<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">I am too old too remember, and too lazy to search in the library ... ;-)<div>Actually, the matter is probably treated in "old" math phys books, such as Courant & Hilbert or Morse & Feshbach. The chapter where this is dealt with is on Sturm Liouville theory (<a href="http://en.wikipedia.org/wiki/Sturm-Liouville_theory)">http://en.wikipedia.org/wiki/Sturm-Liouville_theory)</a>. You do not need to dig very much in math. All I know I learnt why I wrote a solver for the 1D Schroedinger equation, which is the ancestor of the solvers presently included in the QE distribution ... ("practice is the Mother of learning")</div><div>Cheers - SB</div><div><br><div><div>On Aug 11, 2008, at 10:43 PM, Prasenjit Ghosh wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div dir="ltr">Dear Stefano,<br><br>Thanks for the nice explanation.<br>Can you please suggest some "old" quantum mechanics books or some other reading materials where they deal with such issues?<br><br>With regards,<br> Prasenjit.<br><br><div class="gmail_quote">2008/8/11 Stefano Baroni <span dir="ltr"><<a href="mailto:baroni@sissa.it">baroni@sissa.it</a>></span><br><blockquote class="gmail_quote" style="border-left: 1px solid rgb(204, 204, 204); margin: 0pt 0pt 0pt 0.8ex; padding-left: 1ex;"> Dear Wenmei:<br> <br> besides the very sound suggestion by Hande, I would also recommend<br> that you take a glance at any (possibly, old) textbook in elementary<br> quantum mechanics. Why old? Because modern texts often given<br> elementary stuff for granted, which usually it is not.<br> <br> Coming to the specific issue that you raise. The number of nodes in<br> the solution of a 1D Schroedinger equation depends on the energy<br> parameter. The lower the energy, the fewer the nodes. Of course, for<br> an arbitrary energy parameter the resulting wavefunction is not<br> normalizable, that is why it has to be discarded, but for very<br> special, discrete, values, and energy is quantized. It is interesting<br> to understand "intuitively" how quantization of energy comes about. As<br> you rise the energy from any given value, the outer node moves out<br> furthhr and futher from the origin, but the number of nodes stays<br> constant if you do not change the energy too much. If the energy is<br> raised too much, a further node will appear. For some critical value<br> of the energy, the "extra" node will be at infinity: that's when an<br> energy eigenvalue is crossed.<br> <br> I do not expect that, if you have never heard about similar<br> arguments, you will understand mine, which have been stated rather<br> sloppily. Actually, if you decide to study this matter in a serious<br> book, your ability to understand my "sloppy arguments" will be a check<br> of your actual understanding of the serious arguments expounded in the<br> book.<br> <br> Hope this helps, a bit<br> <font color="#888888">Stefano<br> </font><div><div></div><div class="Wj3C7c"><br> On Aug 11, 2008, at 6:36 PM, Hande Ustunel wrote:<br> <br> > Dear Wenmei,<br> ><br> > I usually look at the output of the all-electron calculation and<br> > choose an<br> > energy close to the ae orbital energy. Once I manage to generate the<br> > pseudopotential I go as far away as possible from that reference<br> > without<br> > upsetting the code.<br> ><br> > Hope this helps...<br> > Hande<br> ><br> > On Tue, 12 Aug 2008, [gb2312] 明文美 wrote:<br> ><br> >><br> >> Dear all:<br> >><br> >> I want to generate my own RRKJ type ultrasoft pseudopotential<br> >> with atomic code in PWSCF,<br> >> I encountered a very confusing situation:<br> >><br> >> when I chose -0.20Ry as the second energy used to pseudize the 5s<br> >> orbit, the ld1.x always complains :<br> >> =================================================<br> >> from gener_pseudo : error # 1 too many nodes<br> >> =================================================<br> >> however if I change the energy to be -0.50Ry, the corresponding<br> >> complain disappeared.<br> >><br> >> How should I choose these energy parameters to generate a good<br> >> pseudopotential and how should I have a<br> >> correct understanding of these parameters?<br> >><br> >> Thanks very much .<br> >><br> >> Wenmei<br> >><br> >> My &inputp file is also listed as :<br> >><br> >> 64D 3 2 10 0.00 1.00 2.004D 3 2 0 -0.20 1.00<br> >> 2.005S 1 0 2 0.00 2.00 2.505S 1 0 0 -0.50 2.00<br> >> 2.505P 2 1 3 0.00 2.66 2.985P 2 1 0 -0.20 2.66<br> >> 2.98<br> >><br> >><br> >><br> >><br> >><br> >> _________________________________________________________________<br> >> 快来看看这些猫咪有多逗，爆笑！<br> >> <a href="http://cnweb.search.live.com/video/results.aspx?q=%E5%8F%AF%E7%88%B1%E7%8C%AB%E5%92%AA&Form=MEVHAA" target="_blank">http://cnweb.search.live.com/video/results.aspx?q=%E5%8F%AF%E7%88%B1%E7%8C%AB%E5%92%AA&Form=MEVHAA</a><br> ><br> > --<br> > Hande Toffoli<br> > Department of Physics<br> > Office 439<br> > Middle East Technical University<br> > Ankara 06531, Turkey<br> > Tel : +90 312 210 3264<br> > <a href="http://www.physics.metu.edu.tr/%7Ehande" target="_blank">http://www.physics.metu.edu.tr/~hande</a><br> ><br> > _______________________________________________<br> > Pw_forum mailing list<br> > <a href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</a><br> > <a href="http://www.democritos.it/mailman/listinfo/pw_forum" target="_blank">http://www.democritos.it/mailman/listinfo/pw_forum</a><br> <br> _______________________________________________<br> Pw_forum mailing list<br> <a href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</a><br> <a href="http://www.democritos.it/mailman/listinfo/pw_forum" target="_blank">http://www.democritos.it/mailman/listinfo/pw_forum</a><br> </div></div></blockquote></div><br><br clear="all"><br>-- <br>PRASENJIT GHOSH,<br>POST-DOC,<br>ROOM NO: 265, MAIN BUILDING,<br>CM SECTION, ICTP,<br>STRADA COSTERIA 11,<br>TRIESTE, 34104,<br>ITALY<br>PHONE: +39 040 2240 369 (O)<br> +39 3807528672 (M)<br> </div> _______________________________________________<br>Pw_forum mailing list<br><a href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</a><br>http://www.democritos.it/mailman/listinfo/pw_forum<br></blockquote></div><br><div apple-content-edited="true"> <span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><span class="Apple-style-span" style="border-collapse: separate; color: rgb(0, 0, 0); font-family: Helvetica; font-size: 14px; font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-border-horizontal-spacing: 0px; -webkit-border-vertical-spacing: 0px; -webkit-text-decorations-in-effect: none; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px; "><div style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">---</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">Stefano Baroni - SISSA</span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "> </span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">&</span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "> </span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">DEMOCRITOS National Simulation Center - Trieste</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "><a href="http://www.sissa.it/~baroni">http://www.sissa.it/~baroni</a> / [+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; font-size: 12px; "><br style="font-size: 12px; "></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; font: normal normal normal 12px/normal Helvetica; min-height: 14px; font-size: 12px; "><span class="Apple-style-span" style="font-size: 14px; "><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">La morale est une logique de l'action comme la logique est une morale de la pensée - Jean Piaget</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "><br></span></font></div></span></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">Please, if possible, don't</span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "> </span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">send me MS Word or PowerPoint attachments</span></font></div><div style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; ">Why? See:</span></font><font class="Apple-style-span" size="3"><span class="Apple-style-span" style="font-size: 12px; "> </span></font><font class="Apple-style-span" color="#0023E9" size="3"><span class="Apple-style-span" style="font-size: 12px; "><a href="http://www.gnu.org/philosophy/no-word-attachments.html">http://www.gnu.org/philosophy/no-word-attachments.html</a></span></font></div><div><font class="Apple-style-span" color="#0023E9" size="3"><span class="Apple-style-span" style="font-size: 12px; "><br class="webkit-block-placeholder"></span></font></div></div></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"></div></span><br class="Apple-interchange-newline"> </div><br></div></body></html>