<html><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; ">Alexandra: are you by any chance a student/collaborator of Alfonso's? One of the few persons that I know, still able to teach group theory, and the one who taught me long ago, when group theory was not yet considered archeo-physics in our community. Cheers - Stefano<div><br><div><div>On Feb 10, 2009, at 10:19 AM, <a href="mailto:alexandra.carvalho@epfl.ch">alexandra.carvalho@epfl.ch</a> wrote:</div><br class="Apple-interchange-newline"><blockquote type="cite"><div>Dear Gabriele and Manoj,<br><br>In fact I should clarify that I meant that rotations about directed axes belong<br>to different classes if the axes are not inverted by symmetry.<br>(The word 'equivalent' was a poor choice...)<br><br>Alexandra<br><br>Quoting Gabriele Sclauzero <<a href="mailto:sclauzer@sissa.it">sclauzer@sissa.it</a>>:<br><br><blockquote type="cite">Dear Manoj and Alexandra,<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"> in my limited and maybe naive knowledge of group theory I think that<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><a href="mailto:alexandra.carvalho@epfl.ch">alexandra.carvalho@epfl.ch</a> wrote:<br></blockquote><blockquote type="cite"><blockquote type="cite">Dear Manoj,<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">I do not know about the definitions intrinsic to pwscf,<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">but in group theory directed opperations are not allways equivalent.<br></blockquote></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Rotation of 90 deg about z and about -z are _different_ operations (in every<br></blockquote><blockquote type="cite">group to<br></blockquote><blockquote type="cite">which they belong...). Maybe pwscf uses a naive way to call symmetry<br></blockquote><blockquote type="cite">operations, but you<br></blockquote><blockquote type="cite">may call rotation of 90 deg about -z as "rotation of -90 deg about z" so<br></blockquote><blockquote type="cite">maybe it's more<br></blockquote><blockquote type="cite">clear (do you know the right hand rule?).<br></blockquote><blockquote type="cite">If the rotation of 90 deg about z is present, also the rotation of -90 deg<br></blockquote><blockquote type="cite">about z must be<br></blockquote><blockquote type="cite">present (from the definition of group), as well as the rotation of 180 deg<br></blockquote><blockquote type="cite">about z (which<br></blockquote><blockquote type="cite">you can obtain by applying the rotation of 90 twice). This gives you a C_4<br></blockquote><blockquote type="cite">symmetry about<br></blockquote><blockquote type="cite">the z axis (then if you will add more symmetry operations you will have<br></blockquote><blockquote type="cite">higher symmetry).<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><blockquote type="cite">In this case, the rotations about -z and z will be inequivalent<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">if there is no opperation of the group which reverses the sense of the z<br></blockquote></blockquote><blockquote type="cite">axis.<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">I don't understand what do you mean by "equivalent/inequivalent" in this<br></blockquote><blockquote type="cite">case...<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">Hope it helps,<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">GS<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">For example, in the T group there are 8 C3 opperations although there<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">are only four cube diagonals- see Tinkham "Group Theory and Quantum<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">mechanics" 2003 pp 58.<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">Alexandra Carvalho<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">EPFL Lausanne<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">Quoting Manoj Srivastava <<a href="mailto:manoj@phys.ufl.edu">manoj@phys.ufl.edu</a>>:<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Dear PWSCF users and developers,<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"> I was looking into the code where it calculates symmerty operations for<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">the group. Now the symmetry operations are defined as ' 90 deg rotation -<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">cart. axis [0,0,-1]', & ' 90 deg rotation - cart. axis [0,0,1]<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">I did not understand the point of defining rotation about z and -z<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">axis. Is this because of some numerical constraint? Also, there are<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">elements with 'theta' and '-theta' rotation in the same group. I was just<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">wondering the reason behind this. Any help will be appreciated.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Regards,<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Manoj Srivastava<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Department of Physics<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">University of Florida<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Gainesville, FL, USA.<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">_______________________________________________<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite">Pw_forum mailing list<br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><a href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</a><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><a href="http://www.democritos.it/mailman/listinfo/pw_forum">http://www.democritos.it/mailman/listinfo/pw_forum</a><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">_______________________________________________<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite">Pw_forum mailing list<br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><a href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</a><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><a href="http://www.democritos.it/mailman/listinfo/pw_forum">http://www.democritos.it/mailman/listinfo/pw_forum</a><br></blockquote></blockquote><blockquote type="cite"><blockquote type="cite"><br></blockquote></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">--<br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite"><br></blockquote><blockquote type="cite">o ------------------------------------------------ o<br></blockquote><blockquote type="cite">| Gabriele Sclauzero, PhD Student |<br></blockquote><blockquote type="cite">| c/o: SISSA & CNR-INFM Democritos, |<br></blockquote><blockquote type="cite">| via Beirut 2-4, 34014 Trieste (Italy) |<br></blockquote><blockquote type="cite">| email: <a href="mailto:sclauzer@sissa.it">sclauzer@sissa.it</a> |<br></blockquote><blockquote type="cite">| phone: +39 040 3787 511 |<br></blockquote><blockquote type="cite">| skype: gurlonotturno |<br></blockquote><blockquote type="cite">o ------------------------------------------------ o<br></blockquote><blockquote type="cite">_______________________________________________<br></blockquote><blockquote type="cite">Pw_forum mailing list<br></blockquote><blockquote type="cite"><a href="mailto:Pw_forum@pwscf.org">Pw_forum@pwscf.org</a><br></blockquote><blockquote type="cite"><a href="http://www.democritos.it/mailman/listinfo/pw_forum">http://www.democritos.it/mailman/listinfo/pw_forum</a><br></blockquote><blockquote type="cite"><br></blockquote><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">http://www.democritos.it/mailman/listinfo/pw_forum</a><br></div></blockquote></div><br><div> <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; "><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; ">[+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; "><span class="Apple-style-span" style="font-size: 14px; "><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; "><br></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; ">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></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><br></div></body></html>