The magnitude of the ligand field splitting parameter in the octahedral field can be determined from the frequency of maximum absorption in the optical absorption spectrum.

This absorption arises from an electronic transition from the t2g level to the eg level. This is the most important form of electronic transition in the transition metal complexes, but others are also observed, and these transitions are generally observed in the visible and ultraviolet regions.

The types of interactions are:

Ligand Spectra: Some ligands, such as water and organic molecules, possess characteristic absorption bands, normally in the UV range. These bands are observed in the optical spectra of the complexes, but are somewhat shifted in frequency.

Counter-Ion Spectra: When there is an ionic complex in solution, it must have a counter ion. The spectrum of the counter ion must be known in order to fully interpret the spectrum due to the complex ion.

Charge Transfer Spectra: These spectra arise from electronic transitions between orbitals which are principally those of the metal in character and orbitals which are largely those of the ligands in character.

Ligand Field Spectra: These account for the transition observed above, and arise from transitions between d-orbitals of the metal which have been split due to the ligand field.

Charge Transfer Spectra

Some transition metal complexes have very intense colours, such as the chromate (CrO42-) and permanganate (MnO4) ions, and these colours arise from charge transfer transitions between orbitals of which one is mainly a metal orbital and one is mainly a ligand orbital.

These transitions correspond to metal oxidation, when an electron is transferred from the metal to the ligand, or metal reduction, when an electron is transferred from the ligand to the metal.

Ligand to Metal charge transfer: This corresponds to metal reduction, so a metal which is easily reduced and a ligand which is easily oxidized will result in a low energy transition. Therefore, oxidizable anions like I often form complexes where the charge transfer absorption is in the visible region, eg. TiI4 is violet, HgI2 is red, and AgI is yellow. The trend in frequency of absorption of a range of similar complexes can be explained in terms of the ease of oxidation of the ligand, eg. TiCl62- has a higher absoprtion frequency than TiBr62- because the Br ligand is more easily oxidized than the Cl ligand. Similar trends are observed when the metal cation is strongly oxidizing, where the frequency of absorption follows the oxidizing strength of the metal ion.

Metal to Ligand charge transfer: This corresponds to metal oxidation, so it is necessary for the metal to be easily oxidizable and the ligand to be easily reducible. Easily reducible ligands are those which have a low lying, vacant π* orbital, such as pyridene, and they form strongly coloured complexes with easily oxidizable metal cations such as Fe2+ and Cu+. Depending of the d-number of the cation, two different transitions are possible; the t2g to π* and eg to π* may both be observed.

Metal to Metal charge transfer: Some compounds possess metal ions in two different oxidation states. In these compounds, a charge transfer transition may occur when the electron moves from one metal ion to the other, with one metal ion acting as the reducing agent and the other acting as the oxidizing agent. Compounds of this nature are generally very intensely coloured, such as Prussian Blue, KFeIII[FeII(CN)6].

These charge transfer transitions give intense absorptions in the UV, but which trail into the visible region. Their intensity means that they are strongly coloured, and so often obscure the ligand field spectra.

Ligand Field Spectra

When an electron moves from one d-orbital to another, as in the case of the t2g-to-eg transition, the overall energy of the transition needs to take into account the rearrangement of the other electrons when the transition occurs. The possible transitions which may occur are governed by the possible arrangements of electrons within the d-orbitals before and after the transition takes place.

The possible transitions may be summarized in terms of the selection rules, which come from considering the effects of the coupling of spin and orbital angular momentum in the ion, and the way this changes during the transition on absorption of a photon. The cations of the first row transition metals undergo Russell-Sauders coupling.

Russell Saunders coupling: the d2 ion

When more than one valence electron is present, interactions between the electrons result in couplings between the quantum numbers for the individual electrons. The quantum state of the overall ion depends on the quantum states of the individual electrons.

The quantum state of the electron is determined by the values of n (the principal quantum number), l (the orbital angular momentum quantum number), ml (the magnetic quantum number), and s (the spin quantum number).

There may be coupling between the spin angular momenta of two electrons, spin-spin coupling, the orbital angular momenta of two electrons, orbit-orbit coupling, and the spin and orbital angular momenta of the same electron, spin-orbit coupling.

In the Russell-Saunders scheme, the case for the first row transition elements, and in general for elements up to atomic number 30, the magnitude of coupling is assumed to be in the order:

spin-spin coupling > orbit-orbit coupling > spin-orbit coupling

Spin-spin coupling:

The spin quantum number, S, for a system of electrons is calculated from the spin quantum numbers, s1 and s2, for the separate electrons according to

for the d2 system,

Orbit-orbit coupling:

For two electrons with orbital angular momentum quantum numbers l1 and l2, the total orbital angular momentum quantum number, L, is

This is known as the Clebsh-Gordan Series.

for the d2 system,

l1 = l2 =2, so

Different values of L are referred to by different term letters: S (L=0), P (L=1), D (L=2), F (L=3), G (L=4), H (L=5), …

the d2 system has G, F, D, P, and S states

Spin-orbit coupling:

The total angular momentum quantum number, J, is obtained by coupling the total spin and orbital angular momenta according to:

Different values of S can have different numbers of values of J, or different numbers of levels. The number of levels possible for a given S number is the multiplicity, given by (2S + 1).

the d2 system has multiplicity values

S = 1: (2S+1) = 3 (a triplet)

S = 0: (2S+1) = 1 (a singlet)

The information on the possible values of S, L and J as summarized in the term symbol:

Not all terms are allowed, as some would require electrons with the same spin to occupy the same orbital, in contravention of the Pauli exclusion principle.

the d2 system has the possible terms:

3P, 3F, 1S, 1D, 1G

The relative order of the energies of these terms is given by Hund’s rules:1) The most stable state is the one with the maximum multiplicity

2) For a group of terms with the same multiplicity, the one with the largest value of L lies lowest in energy.

the ground state term for the d2system is:

3F

The selection rules may be summarised:

Spin forbidden transitions: Transitions in which there is a change in the number of unpaired electron spins are forbidden, ie. for a transition to give optical absorption ΔS = 0. Transitions where ΔS is non-zero are spin forbidden.

Orbitally forbidden transitions: Transitions involving the redistribution of electrons within a single quantum shell are forbidden. Thus d-to-d and p-to-p transitions are forbidden but s-to-p and p-to-d transitions are allowed, and correspond to transitions where ΔL = +1 or -1. Transitions of the type g-to-g and u-to-u are said to be parity forbidden.

The fact that these selection rules are not strictly obeyed allows us to see the colours of transition metals and to observe ligand field spectra arising from d-to-d transitions. The selection rules are broken in the following ways:

The spin forbidden rule is relaxed by spin-orbit coupling. The intensities of spin forbidden bands increase with the spin-orbit coupling constant. Spin forbidden bands are, however, extremely weak

The orbital forbidden rule is relaxed by vibronic coupling. A vibrational mode of the complex that is antisymmetric, or of u parity, with respect to the center of symmetry of the complex can mix with the electronic wavefunction. On mixing, the ground state can become mixed with a g-type vibration and the excited state may mix with a u-type vibration, and so a g-to-g transition will acquire some g-to-u character and so become weakly allowed.

Intensity stealing: when a ligand-field transition occurs close to a charge transfer band, mixing of the electronic wavefunctions of the forbidden excited term and the allowed charge transfer level means that electronic transitions to the excited term become allowed.

The intensities of absorption bands are measured experimentally in terms of the molar absorption coefficient (ε). When light shines on a sample, as in a spectrophotometer, the optical density, d, is measured, and this can be converted into the molar absorption coefficient.

I0 is the intensity of light incident on the sample, and I is the intensity of the emergent light.
c is the solution concentration, in gmoldm-3, and l is the path length of the light through the sample, in cm.

Some typical molar absorption coefficients, and the selection rules corresponding to the transition from which they arise are given in the table:

Type of transition Example Typical value of ε
Spin forbidden, Orbital forbidden [Mn(H2O)6]2+ 0.1
Spin allowed, Orbital forbidden [Ti(H2O)6]3+ 10
Spin allowed, Orbital partially allowed by d-p mixing [CoCl4]2- x 102
Spin allowed, Orbital allowed (charge transfer) [TiCl6]2- 104

The values of ε therefore arise from the nature of the transition, in terms of how allowed it is, and account, for example, for the pale pink colour of the aqueous [Mn(H2O)6]2+ ion, and the dark purple colour of the aqueous MnO4 ion, which are familiar from titrations using permanganate as the indicator.