# How to do parity analysis of band extrema?

To check whether electronic transitions are allowed between valence band maximum (VBM) and conduction band minimum (CBM) as per Laporte's rule, the parity of VBM/CBM is needed to be known. This paper discusses this in the second paragraph of page no.5. I would like to know how one can find out the parity of VBM/CBM of the material using the data generated in DFT calculations.

• If a particular energy eigenstate $|n,\mathbf{k}\rangle$ has definite parity, this means it is also an eigenstate of the parity operator P. Then, you can calculate the expectation value $\langle n,\mathbf{k}|P|n,\mathbf{k}\rangle$ and you'll get $+1$ for even parity and $-1$ for odd parity. But note that not all energy eigenstates need to have definite parity. Oct 2, 2022 at 20:21
• @ProfM how this could be obtained for example from a CASTEP output? Thank you.
– Sha
Oct 3, 2022 at 6:19
• You need the wavefunction $|n,\mathbf{k}\rangle$ from the DFT code. I believe that a good way to extract these from CASTEP is to use c2x: c2x.org.uk Oct 4, 2022 at 19:31
• I did find this paper (@ProfM as one of its authors!) using c2x for band parity analysis: doi.org/10.1103/PhysRevB.98.161203. Thank you Professor Monserrat your comment helped me to find this great paper by you.
– Sha
Oct 5, 2022 at 18:22
• Mentioned in chat. Feb 18 at 5:49