<HTML><BODY style="word-wrap: break-word; -khtml-nbsp-mode: space; -khtml-line-break: after-white-space; ">Dear Eric: I feel flattered (and a little guilty) for your taking me so seriously. Also, I may have had a role in starting the discussion taken up by Axel and Fernando which, to tell the truth, I do not fully understand. All in all, forgive my being (involuntarily!) paternalistic, and let's come to the real stuff!<DIV><BR><DIV><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Question: What is the meaning of an eigenvector?</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Answer:<SPAN class="Apple-converted-space"> </SPAN>An eigenvector tells how the atoms are displaced in the vibration.</DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>OK</DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Question: What is the imaginary part?</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Answer: The imaginary part is a phase factor.<SPAN class="Apple-converted-space"> </SPAN></DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>Forgive my fussiness: its is *not* a phase factor. A non zero imaginary part is *due* to a non zero phase.</DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">If one atom has a significant imaginary part with respect to the rest, then then it's displacement in the vibration will be phase shifted with respect to the others.<SPAN class="Apple-converted-space"> </SPAN></DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>OK - More precisely: a phase dfference between two different atomic components of a same eigenvectors imply that the motions of the two atoms are out of phase (i.e. the velocities of the two may vanish at different times)</DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">In the case of a completely imaginary eigenvector, the displacement will be completely out of phase.</DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>This statement, instead, is meaningless. Ou of phase WITH RESPECT TO WHAT? The only meaningful thing is the phase difference between the eigenvector components of two different atoms. An overall phase equal for all the atoms is simply equivalent to a shift of the origin of time. CAN YOU SEE THIS?</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Can you see the analogies with the well know statement that quantum mechanical wavefunctions (a problem which is conceptually TOTALLY different) are defined to within an overall phase factor?</DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Question:<SPAN class="Apple-converted-space"> </SPAN>If the imaginary part is a phase factor, then what does it mean if all of the components are imaginary?</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Answer:<SPAN class="Apple-converted-space"> </SPAN>First answer: nothing.<SPAN class="Apple-converted-space"> </SPAN>If all of the atoms are "out of phase", then they are "in phase" with respect to eachother, therefore, having a completely imaginary or completely real eigenvector should be equivalent.</DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>VERY GOOD. YOU GOT IT!</DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Question:<SPAN class="Apple-converted-space"> </SPAN>Great.<SPAN class="Apple-converted-space"> </SPAN>If this is the case, then why does this eigenvector come up imaginary when all of the other vectors come up real?</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Answer:<SPAN class="Apple-converted-space"> </SPAN>Hmmm.<SPAN class="Apple-converted-space"> </SPAN>That's a good one.<SPAN class="Apple-converted-space"> </SPAN>There must be some reason that the code chooses an imaginary eigenvector for this mode...time to get help on this one.<SPAN class="Apple-converted-space"> </SPAN>We are fresh out of answers.</DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>Very good, Erich. You came to the point. The answer is deceivingly simple. If the the phase is *physically* irrelevant, how could a physically sound mathematical algorithm choose it? Answer (deeper than it may sound at first): AT RANDOM! Actual algorithms may not be really random, but it wouldn't harm if they were!</DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">And with this I come first to the pwscf discussion archive.<SPAN class="Apple-converted-space"> </SPAN>I didn't find any discussion of imaginary eigenvectors with real eigenvalues.<SPAN class="Apple-converted-space"> </SPAN></DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>Hey, hey, hey! Slow down! Real eigenvalues are a consequence of the Hermiticity of the dynamical matrix and have nothing to do with the eigenvector being or not real. A Hermitean matrix may or may not have complex eigenvectors. If the matrix is real, the phases can alwayes (*not* "must": you see the difference?) in such a way that the eigenvectors are real. If the matrix is not real (still being Hermitean), I do not know: eigenvectors are in general complex (i.e. no overall phase can be chosen so as to make all of their components real), but they may also be in some particular instance real ... </DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">I looked to the literature, with the same result.<SPAN class="Apple-converted-space"> </SPAN>There comes a time when one asks himself questions to which he doesn't have the answers, then it comes time to discuss with his peers.<SPAN class="Apple-converted-space"> </SPAN>With that I come to you with my question:</DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; min-height: 14px; "><BR></DIV><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Why did the code "choose" to make this displacement imaginary.<SPAN class="Apple-converted-space"> </SPAN>Is this simply an artifact of the matrix diagonalization, <BR></DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>YES</DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">or is there some physical implications to this?</DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>NO</DIV><DIV><BR><BLOCKQUOTE type="cite"><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; ">Thank you for your time and patience.</DIV></BLOCKQUOTE><DIV><BR class="khtml-block-placeholder"></DIV>Thank you for challenging them! ;-)</DIV><DIV>(I am not kidding: it's refreshing and instructive for us to help, no less that it is for you to be assisted - hopefully ;-)</DIV><DIV><BR class="khtml-block-placeholder"></DIV><DIV>Stefano</DIV><DIV><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; 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; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; 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; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><SPAN class="Apple-style-span" style="border-collapse: separate; border-spacing: 0px 0px; 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; text-align: auto; -khtml-text-decorations-in-effect: none; text-indent: 0px; -apple-text-size-adjust: auto; text-transform: none; orphans: 2; white-space: normal; widows: 2; word-spacing: 0px; "><DIV style="margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px; "><BR class="khtml-block-placeholder"></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 class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">---</SPAN></SPAN></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;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">Stefano Baroni - SISSA</SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">&</SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">DEMOCRITOS National Simulation Center - Trieste</SPAN></SPAN></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;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">[+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)</SPAN></SPAN></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 class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">Please, if possible, don't</SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">send me MS Word or PowerPoint attachments</SPAN></SPAN></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;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; ">Why? See:</SPAN></SPAN></SPAN></FONT><FONT class="Apple-style-span" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;"><SPAN class="Apple-style-span" style="font-size: 12px; "><SPAN class="Apple-style-span" style="font-size: 12px; "> </SPAN></SPAN></SPAN></FONT><A href="http://www.gnu.org/philosophy/no-word-attachments.html"><FONT class="Apple-style-span" color="#0023E9" size="3"><SPAN class="Apple-style-span" style="font-size: 12px;; color: rgb(0, 35, 233); -khtml-text-decorations-in-effect: underline; "><SPAN class="Apple-style-span" style="color: rgb(0, 35, 233); font-size: 12px; -khtml-text-decorations-in-effect: underline; "><SPAN class="Apple-style-span" style="color: rgb(0, 35, 233); font-size: 12px; -khtml-text-decorations-in-effect: underline; ">http://www.gnu.org/philosophy/no-word-attachments.html</SPAN></SPAN></SPAN></FONT></A></DIV><DIV><BR class="khtml-block-placeholder"></DIV><BR class="Apple-interchange-newline"></SPAN></SPAN></SPAN> </DIV><BR></DIV></BODY></HTML>