[Pw_forum] Question on stress in a system with constraints

[Nicola Marzari]

Hi Kostya,

I think that you do not need the response functions if you want to
calculate the bare stress (i.e. derivative with respect to strain,
in which fractional coordinates remain the same). If you want to have 
the bare stress plus constraints, you need to know at least the tensor 
that couples strain and forces.

But keep in mind that the physical stress you want is dressed by the
atominc internal relaxations; those would not be included in your
bare CP stress, or in your bare CP stress plus contraints, and need in
addition the inverse of the dynamical matrix.

Any comments, anyone ?

				nicola


Konstantin Kudin wrote:
>  Dear Nicola and Paolo,
> 
>  Thanks for the comments!
> 
>  However, I do not think that one needs the response functions from
> DFPT  to remove constraints from the stress.
> 
>  What happens is that for homogeneous strain it is probably assumed
> that the fractional coordinates in the cell remain the same, however,
> the lattice vectors change, and so all the atomic Cartesian coordinates
> are updated. Is that how the stress actually defined?
> 
>  With constraints, the change in the lattice vectors should also update
> the fractional coordinates of some atoms, leading to extra derivatives
> which include the usual atomic forces of these atoms times the change
> in the fractional coordinates.
> 
>  Kostya
> 
> 
>  
> 
> --- Nicola Marzari <marzari at MIT.EDU> wrote:
> 
> 
>>
>>Hi Kostya,
>>
>>
>>a quick comment - Don Hamann has written a fairly extensive PRB this
>>year discussing many of these issues: Vol 72, 350105 (2005).
>>
>>CP stresses are (I hope) a derivative with respect to the strain
>>tensor - ie.e they do not take into account that
>>atoms can relax in response to the stress (the paradigmatic case is
>>the
>>response to a strain in the 111 direction in silicon - the internal
>>strain parameter measures how the distance between the two atoms
>>in the unit cell changes in response to the symmetry-breaking
>>stress).
>>So, you have the bare stress, calculated by CP and/or PWSCF, but you
>>want the renormalized one "dressed" by the relaxations (mediated by
>>the
>>inverse of the dynamical matrix, and by the coupling between
>>displacements and strains). The constraint will allow you to
>>renormalize appropriately the bare stress, if you have all the
>>response
>>functions from DFPT (and their correct long-wavelength limit)
>>but it might be easier to do it by finite differeces of the energy
>>along 
>>the strain direction, while constraining the atoms.
>>
>>By the way - the dressing of a perturbation by the ionic relaxations
>>is very relevant for piezoelectricity (e.g. Stefano de Gironcoli
>>1989 PRL) or for the interactions in substitutional alloy
>>(PRL 72 4001 (1994) - some of the issues with the long wavelength
>>limit 
>>are discussed there).
>>
>>Best,
>>
>>			nicola
>>
>>
>>
>>
>>Konstantin Kudin wrote:
>>
>>
>>> Hi there,
>>>
>>> I have a basic question on the stress tensor.
>>>
>>> Presumably, the stress tensor computed in the CP implies
>>
>>continuous
>>
>>>stretching of the underlying system. Is this correct? Is the
>>
>>derivative
>>
>>>with respect to % of stretch, or is it to extra length in Bohrs,
>>
>>for
>>
>>>example?
>>>What are the units of the stress in CP ?
>>>
>>> If one projected out certain forces (for constrained coordinates),
>>>then the stress needs to be corrected for the fact that certain
>>
>>forces
>>
>>>are no longer there. Basically, the original stress was computed as
>>
>>if
>>
>>>all the forces were present, now, however, some of them are
>>
>>missing.
>>
>>>This means that if the system were to be stretched, the constrained
>>>coordinates would still be constrained.
>>>
>>> My question is how I can correct the stress in a system with
>>>constraints if this needs to be done.
>>>
>>> Thanks!
>>> Kostya
>>>
>>>
>>>		
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>>
>>-- 
>>---------------------------------------------------------------------
>>Prof Nicola Marzari   Department of Materials Science and Engineering
>>13-5066   MIT   77 Massachusetts Avenue   Cambridge MA 02139-4307 USA
>>tel 617.4522758  fax 617.2586534  marzari at mit.edu  http://nnn.mit.edu
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> 
> 
> 
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---------------------------------------------------------------------
Prof Nicola Marzari   Department of Materials Science and Engineering
13-5066   MIT   77 Massachusetts Avenue   Cambridge MA 02139-4307 USA
tel 617.4522758  fax 617.2586534  marzari at mit.edu  http://nnn.mit.edu