In this paper "Unified theory of direct or indirect band-gap nature of conventional semiconductors", the authors calculated the irreducible representations of atomic orbitals at three K points, as shown in Table 1


How such irreducible representations are determined? For example, I calculate electronic band structures using VASP, is there any software that can calculate the irreducible representations, particularly, the irreducible representations of atomic orbitals from the WAVECAR file?


I don't know off-hand about software that can do this (though I expect it exists). You can find these irreducible in a character table for that type of k-point and symmetry of the wavevector.

For example Table C10, which I reproduce here with slight edits for notation I'm more accustom to using for molecular point groups, gives the character table for $C_{3v}$ at the point $L$/$\Lambda$.

$$\begin{array}{c|ccc|c} \hline C_\mathrm{3v}(\Lambda) & E & 2C_3 & 3\sigma_\mathrm{v} & \text{Basis} \\ \hline \mathrm{\Lambda_1} & 1 & 1 & 1 & 1, z, x^2+y^2, z^2 \\ \mathrm{\Lambda_2} & 1 & 1 & -1 \\ \mathrm{\Lambda_3} & 2 & -1 & 0 & (x,y), (x^2-y^2,xy), (xz,yz) \\ \hline \end{array}$$

You can then determine the irrep of an orbital based on the symmetry of the basis elements needed to represent it. For example, an $s$ orbital is formed from a totally symmetric function, so it has the totally symmetry irrep $\Lambda_1$. $p_x$, $p_y$, and $p_z$ have a similar totally symmetric component, but also include factors of $x$, $y$, and $z$ respectively, so the $p$ orbitals are $\Lambda_1\oplus\Lambda_3$, with the $\Lambda_1$ due to $p_z$ and the $\Lambda_3$ from the combination of $p_x$ and $p_y$.


For VASP, you may try irvasp or irRep. Or for Quantum espresso, you can directly read the group representations from the output of bands.x.


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