitals^[7]

Ligand-Field scheme summarizing σ-bonding in the octahedral complex [Ti(H₂O)₆]³⁺.

Inmolecular symmetryterms, 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). Theirreducible representationsthat these span area_1g,t_1uande_g.The metal also has six valence orbitals that span theseirreducible representations- the s orbital is labeleda_1g,a set of three p-orbitals is labeledt_1u,and thed_z²andd_x²−y²orbitals are labelede_g.The six σ-bonding molecular orbitals result from the combinations of ligand SALCs with metal orbitals of the same symmetry.^[8]

π-bonding (pi bonding)

π bonding in octahedral complexes occurs in two ways: via any ligandp-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, thep-orbitals of the metal are used for σ bonding (and have the wrongsymmetryto overlap with the ligand p or π or π^*orbitals anyway), so the π interactions take place with the appropriate metald-orbitals, i.e.d_xy,d_xzandd_yz.These are the orbitals that are non-bonding when only σ bonding takes place.

Example of π backbonding withcarbonyl(CO) ligands.

One important π bonding in coordination complexes is metal-to-ligand π bonding, also calledπ backbonding.It occurs when theLUMOs(lowest unoccupied molecular orbitals) of the ligand are anti-bonding π^*orbitals. These orbitals are close in energy to thed_xy,d_xzandd_yzorbitals, with which they combine to form bonding orbitals (i.e. orbitals of lower energy than the aforementioned set ofd-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 metald-orbitals, Δ_Ohas 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-symmetrypor π orbitals on the ligands are filled. They combine with thed_xy,d_xzandd_yzorbitals 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 metald-orbitals, however, becoming theHOMO(highest occupied molecular orbital) of the complex. For that reason, Δ_Odecreases 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 asynergiceffect, 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 oft_2gsymmetry. Thed_xy,d_xzandd_yzorbitals 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 witht_2gsymmetry.)

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 thed-orbitals of the metal ion occupy the non-bonding and, in some cases, anti-bonding MOs. Theenergydifference 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 thed-orbitals on the central atom. As described above, π-donor ligands lead to a small Δ_Oand are called weak- or low-field ligands, whereas π-acceptor ligands lead to a large value of Δ_Oand are called strong- or high-field ligands. Ligands that are neither π-donor nor π-acceptor give a value of Δ_Osomewhere in-between.

The size of Δ_Odetermines the electronic structure of thed⁴-d⁷ions. In complexes of metals with thesed-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_Ocan be overcome by the energetic gain from not pairing the electrons, leading to high-spin. When Δ_Ois large, however, the spin-pairing energy becomes negligible by comparison and a low-spin state arises.

Thespectrochemical seriesis 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

References

^ Ballhausen, Carl Johan, "Introduction to Ligand Field Theory",McGraw-Hill Book Co., New York, 1962
^ Griffith, J.S. (2009).The Theory of Transition-Metal Ions(re-issue ed.). Cambridge University Press.ISBN 978-0521115995.
^ Schläfer, H. L.; Gliemann, G. "Basic Principles of Ligand Field Theory" Wiley Interscience: New York; 1969
^Bhatt, Vasishta (2016-01-01), Bhatt, Vasishta (ed.),"Chapter 1 - Basic Coordination Chemistry",Essentials of Coordination Chemistry,Academic Press, pp. 1–35,doi:10.1016/b978-0-12-803895-6.00001-x,ISBN 978-0-12-803895-6,retrieved2024-06-18
^G. L. Miessler and D. A. Tarr "Inorganic Chemistry" 3rd Ed, Pearson/Prentice Hall,ISBN 0-13-035471-6.
^^a ^b ^cGriffith, J.S. and L.E. Orgel."Ligand Field Theory".Q. Rev. Chem. Soc.1957, 11, 381-393
^"10.3.1: Ligand Field Theory - Molecular Orbitals for an Octahedral Complex".Chemistry LibreTexts.2021-09-03.Retrieved2023-12-07.
^"10.3.1: Ligand Field Theory - Molecular Orbitals for an Octahedral Complex".Chemistry LibreTexts.2021-09-03.Retrieved2023-12-07.

External links

Crystal-field Theory, Tight-binding Method, and Jahn-Teller Effectin E. Pavarini, E. Koch, F. Anders, and M. Jarrell (eds.): Correlated Electrons: From Models to Materials, Jülich 2012,ISBN 978-3-89336-796-2

[1] Ballhausen, Carl Johan, "Introduction to Ligand Field Theory",McGraw-Hill Book Co., New York, 1962

[2] Griffith, J.S. (2009).The Theory of Transition-Metal Ions(re-issue ed.). Cambridge University Press.ISBN 978-0521115995.

[3] Schläfer, H. L.; Gliemann, G. "Basic Principles of Ligand Field Theory" Wiley Interscience: New York; 1969

[4] Bhatt, Vasishta (2016-01-01), Bhatt, Vasishta (ed.),"Chapter 1 - Basic Coordination Chemistry",Essentials of Coordination Chemistry,Academic Press, pp. 1–35,doi:10.1016/b978-0-12-803895-6.00001-x,ISBN 978-0-12-803895-6,retrieved2024-06-18

[5] G. L. Miessler and D. A. Tarr "Inorganic Chemistry" 3rd Ed, Pearson/Prentice Hall,ISBN 0-13-035471-6.

[Griffth_1957-6] Griffith, J.S. and L.E. Orgel."Ligand Field Theory".Q. Rev. Chem. Soc.1957, 11, 381-393

[7] "10.3.1: Ligand Field Theory - Molecular Orbitals for an Octahedral Complex".Chemistry LibreTexts.2021-09-03.Retrieved2023-12-07.

[8] "10.3.1: Ligand Field Theory - Molecular Orbitals for an Octahedral Complex".Chemistry LibreTexts.2021-09-03.Retrieved2023-12-07.

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