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.

## Fatbands M label [SOLVED]

**Moderator:** amadon

### 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

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

### 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

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

Bernard Amadon

CEA

France

CEA

France

### Re: Fatbands M label

Dear ebousquet and Amadon,

thank you for your replies. The question is now clarified.

Best regards.

thank you for your replies. The question is now clarified.

Best regards.