<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:m="http://schemas.microsoft.com/office/2004/12/omml" xmlns="http://www.w3.org/TR/REC-html40">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<meta name="Generator" content="Microsoft Word 15 (filtered medium)">
<style><!--
/* Font Definitions */
@font-face
{font-family:"Cambria Math";
panose-1:2 4 5 3 5 4 6 3 2 4;}
@font-face
{font-family:Calibri;
panose-1:2 15 5 2 2 2 4 3 2 4;}
/* Style Definitions */
p.MsoNormal, li.MsoNormal, div.MsoNormal
{margin:0cm;
margin-bottom:.0001pt;
font-size:12.0pt;
font-family:"Times New Roman",serif;}
a:link, span.MsoHyperlink
{mso-style-priority:99;
color:blue;
text-decoration:underline;}
a:visited, span.MsoHyperlinkFollowed
{mso-style-priority:99;
color:purple;
text-decoration:underline;}
p
{mso-style-priority:99;
mso-margin-top-alt:auto;
margin-right:0cm;
mso-margin-bottom-alt:auto;
margin-left:0cm;
font-size:12.0pt;
font-family:"Times New Roman",serif;}
p.msonormal0, li.msonormal0, div.msonormal0
{mso-style-name:msonormal;
mso-margin-top-alt:auto;
margin-right:0cm;
mso-margin-bottom-alt:auto;
margin-left:0cm;
font-size:12.0pt;
font-family:"Times New Roman",serif;}
span.EmailStyle19
{mso-style-type:personal-reply;
font-family:"Calibri",sans-serif;
color:#1F497D;}
.MsoChpDefault
{mso-style-type:export-only;
font-family:"Calibri",sans-serif;}
@page WordSection1
{size:612.0pt 792.0pt;
margin:70.85pt 70.85pt 2.0cm 70.85pt;}
div.WordSection1
{page:WordSection1;}
--></style><!--[if gte mso 9]><xml>
<o:shapedefaults v:ext="edit" spidmax="1026" />
</xml><![endif]--><!--[if gte mso 9]><xml>
<o:shapelayout v:ext="edit">
<o:idmap v:ext="edit" data="1" />
</o:shapelayout></xml><![endif]-->
</head>
<body lang="EN-US" link="blue" vlink="purple">
<div class="WordSection1">
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Dear Junting<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Did you compare your results with the work of
<a href="https://www.sciencedirect.com/science/article/pii/002199917890092X">McMurchie and Davidson</a> from 1978? The expression for lambda in eq. (3.31) seems to match the implemented ferf for the basic integral.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Best
<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Matthias</span><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif">From:</span></b><span style="font-size:11.0pt;font-family:"Calibri",sans-serif"> cp...@googlegroups.com <...@googlegroups.com>
<b>On Behalf Of </b>Junting<br>
<b>Sent:</b> Montag, 14. September 2020 05:13<br>
<b>To:</b> cp2k <...@googlegroups.com><br>
<b>Subject:</b> [CP2K:13866] A question about a formula in CP2K source code<o:p></o:p></span></p>
<p class="MsoNormal"><o:p> </o:p></p>
<div>
<p class="MsoNormal">Dear CP2K users and developers,<o:p></o:p></p>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">I am reading the source code of CP2K, and I have got lots of insights about formulas and algorithms from it. However, I have encountered a question when reading the aobasis/ai_verfc.F file in src directory.<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">The ai_verfc module is used for calculating the matrix <a|erfc(r)/r|b> in all-electron calculation. This term is divided into a nuclear term <a|1/r|b> and an error function term <a|erf(r)/r|b>. In the code prefactors corresponding these
two terms are calculated by <o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<div>
<p class="MsoNormal">fnuc = 2.0_dp*pi*zetp*f0<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal">ferf = 2.0_dp*SQRT(pi**5*zetw)*zetp*zetq*f0<o:p></o:p></p>
</div>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">and the basic s-orbital integrals are calculated by prefactors and incomplete gamma function<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<div>
<p class="MsoNormal">t = rcp2/zetp<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal">CALL fgamma(nmax-1, t, f)<o:p></o:p></p>
</div>
</div>
<div>
<p class="MsoNormal">vnuc(1, 1, n) = fnuc*f(n-1)<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">t = -f4*rcp2/zetp<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal">CALL fgamma(nmax-1, t, f)<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal">verf(1, 1, n) = ferf*f(n-1)<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">Then the integrals over p or higher orbitals are calculated by a recursion procedure as is discussed in the literature: S. Obara and A. Saika, J. Chem. Phys. 84, 3963 (1986).<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">My question is that according to my derivations, my result for fnuc is consistent with the code, but my ferf is <o:p></o:p></p>
</div>
<div>
<p class="MsoNormal">ferf = 2.0_dp*pi*zetp*f0*(-f4)**(n-0.5)<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal">which is not consistent with the code, and is dependent on the parameter n in the incomplete gamma function f(n).<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">And it is reasonable that on the limit of zetc going to infinity, the error function term <a|erf(r)/r|b> should be the same as the nuclear term <a|1/r|b>. But according to source code, it seems not to be the case.<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">Therefore I want to ask, is this a potential mistake in the source code, or did I miss anything else in the derivations? I really appreciate it if someone could help me solve this question.<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal"><o:p> </o:p></p>
</div>
<div>
<p class="MsoNormal">Best wishes,<o:p></o:p></p>
</div>
<div>
<p class="MsoNormal">Junting<o:p></o:p></p>
</div>
</div>
<p class="MsoNormal">-- <br>
You received this message because you are subscribed to the Google Groups "cp2k" group.<br>
To unsubscribe from this group and stop receiving emails from it, send an email to
<a href="mailto:cp2k+unsu...@googlegroups.com">cp2k+unsu...@googlegroups.com</a>.<br>
To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/cp2k/89673088-6dba-4f9f-9cf8-464569253964o%40googlegroups.com?utm_medium=email&utm_source=footer">
https://groups.google.com/d/msgid/cp2k/89673088-6dba-4f9f-9cf8-464569253964o%40googlegroups.com</a>.<o:p></o:p></p>
</div>
</body>
</html>