I am still puzzled by the results that I obtain with GTPack for the basis functions of irreps of point groups, in particular for the two-dimensional irrep E” of group D3h.
I have followed your comments in a previous post. I see your point and I can reproduce your results, but unfortunately they still don’t solve my problem, because the functions that you propose:
cartf1 –> -y z
cartf2 –> +x z
do not verify the canonical definition of representation basis functions that you introduce in the book –> eq. (5.47) (see an image of my notebook below). They satisfy instead another similar but different relation that you also write later on in the book –> eq. (5.68), but this is not typically the property that one wants!
This is really disturbing, because in every group theory reference one can find that (xz,yz) are indeed the basis functions (with definition eq. (5.47)) of irrep E” of D3h.
Could there be any issue with the way the operator GTTransformationOperator is defined within GTPack?
An independent intriguing result I have found when working on the above task is some difference in the output of commands GTProjectionOperator and GTWignerProjectionOperator, when applied to e.g. an spherical harmonic (see also at the end of my notebook). Here, I am not sure whether this difference
is relevant or not, since I do not know the internal workings of both commands, but I point to it anyway.
Can you help me with this mess, please? Am I misinterpreting something?
thank you for your interest.
Probably the problem is not so serious.
In the meantime I have made extensive tests, and have arrived to
the following conclusion: certainly Eqs. (5.47) and (5.68) are contradictory and one should adhere to the standard definition (5.47),
the problem is that the operator P(T) does not correspond to the action of the command GTTransformationOperator (which instead corresponds to P(T^(-1)) and therefore is consistent with wrong eq. (5.47) ).
Once one takes that into account and define a proper command
representing P(T) (by using T^(-1) in GTTransformationOperator), everything is OK, and the basis functions obtained verify eq. (5.47) ).
So, this trick seems to solve my problem. However, the inconsistency remains in the book and in the definition of GTTransformationOperator.
Thank you again, and regards,
Great, thanks for clarifying! This is consistent with what I found. I agree, there is a bug in GTTransformationOperator. Right now it represents P(T^-1) as you wrote. I will repair this for the next release. As for the book, we are planning to release an erratum (hopefully soon), collecting typos and mistakes we have found so far.
Thanks for looking into this more carefully! Indeed, the equation (5.68) you mention looks wrong, so there might be a (set of) typo(s) in the book. I am currently extensively looking into the irep module, trying to figure out what’s going wrong here. Hope to get back to you soon.