Mulliken labels of -4m2

[thchr]

The choice to deviate from Bilbao's Mulliken labels for -4m2 here:

Crystalline.jl/src/notation.jl

Line 431 in 00682e2

"-4m2" => ImmutableDict("Γ₁"=>"A₁", "Γ₂"=>"B₁", "Γ₃"=>"A₂", "Γ₄"=>"B₂", "Γ₅"=>"E"), # setting 2 *** swapped B₂ and A₂; seems to be a typo in Bilbao? ***

is problematic. Either we need Bilbao to change their labels or we should use their labels, whether right or wrong - it's too hard to compare with Bilbao (e.g., for band representations) if we start deviating.

The reasoning for why Bilbao's Mulliken label assignment is wrong/odd that for -42m, the choice principal rotation is the 4-fold rotation operation, but for -4m2 it is somehow not, which seems wrong. Detailed explanation is recapped in your email to Elcoro of August 21, 2024.

[thchr]

After thinking about this today, and writing with Elcoro from the BCS about it, I've come to the conclusion that it's better to stick to Bilbao's conventions, simply to have one set of conventions.

My considerations from today are best summarized from the emails I wrote to Elcoro and his answer, copied in paraphrased form below for posterity:

Email 1 (to Elcoro)

Summary: I believe that the irrep labels for the A2 and the B2 irreps of the -4m2 point group (i.e., the “setting 2” variant of -42m) are accidentally swapped on the BCS, and wanted to notify you in case this is so.

Reasoning: first, looking at the -42m group, the character table is ((which is consistent with other tables I can find for the D2d group, e.g., from Inui’s book):


Notice that the assignment of A vs. B Mulliken labels is done with reference to the even/odd symmetry under a “principal rotation” associated with the 4-fold rotoinversion (C5).

On the other hand, the character table for -4m2 is:


Now, the assignment of the A vs. B labels is no longer made with reference to the 4-fold rotoinversion, but rather with reference to the symmetry under 21-10 and 2110 (C4).

Since the two groups are related, I had assumed their Mulliken labels would be assigned with reference to the same choice of principal rotation?

On the other hand, the above label assignment is consistent across the single-valued, double-valued, and magnetic point group listings at BCS also, so I imagine it is not a simple typo.

Email #2, shortly after (to Elcoro)

Following up, I think the issue runs deeper than just the assignment of Mulliken labels. Since the -42m and -4m2 groups differ only in the assignment and orientation of the second and third axes, the character tables should be identical under the transformation matrix between the two settings. However, that’s not currently the case.

Current CDML label (wrong)	Corrected CDML label
GM2	GM3
GM3	GM4
GM4	GM2

Equivalently, in terms of the Mulliken label swaps:

Current Mulliken label (wrong)	Corrected Mulliken label
B1	B2
B2	A2
A2	B1

Email 3, shortly after this again (to Elcoro)

A related issue exists for the -6m2 character table (in its comparison to -62m):
Comparing the “character vectors” (sorted by related operations) of -62m and -6m2, the assigned CDML irrep labels of the two differ/agree in the following sense:

-6m2 CDML labels	-62m CDML labels
GM1	GM1
GM2	GM4
GM3	GM3
GM4	GM2
GM5	GM5
GM6	GM6

Email 4 (from Elcoro; relevant bits paraphrased/quoted in isolation)

I agree with you that sometimes the label assignment looks not "logical".

We decided to agree with the tables of representations of the ISOTROPY software suite for space groups and take as labels of the point groups the labels of the irreps at the GM point for the corresponding space group, with P lattice in the standard setting and the same orientations of the axes.

Note that the traces for the rotoinversion -4+ and -4- operations are 1,-1,-1,1 for GM1,GM2,GM3,GM4 in space group 111 (P-42m) and the traces are 1,-1,1,-1 for the same irreps for the space group 115 (P-4m2).

Finally, we assumed the same label transformation between the CDML labels and Mulliken labels (GM1->A1, GM2->B1, GM3->B2, GM4->A2) in both cases.

Of course, we could have taken different decisions. In the end, we try to keep the internal coherence between different programs of the BCS (which has not been always easy!).

Conclusion

Bilbao's choice of GM-labels make sense: it is imperative to keep consistency with the space group labels - it seems in this case, it is the space group labels between P-42m (111) and P-4m2 (115) are the ones guilty of having been assigned in an inconsistent way - but that is a historical thing that cannot be fixed now.
Regarding the Mulliken assignment, I think it would've been nice to make a choice that preserved the physical meaning of the Mulliken labels, i.e., respected their rules and had the same interpretation as the labels in -42m. On the other hand, it does make a certain sense to stick with the assignment convention.