# [Pw_forum] scan the lattice constant

Virginie Quequet virginie.quequet at polytechnique.fr
Fri Sep 9 09:14:57 CEST 2005

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

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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