# Crystal field when inversion is absent

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• ftft
In summary, when expanding the crystal field in spherical harmonics, terms of odd ranks may not appear in the expansion even if the crystal lacks an inversion centre, such as in a tetrahedral crystal field. This is because inversion symmetry is sufficient but not necessary for the absence of odd terms. In general, the crystal field should transform as a totally symmetric representation of the symmetry group, which can be found by looking at how the irreducible representations of the total rotational symmetry group split up in the crystal field subgroup. In cases where the crystal lacks any symmetry at all, such as being classified as C1, all terms in the expansion are important.
ftft
I find in many textbooks that when expanding the crystal field in spherical harmonics, those terms of odd ranks do not appear in the expansion even if the crystal lacks an inversion centre such as the tetrahedral crystal field. Why is that? and when one should include spherical harmonics of odd ranks?

I would expect that in a tetrahedral field, there are no odd terms because of the high remaining symmetry of the tetrahedron. In mathematical language: inversion symmetry is sufficient but not necessary for the absence of odd terms.
In general the crystal field should transform as a totally symmetric representation of the symmetry group. As the symmetry group of the crystal is a subgroup of the total rotational symmetry group SO(3) of which the spherical harmonics span irreducible representations, you can look up how these irreducible representations split up when going to the crystal field subgroup, e.g. using the character table.

Does it mean that when the crystal lacks any symmetry at all, i.e. when it is classified as C1, all terms in the expansion are important?

Yes

Thank you DrDu

## 1. What is "Crystal field when inversion is absent"?

Crystal field when inversion is absent refers to a type of electronic structure in which the energy levels of a molecule or crystal are affected by the surrounding electric field, but not by the inversion of the molecule or crystal. This phenomenon is typically observed in molecules or crystals with an asymmetrical structure.

## 2. How does inversion affect the crystal field?

Inversion refers to the process of exchanging the positions of atoms in a molecule or crystal. In crystals with an inversion center, the electronic structure is not affected by inversion, and the energy levels remain unchanged. However, in crystals without an inversion center, the electronic structure is influenced by the surrounding electric field, resulting in a crystal field when inversion is absent.

## 3. What are the consequences of crystal field when inversion is absent?

The consequences of crystal field when inversion is absent can include changes in the electronic and optical properties of the molecule or crystal. This can lead to different spectroscopic features, such as shifts in absorption and emission energies, as well as changes in the intensity and polarization of light emitted or absorbed by the molecule or crystal.

## 4. What are some examples of molecules or crystals with crystal field when inversion is absent?

One example is the tetrahedral carbon center in certain organic compounds, such as methane. Another example is the lanthanide and actinide ions in crystals, which have an asymmetric electronic structure due to the lack of an inversion center. In both of these cases, the crystal field when inversion is absent can have significant effects on the properties and behavior of the molecules or crystals.

## 5. How is crystal field when inversion is absent studied?

Crystal field when inversion is absent can be studied through various spectroscopic techniques, such as UV-Vis, IR, and Raman spectroscopy. These methods can provide information about the electronic structure and energy levels of the molecule or crystal, as well as the effects of the surrounding electric field. Computational methods, such as density functional theory, can also be used to model and predict the properties of molecules and crystals with crystal field when inversion is absent.

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