[Wien] Multipole moments

[Gavin Abo]

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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