[Wien] Multipole moments

[Laurence Marks]

Dear Gavin,

I believe the paeudo multiple moments are because there are two types of
density inside the sphere:
1. The true density via the LMs.
2. A pseudo charge which is from the part of the PWs that is inside the
spheres. This does not count towards the Coulomb potential, so has to be
subtracted out.

N.B., the pseudo charge is part of the density that the mixer has to
handle, which leads to some issues; an informational digression.

___
Emeritus Professor Laurence Marks (Laurie)
Department of Materials Science and Engineering, Northwestern University
www.numis.northwestern.edu
"Research is to see what everybody else has seen, and to think what nobody
else has thought" Albert Szent-Györgyi

On Sat, Jul 25, 2026, 02:36 Gavin Abo <gabo13279 at gmail.com> wrote:

> Prof. Marks,
>
> Sorry for not having an answer for your question.
>
> While on this topic though, I'm curious. Do you maybe know why in the
> output0 file for lapw0 that the pseudo multipole moment (Q) seems to be
> obtainable by adding (Q-sp + Q-pw) instead of by subtracting (Q-sp -
> Q-pw)?  More about that using output from the Ti example below.
>
> I asked Google's AI, and it told me the Weinert pseudo-charge method [1]
> replaces the real charge density inside each sphere with a smooth,
> Fourier-transformable pseudo-charge density that preserves multipole
> moments.
>
> The WIEN2k 23.2 usersguide [2] on page 134 has:
>
> *By subtracting Q-pw from Q-sp one obtains pseudo-multipole moments Q. *
>
> Equation (16) in Weinert's article [1]:
>
>  q̃_lm^i = q_lm^i - q_lm^Ii (or Q = Q-sp - Q-pw)
>
> Q: The pseudo multipole moment.
>
> Q-sp: The multipole moment for the real charge density inside the sphere.
>
> Q-pw: The "artificial" plane wave multipole moment per [2]. Seems to be
> used mathematically as a background to smooth Q-sp. On slide 3 of [3], I
> believe the *ρ^I(r) has no physical meaning* might be relatable to Q-pw
> being "artificial".
>
> Here is the hand calculator calculation using the values from Ti.output0
> below:
>
> Q_00 = 0.721110477 + 0.846375509 = 1.567485986
>
> Q_40 = -0.391499782 + 0.323553281 = -0.067946501
>
>
> *TiC.output0*
>
> ...
>
>                              ------------------------------
>                               M U L T I P O L M O M E N T S
>                               ------------------------------
>
>
>    ATOM=  1   Ti           Z=22.00   LM=  5     POSITION=  0.000  0.000
> 0.000
>
>    L=  0   M= 0   SPHERE MM =   0.721110477  0.000000000
>    L=  4   M= 0   SPHERE MM =  -0.391499782  0.000000000
> ...
>    ATOM=  1   Ti           Z=22.00   LM=  5     POSITION=  0.000  0.000
> 0.000
>
>    L=  0   M= 0   PLANE WAVE MULTIPOLMOMENT =   0.846375509  0.000000000
>    L=  4   M= 0   PLANE WAVE MULTIPOLMOMENT =   0.323553281  0.000000000
> ...
>    ATOM=  1   Ti           Z=22.00   LM=  5     POSITION=  0.000  0.000
> 0.000
>
>    L=  0   M= 0   PSEUDO MULTIPOLMOMENT     =   1.567485986  0.000000000
>    L=  4   M= 0   PSEUDO MULTIPOLMOMENT     =  -0.067946502  0.000000000
>
> ...
>
> Of possible interest, the article at [4] looks to be referring to the
> charge inside the sphere as a monopole:
>
> *[...] Hammann [5] and Weinert [1] developed the so-called pseudo-charge
> method, making use of the fact that the Coulomb potential outside a sphere
> depends only on the charge outside the sphere and the multipoles of the
> charge inside.  In the spherical charge approximation, this is simply the
> integral of the charge (monopole) inside the sphere.*
> [1] https://doi.org/10.1063/1.524800
> [2] https://susi.theochem.tuwien.ac.at/reg_user/textbooks/usersguide.pdf
> [3] https://www.flapw.de/pm/uploads/User-Documentation/krasovskii.pdf
> [4] http://jswl.xml-journal.net/en/article/id/1895
> [5] Private communications cited in [1]
>
> Thanks,
> Gavin
> WIEN2k user
>
>
> On 7/24/2026 2:00 PM, Laurence Marks wrote:
>
> A quick double check; the multipole moments (MULT in case.in0 is simplest)
> are the dipole from the sphere and the plane waves in some units? (I have
> to work out exactly what the pseudo moments are.) If I am correct then they
> are a quick metric of the on-site polarization, which is a useful number
> (for me).
>
> --
> Emeritus Professor Laurence Marks (Laurie)
> Northwestern University
> Webpage <http://www.numis.northwestern.edu> and Google Scholar link
> <http://scholar.google.com/citations?user=zmHhI9gAAAAJ&hl=en>
> "Research is to see what everybody else has seen, and to think what nobody
> else has thought", Albert Szent-Györgyi
>
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