>From: Stefano Baroni <baroni at sissa.it> >Reply-To: pw_forum at pwscf.org >To: pw_forum at pwscf.org >Subject: Re: [Pw_forum] phonon eigenvectors >Date: Thu, 31 Aug 2006 22:28:04 +0200 > >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!
No worries. It's hard to portray tone over email. I guess I assumed your tone from reading your responses to other postings (where the post-er was clearly not doing their homework) and assumed you thought the same of me...I just wanted to clear that up. > >>Question: What is the meaning of an eigenvector? >>Answer: An eigenvector tells how the atoms are displaced in the >>vibration. > >OK > >>Question: What is the imaginary part? >>Answer: The imaginary part is a phase factor. > >Forgive my fussiness: its is *not* a phase factor. A non zero imaginary >part is *due* to a non zero phase. > >>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. > >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) > >>In the case of a completely imaginary eigenvector, the displacement will >>be completely out of phase. > >This statement, instead, is meaningless. Ou of phase WITH RESPECT TO WHAT? Yeah, poorly chosed words here...I guess I redeem myself below. I really did mean "out of phase with respect to an atom with say a completly real eigenvector". >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? yes. > >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? again, yes. > >>Question: If the imaginary part is a phase factor, then what does it >>mean if all of the components are imaginary? >>Answer: First answer: nothing. 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. > >VERY GOOD. YOU GOT IT! > >>Question: Great. If this is the case, then why does this eigenvector >>come up imaginary when all of the other vectors come up real? >>Answer: Hmmm. That's a good one. There must be some reason that the >>code chooses an imaginary eigenvector for this mode...time to get help on >>this one. We are fresh out of answers. > >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! > >>And with this I come first to the pwscf discussion archive. I didn't >>find any discussion of imaginary eigenvectors with real eigenvalues. > >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 ... OK, that makes sense. > >>I looked to the literature, with the same result. 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. With that I come to you >>with my question: >> >>Why did the code "choose" to make this displacement imaginary. Is this >>simply an artifact of the matrix diagonalization, > >YES > >>or is there some physical implications to this? > >NO > >>Thank you for your time and patience. > >Thank you for challenging them! ;-) >(I am not kidding: it's refreshing and instructive for us to help, no less >that it is for you to be assisted - hopefully ;-) > >Stefano > >--- >Stefano Baroni - SISSA & DEMOCRITOS National Simulation Center - Trieste >[+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype) > >Please, if possible, don't send me MS Word or PowerPoint attachments Amen to that...although, Openoffice does quite well at opening these :) >Why? See: http://www.gnu.org/philosophy/no-word-attachments.html > > >
