# Relating trigonal crystal field parameters ($D_\sigma$ and $D_\tau$) to real space distortion

asked by Hebatalla Elnaggar (2019/06/19 12:12)

Dear all,

I have simulated L-edge XAS data using trigonal distortion and was able to reproduce my spectra for a certain set of $D_q$, $D_\sigma$, and $D_\tau$. Now I am trying to understand what that implies in real space: Is the octahedron compressed or elongated and how much has the bond angles moved. I have checked Ballhausen but the trigonal distortion section is not very detailed. I was able to find only this paper regarding the issue: J. C. Hempel, JCP 64, 11, p4307(1976) however the definitions of the angles are not very clear to me. Any suggestions on how to proceed with this?

Many thanks! Heba

## Answers

Dear Heba,

What you find in your fit are electronic energies of the orbitals, including mixing between the $e_g$ orbitals. The relation between orbital energy and crystal structure is highly non-trivial. Crystal-field splittings are related to the bonding of the transition metal with the neighbouring ligands. This changes the kinetic energy of the orbitals. As hybridisation might be longer ranged than nearest neighbour one needs to consider longer range effects. Even more, wave-functions need to have a node or kink at the nucleus, also at the nucleus of neighbouring atoms. This yields a change in radial functions and a relatively large change in kinetic and potential energy that cancel each other. Last but not least there are potential energies that need to be included. You thus need to carefully include all interactions. A working scheme is based on ligand field theory starting from DFT and using Wannier orbitals. Not the first or the only work that does this, but you can find an example here: Multiplet ligand-field theory using Wannier orbitals, PRB 85 165113 (2012).

For your case, you could do DFT calculations for different structures and get from there a relation between distortion and energy of the orbitals. I.e. pick a structure. Converge a DFT calculation. Create Wannier orbitals. Create the tight binding Hamiltonian on these Wannier functions. Read of the ligand-field Hamiltonian. Determine the effective crystal field splitting. (There are examples how to do this in the tutorials, especially during the workshop(s) in Heidelberg )

Best wishes, Maurits

Dear Maurits,

Many thanks for the explanation and the suggestion. I will run DFT calculations for different structures and try to link the orbital energies to the real space distortion.

Best, Heba

Dear Heba,

I am not sure how it is possible to link structural distortions with the size of the crystal field parameters (or orbital energies). Imagine an atom surrounded by point charges arranged in C3v symmetry. If you only act on the values of the point charges, you will certainly affect your orbital energies, but the structure will be unchanged.

Marius

Dear Marius,

Indeed in that case the structure will be the same although the orbital energies would vary. I had in mind the simplistic example: for a D4h symmetry, if the in-plane orbitals are lower in energy than the out of plane orbitals one can associate that to a compressive distortion in the out of-plane direction. Now I reckon that this is only really correct to say for 1 electron under CF interaction.

Best, Heba