<div dir="auto"><div>Dear Gavin,</div><div dir="auto"><br></div><div dir="auto">I believe the paeudo multiple moments are because there are two types of density inside the sphere:</div><div dir="auto">1. The true density via the LMs.</div><div dir="auto">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.</div><div dir="auto"><br></div><div dir="auto">N.B., the pseudo charge is part of the density that the mixer has to handle, which leads to some issues; an informational digression.</div><div><br></div><div data-smartmail="gmail_signature">___<br>Emeritus Professor Laurence Marks (Laurie)<br>Department of Materials Science and Engineering, Northwestern University<br><a href="http://www.numis.northwestern.edu">www.numis.northwestern.edu</a><br>"Research is to see what everybody else has seen, and to think what nobody else has thought" Albert Szent-Györgyi</div></div><br><div class="gmail_quote gmail_quote_container"><div dir="ltr" class="gmail_attr">On Sat, Jul 25, 2026, 02:36 Gavin Abo <<a href="mailto:gabo13279@gmail.com">gabo13279@gmail.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"><u></u>
<div>
<p>Prof. Marks,</p>
<p>Sorry for not having an answer for your question.</p>
<p>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.</p>
<p>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.</p>
<p>The WIEN2k 23.2 usersguide [2] on page 134 has:</p>
<p><i>By subtracting Q-pw from Q-sp one obtains pseudo-multipole
moments Q. </i></p>
<p>Equation (16) in Weinert's article [1]:</p>
<p> q̃_lm^i = q_lm^i - q_lm^Ii (or Q = Q-sp - Q-pw)</p>
<p>Q: The pseudo multipole moment.</p>
<p>Q-sp: The multipole moment for the real charge density inside the
sphere.</p>
<p>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 <i>ρ^I(r) has no physical meaning</i>
might be relatable to Q-pw being "artificial".</p>
<p>Here is the hand calculator calculation using the values from
Ti.output0 below:</p>
<p>Q_00 = 0.721110477 + 0.846375509 = 1.567485986</p>
<p>Q_40 = -0.391499782 + 0.323553281 = -0.067946501</p>
<p><br>
</p>
<p><b>TiC.output0</b></p>
<p>...</p>
<p> ------------------------------<br>
M U L T I P O L M O M E N T S<br>
------------------------------<br>
<br>
<br>
ATOM= 1 Ti Z=22.00 LM= 5 POSITION= 0.000
0.000 0.000<br>
<br>
L= 0 M= 0 SPHERE MM = 0.721110477 0.000000000<br>
L= 4 M= 0 SPHERE MM = -0.391499782 0.000000000<br>
...<br>
ATOM= 1 Ti Z=22.00 LM= 5 POSITION= 0.000
0.000 0.000<br>
<br>
L= 0 M= 0 PLANE WAVE MULTIPOLMOMENT = 0.846375509
0.000000000<br>
L= 4 M= 0 PLANE WAVE MULTIPOLMOMENT = 0.323553281
0.000000000<br>
...<br>
ATOM= 1 Ti Z=22.00 LM= 5 POSITION= 0.000
0.000 0.000<br>
<br>
L= 0 M= 0 PSEUDO MULTIPOLMOMENT = 1.567485986
0.000000000<br>
L= 4 M= 0 PSEUDO MULTIPOLMOMENT = -0.067946502
0.000000000</p>
<p>...</p>
<p>Of possible interest, the article at [4] looks to be referring to
the charge inside the sphere as a monopole:</p>
<p><i>[...] 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.</i></p>
[1] <a href="https://doi.org/10.1063/1.524800" target="_blank" rel="noreferrer">https://doi.org/10.1063/1.524800</a><br>
[2]
<a href="https://susi.theochem.tuwien.ac.at/reg_user/textbooks/usersguide.pdf" target="_blank" rel="noreferrer">https://susi.theochem.tuwien.ac.at/reg_user/textbooks/usersguide.pdf</a><br>
[3]
<a href="https://www.flapw.de/pm/uploads/User-Documentation/krasovskii.pdf" target="_blank" rel="noreferrer">https://www.flapw.de/pm/uploads/User-Documentation/krasovskii.pdf</a><br>
[4] <a href="http://jswl.xml-journal.net/en/article/id/1895" target="_blank" rel="noreferrer">http://jswl.xml-journal.net/en/article/id/1895</a><br>
[5] Private communications cited in [1]
<p>Thanks,</p>
Gavin<br>
WIEN2k user
<p><br>
</p>
<div>On 7/24/2026 2:00 PM, Laurence Marks
wrote:<br>
</div>
<blockquote type="cite">
<div dir="ltr">
<div>
<div class="gmail_default" style="font-family:arial,sans-serif;font-size:small;color:rgb(0,0,0)">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).</div>
</div>
<div><br>
</div>
<span class="gmail_signature_prefix">-- </span><br>
<div dir="ltr" class="gmail_signature" data-smartmail="gmail_signature">
<div dir="ltr">Emeritus Professor Laurence Marks (Laurie)
<div>Northwestern University<br>
<div><a href="http://www.numis.northwestern.edu" target="_blank" rel="noreferrer">Webpage</a> and <a href="http://scholar.google.com/citations?user=zmHhI9gAAAAJ&hl=en" target="_blank" rel="noreferrer">Google Scholar
link</a></div>
<div>"Research is to see what everybody else has seen, and
to think what nobody else has thought", Albert
Szent-Györgyi</div>
</div>
</div>
</div>
</div>
<br>
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