<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></style> </head> <body lang="EN-US" link="blue" vlink="purple"> <div class="WordSection1"> <p class="MsoNormal"><span lang="DE-CH" style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Dear Junting<o:p></o:p></span></p> <p class="MsoNormal"><span lang="DE-CH" 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">The integral routine ai_verfc works with normalised Cartesian Gaussian functions. CP2K performs the normalisation already during the initialisation phase after the Gaussian basis set information was read.<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<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> Donnerstag, 17. September 2020 19:13<br> <b>To:</b> cp2k <...@googlegroups.com><br> <b>Subject:</b> Re: [CP2K:13916] A question about a formula in CP2K source code<o:p></o:p></span></p> <p class="MsoNormal"><o:p> </o:p></p> <p class="MsoNormal">Dear Matthias,<o:p></o:p></p> <div> <p class="MsoNormal"><o:p> </o:p></p> </div> <div> <p class="MsoNormal">Sorry to bother you again. I have tried both one-electron integral and two-electron integral ways to derive <a|erf(alpha_c*r)/r|b>, and they gave me the same results, which is encouraging. However, there is one remaining question about the normalization factor.<o:p></o:p></p> </div> <div> <p class="MsoNormal"><o:p> </o:p></p> </div> <div> <p class="MsoNormal">The eq.30 of this <a href="https://pubs.rsc.org/en/content/articlelanding/2000/CP/b001167n#!divAbstract"> work</a> shows that there is a nomalization factor N = (alpha_c^2/pi)^(3/2) when rewriting <a|erf(alpha_c*r)/r|b> to a three-center two-electron repulsion integral <ab||c>. But in the work of <a href="https://www.sciencedirect.com/science/article/pii/002199917890092X">McMurchie and Davidson</a>, the expression for lambda in eq. (3.31) seems to be for unnormalized case. If the lambda is multiplied by N = (alpha_c^2/pi)^(3/2), the result will be perfectly in agreement with my result of ferf (i.e. 2*pi*zetp*f0*sqrt(zetc*zetw)).<o:p></o:p></p> </div> <div> <p class="MsoNormal"><o:p> </o:p></p> </div> <div> <p class="MsoNormal">Since the ferf in aobasis/ai_verfc.F matches the the form of lambda, is it supposed to be mutiplied by N? Or has it already been multiplied somewhere else in the code?<o:p></o:p></p> </div> <div> <p class="MsoNormal"><o:p> </o:p></p> </div> <div> <p class="MsoNormal">Best,<o:p></o:p></p> </div> <div> <p class="MsoNormal"><o:p> </o:p></p> </div> <div> <p class="MsoNormal">Junting<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="margin-bottom:12.0pt"><o:p> </o:p></p> </div> <div> <div> <p class="MsoNormal"><span style="font-family:"MS Gothic"">在</span>2020<span style="font-family:"MS Gothic"">年</span>9<span style="font-family:"MS Gothic"">月</span>14<span style="font-family:"MS Gothic"">日星期一</span> UTC+8 <span style="font-family:"MS Gothic"">下午</span>10:29:52<Matthias Krack> <span style="font-family:"MS Gothic""> 写道：</span><o:p></o:p></p> </div> <blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-right:0cm"> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Dear Junting</span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> </span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">The one-electron integral <a_c|-Z_c*erf(alpha*|r - R_c|)/(|r - R_c|)|b> can be rewritten as a three-center two-electron repulsion integral <ab||c> (see eq. 30 of this <a href="https://doi.org/10.1039/B001167N" target="_blank"> work</a>).</span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> </span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">HTH</span><o:p></o:p></p> </div> </div> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> </span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Matthias</span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> </span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><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"> <span class="MsoHyperlink">c...@googlegroups.com</span> <<span class="MsoHyperlink">c...@googlegroups.com</span>> <b>On Behalf Of </b>Junting<br> <b>Sent:</b> Montag, 14. September 2020 16:14<br> <b>To:</b> cp2k <<span class="MsoHyperlink">c...@googlegroups.com</span>><br> <b>Subject:</b> Re: [CP2K:13870] A question about a formula in CP2K source code</span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Dear Matthias,<o:p></o:p></p> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Thank you so much for providing me with this insightful work!<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">However, Eq (3.31) is in Two-electron Integral part, and seems not to involve error function. The <a|erf(r)/r|b> term is one-electron integral, describing the interaction between electron and core, instead of eletron repulsion. I still do not understand how these two are related.<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">And from intuition, if zetc goes to infinity, ferf should become the same as fnuc. But according to source code, they seem quite different, which also makes me confused.<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Best,<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;margin-bottom:12.0pt">Junting<o:p></o:p></p> </div> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-family:"MS Gothic"">在</span>2020<span style="font-family:"MS Gothic"">年</span>9<span style="font-family:"MS Gothic"">月</span>14<span style="font-family:"MS Gothic"">日星期一</span> UTC+8 <span style="font-family:"MS Gothic"">下午</span>4:15:43<Matthias Krack> <span style="font-family:"MS Gothic""> 写道：</span><o:p></o:p></p> </div> <blockquote style="border:none;border-left:solid #CCCCCC 1.0pt;padding:0cm 0cm 0cm 6.0pt;margin-left:4.8pt;margin-top:5.0pt;margin-right:0cm;margin-bottom:5.0pt"> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Dear Junting</span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> </span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><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" target="_blank"> McMurchie and Davidson</a> from 1978? The expression for lambda in eq. (3.31) seems to match the implemented ferf for the basic integral.</span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> </span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Best </span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> </span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Matthias</span><o:p></o:p></p> </div> </div> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"> </span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><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"> <span class="MsoHyperlink">c...@googlegroups.com</span> <<span class="MsoHyperlink">c...@googlegroups.com</span>> <b>On Behalf Of </b>Junting<br> <b>Sent:</b> Montag, 14. September 2020 05:13<br> <b>To:</b> cp2k <<span class="MsoHyperlink">c...@googlegroups.com</span>><br> <b>Subject:</b> [CP2K:13866] A question about a formula in CP2K source code</span><o:p></o:p></p> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Dear CP2K users and developers,<o:p></o:p></p> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">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" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">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" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">fnuc = 2.0_dp*pi*zetp*f0<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">ferf = 2.0_dp*SQRT(pi**5*zetw)*zetp*zetq*f0<o:p></o:p></p> </div> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">and the basic s-orbital integrals are calculated by prefactors and incomplete gamma function<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">t = rcp2/zetp<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">CALL fgamma(nmax-1, t, f)<o:p></o:p></p> </div> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">vnuc(1, 1, n) = fnuc*f(n-1)<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">t = -f4*rcp2/zetp<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">CALL fgamma(nmax-1, t, f)<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">verf(1, 1, n) = ferf*f(n-1)<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">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" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">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" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">ferf = 2.0_dp*pi*zetp*f0*(-f4)**(n-0.5)<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">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" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">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" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">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" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Best wishes,<o:p></o:p></p> </div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Junting<o:p></o:p></p> </div> </div> </div> </div> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">-- <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 <span class="MsoHyperlink">cp2k+...@googlegroups.com</span>.<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" target="_blank"> https://groups.google.com/d/msgid/cp2k/89673088-6dba-4f9f-9cf8-464569253964o%40googlegroups.com</a>.<o:p></o:p></p> </div> </div> </blockquote> </div> </div> </div> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">-- <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 <span class="MsoHyperlink">cp2k+...@googlegroups.com</span>.<o:p></o:p></p> </div> </div> <div> <div> <p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">To view this discussion on the web visit <a href="https://groups.google.com/d/msgid/cp2k/e6dafc7a-0a1b-49ec-bd14-86583294e012n%40googlegroups.com?utm_medium=email&utm_source=footer" target="_blank"> https://groups.google.com/d/msgid/cp2k/e6dafc7a-0a1b-49ec-bd14-86583294e012n%40googlegroups.com</a>.<o:p></o:p></p> </div> </div> </blockquote> </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/4c5353a6-dd2f-48b7-848c-9e8a71bc6f34n%40googlegroups.com?utm_medium=email&utm_source=footer"> https://groups.google.com/d/msgid/cp2k/4c5353a6-dd2f-48b7-848c-9e8a71bc6f34n%40googlegroups.com</a>.<o:p></o:p></p> </div> </body> </html>