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<p>you may want to read and understand the basics of spherical
harmonics, for instance from the wikipedia page</p>
<p><a class="moz-txt-link-freetext" href="https://en.wikipedia.org/wiki/Spherical_harmonics">https://en.wikipedia.org/wiki/Spherical_harmonics</a></p>
<p>in particular see the attached bit.</p>
<p>Stefano deGironcoli<br>
</p>
<p><br>
</p>
<div class="moz-cite-prefix">On 18/12/24 09:47,
<a class="moz-txt-link-abbreviated" href="mailto:toshi1higu@docomo.ne.jp">toshi1higu@docomo.ne.jp</a> wrote:<br>
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<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">Dear
Professor Stefano Baroni,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">Thank
you for your quick response.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">I
think LDOS & ILDOS are defined as,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="ES">LDOS=</span><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">Σ</span><span style="font-size:10.5pt;font-family:游ゴシック" lang="ES">|psi|^2
* </span><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">δ</span><span style="font-size:10.5pt;font-family:游ゴシック" lang="ES">(E-En,k)<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">and
ILDOS=∫(LDOS) dE<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">I
don't understand the relationship between the above ILDOS
and what you said, "x^2 + y^2 + z^2 is spherically
symmetric."<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">Sorry,
but could you please explain?<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">Best
regards,<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US">Toshiharu
Higuchi<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:10.5pt;font-family:游ゴシック" lang="EN-US"><o:p> </o:p></span></p>
<div>
<div style="border:none;border-top:solid #E1E1E1
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<p class="MsoNormal"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US">From:</span></b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif" lang="EN-US"> Stefano de Gironcoli
<a class="moz-txt-link-rfc2396E" href="mailto:degironc@sissa.it"><degironc@sissa.it></a> <br>
<b>Sent:</b> Wednesday, December 18, 2024 5:09 PM<br>
<b>To:</b> <a class="moz-txt-link-rfc2396E" href="mailto:toshi1higu@docomo.ne.jp">"toshi1higu@docomo.ne.jp"</a>
<a class="moz-txt-link-rfc2396E" href="mailto:toshi1higu@docomo.ne.jp"><toshi1higu@docomo.ne.jp></a>; Quantum ESPRESSOusers
Forum <a class="moz-txt-link-rfc2396E" href="mailto:users@lists.quantum-espresso.org"><users@lists.quantum-espresso.org></a>; Stefano
Baroni <a class="moz-txt-link-rfc2396E" href="mailto:baroni@sissa.it"><baroni@sissa.it></a><br>
<b>Subject:</b> Re: [QE-users] What is the difference
between ILDOS and |psi|^2?<o:p></o:p></span></p>
</div>
</div>
<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
<div>
<p class="MsoNormal"><span lang="EN-US"> x^2 + y^2+ z^2 is
spherically symmetric, isn't it ?<o:p></o:p></span></p>
</div>
<div>
<p class="MsoNormal"><span lang="EN-US">Stefano<o:p></o:p></span></p>
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<p class="MsoNormal"><span lang="EN-US"><o:p> </o:p></span></p>
</div>
<p class="MsoNormal"><span lang="EN-US">Get <a href="https://aka.ms/AAb9ysg" moz-do-not-send="true">Outlook
for Android</a><o:p></o:p></span></p>
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<p class="MsoNormal"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:black" lang="EN-US">From:</span></b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:black" lang="EN-US"> users <<a href="mailto:users-bounces@lists.quantum-espresso.org" moz-do-not-send="true" class="moz-txt-link-freetext">users-bounces@lists.quantum-espresso.org</a>>
on behalf of Toshiharu Higuchi via users <<a href="mailto:users@lists.quantum-espresso.org" moz-do-not-send="true" class="moz-txt-link-freetext">users@lists.quantum-espresso.org</a>><br>
<b>Sent:</b> Wednesday, December 18, 2024 8:38:54 AM<br>
<b>To:</b> <a href="mailto:users@lists.quantum-espresso.org" moz-do-not-send="true" class="moz-txt-link-freetext">users@lists.quantum-espresso.org</a>
<<a href="mailto:users@lists.quantum-espresso.org" moz-do-not-send="true" class="moz-txt-link-freetext">users@lists.quantum-espresso.org</a>>;
Stefano Baroni <<a href="mailto:baroni@sissa.it" moz-do-not-send="true" class="moz-txt-link-freetext">baroni@sissa.it</a>><br>
<b>Subject:</b> Re: [QE-users] What is the difference
between ILDOS and |psi|^2?</span><span lang="EN-US"> <o:p></o:p></span></p>
<div>
<p class="MsoNormal"><span lang="EN-US"> <o:p></o:p></span></p>
</div>
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<div>
<p class="MsoNormal"><span style="font-size:11.0pt" lang="EN-US">Dear Professor Stefano Baroni,<br>
Thank you for your quick response.<br>
The |psi|^2 (square of absolute value of wave function)
of the 2p orbital is 2-lobes, and the |psi|^2 of the 3d
orbital is 4-lobes, so they are not spherically
symmetric. I don't quite understand the relationship
between this and what you said, "px, py, and pz are
degenerate, and their square coefficients are
spherically symmetric." <br>
Sorry, but could you please explain?<br>
Best regards,<br>
Toshiharu Higuchi<br>
<br>
-----Original Message-----<br>
From: Stefano Baroni <<a href="mailto:baroni@sissa.it" moz-do-not-send="true" class="moz-txt-link-freetext">baroni@sissa.it</a>>
<br>
Sent: Wednesday, December 18, 2024 3:38 PM<br>
To: "<a href="mailto:toshi1higu@docomo.ne.jp" moz-do-not-send="true" class="moz-txt-link-freetext">toshi1higu@docomo.ne.jp</a>"
<<a href="mailto:toshi1higu@docomo.ne.jp" moz-do-not-send="true" class="moz-txt-link-freetext">toshi1higu@docomo.ne.jp</a>>;
"<a href="mailto:users@lists.quantum-espresso.org" moz-do-not-send="true" class="moz-txt-link-freetext">users@lists.quantum-espresso.org</a>"
<<a href="mailto:users@lists.quantum-espresso.org" moz-do-not-send="true" class="moz-txt-link-freetext">users@lists.quantum-espresso.org</a>><br>
Subject: Re: [QE-users] What is the difference between
ILDOS and |psi|^2?<br>
<br>
Dear Toshiharu: The ILDOS sums the squared moduli of all
the molecular orbitals in a given energy range. px, py,
and pz are degenerate, and their square modulus
spherically symmetric (can you see why?) … Can you now
draw the conclusions yourself? Stefano<br>
<br>
___<br>
Stefano Baroni, Trieste -- <a href="http://stefano.baroni.me" moz-do-not-send="true" class="moz-txt-link-freetext">http://stefano.baroni.me</a><br>
<br>
> On 18 Dec 2024, at 03:56, Toshiharu Higuchi via
users <<a href="mailto:users@lists.quantum-espresso.org" moz-do-not-send="true" class="moz-txt-link-freetext">users@lists.quantum-espresso.org</a>>
wrote:<br>
> <br>
> </span><span style="font-size:11.0pt;font-family:"Tahoma",sans-serif" lang="EN-US"></span><span style="font-size:11.0pt" lang="EN-US">Dear QE Users<br>
> I have a question about the difference between
ILDOS and |psi|^2.<br>
> When I calculated the ILDOS (plot_num=10) and
|psi|^2 (plot_num =7) of <br>
> a hydrogen atom, the patterns were different.<br>
> In the case of |psi|^2, 1s orbital → spherical, 2p
orbital → 2-lobes, <br>
> and 3d orbital → 4-lobes were obtained.<br>
> However, in the case of ILDOS, all patterns were
spherical.<br>
> According to the literature, ILDOS can be
calculated from |psi|^2 and <br>
> Dirac's delta function.<br>
> Even if |psi|^2 is a pattern other than spherical,
ILDOS will be spherical.<br>
> I don't understand the reason for this.<br>
> Please tell me,<br>
> (1) In Quantum-ESPRESSO, are ILDOS and |psi|^2
calculated in separate <br>
> processes? Or is |psi|^2 calculated first and then
ILDOS is calculated <br>
> using this value?<br>
> (2) Does Dirac's delta function have any effect?<br>
> <br>
> Best regards,<br>
> Toshiharu Higuchi<br>
> <br>
>
______________________________________________________________________<br>
> _________ The Quantum ESPRESSO Foundation stands in
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> all civilians worldwide who are victims of
terrorism, military aggression, and indiscriminate
warfare.<br>
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