# How can f-f transitions happen?

The question might seem quite easy to answer but still I am unable to find any useful information. I am aware of laporte- or symmetry-forbidden transitions and for f-f also spin-forbidden transitions in lanthanides but still I can't seem to find any good solution how they can happen.

For a d-d transition there are many exceptions depending on the arguments you come up with. From inverse symmetrical polyhedra that use ligand vibrations to become distorted to s- and p-mixing with the d-orbitals.

But f-orbitals, at least for the lanthanides, are said to not contribute to chemical bonds. Then there is often the argument about spin-orbit coupling but that would probably solve the spin-forbidden transitions. I'm not sure how it would affect the actual f-f transition being forbidden. In an old paper I also read about df- and fg-mixing. That would explain it but I am not sure if this is really the case.

So the question remains can the f-f transition in lanthanide complexes be somehow influenced to become at least a tiny bit more probable and therefore cause the transition to happen?

So I am currently reading the book: Physical Inorganic Chemistry A Coordination Chemistry Approach, S. F. A. Kettle

One of the chapters deals with Lanthanides and Actinides and their spectra. After a long introduction on different energetic levels and spin-orbit coupling there are a few remarks on $$\ce{f->f}$$ transitions although, as the author chooses to explain different ideas and phenomena during his explanation I am not sure if I fully understood his final conclusion. I guess it's probably a sum of many contributions. To summarize it quickly the aspects he is discussing are:

• We can see that the intensities for $$\ce{f->f}$$ transitions in coordination polyhedra with high symmetry tend to be weaker than in low symmetries. That means that there are probably mechanisms such as vibrations and mixing with other orbitals, presumably d-orbitals (like in $$\ce{d->d}$$ transitions).

• For the vibronic coupling (in order to compensate for a parity forbidden transition) he gives $$\ce{NaCs2[YCl6]:M^3+}$$ as an example, where at low temperatures both electronic and vibrational transitions seem to be observable for the trivalent lanthanide dopant.

• Then there are also 'bands' under high-resolution conditions that cannot be explained by vibronic coupling. This is where he introduces magnetic dipole allowed transitions. In his explanation he compares how an electric dipole will invert at an inversion center while, much like in the detectors for mass spectrometers, electrons will be forced to move in a spiral along the magnetic vector and a spiral or circle does not invert here. According to his explanation the magnetic dipole is an even function and as f-orbitals are odd an $$\ce{f->f}$$ transition is $$u * u = g$$ and therefore magnetic dipole allowed.

• Then he also refers to the fact that even though light may not be absorbed in matter it will still travel slower through a medium than through vacuum since it causes oscillations that have to propargate through the medium first. I'll just quote the book at this part:

It has been suggested that the light wave used in absorption studies polarizes the ligands and that the lanthanide ion feels the dipoles created. That is, it is suggested that the ligands provide a non-centric field at the lanthanide ion not by virtue of any vibrational movement of their nuclei but because of the instantaneous polarization of their electron density caused by the light wave itself.

• Then there was a remark on the environment as well. Since the f-orbitals don't really take place in any bonding for the lanthanides, do they actually feel their environment? And for this they considered weak nephelauxetic effects.

• And then there are so called hypersentisitve-transitions (transitions with environment sensitive intensities). And those seem to be related to quadrupole-allowed transitions. The two dipoles in a quadrupole may arise from a polarization by the electric vector as mentioned above. So one dipole is the light vector itself (with the energy corresponding to the hypersentisitve-transitions and the other one being the induced dipole). The difference in environments would arise from different polarizabilities of the ligands in different environments.

The other mechanism seems a bit more complicated. It considers low-symmetry crystal fields with an inherent dipole. So high-symmetry crystal fields distort via a vibration to a low symmetry. The ligand environment is distorted to a point where it become dipolar. And then the light itself is the other dipole.

• The final remark was on the bonding in f-orbitals. Here the complex $$\ce{[UF8]^3-}$$ was considered. By repulsion a square antiprism would be favored if there was no interaction at all with the ligands. What we find however is a cubic arrangemenent. And the $$\ce{f_{xyz}}$$ orbital can contribute to a bond only in the cubic case.

As far as I understand this is explained by Judd–Ofelt theory. When lanthanides are doped in the crystals, the crystal perturbation renders the Laporte selection rule inadequate to explain these transitions. The accumulated influence of the electrons and nucleus of the crystal host creates an electric field called crystal field. This crystal field generates a Stark effect (similar to Zeeman effect, only with an electric field instead of a magnetic one). The perturbation has to be added to the Hamiltonian of the free ion leading to new selection rules, called, Judd–Ofelt selection rules. These selection rules are:

For electric dipole: $$\Delta \textbf{S}=0$$, $$\Delta \textbf{L}\leq 6$$, $$\Delta \textbf{J}\leq 6$$, $$\Delta J=2,4,6$$, opposite parity.

For magnetic dipole: $$\Delta \textbf{S}=0$$, $$\Delta \textbf{L}=0$$, $$\Delta \textbf{J}=0,\pm1$$ , same parity.

For electric quadrupole: $$\Delta \textbf{S}=0$$, $$\Delta \textbf{L}=0,\pm1,\pm2$$, $$\Delta \textbf{J}=0,\pm1,\pm2$$ , same parity.

You can check about the Judd–Ofelt theory in Advances in Spectroscopy for Lasers and Sensing [1, pp. 403–433].

### References

1. Advances in Spectroscopy for Lasers and Sensing; Di Bartolo, B., Forte, O., Eds.; NATO science series II; Springer: Dordrecht, 2006. ISBN 978-1-4020-4787-9.