# pd hybridization parameters of D3d in relation to Vt2g and Veg with Oh

asked by Jun Okamoto (2022/06/27 16:18)

I have questions about pd hybridization values with D3d (without distortion: Ds=0 and Dt = 0) in relation to Vt2g and Veg with Oh to reproduce XAS and RIXS spectra. I calculated L2,3-edges XAS and RIXS spectra of D3d-symmetry Ni2+ including charge transfer without crystal-field effect (Ds = 0, Dt = 0). I input Veg = Veg(Oh), Va1g = Vt2g(Oh), and Veg1 = Vt2g(Oh). Other parameters are the same as those in Oh symmetry. The calculated XAS and RIXS spectral line shapes, however, did not reproduce those with Oh even no trigonal distortion condition. What are the proper values of Veg, Va1g, and Veg1 in relation to Veg(Oh) and Vt2g(Oh) to obtain the same XAS and RIXS spectra with Oh symmetry?

As for D4h without crystal distortion (Ds = 0 and Dt = 0), Ni2+ L2,3-edge XAS and RIXS spectra inputting Va1 = Veg(Oh), Vb1 = Veg(Oh), Ve = Vt2g(Oh), Vb2 = Vt2g(Oh) match with those for Oh symmetry.

Many thanks, Jun Okamoto

, 2022/06/28 02:15, 2022/06/28 19:15

Hi Jun,

I admit, I am not very good with point group symmetries, but I have a hunch.

In Oh symmetry the d gets split into Eg (x2-y2 and z2) and T2g (xy, xz, and yz). In D3d symmetry the d gets split into Eg (x2 - y2 and xy), Eg1 (xz and yz), and A1g (z2).

For D3d, you are trying to set the Eg1 and A1g to have the same hopping parameters as the T2g(Oh), but they refer to different d orbitals. Based on the symmetry, I don't think there it is possible to write the D3d hopping terms such that they reproduce the Oh case.

In D4h symmetry the d gets split into A1g (z2), B1g (x2 - y2), B2g (xy), and Eg (xz and yz). By making the hopping for A1g = B1g = Eg(Oh) and B2g = Eg = T2g(Oh) you are correctly matching the d orbitals for the D4h and Oh symmetries, which is why your XAS and RIXS spectra are being reproduced correctly.

Hopefully this makes some sense. Again, I am not certain that this is the issue, but it is at least something that jumped out at me when I first read your question.

This website is somewhat helpful for visualizing which d orbitals go where in certain symmetries: http://www.gernot-katzers-spice-pages.com/character_tables/Oh.html http://www.gernot-katzers-spice-pages.com/character_tables/D3d.html http://www.gernot-katzers-spice-pages.com/character_tables/D4h.html

Sincerely,

Charles Cardot

, 2022/06/28 05:41

Dear Charles,

I tried several combinations of Veg, Va1g, and Veg1: XAS & RIXS spectra calculated with Veg(D3d) = Veg(Oh), Va1g(D3d) = Vt2g(Oh)*sqrt(2/3) and Veg1(D3d) = Vt2g(Oh)*sqrt(4/3) are close to those in Oh, but still large was discrepancy observed.

In D3d symmetry the d gets split into Eg (x2 - y2 and xy), Eg1 (xz and yz), and A1g (z2).
For D3d, you are trying to set the Eg1 and A1g to have the same hopping parameters as the T2g(Oh), but they refer to different d orbitals.
Based on the symmetry, I don't think there it is possible to write the D3d hopping terms such that they reproduce the Oh case.

Do you mean that basis functions obtained by D3d symmetry Hamiltonian have different symmetries from those calculated by Oh symmetry one, even with canceling crystal field effect (Ds = Dt = 0)?

If so, pd hybridizations in D3d take close but different values from Veg(Oh) and Vt2g(Oh) even under no D3d distortion condition (Ds=Dt=0)?

I'd like to know your suggestions how to obtain proper pd hybridization values. Like, start with pd hybridization values close to Veg and Vt2g, and adjusting these parameters to make the XAS & RIXS spectrum closer to those in Oh symmetry?

Thank you again for replying my questions.

Many thanks, Jun Okamoto

, 2022/06/28 08:46

Dear Jun

Indeed the points raised by Charles are correct (thanks Charles) but there are solutions to this. The Oh point group is given by several symmetry operations. Most noticeably 6 C4 rotations (i.e. rotations by 90 degree or 1/4 of 360) around 3 orthogonal axis. If we take the C4 axis to by the x, y and z axis of an Cartesian coordinate system one obtains a standard setting for the Oh point group. For this setting the $x^2-y^2$ and $z^2$ orbitals can be used to represent the $e_g$ irreducible representation and the $xy$, $yz$ and $zx$ orbitals can be used to represent the $t_{2g}$ irreducible representation. If we rotate the axis we still have $e_g$ and $t_2g$ states, but their wave-functions will look different.

On https://www.quanty.org/physics_chemistry/point_groups/oh you find several possible orientations of the Oh point group. For each orientation (setting of the point group) the page lists the symmetry operations present within the point group. The different settings are linked to a page with more information on the point group for the specific setting.

On https://www.quanty.org/physics_chemistry/point_groups/d3d you find several possible orientations and the symmetry axis of the D3d point group. Again, with a link for each setting to more information.

The standard setting for the Oh group would be to place the C4 axis parallel to x, y, and z and the C3 axis in the 111 direction. For this setting you can find the orbital representations of the irreducible representations with specific total angular momentum $l$ here https://www.quanty.org/physics_chemistry/point_groups/oh/orientation_xyz.

The standard setting for the D3d point is to place the C3 axis in the z direction. See more information for this setting here https://www.quanty.org/physics_chemistry/point_groups/d3d/orientation_zx

The standard orientations of the Oh and D3d point group do not match such that one can not easily transform the orbitals between them. You now need to choose two orientations of the Oh and D3d point group that match with each other. You can go to a specific orientation of the D3d point group and look at the super groups (listed on the site) with the correct orientation.

Sofar so good, but we are not there yet. (sorry, can't make it shorter). If you look at the D3d point group with the C3 axis parallel to the z direction then you find that there are two settings possible for the Oh super group. The D3d group https://www.quanty.org/physics_chemistry/point_groups/d3d/orientation_zx has both the Oh group with settings https://www.quanty.org/physics_chemistry/point_groups/oh/orientation_0sqrt2-1z and https://www.quanty.org/physics_chemistry/point_groups/oh/orientation_0sqrt21z as supergroup. These two different settings will give you a different relation between the orbitals of the D3d pointgroup and Oh point group.

In order to simplify the two possibilities we made the site https://www.quanty.org/physics_chemistry/point_groups/d3d/orientation_zx_a and https://www.quanty.org/physics_chemistry/point_groups/d3d/orientation_zx_b where we discuss the relation between the D3d orbitals and Oh orbitals for the two different Oh supergroup settings with respect to the same D3d setting.

In corundum materials (like Ti2O3) you generally find both supergroup settings within the same crystal for two different sites within the same crystal. The reason I made the pages I listed below was when I worked on V2O3 or Ti2O3 where I ran into exactly the question you just asked.

Hope this helped,
Best wishes,
Maurits

, 2022/06/28 09:11, 2022/06/28 09:44

Dear Maurits,

I need to read and check carefully to understand your explanation about the difference of setting between Oh and D3d symmetries. This difference seems to affect so much on considering orbital anisotropy.

At this moment, I'm considering the case when crystal-field effect is negligibly small case (Ds & Dt = 0). First I check, your latter two simplified possibility can solve our problem or not.

Many thanks,

Jun Okamoto

, 2022/06/29 10:49

Dear Jun,

Yes these difference are very important for the orbital anisotropy!

Best wishes, Maurits