[Pw_forum] Question on stress in a system with constraints

Stefano de Gironcoli degironc at sissa.it
Thu Oct 13 09:53:31 CEST 2005 

Dear Nicola and Kostya,
I did not think deeply about the question, so I may be wrong, but I agree
with Kostya.

Stress is a first order derivative and as such only GS properties should
be needed.

Without constraint it is related to homogeneous deformation, in presence
of constraints it is related to a locally inhomogeneous deformation but
Hellman-Feynman theorem should still apply.
I guess the result will be something like the unconstraint stress plus a
term

\sum_i,gamma F_i,gamma {\partial tau_i,gamma \over \partial
\epsilon_{alpha,beta} } / omega

where F_i,gamma is the (unconstrained) gamma component of the force on
atom i and {\partial tau_i,gamma \over \partial \epsilon_{alpha,beta} }
is the first order variation of the position of atom i with distortion
needed to fulfill the constraint.

In order to compute the stress induced by a displacement one indeed need a
linear response calculation and one could think that also a term involving 
the internal strain parameter (Delta) is needed. something like

\sigma^constr_{alpha,beta} = \sigma^unconstr_{alpha,beta} +
         \sum_i,gamma F_i,gamma *
             {\partial tau_i,gamma \over \partial \epsilon_{alpha,beta} } +
         \sum_i,gamma Delta_{alpha,beta;i,gamma} * delta tau_{i,gamma}

but stress is computed at zero deformation therefore delta tau_{i,gamma} is
actually zero

stefano


On Wed, 12 Oct 2005, Nicola Marzari wrote:

>
>
>
> 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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