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## Fatbands M label [SOLVED]

Total energy, DFT+U, BigDFT,...

Iraola
Posts: 3
Joined: Sun May 20, 2018 6:24 pm

### Fatbands M label

Dear all,

I have performed some fatband calculations with pawfatbnd=2, in order to get the contribution of each |L,M> state belonging to a certain Cu atom to the bands. My question is the following: is M the quantum number corresponding to the Z component of the orbital angular momentum (Lz)?

The question arises from the tutorial https://docs.abinit.org/tutorial/dmft/ about DMFT. There, they mention that each state |L=2,M> corresponds to a d-type orbital of Vanadium. For example, they affirm that the state |L=2,M=-2> corresponds to the orbital d_xy. However, all d orbitals (and the same happens with p and f orbitals) except the one with M=0 are combinations of states |L=2,M> and |L=2, - M> in such a way that the angular part of the resulting wave function is real. Therefore, I suspect that the label M used in abinit is not the quantum number corresponding to Lz.

In the case that M is not the quantum number corresponding to Lz, which criteria should be used to identify each |L,M> with the corresponding atomic orbital?

Thank you.

ebousquet
Posts: 372
Joined: Tue Apr 19, 2011 11:13 am
Location: University of Liege, Belgium

### Re: Fatbands M label  [SOLVED]

Dear Iraola,
The M quantum number corresponds to m_L such that:
-2 --> d_xy
-1 --> d_yz
0 --> d_z^2
1 --> d_xz
2 --> d_x^2-y^2.
However, you have to be careful with the relative orientation between the cartesian axis and bonds between the Cu and the ligands, which could end up to a mixing of the above definition.
Best wishes,
Eric

Posts: 44
Joined: Mon Aug 24, 2009 10:58 am

### Re: Fatbands M label

Hello

Just to complete Eric's answer, the d orbitals are given also in
https://docs.abinit.org/variables/paw/#dmatpawu.
They are in the real harmonics basis (as for prtdosm=2)
and more generally follows the definition found in:
Blanco, Miguel A., Flórez, M., Bermejo, M.
Evaluation of the rotation matrices in the basis of real spherical harmonics
J. Mol. Struct. THEOCHEM 419, 19-27 (1997)
URL: https://doi.org/10.1016/s0166-1280(97)00185-1

Best regards
Bernard
CEA
France

Iraola
Posts: 3
Joined: Sun May 20, 2018 6:24 pm