[Pw_forum] scan the lattice constant

Stefano Baroni baroni at sissa.it
Thu Sep 8 21:44:33 CEST 2005

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  

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.


On Sep 8, 2005, at 9:17 PM, Konstantin Kudin wrote:

>> Basically, as you enalarge your unit cell, your brillouin zone
>> shrinks,
>> and at constant cutoff it means that more lattice vectors (i.e. plane
>> waves) enter into the fixed cutoff sphere. More plane waves means a
>> systematically larger basis set (this is one of the good things
>> of plane waves, it's easy to make the basis set more and more
>> complete -
>> a nightmare in Gaussian), and, variationally, a lower energy. You see
>> in
>> fact your energy drop going to the left.
>  I'd like to chime in here. While indeed reaching the basis set limit
> is sort of difficult with Gaussian basis sets, there is this powerful
> principle called "cancellation of errors". This allows one to chase  
> not
> the full convergence, but convergence with the basis set that is good
> enough. As it turns out, such convergence is actually rather easy to
> achieve.
>  Then one has to care that the energy is fully variational and smooth
> (i.e. differentiable) with respect to changes in atomic coordinates  
> and
> lattice vectors. While indeed PWs do reach the basis set limits  
> easier,
> the smoothness of the potential energy surfaces on the other hand
> leaves something to be desired. The smoothness issue is usually much
> better taken care of with Gaussians. So there you go ... :-)
>  Kostya
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Stefano Baroni - SISSA  &  DEMOCRITOS National Simulation Center -  
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