On Feb 6, 2009, at 10:12 AM, Gabriele Sclauzero wrote: Gabriele:
> Stefano Baroni wrote: >> parameters along energy derivatives (forces and stress). When the >> ground >> state is nondegenerate, energy derivatives have the same symmetry >> as the >> the geometry of the system. > > Please Stefano help me to understand this point, which is not clear > to me. If the ground > state is degenerate (you meant degenerate at a fixed external > potential, i.e. fixed atomic > geometry, am I right?) YES > you could reach a lower energy by lowering the symmetry: do you > think this is possible in practical calculations. This is possible in reality (in fact, the Jahn-Teller effect is exactly this). Whether or not this is possible "in practice" depends on how faithfully "practice" mimics "reality" (out of the metaphor: on how calculations are done). > I thought that you could only end up in > a configuration with higher symmetry than the starting one, not lower. it depends ... when the nuclear geometry is such that the ground state is degenerate, it may happen that two or more Born-Oppenheimer surfaces (each one corresponding to a different degenerate state) intersect at that geometry in such a way that the gradients along each one of the surfaces does not vanish. In the nondegenerate case, symmetry trivially imposes that forces along symmetry-breaking directions vanish. In the degenerate case, the situation is more tricky: symmetry imposes that the SUM of the forces corresponding to different degenerate states vanish. Which implies that you can lower the energy by breaking the symmetry along one of the intersecting BO energy surfaces > From the theoretical point of view, is it correct to apply the > standard formulation of > HK theorem to a system with degenerate GS (is the HK mapping still > valid?) this problem used to be a very popular and controversial one, some time ago. I do not know the "politically correct" answer. Let me give you mine. 1) strictly, the usual derivation of DFT does not hold for degenerate states 2) That derivation can be easily extended to a (arbitrarily small) finite temperature (Mermin's DFT does that) 3) at finite temperature, degeneracy does not matter. DFT holds, the Hellmann-Feynman theorem applies. only one BO surface exists, no symmetry breaking distortions would occur 4) at zero temperature complications arise and the BO surface would develop a cusp, corresponding to different forces along different directions. common sense is restored "in practice" by considering infinitesimal distortions of the high-simmetry geometry. the limit of the forces as the distortion goes to zero would depend on dierction, which is exactly a amnifestation of the Jahn-Teller effect (or at least, so it seems to me) Ciao Stefano --- Stefano Baroni - SISSA & DEMOCRITOS National Simulation Center - Trieste http://www.sissa.it/~baroni / [+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype) La morale est une logique de l'action comme la logique est une morale de la pens?e - Jean Piaget Please, if possible, don't send me MS Word or PowerPoint attachments Why? See: http://www.gnu.org/philosophy/no-word-attachments.html -------------- next part -------------- An HTML attachment was scrubbed... URL: http://www.democritos.it/pipermail/pw_forum/attachments/20090206/a499a8ad/attachment-0001.htm
