 Re: LCAO coefficients ( No.1 ) |
- Date: 2015/08/03 17:41
- Name: Artem Pulkin
- Hi,
First of all I would like to emphasize: there is nothing special about spin. Your wavefunction at a given k and a given E |k,E> is presented in the OpenMX basis as:
|k,E> = sum a_i |i>
where |i> is some basis set. The coefficients a_i are given in the output as (example):
Re(U) Im(U) Re(D) Im(D)
1 Mo 0 s -0.14205 0.00083 -0.98760 -0.00000
1 s -0.00136 0.00001 -0.00943 0.00000
0 px -0.00006 0.00000 -0.00039 0.00000
0 py 0.00003 -0.00000 0.00024 0.00000
0 pz 0.00000 0.00000 -0.00000 0.00000
1 px 0.00016 -0.00000 0.00109 -0.00000
1 py -0.00009 0.00000 -0.00063 -0.00000
1 pz -0.00000 -0.00000 0.00000 -0.00000
(Note that two wavefunctions per row are written in the output). Now let's read it:
|my-wavefunction-at-(k,E)> =
(-0.14205 + i * 0.00083) * |'s' n=0 up-spin orbital of atom 1 (Mo)> +
(-0.98760 + i * -0.00000) * |'s' n=0 down-spin orbital of atom 1 (Mo)> +
(-0.00136 + i * 0.00001) * |'s' n=1 up-spin orbital of atom 1 (Mo)> +
(-0.00943 + i * 0.00000) * |'s' n=1 down-spin orbital of atom 1 (Mo)> +
( 0.00006 + i * 0.00000) * |'px' n=0 up-spin orbital of atom 1 (Mo)> +
...
And so on. Coming to your question,
'how I should relate the two complex LCAO coefficient to the \phi_\nu^\alpha and \phi_\nu^\beta'
The answer is '\phi_\nu^\alpha' corresponds to the first and second columns while '\phi_\nu^\beta' is both third and fourth columns. I.e. '\phi_\nu^\alpha' is an up-spin component of the wavefunction and consists of all up-spin orbitals of all atoms in your calculation.
It will be hard to do projected band structure (what you call 'orbital composition of my band structure') from this data, though:
1) A lot of textual output has to be generated and processed
2) LCAO basis is non-orthogonal which leaves you two options to my knowledge: Mulliken and Lowdin analysis. As far as I remember both require overlap matrix in addition to the LCAO coefficients.
Regards,
Artem
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| Re: LCAO coefficients ( No.2 ) |
- Date: 2015/08/05 13:15
- Name: Eike F. Schwier <schwier@hiroshima-u.ac.jp>
- Dear Artem,
thank you for the detailed explanations, it helps me a lot in understanding the LCAO output and what I may or may not use it for.
I would like to confirm one thing: You say that the LCAO basis functions form a non-orthogonal basis. I thought that the irreducible representation of the cubic lattice (p_x, p_y, p_z, d_x2-y2, ...) used in the DOS and LCAO calculations should always form an orthogonal basis set regardless of the actual lattice symmetry and that as long as I have the complex LCAO coefficients I retain the full amplitude and phase information of the wavefunction at any |eB,k>. That way I though I may project the wavefunction on any given basis later on. Or just calculate the norm for the orbital projected band structure. Is this wrong? Or is it just not applicable for the spin analysis and I may still for both collinear and non-collinear LCAO calculate the norm for projected band structure?
best regards,
Eike
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| Re: LCAO coefficients ( No.3 ) |
- Date: 2015/08/06 11:25
- Name: T. Ozaki
- Hi,
A proper projection onto orbitals in a sense that the sum of weights over orbitals becomes unity
actually requires both the LCAO coefficients and overlap matrix elements.
Both the information are available from the output, while you need to write a couple of codes
to gather all the information and process them.
Regards,
TO
P.S.
We are planning to release Ver. 3.8 in Feb. 2016. The new version will include an unfolding
method (http://iopscience.iop.org/0953-8984/25/34/345501/) which supports orbital
decomposition of band structure.
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| Re: LCAO coefficients ( No.4 ) |
- Date: 2015/08/06 18:07
- Name: Artem Pulkin
- Eike,
You can easily project your wavefunction on whatever you want, there is no conceptual problem with that. However if you choose to project your wavefunction onto OpenMX basis (which, of course, makes sense) you have to interpret it correctly (Mulliken, Lowdin analysis).
However the LCAO basis is almost orthogonal, the overlaps are usually small. If you take coefficients from the output, square them and renormalize (as if the basis is exactly orthogonal) you will get reasonable results.
The non-orthogonality comes from the overlaps between two LCAOs centered at DIFFERENT atoms.
Again, I would like to emphasize that the statement 'the complex LCAO coefficients give the full amplitude and phase information of the wavefunction at any |eB,k>' is absolutely true. The take-away message is that the wavefunction is OK, but the basis is bad.
Regards,
Artem
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| Re: LCAO coefficients ( No.5 ) |
- Date: 2015/08/07 12:14
- Name: Eike F. Schwier <schwier@hiroshima-u.ac.jp>
- Dear Ozaki-sensei,
dear Artem,
thanks again for the explanation. If I understand you correctly, then the norm of the LCAO coefficients alone becomes problematic to use, once I compare the absolute weight from different atoms in the band structure due to the missing overlap matrix and the deviation from unity of the norm. However, if I only want to compare the contribution of a single atom to the overall bare band structure the LCAO norm alone would be reasonable to use because the basis is almost orthogonal.
best regards,
Eike
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| Re: LCAO coefficients ( No.6 ) |
- Date: 2015/08/07 12:42
- Name: T. Ozaki
- Hi,
You are totally right. The result calculated by only the LCAO coefficients would be
enough to qualitatively know what orbitals contribute to specific bands.
Regards,
TO
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LCAO coefficients