Orbital level split (2)

(Under construction, sorry!)

Keyword: Jahn-Teller distortion, Spin-orbit coupling, Orbital energy split, etc.

Continuation of previous document. Recommend reading it first, if you are not familiar on these topics. :)
(Again many figures are adopted from random internet documents.. Thanks to all of them.)

Typo, correction & suggestion welcome, email: yeschowh@snu.ac.kr

Beyond perfect octahedron: Lattice Distortion and Spin-Orbit Coupling

While crystal field split gives a clear starting point on orbital split, 10 Dq and U, sometimes this is not enough.
In some case, it is not sufficient to capture the full complexity of the system.

To fully describe the subtle physical phenomena observed in real materials, we must also consider effects such as lattice distortions (e.g., Jahn-Teller distortions) and spin-orbit coupling, both of which contribute to further splitting and fine structure in the d-electron energy levels.

What is Jahn-Teller Distortion and Spin-Orbit Coupling? 

When describing d-electron systems beyond the basic 10Dq crystal field splitting, two key effects must be taken into account: Jahn-Teller distortion and spin-orbit coupling.

• Jahn-Teller distortion arises in systems with electronic degeneracy (e.g., partially filled Eg levels) because the system can lower its overall energy by distorting the lattice, by removing the orbital degeneracy.

• Spin-orbit coupling connects an electron's spin and orbital angular momenta, leading to further splitting of energy levels by constant j manifold. This coupling is also the primary origin of leading term in magnetic anisotropy.

These effects provide the essential motivation for moving beyond simple crystal field theory to a more complete description of the physics in transition metal systems.

Let's start from how Jahn-Teller distortion changes the current picture.

Jahn-Teller distortion: Stabilization by Degeneracy Split

Motivation for the Jahn-Teller distortion is like this:
(1) Consider well-defined T2g - Eg level split on perfect octahedron.
(2) Electrons will be filled according to valance between 10Dq and U.
(3) After fully arranging the electronic configuration, consider perfect octahedron elongated, (or compressed if you want).
(4) Since lattice symmetry changed, T2g / Eg will be not perfectly degenerated. Their level would be splitted also.
(5) Due to degeneracy lifting, orbital filling also changes.
(6) If Eg level splits and there is only one electron in Eg, then total energy would be lowered compared to before elongation.

So, Jahn-Teller distortion is all about spontaneous lattice distortion to make total electronic energy lower and lower.

Let's consider Mn3+ case with small 10Dq as example.
It has 4 electrons in 3d orbital and electrons will be preferred to occupy different orbitals to avoid Coulomb repulsion +U.

Above picture contains all information to understand Jahn-Teller effect.

• Elongation means ligands coordinated from z-axis are getting further.

• It leads to dz² / dx²-y² degeneracy lift. 

  • dz² will have less energy than dx²-y² since direct repulsion from z-axis gets weaker.

• Similar discussion would be applicable to dxy, dyz and dzx.

  • dxy would be less affected by z-axis ligand, therefore its energy would be higher compared to dyz and dzx.

Of course, to compare the two states, we need to set energy reference to the same point.
It means total orbital level should be shifted such that summation of all levels leads to zero (or some constant).
This should be always considered, whether you are dealing with 10Dq split, Jahn-Teller split or any other.

Without appropriate energy zero, when you remove JT-distortion and 10Dq split, you leads to constant shift of energy level for orbitals in free space.

"Why energy zero is important" is easy to understand also when we consider Cu2+ Jahn-Teller case.
Cu2+ has 9 electrons in d-orbital.

Therefore initially looking, it seems like energy is increased by just watching the energy level diagram.
But considering the energy zero issue and another Eg level, total energy is lowered and all makes sense.

Above formalism is the basic mechanism of Jahn-Teller distortion, spontaneous crunch of lattice structure.

With this picture in our mind, we can consider why this kind of spontaneous lattice distortion could even happen.
I mean, why we do not have to consider elastic energy in first step many cases.

Jahn - Teller effect is basically consideration of lattice distortion. Therefore it would accompany elastic deformation.
This elastic deformation leads to degeneracy lift in orbital level, which leads to energy stabilization in electronic configuration.
Considering (1) simple spring model for deformation and (2) linear relation between deformation and orbital level shift, we can draw this kind of pictures.

Spin-orbit coupling (SOC): Relativistic correction

Spin-orbit coupling (SOC) is a relativistic effect that universally appears in both atomic and condensed matter systems.
Even in condensed matter physics, SOC emerges in a wide range of contexts—band topology to magnetic anisotropy.
Thus it is crucial to specify which context this effect is considered.

In this discussion, we focus specifically on the impact of SOC on orbital-level splitting.
This can generally explains why magnetic anisotropy is strong for some electronic configuration.
While the Jahn-Teller effect also plays a significant role in lifting orbital degeneracies, let's just focus on SOC only here.
This simplification allows us to understand the fundamental mechanisms without conflating multiple effects.

So starting point is octahedron geometry.
T2g - Eg split happens as initial state and SOC plays perturbative role also here.
Especially, SOC changes the energy landscape of T2g level. Let's see how this happens.