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<div><span></span>Dear <span style="font-size: 10.5pt; line-height: 1.5; background-color: window;">Robert Hembree,</span></div><div><span style="font-size: 10.5pt; line-height: 1.5; background-color: window;"> I am sorry that I had something wrong in my expression before.</span></div><div><span style="font-size: 10.5pt; line-height: 1.5; background-color: window;">1) Actually, I focus on the result of </span><span style="background-color: window; font-size: 10.5pt; line-height: 1.5;"><font color="#ff0000">conjg(psi_1(r)) * psi_2(r)</font>, here both wavefunction belong to the same k point and 'conjg' is complex conjugate operation. I agree with your conclusion that < psi_1(r) | psi_2(r) is a real number, which is also checked before by me. However, conjg(psi_1(r)) * psi_2(r) is the array but not a number, and I want to clarify whether the array is real array.</span></div><div><span style="background-color: window; font-size: 10.5pt; line-height: 1.5;">2) We share the same result.</span></div><div><span style="background-color: window; font-size: 10.5pt; line-height: 1.5;"><br></span></div><div>Best regards</div><div>Jiqiang Li</div><div>Fudan University, China</div>
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<blockquote style="margin-top: 0px; margin-bottom: 0px; margin-left: 0.5em;"><div> </div><div style="border:none;border-top:solid #B5C4DF 1.0pt;padding:3.0pt 0cm 0cm 0cm"><div style="PADDING-RIGHT: 8px; PADDING-LEFT: 8px; FONT-SIZE: 12px;FONT-FAMILY:tahoma;COLOR:#000000; BACKGROUND: #efefef; PADDING-BOTTOM: 8px; PADDING-TOP: 8px"><div><b>From:</b> <a href="mailto:hembreerofphysics@gmail.com" style="color: rgb(5, 99, 193); text-decoration: underline;">Robert Hembree</a></div><div><b>Date:</b> 2015-04-17 14:07</div><div><b>To:</b> <a href="mailto:pw_forum@pwscf.org" style="color: rgb(5, 99, 193); text-decoration: underline;">'PWSCF Forum'</a></div><div><b>Subject:</b> Re: [Pw_forum] About wavefunction</div></div></div><div><div class="FoxDiv20150417144116162452"><!--[if gte mso 9]><xml>
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</o:shapelayout></xml><![endif]--><div class="WordSection1" style="page: WordSection1;"><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Dear Jiqiang Li,<o:p></o:p></span></p><p class="MsoListParagraph" style="text-indent: -0.25in; margin: 0in 0in 0.0001pt 0.5in; font-size: 12pt; font-family: 'Times New Roman', serif;"><!--[if !supportLists]--><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><span style="mso-list:Ignore">1)<span style="font:7.0pt "Times New Roman""> </span></span></span><!--[endif]--><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">If you compute <psi_i|psi_j> at the same point then the result is delta_ij. If the bands are different at the same kpoint then you will get zero for this value. <o:p></o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">If you are outputting the wavefunction at <psi_1|psi_2> and getting a complex number other than 0+0i or some other sufficiently small number I suspect you are doing it incorrectly. For example when I compute these values I get a value on the order of 10^-17. <o:p></o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Perhaps if you modified your wavefunction. It should instead read psi_j(k,r) = sum_G C_j,G(k) exp(-(G+k)*r)) as a plane wave description would generally require. <o:p></o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">If you work it out by hand then the overlap will be <psi_i|psi_j> = sum_G C*_i,G(k) C_j,G(k)<o:p></o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">My suspicion is that you are doing something incorrectly. <o:p></o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p><p class="MsoListParagraph" style="text-indent: -0.25in; margin: 0in 0in 0.0001pt 0.5in; font-size: 12pt; font-family: 'Times New Roman', serif;"><!--[if !supportLists]--><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><span style="mso-list:Ignore">2)<span style="font:7.0pt "Times New Roman""> </span></span></span><!--[endif]--><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">In my experience no, not unless you force the Gamma point only calculation. <o:p></o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">I apologize if the notation here doesn’t quite work out so well. Writing equations in text like this does not work well. <o:p></o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Robert Hembree<o:p></o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D">Quantum Theory project. <o:p></o:p></span></p><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:11.0pt;font-family:"Calibri",sans-serif;color:#1F497D"><o:p> </o:p></span></p><div><div style="border:none;border-top:solid #E1E1E1 1.0pt;padding:3.0pt 0in 0in 0in"><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><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"> pw_forum-bounces@pwscf.org [mailto:pw_forum-bounces@pwscf.org] <b>On Behalf Of </b>Jiqiang Li<br><b>Sent:</b> Friday, April 17, 2015 12:59 AM<br><b>To:</b> pw_forum<br><b>Subject:</b> [Pw_forum] About wavefunction<o:p></o:p></span></p></div></div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><o:p> </o:p></p><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black">Dear all,<o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black"> 1. In Quantum Espresso, do wavefunction in real space form at the same k point share the same phase? i.e., psi_1(r) = A(r)exp(ikr) and psi_2(r)=B(r)exp(ikr), psi_1 and psi_2 belong to different bands at same k point.<o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black"> If so, we should find the vaule < psi_1 | psi_2 > = A(r)exp(-ikr)*B(r)exp(ikr)=A(r)B(r), which is a series of real number. However, I output the wavefunctions and found <span style="background:white">< psi_1 | psi_2 > was also a complex number.</span><o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black;background:white"> Could you tell me the reason?</span><span style="font-size:10.5pt;font-family:"????",serif;color:black"><o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black;background:white"> 2. Is the wavefunction in real space form at Gamma point a series of real number? ( not only for the gamma point only calculation)</span><span style="font-size:10.5pt;font-family:"????",serif;color:black"><o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black;background:white"> Thanks for your consideration!</span><span style="font-size:10.5pt;font-family:"????",serif;color:black"><o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black"><o:p> </o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black;background:white">Best regards!</span><span style="font-size:10.5pt;font-family:"????",serif;color:black"><o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black"><o:p> </o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black;background:white">Jiqiang Li</span><span style="font-size:10.5pt;font-family:"????",serif;color:black"><o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black">Fudan University, China<o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black"> <o:p></o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black"><o:p> </o:p></span></p></div><div><p class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black"><o:p> </o:p></span></p></div><div class="MsoNormal" style="margin: 0in 0in 0.0001pt; font-size: 12pt; font-family: 'Times New Roman', serif;"><span style="font-size:10.5pt;font-family:"????",serif;color:black"><hr size="1" width="210" style="width:157.5pt" noshade="" align="left"></span></div></div></div></div></blockquote>
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