Ligand-Field Theory

This can be directly inserted into the Zeeman term B

yielding the effective Hamiltonian for a J multiplet B

Excursion: Calculation of the energies af a term such as Sˆ Lˆr r

V.3. Ligand-Field Theory

- 3 - Vorl. #6 (12. Nov. 2010) V.3.a. Introduction

Complex Bonds

The obvious goal of ligand-fiel theory is to describe the impact of the ligands on the behavior, e.g., the energy spectrum, of an ion in a metal cluster. To tackle this problem, it is useful to fist understand the nature of the chemical bonding in such clusters better, which is by complex bonding. In contrast to a covalent bond, in which unpaired electrons in the orbitals of two atoms/ions are so to say shared by both partners to form a chemical bond, it are the paired electrons of the ligands which are involved in the complex bonding. These electrons, also called lone pairs, stay near to their ligand, but due to their negative charges create an electric Coulomb field at the position of the metal ion. E.g. of the lone pairs in N and O are shown in this sketch:

O O O O N

N

may be involved in a complex bond covalent bond

The negatively charged lone pairs at the ligands result in a Coulomb field at the position of the metal ion.

Hence, the Hamiltonian to describe the metal ion needs to be complemented by an additional term, the ligand-field term Hˆ_LF, which is essentially nothing else than another coulomb term:

Hˆ LF

Hˆ Hˆ → +

Coordination Types

It may be anticipated already at this point, that the characteristics of the arrangement of the ligands around a metal ion, that is, the coordination sphere, may be crucial in determining the effect of the ligand field. depending on the number of coordinated ligands, three main types occur:

coordination number type

6 octahedral

8 cubic

4 tetrahedral

Metalion

„Ligand“

Example:

the coordination spheres of the Mn ions in the molecule Mn₁₂ is six-fold and quasi-octahedral, for one of its Mn ions it is shown to the right.

Usually, in particular in molecules (this may be different in inorganic crystals), the coordination spheres are not exactly of one of the main types, but significant distortions are present. These may be seen for instance also in the given example for Mn12.

In this lecture, we will mainly be concerned with octahedral coordination spheres. Generic distortion modes are then:

- tetragonal (axial)

here the octahedron is stretched/compressed along one of the C4 symmetry axes

- trigonal

here the octahedron is stretched/compressed along one of the C3 symmetry axes

Next to the six-fold octahedral coordination, sometimes also five-fold coordination is observed, in particular in Cu(II) compounds. Here two main types appear:

- quadratic pyrimidal

this is so to say an octahedron with on center missing - trigonal bipyrimidal

this is similar to an "octahedron" but only constructed from a triangular base plane

However, despite the distortions, the main coordination type is still present, that is, the Coulomb field may be distinguished into the dominant octahedral part, and the weaker distorted part. We write

with

Vˆ Tˆ Hˆ

LF = + Tˆ >>Vˆ

Tˆ : octahedral contribution of the ligand field

Vˆ : remaining contribution of the ligand field due to the distortion Energy Considerations

As for the atom, it will now also be important to have an idea of the energy scale of the ligand fields, since this may determine the proper solution procedure.

Since the ligand field is also a Coulomb field, we may expect it to be rather strong, stronger than the spin-orbit coupling, but since the distance of the electrons which produce the fields are farther apart than the electrons at the metal ion, we expect this fields to be weaker than the dominant part in the Hamiltonian of a free atom, namely the Hartree-Fock part.

≈ 10⁴ cm^-1 HˆLF

This brings the ligand-field part to be on the same order as the residual electron-electron interaction, , and two potentially fundamentally different situations need to be distinguished, namely the weak and strong ligand-field cases

Hˆee

Δ

- 5 - Vorl. #6 (12. Nov. 2010)

weak ligand field ΔHˆ_ee >Hˆ_LF strong ligand field ΔHˆ_ee <Hˆ _LF

Crystal-Field Theory

This is a simplified version of the more complete ligand-field theory, in as much as it ignores the fact that the electrons of the ligand do not only provide charges, but are in fact situated in orbitals, which may overlap, if only slightly, with the metal orbitals. In the crystal-field theory such chemical effects are ignored, that is, the situation is treated as

free ion + electric Coulomb field due to the ligands Ligand-Field Theory

Here, all chemical details are in principle included, resulting in a much more sophisticated theory. To give an estimate of these effects it may be stated, that covalency of complex bonds is about 5-10%, i.e., the electrons of the metal ion reside for few percents of their time on the ligands.

Although crystal-field theory cannot describe many important effects, it is highly successful if used as a phenomenological theory, which will completely be satisfactory for our purposes here.

Molecular Nanomagnets ©Oliver Waldmann