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I wanted to calculate the normal modes of some particular material using the Quantum ESPRESSO. Everything went fairly well, but there is a couple of lines in the output that I do not understand. In particular, one of the lines that came out in the ph.out file reads as:

freq ( 4 - 5) = 62.3 [cm-1] --> E_u X_5' M_5' I

From here we can read off the phonon frequency $\left(62.3 \,\textrm{cm}^{-1}\right)$ and we can also see that this particular vibration belongs to $E_u$ irreducible representations. It is also possible to see that this mode is IR active. However, I do not understand this piece of information:

X_5' M_5'

What does this tell us?

PS: We are working with point group $D_{4h}$ (4/mmm).

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For questions like these where the answer is not provided by the Quantum ESPRESSO user guide, it usually pays off to search the source code on the QE gitlab repository.

Searching for "freq" turns up find_mode_sym.f90, which shows that this information is stored in the name_rap variable. Searching for "name_rap(1)" turns up PW/src/divide_class.f90

The comment documenting the set_irr_rap subroutine mentions

Various names have been used in the litterature to identify the irreducible representations. Several equivalent names are collected in this routine. The first name is taken from the book of P.W. Atkins, M.S. Child, and C.S.G. Phillips, "Tables for group theory". D, G, L, S are used for Delta, Gamma, Lambda and Sigma. Representations which correspond to infrared or raman active modes are identified with the string in ir_ram: I (infrared active), R (Raman active), I+R (Infrared and Raman active).

I.e. X_5' and M_5' are just labels for the E_u representation in alternative labelling schemes [1].

[1] Which labelling schemes I don't know but I'm also not an expert in point groups. Clicking on blame will tell you that the last person who touched this code is Andrea dal Corso in 2006. If you would like to contribute to improving the documentation, I suggest you open an issue and ping @dalcorso for a reference concerning these symbols.

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