Orbital level splits

It seemed simple and straight forward, but it contains far more than that.

And I always forget about the details.

(A lot of image cropped from somewhere, this document is far from original)

Feedback welcome: yeschowh@snu.ac.kr :)

📌 A Gateway to 3d Transition Metal Magnetism: 

3d transition metals (and not exclusively limited to 3d, but let’s start here) tend to exhibit magnetism.
This magnetism originates from unpaired electrons in d-orbitals, which contribute to a net spin moment.
These 3d transition metals (such as V, Cr, Mn, Fe, Co, Ni, and sometimes Cu) are often surrounded by chalcogens, like oxygen or sulfur.
The five distinct d-orbitals experience a lifting of degeneracy due to the presence of these chalcogens.

But here's a puzzle:

why do elements like Co and Fe exhibit strong magnetic anisotropy, while others like Mn, Ni, and Cr display much weaker anisotropy in general?  Is it purely due to electron count, or is there something more subtle at play?

Here, let’s focus on the octahedral environment and explore how chalcogens influence the d-orbital splitting and how this results in the magnetic properties observed in these compounds.

✅ Perfect Octahedral Environment and Orbital Split

3d orbitals exhibit two spin degrees of freedom and adopt various spatial configurations.
While these may not resemble the familiar hydrogen atom solutions, they represent the same quantum mechanical states.
As can be checked from below figure, they are just linear combination of eigenstate of angular momentum.

However, it’s important to note that the orbitals we are working with are not eigenstates of angular momentum.
Instead of pure Y(l, m) functions characterized by well-defined angular momentum quantum numbers, we are using real orbitals, which are superpositions of +m and –m components of these eigenstates.

This arises because the continuous rotational symmetry is broken by the surrounding ligands in the crystal field environment.
As a result, electrons no longer occupy freely rotating states but are confined to fixed orientations determined by the point-group symmetry of the coordination environment.

This explains why, in our analysis of d-electron systems in octahedral fields, we use these real, spatially fixed orbitals as our basis.
They reflect the true symmetry of the system and provide a more accurate starting point for describing the electronic structure and magnetic properties.

🧲 Leading Factor Behind Orbital Degeneracy Splitting: Coulomb Repulsion

The primary driving force behind the splitting of d-orbital degeneracy is Coulomb repulsion.

A simplified conceptual picture unfolds as follows (discussed in detail later part):

1. Surrounding ligands are negatively charged, while the central transition metal ion is positively charged.
   This leads to partial charge transfer from the ligands to the metal center.

2. Electrons in the d-orbitals experience electrostatic repulsion from these negatively charged ligands.

3. As a result, orbitals pointing directly towards the ligands (dx²−y², dz²) experience greater energy repulsion, while those between ligands (dxy, dxz, dyz) are stabilized. This creates the characteristic T2g–Eg splitting for octahedral geometry.

This splitting pattern forms the basis of the electronic configuration and magnetic properties of transition metal compounds.

So here is a quick takeaways:

The five d-orbital states we start with serve as the basis for perturbation theory—with the crystal field from ligands acting as the perturbing potential. Every complex interaction we consider later stems from this starting point.

And the underlying cause? Coulomb repulsion between electrons.

In essence, the magnetism we observe is a consequence of Coulomb repulsion, with all subsequent interactions adding layers of subtlety to this foundational principle.

Let's take a details on this process.

🧩 Octahedron (Oₕ) symmetry: Crystal field splitting

Let’s imagine the five d-orbitals sitting inside a perfectly symmetric octahedral cage of ligands.
At first glance, these orbitals seem degenerate—sharing the same energy. But symmetry has its own way of breaking this. 

🔸 Why does this happen?

The Oₕ point group has a rich collection of symmetry operations—rotations, reflections, inversions—that divide space into symmetry classes. 

When we analyze how the d-orbitals transform under these operations, a clear pattern emerges:

• (dxy, dyz, dzx) form one set (T2g)

• (dx²–y², dz²) form another set (Eg)

These groupings aren’t arbitrary—they arise because the crystal field potential interacts differently with orbitals pointing along ligand axes (Eg) versus those lying between them (T2g).

⚡ The punchline: splitting made simple

In crystal field theory, this means: (dx²–y², dz²) split from (dxy, dyz, dzx) because symmetry demands it.

If you are still not confident, take an example, it may help us to understand the situation.

Let’s consider the dz² orbital case for the example. It can be characterized as dz² ~ (2z²–x²–y²)/r², in terms of Cartesian coordinates.

No matter which operation is applied that preserves Oₕ symmetry (i.e., octahedral invariance), the transformed dz² will always be expressed as a linear combination of dz² and dx²–y². It will never mix with dxy, dyz, or dzx. The same argument applies to T2g members.

A careful analysis of this argument leads to the following conclusion:

(dz², dx²–y²) and (dxy, dyz, dzx) will each mix within their respective groups without crossing boundaries under symmetry operations that preserve the crystal structure. 

Therefore, the environment before and after the transformation remains the same, rendering them energetically indistinguishable. 

This means that (dz², dx²–y²) and (dxy, dyz, dzx) should be energetically degenerate.

Of course, we don't know whether Eg is energetically higher or T2g is.
This distinction arises from the detailed crystal field picture we are constructing; in this case, Eg is disfavored

📚 Electron Filling in d-Orbitals: The Role of 10Dq and Coulomb Repulsion

Once the T2g and Eg orbitals are split by the crystal field (often quantified as 10Dq, where Dq denotes the crystal field splitting parameter), the next question is how electrons occupy these levels.

There is one more thing to take into account in electron filling, Coulomb repulsion U between the electrons in the same orbital.

In an crystal field, electrons are filled according to both the energy splitting (10Dq) and the intra-atomic Coulomb repulsion (U).

Sometimes, these two energy scale is not so different. Therefore two criteria should be considered from the experimental result.

• If the crystal field splitting (10Dq) is large compared to the coulomb repulsion between the electrons in the same orbital, a low-spin configuration forms where electrons pair up in the lower-energy T2g orbitals before occupying the higher-energy Eg levels. 

• If the crystal field splitting is small (weak-field ligands), a high-spin configuration results, with electrons maximizing unpaired occupancy across T2g and Eg.

This delicate balance between 10Dq and electron-electron Coulomb repulsion defines the electronic configuration, magnetic moment, and overall properties of the transition metal complex.

Let's take a look at d7 configuration and understand how octahedron geometry leads to low-spin or high-spin.

🧲 High Spin and Low Spin Configurations: U and 10Dq.

The distinction between high-spin and low-spin states hinges on the competition between the magnitude of 10Dq and the Coulomb repulsion energy U between the same orbital with different spin (pairing energy). 

Coulomb repulsion is intrinsic to electron interactions within the same d-orbital type and remains relatively constant across different environments.

However, 10Dq depends on the nature of the surrounding ligands: strong-field ligands increase 10Dq, favoring low-spin configurations, while weak-field ligands result in a smaller 10Dq, leading to high-spin configurations.

As an example, consider a d7 system such as Co2+, where the arrangement of 7 electrons can vary between high-spin and low-spin states depending on the relative magnitudes of 10Dq and U. Below figure illustrates this behavior.

Understanding this interplay helps predict and rationalize the magnetic and electronic behaviors of 3d transition metal complexes in octahedral fields.

Of course, we can't determine whether it adopts a high-spin or low-spin configuration solely based on the fact that the cobalt ion has 7 d-electrons. 

This depends on the strength of the ligand field, which must be inferred from experimental data rather than theoretical reasoning alone.

Theory is there to explain the mess, not to make a bigger one! 

😊 Welcome to Starting Line, Are You Ready to Run?

This is the basic starting point of orbital split and electron filling.

Of course, there are much more things to be considered. Octahedron distortion, spin orbit coupling, multiplet effect or so.

But it seems I already spent too much scroll on this page.

Further part will be discussed in next document..