[Pw_forum] scan the lattice constant

Stefano Baroni baroni at sissa.it
Fri Sep 9 09:51:34 CEST 2005

Dear Virginie: the only reason I can see why E(V) may be  
discontinuous in a metal while keeping the number of PW's fixed is  
the finite number of calculated energy bands above the Fermi energy.  
It is perfectly possible that this number varies with volume (if  
bands are calculated up to some prescribed cutoff energy), or else  
that the character of some of the calculated bands changes with  
volume. In either cases, I would expect discontinuities in E(V) if  
the range of bands above Ef is not large enough with respect to the  
(Gaussian) smearing. I do not think that these discontinuities have  
much to do with the number of k points. This can be easily checked by  
alternatively increasing the number of k points or the number of  
extra bands above Ef. Stefano

On Sep 9, 2005, at 9:14 AM, Virginie Quequet wrote:

> Dear Stefano
> Are you sure, that, even for metal, the potential energy surfaces are
> smooth when we use fixed number of PW's? Because some time ago, I have
> problem of discontinuities with Ti and I try to impose fixed number of
> PW's, but the discontinnuities did not disappeared. I have thought  
> that
> not good enough convergence in k-point was the reason, and that the
> potential energy surface is perfectly smooth only in the limit of an
> infinit number of k-point for metal. Is it wrong?
> -- 
> Virginie Quequet <virginie.quequet at polytechnique.fr>
> From: Stefano Baroni <baroni at sissa.it>
> Subject: Re: [Pw_forum] scan the lattice constant
> Date: Thu, 8 Sep 2005 21:44:33 +0200
> To: pw_forum at pwscf.org
> Reply-To: pw_forum at pwscf.org
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> Kostya: I beg to differ here. I do not want to open a thread "PW's
> vs. Gaussians" which would have the same chances to be settled as
> "coca vs. pepsi", but what  you say, while perfectly (kind of) clear
> to me, may be misleading to the less experienced.
> PW potential energy surfaces (PES) are as smooth as they can be (i.e.
> *perfectly smooth*, or of class C(\infty), if you prefer) *with fixed
> boundary conditions* (i.e. for a fixed shape/size of the unit cell
> which determines the periodic boundary conditions). This smoothness
> has strictly nothing to do with any cancellation of errors. With a
> fixed number of PW's, PES's are perfectly smooth, but the
> (variational) accuracy would depend on the size/shape of the unit
> cell. Discontinuities come in when varying the boundary conditions
> because PW basis sets depend on boundary conditions (conceptually)
> for exactly the same reasons why localized basis sets depend on
> nuclear positions. With Gaussians you have "Pulay forces" (which
> require a special treatment going beyond Helmann Feynman), with PW's
> you have "Pulay stresses" (which require an equally special ad hoc
> treatment).
> It is indeed the ease with which accuracy can be assessed and
> improved with PW's which make the problem appear. As said, if one was
> ready to use a fixed basis set for different geometries (as one does
> with localized basis sets without really caring about the dependence
> of the accuracy on the geometry), the potential energy surface would
> be perfectly smooth, though less accurate.
> As simple as that. No misterious powerful cancellation principles  
> here.
> Stefano
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Stefano Baroni - SISSA  &  DEMOCRITOS National Simulation Center -  
[+39] 040 3787 406 (tel) -528 (fax) / stefanobaroni (skype)

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