[Pw_forum] elastic constants and internal strain

[Stefano Baroni]

On Jan 24, 2010, at 4:08 PM, Eduardo Ariel Menendez Proupin wrote:

> Hi all,

Hi Edoardo:

> As far as I understand, PWSCF calculates the stress following Nielsen and Martin, PRL 50, 697 (1985), with expressions updated for ultrasoft pseudopotentials and maybe other thechnical upgrades. In the formalism of Nielsen and Martin, the stress is calculated as a partial derivative of the energy versus the strain tensor, hence keeping frozen the atomic coordinates.

to be more precise, scaling all the atomic coordinates homogeneously

> The effect of keeping frosen the atomic fractional coordinates have a large effect in the paradigmatic case of the C44 elastic modulus of silicon, where there is a large internal strain when the cristal is strained along the [111] direction. In the times of 1985, the internal strain contribution to the stress was evaluated using more theory and the (I guess, experimental) values of the TO phonon frequency.

actually, for that matter, no need to use experimental values. 

> In modern times, I would instead relax the atomos in the strained cell, and take the value of the stress tensor at the relaxed geometry.

free to do so, but I would still prefer to combine three ingredients that are easier to obtain, namely the "bare" lattice constan c^0_{44}, the TO frequency, and the internal strain parameter. These three quantities are the second derivatives of the energy with respect to: 1) a macroscopic strain, keeping the atoms frozen at their homoeneously strained positions; 2) the atomic displacents at zero strain; 3) (1) and (2) mixed derivative. (1) and (2) are obvious to obtain, (3) is simply the force on atoms linearly indiced by an applied strain at zero microscopic distortion, or the stress linearly induced by the displacement of the ions at zero strain.

> Doing so, I think I still lose 
> part of the stress, related with the derivative of the atomic positions with respect to the strain.  

If you do things properly, I do not think you loose anything

> On the other hand, the internal strain can be accounted for, calculating the elastic moduli from the second derivative of the energy with respect to the strain. 

correct

> I tested both methods to obtain the C44 constant of silicon, using the strain long the [111] direction. With both methods I obtain the same value of 77.1 GPa. I expected to obtain different values. What am I missing?

you miss trust in your results. you obtain the same results because they ought to be the same ...

>  I checked that not allowing relaxation, I also obtained the same value for the C44(0) using the energy and the stress data. I also checked that the constant C11 give consistent values.

no surprise ...

take care,
Stefano

---
Stefano Baroni - SISSA  &  DEMOCRITOS National Simulation Center - Trieste
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