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This question is coming from this paper, which investigated the spontaneous spin and valley polarizations of monolayer LaBr2.

The geometric information of monolayer LaBr2 is displayed as following (Br: black, La: orange):

enter image description here

Then the author argues that:

As shown in (a), each La atom is trigonal prismatic coordinated to six Br atoms. Under such a condition, the d orbitals of La split into three groups: $a$ ($d_{z^2}$), $e_1$ ($d_{xz}$, $d_{yz}$), and $e_2$ ($d_{xy}$, $d_{x^2-y^2}$), see (d):

enter image description here

  • How may I understand this partition of five d-orbitals under such a trigonal prismatic geometry?
  • Is that related to crystal splitting?
  • How to explain this?
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    $\begingroup$ This looks like crystal field splitting, see en.wikipedia.org/wiki/Crystal_field_theory $\endgroup$ Sep 5 '20 at 12:35
  • $\begingroup$ +1. You might have to give more detail about what you need, rather than just saying "how do I understand this" or "how to explain this". Is the problem that the Wikipedia page Susi gave has things like "trigonal bipyramidal" and "square planar" but not explicitly "trigonal prismatic" (the term you used in your title)? $\endgroup$ Sep 5 '20 at 16:31
  • $\begingroup$ i am very not confident that I can explain this correctly, but in my coursework we never actually addressed trigonal prismatic configurations in inorganic chemistry. It seems likely that Jack simply would like some good resource to understand this splitting. My book I have available also does not include this. Inorganic Chemistry by Gary L. Miessler (5th edition). $\endgroup$ Sep 5 '20 at 17:44
  • $\begingroup$ @Nike Dattani The term "trigonal prismatic" means the inset in FIG.(a) and is used in the cited paper. $\endgroup$
    – Jack
    Sep 5 '20 at 22:56
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  • Is that related to crystal splitting?

Yes. In case there are any doubts, the caption to the figure does make it clear that it's related to crystal field splitting, "(d) The splitting of d orbitals under the trigonal prismatic crystal field".

  • How may I understand this partition of five d-orbitals under such a trigonal prismatic geometry?

If you haven't already done so, I would recommend first working through the more symmetric case of an octahedral environment (cubic crystal field). My go-to reference for this would be the book "Physics of Transition Metal Oxides" edited by Maekawa et al., Springer (2010), but it's also treated in many other books. If you have worked through that case, it should be clear that reducing the symmetry from that of a free atom to an octahedral environment introduces an energy splitting between the $d$ orbitals, and that further reducing the symmetry may introduce further splitting. The details of how this manifests in the trigonal prismatic environment was worked out in R. Huisman, R. de Jonge, C. Haas, and F. Jellinek, "Trigonal-prismatic coordination in solid compounds of transition metals," Journal of Solid State Chemistry 3, 56-66 (1971).

Unfortunately, I'm not aware of a more pedagogical reference that covers the details for the trigonal-prismatic case. (If anyone does, please post an answer.) Hence the separate recommendation to cover the cubic crystal field first. However, if you end up going this route, do note that there is a change of coordinate system. Maekawa et al. have the $z$ axis point, towards a ligand, but Huisman et al. does not, as shown in their Fig. 1. This does change the $d$-orbital labels and the geometry, but not the overall logic.

enter image description here

With that out of the way, let's jump straight to Huisman et al.'s results in their figure 3 (shown below). The lower states are related to $p$-orbitals, and can be ignored for our current purposes. The upper levels are related to the $d$-orbitals. On the left you have the free ion case, where there is perfect degeneracy. On the right is the octahedral case, where the orbitals split into a lower three-fold degenerate $t_{2g}^*$ manifold (involving $d_{z^2}$, $d_{xy}$ and $d_{x^2-y^2}$) and an upper two-fold $e_g^*$ manifold (with $d_{xz}$, $d_{yz}$). (I think the asterisk is just used to distinguish these manifolds/subspaces from the $p$-orbital ones.) To see that this makes sense, you can think about crystal field splitting as an electrostatics problem. The orbital has to pay an energy penalty if it points towards another ion with other electrons present. Hence the orbitals that point "between" nearby ligands have the lowest energy.

Then, in the tetragonal prismatic case, $t_{2g}*$ splits into a single-degenerate $a_1'^*$ level ($d_{z^2}$), and a double-degenerate $e'^*$ manifold ($d_{xy}$, $d_{x^2-y^2}$). ($e''^*$ has the same orbitals as $e_g^*$.) Again we can use electrostatics for a quick sanity check. By tuning $\theta_0$ we do change the symmetry between $d_{z^2}$ and ${d_{xy},\,d_{x^2-y^2}}$ but not between $d_{xz}$ and $d_{yz}$. Hence, the latter can change in energy, but not split up. However, I don't have a similar hand-waving argument to determine whether $d_{z^2}$ is the lowest in energy or not. I think this depends on the lattice details.

Crystal field splitting in ideal trigonal prism and ideal octahedron

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