[Wien] Multipole moments

[Peter Blaha]

If you look into lapw0.F you see:

         DO LM1=1,LMMAX(JATOM)
           L=IABS( LM(1,LM1,JATOM) )
           DO J=1,JRI(JATOM)
             VALUE(J)=CLM(J,LM1,JATOM)*R(J)**L
           ENDDO
           CALL CHARGE(R,DX(JATOM),VALUE,1,JRI(JATOM),QIN)
           QQ(LM1,JATOM)=-QIN
           IF(L.EQ.0) THEN
             QEL(JATOM) =QIN
             QQ(1,JATOM)=QQ(1,JATOM)/SQFP + ZZ(JATOM)/SQFP

so it is an integral of the Clm-expansion coefficients of the electronic 
charge density times   r^L.

But in addition, this integral is taken negative (electronic charge is 
negative, but of course in the calculations rho is positive) and for L=0 
the nuclear charge Z (which is a positive number) divided by sqrt(4 pi) 
= Y_00 is added.

For the PW multipole moments, there is no sign change, thus these 
numbers are added (and not subtracted).

So the units of the sphere MM should be -e . bohr**L, the PW MM are of 
opposite sign.

Regards


Am 25.07.2026 um 09:36 schrieb 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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> 
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-- 
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Peter Blaha,  Inst. f. Materials Chemistry, TU Vienna, A-1060 Vienna
Email: peter.blaha at tuwien.ac.at
WWW:   http://www.imc.tuwien.ac.at      WIEN2k: http://www.wien2k.at
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